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Switch to NIPS style; Reflow the mathematical equations

arxiv-v3
Markus Kaiser 1 year ago
parent
commit
365b72d0e6
6 changed files with 906 additions and 2373 deletions
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bayesian_warped_dependent_gp.tex View File

@@ -19,46 +19,28 @@
19 19
 \title{Bayesian Alignments of Warped Multi-Output Gaussian Processes}
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 \author{\href{mailto:markus.kaiser@siemens.com}{Markus Kaiser}}
21 21
 
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+\author{
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+    Markus Kaiser\\
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+    Siemens AG\\
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+    Technical University of Munich\\
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+    \texttt{markus.kaiser@siemens.com}\\
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+    \And
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+    Clemens Otte\\
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+    Siemens AG\\
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+    \texttt{clemens.otte@siemens.com}\\
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+    \And
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+    Thomas Runkler\\
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+    Siemens AG\\
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+    Technical University of Munich\\
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+    \texttt{thomas.runkler@siemens.com}\\
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+    \And
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+    Carl Henrik Ek\\
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+    University of Bristol\\
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+    \texttt{carlhenrik.ek@bristol.ac.uk}\\
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+}
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+
22 42
 \begin{document}
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-\twocolumn[
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-\icmltitle{Bayesian Alignments of Warped Multi-Output Gaussian Processes}
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-
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-% It is OKAY to include author information, even for blind
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-% submissions: the style file will automatically remove it for you
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-% unless you've provided the [accepted] option to the icml2017
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-% package.
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-
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-% list of affiliations. the first argument should be a (short)
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-% identifier you will use later to specify author affiliations
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-% Academic affiliations should list Department, University, City, Region, Country
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-% Industry affiliations should list Company, City, Region, Country
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-
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-% you can specify symbols, otherwise they are numbered in order
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-% ideally, you should not use this facility. affiliations will be numbered
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-% in order of appearance and this is the preferred way.
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-\icmlsetsymbol{equal}{*}
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-
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-\begin{icmlauthorlist}
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-\icmlauthor{Markus Kaiser}{tum,siemens}
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-\icmlauthor{Clemens Otte}{siemens}
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-\icmlauthor{Thomas Runkler}{tum,siemens}
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-\icmlauthor{Carl Henrik Ek}{bristol}
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-\end{icmlauthorlist}
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-
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-\icmlaffiliation{tum}{Department of Computer Science, Technical University of Munich, Germany}
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-\icmlaffiliation{siemens}{Siemens AG, Corporate Technology, Munich, Germany}
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-\icmlaffiliation{bristol}{University of Bristol, United Kingdom}
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-
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-\icmlcorrespondingauthor{Markus Kaiser}{markus.kaiser@siemens.com}
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-
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-% You may provide any keywords that you
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-% find helpful for describing your paper; these are used to populate
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-% the "keywords" metadata in the PDF but will not be shown in the document
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-% \icmlkeywords{}
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-
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-\vskip 0.3in
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-]
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-\printAffiliationsAndNotice{}
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+\maketitle
62 44
 
63 45
 \begin{abstract}
64 46
     We present a Bayesian extension to convolution processes which defines a representation between multiple functions via an embedding in a shared latent space.
@@ -129,8 +111,7 @@ Similar to \textcite{boyle_dependent_2004}, we assume the latent processes to be
129 111
 Under this prior, the $f_d$ are also GPs  with zero mean and
130 112
 \begin{align}
131 113
 \begin{split}
132
-    \MoveEqLeft \Moment{\cov}{f_d(\mat{x}), f_{d^\prime}(\mat{x^\prime})} = \\
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-    &\sum_{r=1}^R \int T_{d,r}(\mat{x} - \mat{z}) T_{d^\prime,r}(\mat{x^\prime} - \mat{z}) \diff \mat{z}.
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+    \MoveEqLeft \Moment{\cov}{f_d(\mat{x}), f_{d^\prime}(\mat{x^\prime})} = \sum_{r=1}^R \int T_{d,r}(\mat{x} - \mat{z}) T_{d^\prime,r}(\mat{x^\prime} - \mat{z}) \diff \mat{z}.
134 115
 \end{split}
135 116
 \end{align}
136 117
 
@@ -146,8 +127,7 @@ With $\Set{\sigma_{d,r}, \mat{\ell_{d, r}}}$ denoting the set of kernel hyper pa
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 \begin{align}
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 \label{eq:dependent_kernel}
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 \begin{split}
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-    \MoveEqLeft[1] \Moment{\cov}{f_d(\mat{x}), f_{d^\prime}(\mat{x^\prime})} = \\
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-    &\sum_{r=1}^R \frac{(2\pi)^{\frac{K}{2}}\sigma_{d, r}\sigma_{d^\prime, r}}{\prod_{k=1}^K \hat{\ell}_{d, d^\prime, r, k}\inv} \Fun*{\exp}{-\frac{1}{2} \sum_{k=1}^K \frac{(x_k - x^\prime_k)^2}{\hat{\ell}_{d, d^\prime, r, k}^2}},
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+    \MoveEqLeft[1] \Moment{\cov}{f_d(\mat{x}), f_{d^\prime}(\mat{x^\prime})} = \sum_{r=1}^R \frac{(2\pi)^{\frac{K}{2}}\sigma_{d, r}\sigma_{d^\prime, r}}{\prod_{k=1}^K \hat{\ell}_{d, d^\prime, r, k}\inv} \Fun*{\exp}{-\frac{1}{2} \sum_{k=1}^K \frac{(x_k - x^\prime_k)^2}{\hat{\ell}_{d, d^\prime, r, k}^2}},
151 131
 \end{split}
152 132
 \end{align}
153 133
 with $\hat{\ell}_{d, d^\prime, r, k} = \sqrt{\ell_{d, r, k}^2 + \ell_{d^\prime, r, k}^2}$.
@@ -203,8 +183,8 @@ It can be written using its marginals \parencite{titsias_variational_2009} as
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 \label{eq:augmented_joint}
204 184
 \begin{split}
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     \Prob{\rv{\hat{a}}, \rv{u}} &= \Gaussian{\rv{\hat{a}} \given \mat{\mu_a}, \mat{\Sigma_a}}\Gaussian{\rv{u} \given \rv{Z}, \mat{K_{uu}}}\text{, with} \\
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-    \mat{\mu_a} &= \mat{X} + \mat{K_{au}}\mat{K_{uu}}\inv(\rv{u} - \mat{Z}), \\
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-    \mat{\Sigma_a} &= \mat{K_{aa}} - \mat{K_{au}}\mat{K_{uu}}\inv\mat{K_{ua}},
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+    \mat{\mu_a} &= \mat{X} + \mat{K_{au}}\mat{K_{uu}}\inv(\rv{u} - \mat{Z}), \qquad
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+    \mat{\Sigma_a} = \mat{K_{aa}} - \mat{K_{au}}\mat{K_{uu}}\inv\mat{K_{ua}},
208 188
 \end{split}
209 189
 \end{align}
210 190
 where, after dropping some indices and explicit conditioning on $\mat{X}$ and $\mat{Z}$ for clarity, $\rv{\hat{a}}$ denotes the function values $a_d(\mat{X})$ without noise and we write the Gram matrices as $\mat{K_{au}} = k_{a, d}(\mat{X}, \mat{Z})$.
@@ -233,8 +213,8 @@ After dropping the indices again, using Jensen's inequality we get
233 213
 \begin{align}
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 \label{eq:svgp_log_likelihood}
235 215
 \begin{split}
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-    \log \Prob{\rv{a} \given \mat{X}} &= \log \int \Prob{\rv{a} \given \rv{u}} \Prob{\rv{u}} \diff \rv{u} \\
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-    \MoveEqLeft\geq \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}} - \KL{\Variat{\rv{u}}}{\Prob{\rv{u}}},
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+    \log \Prob{\rv{a} \given \mat{X}} &= \log \int \Prob{\rv{a} \given \rv{u}} \Prob{\rv{u}} \diff \rv{u}
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+    \geq \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}} - \KL{\Variat{\rv{u}}}{\Prob{\rv{u}}},
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 \end{split}
239 219
 \end{align}
240 220
 where $\Moment{\E_{\Variat{\rv{u}}}}{{}\cdot{}}$ denotes the expected value with respect to the distribution $\Variat{\rv{u}}$ and $\KL{{}\cdot{}}{{}\cdot{}}$ denotes the KL divergence, which can be evaluated analytically.
@@ -249,24 +229,25 @@ To bound the required expectation, we use Jensen's inequality again together wit
249 229
 We apply this bound to the expectation to get
250 230
 \begin{align}
251 231
 \begin{split}
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-    \MoveEqLeft[4] \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}}
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-    \geq \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}} \\
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-    &- \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}\text{, with}
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+    \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}}
233
+    \geq \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}}
234
+    - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}\text{, with}
255 235
 \end{split} \displaybreak[0]\\
256 236
 \begin{split}
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-    \MoveEqLeft[4] \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}} = \\
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-    &\log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye} \\
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-    &+ \frac{1}{2\sigma_a^2}\Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
237
+    \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}} =
238
+    \log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye}
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+    + \frac{1}{2\sigma_a^2}\Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
260 240
 \end{split}
261 241
 \end{align}
262 242
 Resubstituting this result into \cref{eq:svgp_log_likelihood} yields the final bound
263 243
 \begin{align}
264 244
 \label{eq:svgp_bound}
265 245
 \begin{split}
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-    \MoveEqLeft\log \Prob{\rv{a} \given \rv{X}} \geq \\
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-    &\log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye} \\
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-    &- \vphantom{\frac{1}{2\sigma_a^2}} \KL*{\Variat{\rv{u}}}{\Prob{\rv{u}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
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-    &- \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
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+    \log \Prob{\rv{a} \given \rv{X}} \geq
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+    &\log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye}
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+    - \vphantom{\frac{1}{2\sigma_a^2}} \KL*{\Variat{\rv{u}}}{\Prob{\rv{u}}} \\
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+    &- \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}
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+    - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
270 251
 \end{split}
271 252
 \end{align}
272 253
 This bound, which depends on the hyper parameters of the kernel and likelihood $\left\{ \mat{\theta}, \sigma_a \right\}$ and the variational parameters $\left\{\mat{Z}, \mat{m}, \mat{S} \right\}$, can be calculated in $\Oh(NM^2)$ time.
@@ -292,8 +273,8 @@ We will therefore need to handle all of the different $\rv{f_d}$, together denot
292 273
 We start by considering the relevant terms from \cref{eq:full_model} and apply \cref{eq:svgp_log_marginal_likelihood} to marginalize $\rv{a}$ in
293 274
 \begin{align}
294 275
 \begin{split}
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-    \log\Prob{\rv{f} \given \rv{u_f}, \rv{u_a}} &\geq \Moment{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}} \\
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-    \MoveEqLeft[5] - \Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}}
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+    \log\Prob{\rv{f} \given \rv{u_f}, \rv{u_a}} &\geq \Moment{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}}
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+    - \Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}}
297 278
     - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}},
298 279
 \end{split}
299 280
 \end{align}
@@ -301,42 +282,38 @@ where we write $\aProb{\rv{a} \given \rv{u_a}} = \Gaussian*{\rv{a} \given \mat{\
301 282
 Due to our assumption that $\rv{u_a}$ is a sufficient statistic for $\rv{a}$ we choose
302 283
 \begin{align}
303 284
 \label{eq:variational_assumption}
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-\begin{split}
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-\Variat{\rv{a} \given \rv{u_a}} &= \aProb{\rv{a} \given \rv{u_a}}\text{, and}\\
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-\Variat{\rv{a}} &= \int \aProb{\rv{a} \given \rv{u_a}} \Variat{\rv{u_a}} \diff \rv{u_a},
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-\end{split}
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+    \Variat{\rv{a} \given \rv{u_a}} &= \aProb{\rv{a} \given \rv{u_a}}\text{, and} &
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+    \Variat{\rv{a}} &= \int \aProb{\rv{a} \given \rv{u_a}} \Variat{\rv{u_a}} \diff \rv{u_a},
308 287
 \end{align}
309 288
 and use another variational approximation to marginalize $\rv{u_a}$.
310 289
 This yields
311 290
 \begin{align}
312 291
 \begin{split}
313 292
     \label{eq:f_marginal_likelihood}
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-    \MoveEqLeft \log \Prob{\rv{f} \given \rv{u_f}} \geq \\
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-    &- \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
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-    &- \frac{1}{2\sigma_f^2} \Moment*{\E_{\Variat{\rv{a}}}}{\Fun*{\tr}{\mat{\Sigma_f}}} \\
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-    &+ \Moment*{\E_{\Variat{\rv{a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}}.
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+    \log \Prob{\rv{f} \given \rv{u_f}} \geq
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+    &\Moment*{\E_{\Variat{\rv{a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}}
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+    - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} \\
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+    &- \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}
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+    - \frac{1}{2\sigma_f^2} \Moment*{\E_{\Variat{\rv{a}}}}{\Fun*{\tr}{\mat{\Sigma_f}}}.
318 298
 \end{split}
319 299
 \end{align}
320 300
 The expectations with respect to $\Variat{\rv{a}}$ involve expectations of kernel matrices, also called $\Psi$-statistics, in the same way as in \parencites{damianou_deep_2012} and are given by
321 301
 \begin{align}
322
-\begin{split}
323 302
     \label{eq:psi_statistics}
324
-    \psi_f &= \Moment*{\E_{\Variat{\rv{a}}}}{\Fun*{\tr}{\mat{K_{ff}}}}, \\
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-    \mat{\Psi_f} &= \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{fu}}}, \\
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-    \mat{\Phi_f} &= \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{uf}}\mat{K_{fu}}}. \\
327
-\end{split}
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+    \psi_f &= \Moment*{\E_{\Variat{\rv{a}}}}{\Fun*{\tr}{\mat{K_{ff}}}}, &
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+    \mat{\Psi_f} &= \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{fu}}}, &
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+    \mat{\Phi_f} &= \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{uf}}\mat{K_{fu}}}.
328 306
 \end{align}
329 307
 These $\Psi$-statistics can be computed analytically for multiple kernels, including the squared exponential kernel in \cref{eq:rbf_kernel}.
330 308
 In \cref{subsec:kernel_expectations} we show closed-form solutions for these $\Psi$-statistics for the implicit kernel defined in the CP layer.
331 309
 To obtain the final formulation of the desired bound for $\log \Prob{\rv{f} \given \rv{u_f}}$ we substitute \cref{eq:psi_statistics} into \cref{eq:f_marginal_likelihood} and get the analytically tractable bound
332 310
 \begin{align}
333 311
 \begin{split}
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-    \MoveEqLeft \log \Prob{\rv{f} \given \rv{u_f}} \geq \\
335
-    &- \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
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+    \log \Prob{\rv{f} \given \rv{u_f}} \geq
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+    &\log\Gaussian*{\rv{f} \given \mat{\Psi_f}\mat{K_{u_fu_f}}\inv \mat{m_f}, \sigma_f^2\Eye}
314
+    - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
336 315
     &- \frac{1}{2\sigma_f^2} \left( \psi_f - \Fun*{\tr}{\mat{\Psi_f}\mat{K_{u_fu_f}}\inv} \right) \\
337
-    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \vphantom{\mat{K_{u_f}}\inv}\right. \\
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-    &\qquad\qquad \left. \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right) \\
339
-    &+ \log\Gaussian*{\rv{f} \given \mat{\Psi_f}\mat{K_{u_fu_f}}\inv \mat{m_f}, \sigma_f^2\Eye}.
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+    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right)
340 317
 \end{split}
341 318
 \end{align}
342 319
 The uncertainties in the first layer have been propagated variationally to the second layer.
@@ -347,17 +324,17 @@ In order to derive a bound for $\log \Prob{\rv{y} \given \rv{u_g}}$ we apply the
347 324
 \begin{align}
348 325
 \label{eq:full_bound}
349 326
 \begin{split}
350
-    \MoveEqLeft[1]\log \Prob{\rv{y}\given \mat{X}} \geq - \sum_{d=1}^D \frac{1}{2\sigma_{a, d}^2} \Fun{\tr}{\mat{\Sigma_{a, d}}} \\
351
-    &- \frac{1}{2\sigma_f^2} \left( \psi_{f} - \Fun*{\tr}{\mat{\Phi_f} \mat{K_{u_fu_f}}\inv} \right) \\
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-    &- \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \left( \psi_{g, d} - \Fun*\tr{\mat{\Phi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv} \right) \\
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-    &- \sum_{d=1}^D \KL{\Variat{\rv{u_{a, d}}}}{\Prob{\rv{u_{a, d}}}} \\
354
-    &- \KL{\Variat{\rv{u_f}}}{\Prob{\rv{u_f}}} \\
355
-    &- \sum_{d=1}^D \KL{\Variat{\rv{u_{y, d}}}}{\Prob{\rv{u_{y, d}}}} \\
356
-    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \vphantom{\mat{K_{u_f}}\inv}\right. \\
357
-    &\quad \left. \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right) \\
358
-    &- \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \tr\left(\left(\mat{\Phi_{g, d}} - \mat{\Psi_{g, d}}\tran\mat{\Psi_{g, d}}\right) \vphantom{\mat{K_{u_f}}\inv}\right. \\
359
-    &\quad \left. \mat{K_{u_{g, d}u_{g, d}}}\inv \left(\mat{m_{g, d}}\mat{m_{g, d}}\tran + \mat{S_{g, d}}\right) \right. \left. \mat{K_{u_{g, d}u_{g, d}}}\inv\right) \\
360
-    &+ \sum_{d=1}^D \log\Gaussian*{\rv{y_d} \given \mat{\Psi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv \mat{m_{g, d}}, \sigma_{y, d}^2 \Eye}
327
+    \MoveEqLeft\log \Prob{\rv{y}\given \mat{X}} \geq
328
+    \sum_{d=1}^D \log\Gaussian*{\rv{y_d} \given \mat{\Psi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv \mat{m_{g, d}}, \sigma_{y, d}^2 \Eye}
329
+    - \sum_{d=1}^D \frac{1}{2\sigma_{a, d}^2} \Fun{\tr}{\mat{\Sigma_{a, d}}} \\
330
+    &- \frac{1}{2\sigma_f^2} \left( \psi_{f} - \Fun*{\tr}{\mat{\Phi_f} \mat{K_{u_fu_f}}\inv} \right)
331
+    - \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \left( \psi_{g, d} - \Fun*\tr{\mat{\Phi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv} \right) \\
332
+    &- \sum_{d=1}^D \KL{\Variat{\rv{u_{a, d}}}}{\Prob{\rv{u_{a, d}}}}
333
+    - \KL{\Variat{\rv{u_f}}}{\Prob{\rv{u_f}}}
334
+    - \sum_{d=1}^D \KL{\Variat{\rv{u_{y, d}}}}{\Prob{\rv{u_{y, d}}}} \\
335
+    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right) \\
336
+    &- \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \tr\left(\left(\mat{\Phi_{g, d}} - \mat{\Psi_{g, d}}\tran\mat{\Psi_{g, d}}\right)
337
+    \mat{K_{u_{g, d}u_{g, d}}}\inv \left(\mat{m_{g, d}}\mat{m_{g, d}}\tran + \mat{S_{g, d}}\right) \mat{K_{u_{g, d}u_{g, d}}}\inv\right)
361 338
 \end{split}
362 339
 \end{align}
363 340
 
@@ -380,18 +357,18 @@ Similar to the notation $\Fun{\hat{f}}{\cdot}$, we use the notation $\hat{\sigma
380 357
 The expectation $\mat{\Psi_f} = \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{fu}}}$ connecting the alignments and the pseudo inputs is given by
381 358
 \begin{align}
382 359
 \begin{split}
383
-    \left( \mat{\Psi_f} \right)_{ni} &= \hat{\sigma}_{ni}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + (\mat{\Sigma_a})_{nn}\inv}} \\
384
-    \MoveEqLeft \exp\left(-\frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv\hat{\ell}_{ni}}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni}} \left((\mat{\mu_a})_n - \mat{u_i}\right)^2\right),
360
+    \left( \mat{\Psi_f} \right)_{ni} &= \hat{\sigma}_{ni}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + (\mat{\Sigma_a})_{nn}\inv}}
361
+    \exp\left(-\frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv\hat{\ell}_{ni}}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni}} \left((\mat{\mu_a})_n - \mat{u_i}\right)^2\right),
385 362
 \end{split}
386 363
 \end{align}
387 364
 where $\hat{\ell}_{ni}$ is the combined length scale corresponding to the same kernel as $\hat{\sigma}_{ni}$.
388 365
 Lastly, $\mat{\Phi_f} = \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{uf}}\mat{K_{fu}}}$ connects alignments and pairs of pseudo inputs with the closed form
389 366
 \begin{align}
390 367
 \begin{split}
391
-    \left( \mat{\Phi_f} \right)_{ij} &= \sum_{n=1}^N \hat{\sigma}_{ni}^2 \hat{\sigma}_{nj}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + \hat{\ell}_{nj} + (\mat{\Sigma_a})_{nn}\inv}} \\
392
-    &\quad \exp\left( -\frac{1}{2} \vphantom{\left(\frac{\hat{\ell}}{\hat{\ell}}\right)^2} \frac{\hat{\ell}_{ni}\hat{\ell}_{nj}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} (\mat{u_i} - \mat{u_j})^2 \right. \\
393
-    &\quad {} - \frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv(\hat{\ell}_{ni} + \hat{\ell}_{nj})}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni} + \hat{\ell}_{nj}} \\
394
-    &\quad \left.\left( (\mat{\mu_a})_n - \frac{\hat{\ell}_{ni} \mat{u_i} + \hat{\ell}_{nj} \mat{u_j}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} \right)^2 \right).
368
+    \left( \mat{\Phi_f} \right)_{ij} &= \sum_{n=1}^N \hat{\sigma}_{ni}^2 \hat{\sigma}_{nj}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + \hat{\ell}_{nj} + (\mat{\Sigma_a})_{nn}\inv}}
369
+    \exp\left( -\frac{1}{2} \vphantom{\left(\frac{\hat{\ell}}{\hat{\ell}}\right)^2} \frac{\hat{\ell}_{ni}\hat{\ell}_{nj}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} (\mat{u_i} - \mat{u_j})^2 \right. \\
370
+    &\quad {} - \frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv(\hat{\ell}_{ni} + \hat{\ell}_{nj})}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni} + \hat{\ell}_{nj}}
371
+    \left.\left( (\mat{\mu_a})_n - \frac{\hat{\ell}_{ni} \mat{u_i} + \hat{\ell}_{nj} \mat{u_j}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} \right)^2 \right).
395 372
 \end{split}
396 373
 \end{align}
397 374
 
@@ -611,8 +588,8 @@ The joint probability distribution of the data can then be written as
611 588
 \begin{align}
612 589
 \begin{split}
613 590
     \label{app:eq:full_model}
614
-    \Prob{\rv{y}, \rv{f}, \rv{a} \given \mat{X}} &= \\
615
-    \MoveEqLeft\Prob{\rv{f} \given \rv{a}} \prod_{d=1}^D \Prob{\rv{y_d} \given \rv{f_d}}\Prob{\rv{a_d} \given \rv{X}}, \\
591
+    \Prob{\rv{y}, \rv{f}, \rv{a} \given \mat{X}} &=
592
+    \Prob{\rv{f} \given \rv{a}} \prod_{d=1}^D \Prob{\rv{y_d} \given \rv{f_d}}\Prob{\rv{a_d} \given \rv{X}}, \\
616 593
     \rv{a_d} \mid \mat{X} &\sim \Gaussian{\mat{X}, \mat{K_{a, d}} + \sigma^2_{a, d}\Eye}, \\
617 594
     \rv{f} \mid \mat{a} &\sim \Gaussian{\mat{0}, \mat{K_f} + \sigma^2_f\Eye}, \\
618 595
     \rv{y_d} \mid \mat{f_d} &\sim \Gaussian{\mat{f_d}, \mat{K_{g, d}} + \sigma^2_{y, d}\Eye}.
@@ -667,10 +644,8 @@ After dropping the indices again, using Jensen's inequality we get
667 644
     \log \Prob{\rv{a} \given \mat{X}} &= \log \int \Prob{\rv{a} \given \rv{u}} \Prob{\rv{u}} \diff \rv{u} \\
668 645
     &= \log \int \Variat{\rv{u}} \frac{\Prob{\rv{a} \given \rv{u}} \Prob{\rv{u}}}{\Variat{\rv{u}}} \diff \rv{u} \\
669 646
     &\geq \int \Variat{\rv{u}} \log \frac{\Prob{\rv{a} \given \rv{u}} \Prob{\rv{u}}}{\Variat{\rv{u}}} \diff \rv{u} \\
670
-    &= \int \log \Prob{\rv{a} \given \rv{u}} \Variat{\rv{u}} \diff \rv{u} \\
671
-    &\quad {} - \int \Variat{\rv{u}} \log \frac{\Variat{\rv{u}}}{\Prob{\rv{u}}} \diff \rv{u} \\
672
-    &= \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}} \\
673
-    &\quad {} - \KL{\Variat{\rv{u}}}{\Prob{\rv{u}}},
647
+    &= \int \log \Prob{\rv{a} \given \rv{u}} \Variat{\rv{u}} \diff \rv{u} - \int \Variat{\rv{u}} \log \frac{\Variat{\rv{u}}}{\Prob{\rv{u}}} \diff \rv{u} \\
648
+    &= \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}} - \KL{\Variat{\rv{u}}}{\Prob{\rv{u}}},
674 649
 \end{split}
675 650
 \end{align}
676 651
 where $\Moment{\E_{\Variat{\rv{u}}}}{{}\cdot{}}$ denotes the expected value with respect to the distribution $\Variat{\rv{u}}$ and $\KL{{}\cdot{}}{{}\cdot{}}$ denotes the KL divergence, which can be evaluated analytically.
@@ -679,7 +654,7 @@ To bound the required expectation, we use Jensen's inequality again together wit
679 654
 \begin{align}
680 655
 \label{app:eq:svgp_log_marginal_likelihood}
681 656
 \begin{split}
682
-    \MoveEqLeft\log\Prob{\rv{a} \given \rv{u}} = \\
657
+    \log\Prob{\rv{a} \given \rv{u}}
683 658
     &= \log\int \Prob{\rv{a} \given \rv{\hat{a}}} \Prob{\rv{\hat{a}} \given \rv{u}} \diff \rv{\hat{a}} \\
684 659
     &= \log\int \Gaussian{\rv{a} \given \rv{\hat{a}}, \sigma_a^2 \Eye} \Gaussian{\rv{\hat{a}} \given \mat{\mu_a}, \mat{\Sigma_a}} \diff \rv{\hat{a}} \\
685 660
     &\geq \int \log\Gaussian{\rv{a} \given \rv{\hat{a}}, \sigma_a^2 \Eye} \Gaussian{\rv{\hat{a}} \given \mat{\mu_a}, \mat{\Sigma_a}} \diff \rv{\hat{a}} \\
@@ -689,24 +664,25 @@ To bound the required expectation, we use Jensen's inequality again together wit
689 664
 We apply this bound to the expectation to get
690 665
 \begin{align}
691 666
 \begin{split}
692
-    \MoveEqLeft[4] \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}}
693
-    \geq \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}} \\
694
-    &- \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}\text{, with}
667
+    \Moment{\E_{\Variat{\rv{u}}}}{\log \Prob{\rv{a} \given \rv{u}}}
668
+    &\geq \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}}
669
+    - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}\text{, with}
695 670
 \end{split} \\
696 671
 \begin{split}
697
-    \MoveEqLeft[4] \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}} = \\
698
-    &\log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye} \\
699
-    &+ \frac{1}{2\sigma_a^2}\Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
672
+    \Moment{\E_{\Variat*{\rv{u}}}}{\log\Gaussian{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}}
673
+    &= \log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye} \\
674
+    &\quad {} + \frac{1}{2\sigma_a^2}\Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
700 675
 \end{split}
701 676
 \end{align}
702 677
 Resubstituting this result into \cref{app:eq:svgp_log_likelihood} yields the final bound
703 678
 \begin{align}
704 679
 \label{app:eq:svgp_bound}
705 680
 \begin{split}
706
-    \MoveEqLeft\log \Prob{\rv{a} \given \rv{X}} \geq \\
707
-    &\log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye} \\
708
-    &- \vphantom{\frac{1}{2\sigma_a^2}} \KL*{\Variat{\rv{u}}}{\Prob{\rv{u}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
709
-    &- \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
681
+    \log \Prob{\rv{a} \given \rv{X}}
682
+    &\geq \log \Gaussian{\rv{a} \given \mat{K_{au}}\mat{K_{uu}}\inv\mat{m}, \sigma_a^2 \Eye}
683
+    - \vphantom{\frac{1}{2\sigma_a^2}} \KL*{\Variat{\rv{u}}}{\Prob{\rv{u}}} \\
684
+    &\quad {} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}
685
+    - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{K_{au}}\mat{K_{uu}}\inv\mat{S}\mat{K_{uu}}\inv\mat{K_{ua}}}.
710 686
 \end{split}
711 687
 \end{align}
712 688
 This bound, which depends on the hyper parameters of the kernel and likelihood $\left\{ \mat{\theta}, \sigma_a \right\}$ and the variational parameters $\left\{\mat{Z}, \mat{m}, \mat{S} \right\}$, can be calculated in $\Oh(NM^2)$ time.
@@ -732,13 +708,13 @@ We will therefore need to handle all of the different $\rv{f_d}$, together denot
732 708
 We start by considering the relevant terms from \cref{app:eq:full_model} and apply \cref{app:eq:svgp_log_marginal_likelihood} to marginalize $\rv{a}$ in
733 709
 \begin{align}
734 710
 \begin{split}
735
-    \MoveEqLeft \log\Prob{\rv{f} \given \rv{u_f}, \rv{u_a}} = \\
711
+    \log\Prob{\rv{f} \given \rv{u_f}, \rv{u_a}}
736 712
     &= \log\int\Prob{\rv{f}, \rv{a} \given \rv{u_f}, \rv{u_a}}\diff\rv{a} \\
737
-    &\geq \log\int \aProb{\rv{f} \given \rv{u_f}, \rv{a}} \aProb{\rv{a} \given \rv{u_a}} \\
738
-    &\quad {} \cdot \Fun*{\exp}{-\frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} - \frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}} \diff \rv{a} \\
739
-    &\geq \Moment{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}} \\
740
-    &\quad {} - \Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}} \\
741
-    &\quad {} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}},
713
+    &\geq \log\int \aProb{\rv{f} \given \rv{u_f}, \rv{a}} \aProb{\rv{a} \given \rv{u_a}}
714
+    \cdot \Fun*{\exp}{-\frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} - \frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}} \diff \rv{a} \\
715
+    &\geq \Moment{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}}
716
+    - \Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}}
717
+    - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}},
742 718
 \end{split}
743 719
 \end{align}
744 720
 where we write $\aProb{\rv{a} \given \rv{u_a}} = \Gaussian*{\rv{a} \given \mat{\mu_a}, \sigma_a^2 \Eye}$ to incorporate the Gaussian noise in the latent space.
@@ -755,18 +731,20 @@ This yields
755 731
 \begin{align}
756 732
 \begin{split}
757 733
     \label{app:eq:f_marginal_likelihood}
758
-    \MoveEqLeft \log \Prob{\rv{f} \given \rv{u_f}} = \\
734
+    \log \Prob{\rv{f} \given \rv{u_f}}
759 735
     &= \log \int \Prob{\rv{f}, \rv{u_a} \given \rv{u_f}} \diff \rv{u_a} \\
760 736
     &= \log \int \Prob{\rv{f} \given \rv{u_f}, \rv{u_a}} \Prob{\rv{u_a}} \diff \rv{u_a} \\
761 737
     &\geq \int \Variat{\rv{u_a}} \log\frac{\Prob{\rv{f} \given \rv{u_f}, \rv{u_a}} \Prob{\rv{u_a}}}{\Variat{\rv{u_a}}} \diff \rv{u_a} \\
762
-    &= \Moment*{\E_{\Variat{\rv{u_a}}}}{\log \Prob{\rv{f} \given \rv{u_a}, \rv{u_f}}} \\
763
-    &\quad {} - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} \\
764
-    &\geq - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
765
-    &\quad {} - \Moment*{\E_{\Variat{\rv{u_a}}}}{\Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}}} \\
766
-    &\quad {} + \Moment*{\E_{\Variat{\rv{u_a}}}}{\Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}}} \\
767
-    &\geq - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
768
-    &\quad {} - \frac{1}{2\sigma_f^2} \Moment*{\E_{\Variat{\rv{a}}}}{\Fun*{\tr}{\mat{\Sigma_f}}} \\
769
-    &\quad {} + \Moment*{\E_{\Variat{\rv{a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}},
738
+    &= \Moment*{\E_{\Variat{\rv{u_a}}}}{\log \Prob{\rv{f} \given \rv{u_a}, \rv{u_f}}}
739
+    - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} \\
740
+    &\geq \Moment*{\E_{\Variat{\rv{u_a}}}}{\Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}}}
741
+    - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} \\
742
+    &\quad {} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}
743
+    - \Moment*{\E_{\Variat{\rv{u_a}}}}{\Moment*{\E_{\aProb{\rv{a} \given \rv{u_a}}}}{\frac{1}{2\sigma_f^2} \Fun*{\tr}{\mat{\Sigma_f}}}} \\
744
+    &\geq \Moment*{\E_{\Variat{\rv{a}}}}{\log \aProb{\rv{f} \given \rv{u_f}, \rv{a}}},
745
+    - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} \\
746
+    &\quad {} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}}
747
+    - \frac{1}{2\sigma_f^2} \Moment*{\E_{\Variat{\rv{a}}}}{\Fun*{\tr}{\mat{\Sigma_f}}},
770 748
 \end{split}
771 749
 \end{align}
772 750
 where we apply Fubini's theorem to exchange the order of integration in the expected values.
@@ -784,12 +762,11 @@ In \cref{app:subsec:kernel_expectations} we show closed-form solutions for these
784 762
 To obtain the final formulation of the desired bound for $\log \Prob{\rv{f} \given \rv{u_f}}$ we substitute \cref{app:eq:psi_statistics} into \cref{app:eq:f_marginal_likelihood} and get the analytically tractable bound
785 763
 \begin{align}
786 764
 \begin{split}
787
-    \MoveEqLeft \log \Prob{\rv{f} \given \rv{u_f}} \geq \\
788
-    &- \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
765
+    \log \Prob{\rv{f} \given \rv{u_f}} \geq
766
+    &\log\Gaussian*{\rv{f} \given \mat{\Psi_f}\mat{K_{u_fu_f}}\inv \mat{m_f}, \sigma_f^2\Eye}
767
+    - \KL{\Variat{\rv{u_a}}}{\Prob{\rv{u_a}}} - \frac{1}{2\sigma_a^2} \Fun*{\tr}{\mat{\Sigma_a}} \\
789 768
     &- \frac{1}{2\sigma_f^2} \left( \psi_f - \Fun*{\tr}{\mat{\Psi_f}\mat{K_{u_fu_f}}\inv} \right) \\
790
-    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \vphantom{\mat{K_{u_f}}\inv}\right. \\
791
-    &\qquad\qquad \left. \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right) \\
792
-    &+ \log\Gaussian*{\rv{f} \given \mat{\Psi_f}\mat{K_{u_fu_f}}\inv \mat{m_f}, \sigma_f^2\Eye}.
769
+    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right)
793 770
 \end{split}
794 771
 \end{align}
795 772
 The uncertainties in the first layer have been propagated variationally to the second layer.
@@ -800,19 +777,17 @@ In order to derive a bound for $\log \Prob{\rv{y} \given \rv{u_g}}$ we apply the
800 777
 \begin{align}
801 778
 \label{app:eq:full_bound}
802 779
 \begin{split}
803
-    \MoveEqLeft[1]\log \Prob{\rv{y}\given \mat{X}} \geq \\
804
-    &- \sum_{d=1}^D \frac{1}{2\sigma_{a, d}^2} \Fun{\tr}{\mat{\Sigma_{a, d}}} \\
805
-    &- \frac{1}{2\sigma_f^2} \left( \psi_{f} - \Fun*{\tr}{\mat{\Phi_f} \mat{K_{u_fu_f}}\inv} \right) \\
806
-    &- \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \left( \psi_{g, d} - \Fun*\tr{\mat{\Phi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv} \right) \\
807
-    &- \sum_{d=1}^D \KL{\Variat{\rv{u_{a, d}}}}{\Prob{\rv{u_{a, d}}}} \\
808
-    &- \KL{\Variat{\rv{u_f}}}{\Prob{\rv{u_f}}} \\
809
-    &- \sum_{d=1}^D \KL{\Variat{\rv{u_{y, d}}}}{\Prob{\rv{u_{y, d}}}} \\
810
-    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \vphantom{\mat{K_{u_f}}\inv}\right. \\
811
-    &\qquad\qquad \left. \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right) \\
812
-    &- \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \tr\left(\left(\mat{\Phi_{g, d}} - \mat{\Psi_{g, d}}\tran\mat{\Psi_{g, d}}\right) \vphantom{\mat{K_{u_f}}\inv}\right. \\
813
-    &\qquad\qquad \left. \mat{K_{u_{g, d}u_{g, d}}}\inv \left(\mat{m_{g, d}}\mat{m_{g, d}}\tran + \mat{S_{g, d}}\right) \right. \\
814
-    &\qquad\qquad \left. \mat{K_{u_{g, d}u_{g, d}}}\inv\right) \\
815
-    &+ \sum_{d=1}^D \log\Gaussian*{\rv{y_d} \given \mat{\Psi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv \mat{m_{g, d}}, \sigma_{y, d}^2 \Eye}
780
+    \MoveEqLeft\log \Prob{\rv{y}\given \mat{X}} \geq
781
+    \sum_{d=1}^D \log\Gaussian*{\rv{y_d} \given \mat{\Psi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv \mat{m_{g, d}}, \sigma_{y, d}^2 \Eye}
782
+    - \sum_{d=1}^D \frac{1}{2\sigma_{a, d}^2} \Fun{\tr}{\mat{\Sigma_{a, d}}} \\
783
+    &- \frac{1}{2\sigma_f^2} \left( \psi_{f} - \Fun*{\tr}{\mat{\Phi_f} \mat{K_{u_fu_f}}\inv} \right)
784
+    - \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \left( \psi_{g, d} - \Fun*\tr{\mat{\Phi_{g, d}} \mat{K_{u_{g, d}u_{g, d}}}\inv} \right) \\
785
+    &- \sum_{d=1}^D \KL{\Variat{\rv{u_{a, d}}}}{\Prob{\rv{u_{a, d}}}}
786
+    - \KL{\Variat{\rv{u_f}}}{\Prob{\rv{u_f}}}
787
+    - \sum_{d=1}^D \KL{\Variat{\rv{u_{y, d}}}}{\Prob{\rv{u_{y, d}}}} \\
788
+    &- \frac{1}{2\sigma_f^2} \tr\left(\left(\mat{\Phi_f} - \mat{\Psi_f}\tran\mat{\Psi_f}\right) \mat{K_{u_fu_f}}\inv \left(\mat{m_f}\mat{m_f}\tran + \mat{S_f}\right)\mat{K_{u_fu_f}}\inv\right) \\
789
+    &- \sum_{d=1}^D\frac{1}{2\sigma_{y, d}^2} \tr\left(\left(\mat{\Phi_{g, d}} - \mat{\Psi_{g, d}}\tran\mat{\Psi_{g, d}}\right)
790
+    \mat{K_{u_{g, d}u_{g, d}}}\inv \left(\mat{m_{g, d}}\mat{m_{g, d}}\tran + \mat{S_{g, d}}\right) \mat{K_{u_{g, d}u_{g, d}}}\inv\right)
816 791
 \end{split}
817 792
 \end{align}
818 793
 
@@ -841,10 +816,10 @@ The expectation $\mat{\Psi_f} = \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{fu}}}$ co
841 816
 \begin{align}
842 817
 \begin{split}
843 818
     \mat{\Psi_f} &= \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{fu}}}\text{, with} \\
844
-    \MoveEqLeft\left( \mat{\Psi_f} \right)_{ni} = \\
819
+    \left( \mat{\Psi_f} \right)_{ni}
845 820
     &= \int \Moment*{\cov}{\Fun{\hat{f}}{\mat{a_n}}, \Fun{\hat{f}}{\mat{u_i}}} \Variat{\rv{a_n}} \diff \rv{a_n} \\
846
-    &= \hat{\sigma}_{ni}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + (\mat{\Sigma_a})_{nn}\inv}} \\
847
-    &\quad {} \cdot \exp\left(-\frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv\hat{\ell}_{ni}}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni}} \left((\mat{\mu_a})_n - \mat{u_i}\right)^2\right)
821
+    &= \hat{\sigma}_{ni}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + (\mat{\Sigma_a})_{nn}\inv}}
822
+    \cdot \exp\left(-\frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv\hat{\ell}_{ni}}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni}} \left((\mat{\mu_a})_n - \mat{u_i}\right)^2\right)
848 823
 \end{split}
849 824
 \end{align}
850 825
 where $\hat{\ell}_{ni}$ is the combined length scale corresponding to the same kernel as $\hat{\sigma}_{ni}$.
@@ -852,12 +827,12 @@ Lastly, $\mat{\Phi_f} = \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{uf}}\mat{K_{fu}}}
852 827
 \begin{align}
853 828
 \begin{split}
854 829
     \mat{\Phi_f} &= \Moment*{\E_{\Variat{\rv{a}}}}{\mat{K_{uf}}\mat{K_{fu}}}\text{, with} \\
855
-    \left( \mat{\Phi_f} \right)_{ij} &= \sum_{n=1}^N \int \Moment*{\cov}{\Fun{\hat{f}}{\mat{a_n}}, \Fun{\hat{f}}{\mat{u_i}}} \\
856
-    &\quad {} \cdot \Moment*{\cov}{\Fun{\hat{f}}{\mat{a_n}}, \Fun{\hat{f}}{\mat{u_j}}} \Variat{\rv{a_n}} \diff \rv{a_n} \\
857
-    &= \sum_{n=1}^N \hat{\sigma}_{ni}^2 \hat{\sigma}_{nj}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + \hat{\ell}_{nj} + (\mat{\Sigma_a})_{nn}\inv}} \\
858
-    &\quad {} \cdot \exp\left( -\frac{1}{2} \frac{\hat{\ell}_{ni}\hat{\ell}_{nj}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} (\mat{u_i} - \mat{u_j})^2 \right. \\
859
-    &\quad {} - \frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv(\hat{\ell}_{ni} + \hat{\ell}_{nj})}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni} + \hat{\ell}_{nj}} \\
860
-    &\quad {} \cdot \left.\left( (\mat{\mu_a})_n - \frac{\hat{\ell}_{ni} \mat{u_i} + \hat{\ell}_{nj} \mat{u_j}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} \right)^2 \right).
830
+    \left(\mat{\Phi_f} \right)_{ij} &= \sum_{n=1}^N \int \Moment*{\cov}{\Fun{\hat{f}}{\mat{a_n}}, \Fun{\hat{f}}{\mat{u_i}}}
831
+    \cdot \Moment*{\cov}{\Fun{\hat{f}}{\mat{a_n}}, \Fun{\hat{f}}{\mat{u_j}}} \Variat{\rv{a_n}} \diff \rv{a_n} \\
832
+    &= \sum_{n=1}^N \hat{\sigma}_{ni}^2 \hat{\sigma}_{nj}^2 \sqrt{\frac{(\mat{\Sigma_a})_{nn}\inv}{\hat{\ell}_{ni} + \hat{\ell}_{nj} + (\mat{\Sigma_a})_{nn}\inv}}
833
+    \cdot \exp\left( -\frac{1}{2} \frac{\hat{\ell}_{ni}\hat{\ell}_{nj}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} (\mat{u_i} - \mat{u_j})^2 \right. \\
834
+    &\quad {} - \frac{1}{2} \frac{(\mat{\Sigma_a})_{nn}\inv(\hat{\ell}_{ni} + \hat{\ell}_{nj})}{(\mat{\Sigma_a})_{nn}\inv + \hat{\ell}_{ni} + \hat{\ell}_{nj}}
835
+    \cdot \left.\left( (\mat{\mu_a})_n - \frac{\hat{\ell}_{ni} \mat{u_i} + \hat{\ell}_{nj} \mat{u_j}}{\hat{\ell}_{ni} + \hat{\ell}_{nj}} \right)^2 \right).
861 836
 \end{split}
862 837
 \end{align}
863 838
 

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-% File: icml2018.sty (LaTeX style file for ICML-2018, version of 2017-10-28)
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-% Modified by Iain Murray 2018: changed years, location. Remove affiliation notes when anonymous.
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-%     Move times dependency from .tex to .sty so fewer people delete it.
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-% Modified by Daniel Roy 2017: changed byline to use footnotes for affiliations, and removed emails
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-% 
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-% Modified by Fei Sha 9/2/2013: changed the year, location form the previous template for ICML 2013
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-% Modified by Fei Sha 4/24/2013: (1) remove the extra whitespace after the first author's email address (in %the camera-ready version) (2) change the Proceeding ... of ICML 2010 to 2014 so PDF's metadata will show up % correctly
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-% - running title. If the original title is to long or is breaking a line,
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-%   Added fancyhdr package to get a running head. 
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-% - Updated to store the page size because pdflatex does compile the 
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-% - Added a package option [accepted] that selects between the ``Under
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-%\addtofullauthorlist{#1}%
445
-%#1\@for\theaffil:=#2\do{\pa{\theaffil}}%
446
-%}
447
-\newcommand{\icmlauthor}[2]{%
448
-  \ifdefined\isaccepted
449
-    \mbox{\bf #1}\,\@for\theaffil:=#2\do{\@pa{\theaffil}} \addtofullauthorlist{#1}%
450
-   \else
451
-    \ifdefined\@icmlfirsttime
452
-    \else
453
-      \gdef\@icmlfirsttime{1}
454
-      \mbox{\bf Anonymous Authors}\@pa{@anon} \addtofullauthorlist{Anonymous Authors}
455
-     \fi
456
-    \fi
457
-}
458
-
459
-\newcommand{\icmlsetsymbol}[2]{%
460
-  \expandafter\gdef\csname @icmlsymbol#1\endcsname{#2}
461
- }
462
-   
463
-
464
-\newcommand{\icmlaffiliation}[2]{%
465
-\ifdefined\isaccepted
466
-\ifcsname the@affil#1\endcsname
467
- \expandafter\gdef\csname @affilname\csname the@affil#1\endcsname\endcsname{#2}%
468
-\else
469
-  {\bf AUTHORERR: Error in use of \textbackslash{}icmlaffiliation command. Label ``#1'' not mentioned in some \textbackslash{}icmlauthor\{author name\}\{labels here\} command beforehand. }
470
-  \typeout{}%
471
-  \typeout{}%
472
-  \typeout{*******************************************************}%
473
-  \typeout{Affiliation label undefined. }%
474
-  \typeout{Make sure \string\icmlaffiliation\space follows }
475
-  \typeout{all of \string\icmlauthor\space commands}%
476
-  \typeout{*******************************************************}%
477
-  \typeout{}%
478
-  \typeout{}%
479
-\fi
480
-\else % \isaccepted
481
- % can be called multiple times... it's idempotent
482
- \expandafter\gdef\csname @affilname1\endcsname{Anonymous Institution, Anonymous City, Anonymous Region, Anonymous Country}
483
-\fi
484
-}
485
-
486
-\newcommand{\icmlcorrespondingauthor}[2]{
487
-\ifdefined\isaccepted
488
- \ifdefined\icmlcorrespondingauthor@text
489
-   \g@addto@macro\icmlcorrespondingauthor@text{, #1 \textless{}#2\textgreater{}}
490
- \else
491
-   \gdef\icmlcorrespondingauthor@text{#1 \textless{}#2\textgreater{}}
492
- \fi
493
-\else
494
-\gdef\icmlcorrespondingauthor@text{Anonymous Author \textless{}anon.email@domain.com\textgreater{}}
495
-\fi
496
-}
497
-
498
-\newcommand{\icmlEqualContribution}{\textsuperscript{*}Equal contribution }
499
-
500
-\newcounter{@affilnum}
501
-\newcommand{\printAffiliationsAndNotice}[1]{%
502
-\stepcounter{@affiliationcounter}%
503
-{\let\thefootnote\relax\footnotetext{\hspace*{-\footnotesep}\ifdefined\isaccepted #1\fi%
504
-\forloop{@affilnum}{1}{\value{@affilnum} < \value{@affiliationcounter}}{
505
-\textsuperscript{\arabic{@affilnum}}\ifcsname @affilname\the@affilnum\endcsname%
506
-\csname @affilname\the@affilnum\endcsname%
507
-\else
508
-{\bf AUTHORERR: Missing \textbackslash{}icmlaffiliation.}
509
-\fi
510
-}.
511
-\ifdefined\icmlcorrespondingauthor@text
512
-Correspondence to: \icmlcorrespondingauthor@text.
513
-\else
514
-{\bf AUTHORERR: Missing \textbackslash{}icmlcorrespondingauthor.}
515
-\fi
516
-
517
-\ \\
518
-\Notice@String
519
-}
520
-}
521
-}
522
-
523
-%\makeatother
524
-
525
-\long\def\icmladdress#1{%
526
- {\bf The \textbackslash{}icmladdress command is no longer used.  See the example\_paper PDF .tex for usage of \textbackslash{}icmlauther and \textbackslash{}icmlaffiliation.}
527
-}
528
-
529
-%% keywords as first class citizens
530
-\def\icmlkeywords#1{%
531
-%  \ifdefined\isaccepted \else
532
-%    \par {\bf Keywords:} #1%
533
-%  \fi
534
-%  \ifdefined\nohyperref\else\ifdefined\hypersetup
535
-%    \hypersetup{pdfkeywords={#1}}
536
-%  \fi\fi
537
-%  \ifdefined\isaccepted \else
538
-%    \par {\bf Keywords:} #1%
539
-%  \fi
540
-  \ifdefined\nohyperref\else\ifdefined\hypersetup
541
-    \hypersetup{pdfkeywords={#1}}
542
-  \fi\fi
543
-}
544
-
545
-% modification to natbib citations
546
-% \setcitestyle{authoryear,round,citesep={;},aysep={,},yysep={;}}
547
-
548
-% Redefinition of the abstract environment. 
549
-\renewenvironment{abstract}
550
-   {%
551
-% Insert the ``appearing in'' copyright notice.
552
-%\@copyrightspace
553
-\centerline{\large\bf Abstract}
554
-    \vspace{-0.12in}\begin{quote}}
555
-   {\par\end{quote}\vskip 0.12in}
556
-
557
-% numbered section headings with different treatment of numbers
558
-
559
-\def\@startsection#1#2#3#4#5#6{\if@noskipsec \leavevmode \fi
560
-   \par \@tempskipa #4\relax
561
-   \@afterindenttrue
562
-% Altered the following line to indent a section's first paragraph. 
563
-%  \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi
564
-   \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \fi
565
-   \if@nobreak \everypar{}\else
566
-     \addpenalty{\@secpenalty}\addvspace{\@tempskipa}\fi \@ifstar
567
-     {\@ssect{#3}{#4}{#5}{#6}}{\@dblarg{\@sict{#1}{#2}{#3}{#4}{#5}{#6}}}}
568
-
569
-\def\@sict#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth
570
-     \def\@svsec{}\else 
571
-     \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname}\fi
572
-     \@tempskipa #5\relax
573
-      \ifdim \@tempskipa>\z@
574
-        \begingroup #6\relax
575
-          \@hangfrom{\hskip #3\relax\@svsec.~}{\interlinepenalty \@M #8\par}
576
-        \endgroup
577
-       \csname #1mark\endcsname{#7}\addcontentsline
578
-         {toc}{#1}{\ifnum #2>\c@secnumdepth \else
579
-                      \protect\numberline{\csname the#1\endcsname}\fi
580
-                    #7}\else
581
-        \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname
582
-                      {#7}\addcontentsline
583
-                           {toc}{#1}{\ifnum #2>\c@secnumdepth \else
584
-                             \protect\numberline{\csname the#1\endcsname}\fi
585
-                       #7}}\fi
586
-     \@xsect{#5}}
587
-
588
-\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth
589
-     \def\@svsec{}\else 
590
-     \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname\hskip 0.4em }\fi
591
-     \@tempskipa #5\relax
592
-      \ifdim \@tempskipa>\z@ 
593
-        \begingroup #6\relax
594
-          \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par}
595
-        \endgroup
596
-       \csname #1mark\endcsname{#7}\addcontentsline
597
-         {toc}{#1}{\ifnum #2>\c@secnumdepth \else
598
-                      \protect\numberline{\csname the#1\endcsname}\fi
599
-                    #7}\else
600
-        \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname
601
-                      {#7}\addcontentsline
602
-                           {toc}{#1}{\ifnum #2>\c@secnumdepth \else
603
-                             \protect\numberline{\csname the#1\endcsname}\fi
604
-                       #7}}\fi
605
-     \@xsect{#5}}
606
-
607
-% section headings with less space above and below them
608
-\def\thesection {\arabic{section}}
609
-\def\thesubsection {\thesection.\arabic{subsection}}
610
-\def\section{\@startsection{section}{1}{\z@}{-0.12in}{0.02in}
611
-             {\large\bf\raggedright}}
612
-\def\subsection{\@startsection{subsection}{2}{\z@}{-0.10in}{0.01in}
613
-                {\normalsize\bf\raggedright}}
614
-\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-0.08in}{0.01in}
615
-                {\normalsize\sc\raggedright}}
616
-\def\paragraph{\@startsection{paragraph}{4}{\z@}{1.5ex plus
617
-  0.5ex minus .2ex}{-1em}{\normalsize\bf}}
618
-\def\subparagraph{\@startsection{subparagraph}{5}{\z@}{1.5ex plus
619
-  0.5ex minus .2ex}{-1em}{\normalsize\bf}}
620
- 
621
-% Footnotes 
622
-\footnotesep 6.65pt % 
623
-\skip\footins 9pt 
624
-\def\footnoterule{\kern-3pt \hrule width 0.8in \kern 2.6pt } 
625
-\setcounter{footnote}{0} 
626
- 
627
-% Lists and paragraphs 
628
-\parindent 0pt 
629
-\topsep 4pt plus 1pt minus 2pt 
630
-\partopsep 1pt plus 0.5pt minus 0.5pt 
631
-\itemsep 2pt plus 1pt minus 0.5pt 
632
-\parsep 2pt plus 1pt minus 0.5pt 
633
-\parskip 6pt
634
- 
635
-\leftmargin 2em \leftmargini\leftmargin \leftmarginii 2em 
636
-\leftmarginiii 1.5em \leftmarginiv 1.0em \leftmarginv .5em  
637
-\leftmarginvi .5em 
638
-\labelwidth\leftmargini\advance\labelwidth-\labelsep \labelsep 5pt 
639
- 
640
-\def\@listi{\leftmargin\leftmargini} 
641
-\def\@listii{\leftmargin\leftmarginii 
642
-   \labelwidth\leftmarginii\advance\labelwidth-\labelsep 
643
-   \topsep 2pt plus 1pt minus 0.5pt 
644
-   \parsep 1pt plus 0.5pt minus 0.5pt 
645
-   \itemsep \parsep} 
646
-\def\@listiii{\leftmargin\leftmarginiii 
647
-    \labelwidth\leftmarginiii\advance\labelwidth-\labelsep 
648
-    \topsep 1pt plus 0.5pt minus 0.5pt  
649
-    \parsep \z@ \partopsep 0.5pt plus 0pt minus 0.5pt 
650
-    \itemsep \topsep} 
651
-\def\@listiv{\leftmargin\leftmarginiv 
652
-     \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} 
653
-\def\@listv{\leftmargin\leftmarginv 
654
-     \labelwidth\leftmarginv\advance\labelwidth-\labelsep} 
655
-\def\@listvi{\leftmargin\leftmarginvi 
656
-     \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} 
657
- 
658
-\abovedisplayskip 7pt plus2pt minus5pt% 
659
-\belowdisplayskip \abovedisplayskip 
660
-\abovedisplayshortskip  0pt plus3pt%    
661
-\belowdisplayshortskip  4pt plus3pt minus3pt% 
662
- 
663
-% Less leading in most fonts (due to the narrow columns) 
664
-% The choices were between 1-pt and 1.5-pt leading 
665
-\def\@normalsize{\@setsize\normalsize{11pt}\xpt\@xpt} 
666
-\def\small{\@setsize\small{10pt}\ixpt\@ixpt} 
667
-\def\footnotesize{\@setsize\footnotesize{10pt}\ixpt\@ixpt} 
668
-\def\scriptsize{\@setsize\scriptsize{8pt}\viipt\@viipt} 
669
-\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt} 
670
-\def\large{\@setsize\large{14pt}\xiipt\@xiipt} 
671
-\def\Large{\@setsize\Large{16pt}\xivpt\@xivpt} 
672
-\def\LARGE{\@setsize\LARGE{20pt}\xviipt\@xviipt} 
673
-\def\huge{\@setsize\huge{23pt}\xxpt\@xxpt} 
674
-\def\Huge{\@setsize\Huge{28pt}\xxvpt\@xxvpt} 
675
-
676
-% Revised formatting for figure captions and table titles. 
677
-\newsavebox\newcaptionbox\newdimen\newcaptionboxwid
678
-
679
-\long\def\@makecaption#1#2{
680
- \vskip 10pt 
681
-        \baselineskip 11pt
682
-        \setbox\@tempboxa\hbox{#1. #2}
683
-        \ifdim \wd\@tempboxa >\hsize
684
-        \sbox{\newcaptionbox}{\small\sl #1.~}
685
-        \newcaptionboxwid=\wd\newcaptionbox
686
-        \usebox\newcaptionbox {\footnotesize #2}
687
-%        \usebox\newcaptionbox {\small #2}
688
-        \else 
689
-          \centerline{{\small\sl #1.} {\small #2}} 
690
-        \fi}
691
-
692
-\def\fnum@figure{Figure \thefigure}
693
-\def\fnum@table{Table \thetable}
694
-
695
-% Strut macros for skipping spaces above and below text in tables. 
696
-\def\abovestrut#1{\rule[0in]{0in}{#1}\ignorespaces}
697
-\def\belowstrut#1{\rule[-#1]{0in}{#1}\ignorespaces}
698
-
699
-\def\abovespace{\abovestrut{0.20in}}
700
-\def\aroundspace{\abovestrut{0.20in}\belowstrut{0.10in}}
701
-\def\belowspace{\belowstrut{0.10in}}
702
-
703
-% Various personal itemization commands. 
704
-\def\texitem#1{\par\noindent\hangindent 12pt
705
-               \hbox to 12pt {\hss #1 ~}\ignorespaces}
706
-\def\icmlitem{\texitem{$\bullet$}}
707
-
708
-% To comment out multiple lines of text.
709
-\long\def\comment#1{}
710
-
711
-
712
-
713
-
714
-%% Line counter (not in final version). Adapted from NIPS style file by Christoph Sawade
715
-
716
-% Vertical Ruler
717
-% This code is, largely, from the CVPR 2010 conference style file
718
-% ----- define vruler
719
-\makeatletter
720
-\newbox\icmlrulerbox
721
-\newcount\icmlrulercount
722
-\newdimen\icmlruleroffset
723
-\newdimen\cv@lineheight
724
-\newdimen\cv@boxheight
725
-\newbox\cv@tmpbox
726
-\newcount\cv@refno
727
-\newcount\cv@tot
728
-% NUMBER with left flushed zeros  \fillzeros[<WIDTH>]<NUMBER>
729
-\newcount\cv@tmpc@ \newcount\cv@tmpc
730
-\def\fillzeros[#1]#2{\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi
731
-\cv@tmpc=1 %
732
-\loop\ifnum\cv@tmpc@<10 \else \divide\cv@tmpc@ by 10 \advance\cv@tmpc by 1 \fi
733
-   \ifnum\cv@tmpc@=10\relax\cv@tmpc@=11\relax\fi \ifnum\cv@tmpc@>10 \repeat
734
-\ifnum#2<0\advance\cv@tmpc1\relax-\fi
735
-\loop\ifnum\cv@tmpc<#1\relax0\advance\cv@tmpc1\relax\fi \ifnum\cv@tmpc<#1 \repeat
736
-\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi \relax\the\cv@tmpc@}%
737
-% \makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]
738
-\def\makevruler[#1][#2][#3][#4][#5]{
739
-	\begingroup\offinterlineskip
740
-		\textheight=#5\vbadness=10000\vfuzz=120ex\overfullrule=0pt%
741
-		\global\setbox\icmlrulerbox=\vbox to \textheight{%
742
-			{
743
-				\parskip=0pt\hfuzz=150em\cv@boxheight=\textheight
744
-				\cv@lineheight=#1\global\icmlrulercount=#2%
745
-				\cv@tot\cv@boxheight\divide\cv@tot\cv@lineheight\advance\cv@tot2%
746
-				\cv@refno1\vskip-\cv@lineheight\vskip1ex%
747
-				\loop\setbox\cv@tmpbox=\hbox to0cm{					 % side margin
748
-					\hfil {\hfil\fillzeros[#4]\icmlrulercount}
749
-				}%
750
-				\ht\cv@tmpbox\cv@lineheight\dp\cv@tmpbox0pt\box\cv@tmpbox\break
751
-				\advance\cv@refno1\global\advance\icmlrulercount#3\relax
752
-				\ifnum\cv@refno<\cv@tot\repeat
753
-			}
754
-		}
755
-	\endgroup
756
-}%
757
-\makeatother
758
-% ----- end of vruler
759
-
760
-
761
-% \makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]
762
-\def\icmlruler#1{\makevruler[12pt][#1][1][3][\textheight]\usebox{\icmlrulerbox}}
763
-\AddToShipoutPicture{%
764
-\icmlruleroffset=\textheight
765
-\advance\icmlruleroffset by 5.2pt % top margin
766
-  \color[rgb]{.7,.7,.7}
767
-  \ifdefined\isaccepted \else
768
-	  \AtTextUpperLeft{%
769
-	    \put(\LenToUnit{-35pt},\LenToUnit{-\icmlruleroffset}){%left ruler
770
-	      \icmlruler{\icmlrulercount}}
771
-%	    \put(\LenToUnit{1.04\textwidth},\LenToUnit{-\icmlruleroffset}){%right ruler
772
-%	      \icmlruler{\icmlrulercount}}
773
-	  }
774
-	 \fi
775
-}
776
-\endinput

+ 357
- 0
nips_2018.sty View File

@@ -0,0 +1,357 @@
1
+% partial rewrite of the LaTeX2e package for submissions to the
2
+% Conference on Neural Information Processing Systems (NIPS):
3
+%
4
+% - uses more LaTeX conventions
5
+% - line numbers at submission time replaced with aligned numbers from
6
+%   lineno package
7
+% - \nipsfinalcopy replaced with [final] package option
8
+% - automatically loads times package for authors
9
+% - loads natbib automatically; this can be suppressed with the
10
+%   [nonatbib] package option
11
+% - adds foot line to first page identifying the conference
12
+% - adds preprint option for submission to e.g. arXiv
13
+%
14
+% Roman Garnett (garnett@wustl.edu) and the many authors of
15
+% nips15submit_e.sty, including MK and drstrip@sandia
16
+%
17
+% last revision: March 2018
18
+
19
+\NeedsTeXFormat{LaTeX2e}
20
+\ProvidesPackage{nips_2018}[2018/03/03 NIPS 2018 submission/camera-ready style file]
21
+
22
+% declare final option, which creates camera-ready copy
23
+\newif\if@nipsfinal\@nipsfinalfalse
24
+\DeclareOption{final}{
25
+  \@nipsfinaltrue
26
+}
27
+
28
+% declare nonatbib option, which does not load natbib in case of
29
+% package clash (users can pass options to natbib via
30
+% \PassOptionsToPackage)
31
+\newif\if@natbib\@natbibtrue
32
+\DeclareOption{nonatbib}{
33
+  \@natbibfalse
34
+}
35
+
36
+% declare preprint option, which creates a preprint version ready for
37
+% upload to, e.g., arXiv
38
+\newif\if@preprint\@preprintfalse
39
+\DeclareOption{preprint}{
40
+  \@preprinttrue
41
+}
42
+
43
+\ProcessOptions\relax
44
+
45
+% determine whether this is an anonymized submission
46
+\newif\if@submission\@submissiontrue
47
+\if@nipsfinal\@submissionfalse\fi
48
+\if@preprint\@submissionfalse\fi
49
+
50
+% fonts
51
+\renewcommand{\rmdefault}{ptm}
52
+\renewcommand{\sfdefault}{phv}
53
+
54
+% change this every year for notice string at bottom
55
+\newcommand{\@nipsordinal}{32nd}
56
+\newcommand{\@nipsyear}{2018}
57
+\newcommand{\@nipslocation}{Montr\'{e}al, Canada}
58
+
59
+% handle tweaks for camera-ready copy vs. submission copy
60
+\if@preprint
61
+  \newcommand{\@noticestring}{%
62
+    Preprint. Work in progress.%
63
+  }
64
+\else
65
+  \if@nipsfinal
66
+    \newcommand{\@noticestring}{%
67
+      \@nipsordinal\/ Conference on Neural Information Processing Systems
68
+      (NIPS \@nipsyear), \@nipslocation.%
69
+    }
70
+  \else
71
+    \newcommand{\@noticestring}{%
72
+      Submitted to \@nipsordinal\/ Conference on Neural Information
73
+      Processing Systems (NIPS \@nipsyear). Do not distribute.%
74
+    }
75
+
76
+    % line numbers for submission
77
+    \RequirePackage{lineno}
78
+    \linenumbers
79
+
80
+    % fix incompatibilities between lineno and amsmath, if required, by
81
+    % transparently wrapping linenomath environments around amsmath
82
+    % environments
83
+    \AtBeginDocument{%
84
+      \@ifpackageloaded{amsmath}{%
85
+        \newcommand*\patchAmsMathEnvironmentForLineno[1]{%
86
+          \expandafter\let\csname old#1\expandafter\endcsname\csname #1\endcsname
87
+          \expandafter\let\csname oldend#1\expandafter\endcsname\csname end#1\endcsname
88
+          \renewenvironment{#1}%
89
+                           {\linenomath\csname old#1\endcsname}%
90
+                           {\csname oldend#1\endcsname\endlinenomath}%
91
+        }%
92
+        \newcommand*\patchBothAmsMathEnvironmentsForLineno[1]{%
93
+          \patchAmsMathEnvironmentForLineno{#1}%
94
+          \patchAmsMathEnvironmentForLineno{#1*}%
95
+        }%
96
+        \patchBothAmsMathEnvironmentsForLineno{equation}%
97
+        \patchBothAmsMathEnvironmentsForLineno{align}%
98
+        \patchBothAmsMathEnvironmentsForLineno{flalign}%
99
+        \patchBothAmsMathEnvironmentsForLineno{alignat}%
100
+        \patchBothAmsMathEnvironmentsForLineno{gather}%
101
+        \patchBothAmsMathEnvironmentsForLineno{multline}%
102
+      }{}
103
+    }
104
+  \fi
105
+\fi
106
+
107
+% load natbib unless told otherwise
108
+\if@natbib
109
+  \RequirePackage{natbib}
110
+\fi
111
+
112
+% set page geometry
113
+\usepackage[verbose=true,letterpaper]{geometry}
114
+\AtBeginDocument{
115
+  \newgeometry{
116
+    textheight=9in,
117
+    textwidth=5.5in,
118
+    top=1in,
119
+    headheight=12pt,
120
+    headsep=25pt,
121
+    footskip=30pt
122
+  }
123
+  \@ifpackageloaded{fullpage}
124
+    {\PackageWarning{nips_2018}{fullpage package not allowed! Overwriting formatting.}}
125
+    {}
126
+}
127
+
128
+\widowpenalty=10000
129
+\clubpenalty=10000
130
+\flushbottom
131
+\sloppy
132
+
133
+% font sizes with reduced leading
134
+\renewcommand{\normalsize}{%
135
+  \@setfontsize\normalsize\@xpt\@xipt
136
+  \abovedisplayskip      7\p@ \@plus 2\p@ \@minus 5\p@
137
+  \abovedisplayshortskip \z@ \@plus 3\p@
138
+  \belowdisplayskip      \abovedisplayskip
139
+  \belowdisplayshortskip 4\p@ \@plus 3\p@ \@minus 3\p@
140
+}
141
+\normalsize
142
+\renewcommand{\small}{%
143
+  \@setfontsize\small\@ixpt\@xpt
144
+  \abovedisplayskip      6\p@ \@plus 1.5\p@ \@minus 4\p@
145
+  \abovedisplayshortskip \z@  \@plus 2\p@
146
+  \belowdisplayskip      \abovedisplayskip
147
+  \belowdisplayshortskip 3\p@ \@plus 2\p@   \@minus 2\p@
148
+}
149
+\renewcommand{\footnotesize}{\@setfontsize\footnotesize\@ixpt\@xpt}
150
+\renewcommand{\scriptsize}{\@setfontsize\scriptsize\@viipt\@viiipt}
151
+\renewcommand{\tiny}{\@setfontsize\tiny\@vipt\@viipt}
152
+\renewcommand{\large}{\@setfontsize\large\@xiipt{14}}
153
+\renewcommand{\Large}{\@setfontsize\Large\@xivpt{16}}
154
+\renewcommand{\LARGE}{\@setfontsize\LARGE\@xviipt{20}}
155
+\renewcommand{\huge}{\@setfontsize\huge\@xxpt{23}}
156
+\renewcommand{\Huge}{\@setfontsize\Huge\@xxvpt{28}}
157
+
158
+% sections with less space
159
+\providecommand{\section}{}
160
+\renewcommand{\section}{%
161
+  \@startsection{section}{1}{\z@}%
162
+                {-2.0ex \@plus -0.5ex \@minus -0.2ex}%
163
+                { 1.5ex \@plus  0.3ex \@minus  0.2ex}%
164
+                {\large\bf\raggedright}%
165
+}
166
+\providecommand{\subsection}{}
167
+\renewcommand{\subsection}{%
168
+  \@startsection{subsection}{2}{\z@}%
169
+                {-1.8ex \@plus -0.5ex \@minus -0.2ex}%
170
+                { 0.8ex \@plus  0.2ex}%
171
+                {\normalsize\bf\raggedright}%
172
+}
173
+\providecommand{\subsubsection}{}
174
+\renewcommand{\subsubsection}{%
175
+  \@startsection{subsubsection}{3}{\z@}%
176
+                {-1.5ex \@plus -0.5ex \@minus -0.2ex}%
177
+                { 0.5ex \@plus  0.2ex}%
178
+                {\normalsize\bf\raggedright}%
179
+}
180
+\providecommand{\paragraph}{}
181
+\renewcommand{\paragraph}{%
182
+  \@startsection{paragraph}{4}{\z@}%
183
+                {1.5ex \@plus 0.5ex \@minus 0.2ex}%
184
+                {-1em}%
185
+                {\normalsize\bf}%
186
+}
187
+\providecommand{\subparagraph}{}
188
+\renewcommand{\subparagraph}{%
189
+  \@startsection{subparagraph}{5}{\z@}%
190
+                {1.5ex \@plus 0.5ex \@minus 0.2ex}%
191
+                {-1em}%
192
+                {\normalsize\bf}%
193
+}
194
+\providecommand{\subsubsubsection}{}
195
+\renewcommand{\subsubsubsection}{%
196
+  \vskip5pt{\noindent\normalsize\rm\raggedright}%
197
+}
198
+
199
+% float placement
200
+\renewcommand{\topfraction      }{0.85}
201
+\renewcommand{\bottomfraction   }{0.4}
202
+\renewcommand{\textfraction     }{0.1}
203
+\renewcommand{\floatpagefraction}{0.7}
204
+
205
+\newlength{\@nipsabovecaptionskip}\setlength{\@nipsabovecaptionskip}{7\p@}
206
+\newlength{\@nipsbelowcaptionskip}\setlength{\@nipsbelowcaptionskip}{\z@}
207
+
208
+\setlength{\abovecaptionskip}{\@nipsabovecaptionskip}
209
+\setlength{\belowcaptionskip}{\@nipsbelowcaptionskip}
210
+
211
+% swap above/belowcaptionskip lengths for tables
212
+\renewenvironment{table}
213
+  {\setlength{\abovecaptionskip}{\@nipsbelowcaptionskip}%
214
+   \setlength{\belowcaptionskip}{\@nipsabovecaptionskip}%
215
+   \@float{table}}
216
+  {\end@float}
217
+
218
+% footnote formatting
219
+\setlength{\footnotesep }{6.65\p@}
220
+\setlength{\skip\footins}{9\p@ \@plus 4\p@ \@minus 2\p@}
221
+\renewcommand{\footnoterule}{\kern-3\p@ \hrule width 12pc \kern 2.6\p@}
222
+\setcounter{footnote}{0}
223
+
224
+% paragraph formatting
225
+\setlength{\parindent}{\z@}
226
+\setlength{\parskip  }{5.5\p@}
227
+
228
+% list formatting
229
+\setlength{\topsep       }{4\p@ \@plus 1\p@   \@minus 2\p@}
230
+\setlength{\partopsep    }{1\p@ \@plus 0.5\p@ \@minus 0.5\p@}
231
+\setlength{\itemsep      }{2\p@ \@plus 1\p@   \@minus 0.5\p@}
232
+\setlength{\parsep       }{2\p@ \@plus 1\p@   \@minus 0.5\p@}
233
+\setlength{\leftmargin   }{3pc}
234
+\setlength{\leftmargini  }{\leftmargin}
235
+\setlength{\leftmarginii }{2em}
236
+\setlength{\leftmarginiii}{1.5em}
237
+\setlength{\leftmarginiv }{1.0em}
238
+\setlength{\leftmarginv  }{0.5em}
239
+\def\@listi  {\leftmargin\leftmargini}
240
+\def\@listii {\leftmargin\leftmarginii
241
+              \labelwidth\leftmarginii
242
+              \advance\labelwidth-\labelsep
243
+              \topsep  2\p@ \@plus 1\p@    \@minus 0.5\p@
244
+              \parsep  1\p@ \@plus 0.5\p@ \@minus 0.5\p@
245
+              \itemsep \parsep}
246
+\def\@listiii{\leftmargin\leftmarginiii
247
+              \labelwidth\leftmarginiii
248
+              \advance\labelwidth-\labelsep
249
+              \topsep    1\p@ \@plus 0.5\p@ \@minus 0.5\p@
250
+              \parsep    \z@
251
+              \partopsep 0.5\p@ \@plus 0\p@ \@minus 0.5\p@
252
+              \itemsep \topsep}
253
+\def\@listiv {\leftmargin\leftmarginiv
254
+              \labelwidth\leftmarginiv
255
+              \advance\labelwidth-\labelsep}
256
+\def\@listv  {\leftmargin\leftmarginv
257
+              \labelwidth\leftmarginv
258
+              \advance\labelwidth-\labelsep}
259
+\def\@listvi {\leftmargin\leftmarginvi
260
+              \labelwidth\leftmarginvi
261
+              \advance\labelwidth-\labelsep}
262
+
263
+% create title
264
+\providecommand{\maketitle}{}
265
+\renewcommand{\maketitle}{%
266
+  \par
267
+  \begingroup
268
+    \renewcommand{\thefootnote}{\fnsymbol{footnote}}
269
+    % for perfect author name centering
270
+    \renewcommand{\@makefnmark}{\hbox to \z@{$^{\@thefnmark}$\hss}}
271
+    % The footnote-mark was overlapping the footnote-text,
272
+    % added the following to fix this problem               (MK)
273
+    \long\def\@makefntext##1{%
274
+      \parindent 1em\noindent
275
+      \hbox to 1.8em{\hss $\m@th ^{\@thefnmark}$}##1
276
+    }
277
+    \thispagestyle{empty}
278
+    \@maketitle
279
+    \@thanks
280
+    \@notice
281
+  \endgroup
282
+  \let\maketitle\relax
283
+  \let\thanks\relax
284
+}
285
+
286
+% rules for title box at top of first page
287
+\newcommand{\@toptitlebar}{
288
+  \hrule height 4\p@
289
+  \vskip 0.25in
290
+  \vskip -\parskip%
291
+}
292
+\newcommand{\@bottomtitlebar}{
293
+  \vskip 0.29in
294
+  \vskip -\parskip
295
+  \hrule height 1\p@
296
+  \vskip 0.09in%
297
+}
298
+
299
+% create title (includes both anonymized and non-anonymized versions)
300
+\providecommand{\@maketitle}{}
301
+\renewcommand{\@maketitle}{%
302
+  \vbox{%
303
+    \hsize\textwidth
304
+    \linewidth\hsize
305
+    \vskip 0.1in
306
+    \@toptitlebar
307
+    \centering
308
+    {\LARGE\bf \@title\par}
309
+    \@bottomtitlebar
310
+    \if@submission
311
+      \begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}
312
+        Anonymous Author(s) \\
313
+        Affiliation \\
314
+        Address \\
315
+        \texttt{email} \\
316
+      \end{tabular}%
317
+    \else
318
+      \def\And{%
319
+        \end{tabular}\hfil\linebreak[0]\hfil%
320
+        \begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}\ignorespaces%
321
+      }
322
+      \def\AND{%
323
+        \end{tabular}\hfil\linebreak[4]\hfil%
324
+        \begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}\ignorespaces%
325
+      }
326
+      \begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}\@author\end{tabular}%
327
+    \fi
328
+    \vskip 0.3in \@minus 0.1in
329
+  }
330
+}
331
+
332
+% add conference notice to bottom of first page
333
+\newcommand{\ftype@noticebox}{8}
334
+\newcommand{\@notice}{%
335
+  % give a bit of extra room back to authors on first page
336
+  \enlargethispage{2\baselineskip}%
337
+  \@float{noticebox}[b]%
338
+    \footnotesize\@noticestring%
339
+  \end@float%
340
+}
341
+
342
+% abstract styling
343
+\renewenvironment{abstract}%
344
+{%
345
+  \vskip 0.075in%
346
+  \centerline%
347
+  {\large\bf Abstract}%
348
+  \vspace{0.5ex}%
349
+  \begin{quote}%
350
+}
351
+{
352
+  \par%
353
+  \end{quote}%
354
+  \vskip 1ex%
355
+}
356
+
357
+\endinput

+ 419
- 0
nips_2018.tex View File

@@ -0,0 +1,419 @@
1
+\documentclass{article}
2
+
3
+% if you need to pass options to natbib, use, e.g.:
4
+% \PassOptionsToPackage{numbers, compress}{natbib}
5
+% before loading nips_2018
6
+
7
+% ready for submission
8
+\usepackage{nips_2018}
9
+
10
+% to compile a preprint version, e.g., for submission to arXiv, add
11
+% add the [preprint] option:
12
+% \usepackage[preprint]{nips_2018}
13
+
14
+% to compile a camera-ready version, add the [final] option, e.g.:
15
+% \usepackage[final]{nips_2018}
16
+
17
+% to avoid loading the natbib package, add option nonatbib:
18
+% \usepackage[nonatbib]{nips_2018}
19
+
20
+\usepackage[utf8]{inputenc} % allow utf-8 input
21
+\usepackage[T1]{fontenc}    % use 8-bit T1 fonts
22
+\usepackage{hyperref}       % hyperlinks
23
+\usepackage{url}            % simple URL typesetting
24
+\usepackage{booktabs}       % professional-quality tables
25
+\usepackage{amsfonts}       % blackboard math symbols
26
+\usepackage{nicefrac}       % compact symbols for 1/2, etc.
27
+\usepackage{microtype}      % microtypography
28
+
29
+\title{Formatting instructions for NIPS 2018}
30
+
31
+% The \author macro works with any number of authors. There are two
32
+% commands used to separate the names and addresses of multiple
33
+% authors: \And and \AND.
34
+%
35
+% Using \And between authors leaves it to LaTeX to determine where to
36
+% break the lines. Using \AND forces a line break at that point. So,
37
+% if LaTeX puts 3 of 4 authors names on the first line, and the last
38
+% on the second line, try using \AND instead of \And before the third
39
+% author name.
40
+
41
+\author{
42
+  David S.~Hippocampus\thanks{Use footnote for providing further
43
+    information about author (webpage, alternative
44
+    address)---\emph{not} for acknowledging funding agencies.} \\
45
+  Department of Computer Science\\
46
+  Cranberry-Lemon University\\
47
+  Pittsburgh, PA 15213 \\
48
+  \texttt{hippo@cs.cranberry-lemon.edu} \\
49
+  %% examples of more authors
50
+  %% \And
51
+  %% Coauthor \\
52
+  %% Affiliation \\
53
+  %% Address \\
54
+  %% \texttt{email} \\
55
+  %% \AND
56
+  %% Coauthor \\
57
+  %% Affiliation \\
58
+  %% Address \\
59
+  %% \texttt{email} \\
60
+  %% \And
61
+  %% Coauthor \\
62
+  %% Affiliation \\
63
+  %% Address \\
64
+  %% \texttt{email} \\
65
+  %% \And
66
+  %% Coauthor \\
67
+  %% Affiliation \\
68
+  %% Address \\
69
+  %% \texttt{email} \\
70
+}
71
+
72
+\begin{document}
73
+% \nipsfinalcopy is no longer used
74
+
75
+\maketitle
76
+
77
+\begin{abstract}
78
+  The abstract paragraph should be indented \nicefrac{1}{2}~inch
79
+  (3~picas) on both the left- and right-hand margins. Use 10~point
80
+  type, with a vertical spacing (leading) of 11~points.  The word
81
+  \textbf{Abstract} must be centered, bold, and in point size 12. Two
82
+  line spaces precede the abstract. The abstract must be limited to
83
+  one paragraph.
84
+\end{abstract}
85
+
86
+\section{Submission of papers to NIPS 2018}
87
+
88
+NIPS requires electronic submissions.  The electronic submission site
89
+is
90
+\begin{center}
91
+  \url{https://cmt.research.microsoft.com/NIPS2018/}
92
+\end{center}
93
+
94
+Please read the instructions below carefully and follow them faithfully.
95
+
96
+\subsection{Style}
97
+
98
+Papers to be submitted to NIPS 2018 must be prepared according to the
99
+instructions presented here. Papers may only be up to eight pages
100
+long, including figures. Additional pages \emph{containing only
101
+  acknowledgments and/or cited references} are allowed. Papers that
102
+exceed eight pages of content (ignoring references) will not be
103
+reviewed, or in any other way considered for presentation at the
104
+conference.
105
+
106
+The margins in 2018 are the same as since 2007, which allow for
107
+$\sim$$15\%$ more words in the paper compared to earlier years.
108
+
109
+Authors are required to use the NIPS \LaTeX{} style files obtainable
110
+at the NIPS website as indicated below. Please make sure you use the
111
+current files and not previous versions. Tweaking the style files may
112
+be grounds for rejection.
113
+
114
+\subsection{Retrieval of style files}
115
+
116
+The style files for NIPS and other conference information are
117
+available on the World Wide Web at
118
+\begin{center}
119
+  \url{http://www.nips.cc/}
120
+\end{center}
121
+The file \verb+nips_2018.pdf+ contains these instructions and
122
+illustrates the various formatting requirements your NIPS paper must
123
+satisfy.
124
+
125
+The only supported style file for NIPS 2018 is \verb+nips_2018.sty+,
126
+rewritten for \LaTeXe{}.  \textbf{Previous style files for \LaTeX{}
127
+  2.09, Microsoft Word, and RTF are no longer supported!}
128
+
129
+The \LaTeX{} style file contains three optional arguments: \verb+final+,
130
+which creates a camera-ready copy, \verb+preprint+, which creates a
131
+preprint for submission to, e.g., arXiv, and \verb+nonatbib+, which will
132
+not load the \verb+natbib+ package for you in case of package clash.
133
+
134
+\paragraph{New preprint option for 2018}
135
+If you wish to post a preprint of your work online, e.g., on arXiv,
136
+using the NIPS style, please use the \verb+preprint+ option. This will
137
+create a nonanonymized version of your work with the text
138
+``Preprint. Work in progress.''  in the footer. This version may be
139
+distributed as you see fit. Please \textbf{do not} use the
140
+\verb+final+ option, which should \textbf{only} be used for papers
141
+accepted to NIPS.
142
+
143
+At submission time, please omit the \verb+final+ and \verb+preprint+
144
+options. This will anonymize your submission and add line numbers to aid
145
+review. Please do \emph{not} refer to these line numbers in your paper
146
+as they will be removed during generation of camera-ready copies.
147
+
148
+The file \verb+nips_2018.tex+ may be used as a ``shell'' for writing
149
+your paper. All you have to do is replace the author, title, abstract,
150
+and text of the paper with your own.
151
+
152
+The formatting instructions contained in these style files are
153
+summarized in Sections \ref{gen_inst}, \ref{headings}, and
154
+\ref{others} below.
155
+
156
+\section{General formatting instructions}
157
+\label{gen_inst}
158
+
159
+The text must be confined within a rectangle 5.5~inches (33~picas)
160
+wide and 9~inches (54~picas) long. The left margin is 1.5~inch
161
+(9~picas).  Use 10~point type with a vertical spacing (leading) of
162
+11~points.  Times New Roman is the preferred typeface throughout, and
163
+will be selected for you by default.  Paragraphs are separated by
164
+\nicefrac{1}{2}~line space (5.5 points), with no indentation.
165
+
166
+The paper title should be 17~point, initial caps/lower case, bold,
167
+centered between two horizontal rules. The top rule should be 4~points
168
+thick and the bottom rule should be 1~point thick. Allow
169
+\nicefrac{1}{4}~inch space above and below the title to rules. All
170
+pages should start at 1~inch (6~picas) from the top of the page.
171
+
172
+For the final version, authors' names are set in boldface, and each
173
+name is centered above the corresponding address. The lead author's
174
+name is to be listed first (left-most), and the co-authors' names (if
175
+different address) are set to follow. If there is only one co-author,
176
+list both author and co-author side by side.
177
+
178
+Please pay special attention to the instructions in Section \ref{others}
179
+regarding figures, tables, acknowledgments, and references.
180
+
181
+\section{Headings: first level}
182
+\label{headings}
183
+
184
+All headings should be lower case (except for first word and proper
185
+nouns), flush left, and bold.
186
+
187
+First-level headings should be in 12-point type.
188
+
189
+\subsection{Headings: second level}
190
+
191
+Second-level headings should be in 10-point type.
192
+
193
+\subsubsection{Headings: third level}
194
+
195
+Third-level headings should be in 10-point type.
196
+
197
+\paragraph{Paragraphs}
198
+
199
+There is also a \verb+\paragraph+ command available, which sets the
200
+heading in bold, flush left, and inline with the text, with the
201
+heading followed by 1\,em of space.
202
+
203
+\section{Citations, figures, tables, references}
204
+\label{others}
205
+
206
+These instructions apply to everyone.
207
+
208
+\subsection{Citations within the text}
209
+
210
+The \verb+natbib+ package will be loaded for you by default.
211
+Citations may be author/year or numeric, as long as you maintain
212
+internal consistency.  As to the format of the references themselves,
213
+any style is acceptable as long as it is used consistently.
214
+
215
+The documentation for \verb+natbib+ may be found at
216
+\begin{center}
217
+  \url{http://mirrors.ctan.org/macros/latex/contrib/natbib/natnotes.pdf}
218
+\end{center}
219
+Of note is the command \verb+\citet+, which produces citations
220
+appropriate for use in inline text.  For example,
221
+\begin{verbatim}
222
+   \citet{hasselmo} investigated\dots
223
+\end{verbatim}
224
+produces
225
+\begin{quote}
226
+  Hasselmo, et al.\ (1995) investigated\dots
227
+\end{quote}
228
+
229
+If you wish to load the \verb+natbib+ package with options, you may
230
+add the following before loading the \verb+nips_2018+ package:
231
+\begin{verbatim}
232
+   \PassOptionsToPackage{options}{natbib}
233
+\end{verbatim}
234
+
235
+If \verb+natbib+ clashes with another package you load, you can add
236
+the optional argument \verb+nonatbib+ when loading the style file:
237
+\begin{verbatim}
238
+   \usepackage[nonatbib]{nips_2018}
239
+\end{verbatim}
240
+
241
+As submission is double blind, refer to your own published work in the
242
+third person. That is, use ``In the previous work of Jones et
243
+al.\ [4],'' not ``In our previous work [4].'' If you cite your other
244
+papers that are not widely available (e.g., a journal paper under
245
+review), use anonymous author names in the citation, e.g., an author
246
+of the form ``A.\ Anonymous.''
247
+
248
+\subsection{Footnotes}
249
+
250
+Footnotes should be used sparingly.  If you do require a footnote,
251
+indicate footnotes with a number\footnote{Sample of the first
252
+  footnote.} in the text. Place the footnotes at the bottom of the
253
+page on which they appear.  Precede the footnote with a horizontal
254
+rule of 2~inches (12~picas).
255
+
256
+Note that footnotes are properly typeset \emph{after} punctuation
257
+marks.\footnote{As in this example.}
258
+
259
+\subsection{Figures}
260
+
261
+\begin{figure}
262
+  \centering
263
+  \fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{4cm}{0cm}}
264
+  \caption{Sample figure caption.}
265
+\end{figure}
266
+
267
+All artwork must be neat, clean, and legible. Lines should be dark
268
+enough for purposes of reproduction. The figure number and caption
269
+always appear after the figure. Place one line space before the figure
270
+caption and one line space after the figure. The figure caption should
271
+be lower case (except for first word and proper nouns); figures are
272
+numbered consecutively.
273
+
274
+You may use color figures.  However, it is best for the figure
275
+captions and the paper body to be legible if the paper is printed in
276
+either black/white or in color.
277
+
278
+\subsection{Tables}
279
+
280
+All tables must be centered, neat, clean and legible.  The table
281
+number and title always appear before the table.  See
282
+Table~\ref{sample-table}.
283
+
284
+Place one line space before the table title, one line space after the
285
+table title, and one line space after the table. The table title must
286
+be lower case (except for first word and proper nouns); tables are
287
+numbered consecutively.
288
+
289
+Note that publication-quality tables \emph{do not contain vertical
290
+  rules.} We strongly suggest the use of the \verb+booktabs+ package,
291
+which allows for typesetting high-quality, professional tables:
292
+\begin{center}
293
+  \url{https://www.ctan.org/pkg/booktabs}
294
+\end{center}
295
+This package was used to typeset Table~\ref{sample-table}.
296
+
297
+\begin{table}
298
+  \caption{Sample table title}
299
+  \label{sample-table}
300
+  \centering
301
+  \begin{tabular}{lll}
302
+    \toprule
303
+    \multicolumn{2}{c}{Part}                   \\
304
+    \cmidrule(r){1-2}
305
+    Name     & Description     & Size ($\mu$m) \\
306
+    \midrule
307
+    Dendrite & Input terminal  & $\sim$100     \\
308
+    Axon     & Output terminal & $\sim$10      \\
309
+    Soma     & Cell body       & up to $10^6$  \\
310
+    \bottomrule
311
+  \end{tabular}
312
+\end{table}
313
+
314
+\section{Final instructions}
315
+
316
+Do not change any aspects of the formatting parameters in the style
317
+files.  In particular, do not modify the width or length of the
318
+rectangle the text should fit into, and do not change font sizes
319
+(except perhaps in the \textbf{References} section; see below). Please
320
+note that pages should be numbered.
321
+
322
+\section{Preparing PDF files}
323
+
324
+Please prepare submission files with paper size ``US Letter,'' and
325
+not, for example, ``A4.''
326
+
327
+Fonts were the main cause of problems in the past years. Your PDF file
328
+must only contain Type 1 or Embedded TrueType fonts. Here are a few
329
+instructions to achieve this.
330
+
331
+\begin{itemize}
332
+
333
+\item You should directly generate PDF files using \verb+pdflatex+.
334
+
335
+\item You can check which fonts a PDF files uses.  In Acrobat Reader,
336
+  select the menu Files$>$Document Properties$>$Fonts and select Show
337
+  All Fonts. You can also use the program \verb+pdffonts+ which comes
338
+  with \verb+xpdf+ and is available out-of-the-box on most Linux
339
+  machines.
340
+
341
+\item The IEEE has recommendations for generating PDF files whose
342
+  fonts are also acceptable for NIPS. Please see
343
+  \url{http://www.emfield.org/icuwb2010/downloads/IEEE-PDF-SpecV32.pdf}
344
+
345
+\item \verb+xfig+ "patterned" shapes are implemented with bitmap
346
+  fonts.  Use "solid" shapes instead.
347
+
348
+\item The \verb+\bbold+ package almost always uses bitmap fonts.  You
349
+  should use the equivalent AMS Fonts:
350
+\begin{verbatim}
351
+   \usepackage{amsfonts}
352
+\end{verbatim}
353
+followed by, e.g., \verb+\mathbb{R}+, \verb+\mathbb{N}+, or
354
+\verb+\mathbb{C}+ for $\mathbb{R}$, $\mathbb{N}$ or $\mathbb{C}$.  You
355
+can also use the following workaround for reals, natural and complex:
356
+\begin{verbatim}
357
+   \newcommand{\RR}{I\!\!R} %real numbers
358
+   \newcommand{\Nat}{I\!\!N} %natural numbers
359
+   \newcommand{\CC}{I\!\!\!\!C} %complex numbers
360
+\end{verbatim}
361
+Note that \verb+amsfonts+ is automatically loaded by the
362
+\verb+amssymb+ package.
363
+
364
+\end{itemize}
365
+
366
+If your file contains type 3 fonts or non embedded TrueType fonts, we
367
+will ask you to fix it.
368
+
369
+\subsection{Margins in \LaTeX{}}
370
+
371
+Most of the margin problems come from figures positioned by hand using
372
+\verb+\special+ or other commands. We suggest using the command
373
+\verb+\includegraphics+ from the \verb+graphicx+ package. Always
374
+specify the figure width as a multiple of the line width as in the
375
+example below:
376
+\begin{verbatim}
377
+   \usepackage[pdftex]{graphicx} ...
378
+   \includegraphics[width=0.8\linewidth]{myfile.pdf}
379
+\end{verbatim}
380
+See Section 4.4 in the graphics bundle documentation
381
+(\url{http://mirrors.ctan.org/macros/latex/required/graphics/grfguide.pdf})
382
+
383
+A number of width problems arise when \LaTeX{} cannot properly
384
+hyphenate a line. Please give LaTeX hyphenation hints using the
385
+\verb+\-+ command when necessary.
386
+
387
+\subsubsection*{Acknowledgments}
388
+
389
+Use unnumbered third level headings for the acknowledgments. All
390
+acknowledgments go at the end of the paper. Do not include
391
+acknowledgments in the anonymized submission, only in the final paper.
392
+
393
+\section*{References}
394
+
395
+References follow the acknowledgments. Use unnumbered first-level
396
+heading for the references. Any choice of citation style is acceptable
397
+as long as you are consistent. It is permissible to reduce the font
398
+size to \verb+small+ (9 point) when listing the references. {\bf
399
+  Remember that you can use more than eight pages as long as the
400
+  additional pages contain \emph{only} cited references.}
401
+\medskip
402
+
403
+\small
404
+
405
+[1] Alexander, J.A.\ \& Mozer, M.C.\ (1995) Template-based algorithms
406
+for connectionist rule extraction. In G.\ Tesauro, D.S.\ Touretzky and
407
+T.K.\ Leen (eds.), {\it Advances in Neural Information Processing
408
+  Systems 7}, pp.\ 609--616. Cambridge, MA: MIT Press.
409
+
410
+[2] Bower, J.M.\ \& Beeman, D.\ (1995) {\it The Book of GENESIS:
411
+  Exploring Realistic Neural Models with the GEneral NEural SImulation
412
+  System.}  New York: TELOS/Springer--Verlag.
413
+
414
+[3] Hasselmo, M.E., Schnell, E.\ \& Barkai, E.\ (1995) Dynamics of
415
+learning and recall at excitatory recurrent synapses and cholinergic
416
+modulation in rat hippocampal region CA3. {\it Journal of
417
+  Neuroscience} {\bf 15}(7):5249-5262.
418
+
419
+\end{document}

+ 2
- 3
preamble/packages_paper.tex View File

@@ -14,7 +14,7 @@
14 14
 
15 15
 % Style
16 16
 \usepackage[hypertexnames=false, unicode, pdfusetitle]{hyperref}
17
-\usepackage[nohyperref]{icml2018}
17
+\usepackage[nonatbib]{nips_2018}
18 18
 
19 19
 % Fonts
20 20
 % \usepackage{lmodern} # :(
@@ -35,8 +35,7 @@
35 35
 \usepackage{url}
36 36
 \usepackage[capitalise, nameinlink, noabbrev]{cleveref}
37 37
 \crefformat{equation}{(#2#1#3)}
38
-\usepackage[style=apa, backend=biber, url=false]{biblatex}
39
-\DeclareLanguageMapping{english}{english-apa}
38
+\usepackage[style=numeric, backend=biber, url=false]{biblatex}
40 39
 
41 40
 % Misc
42 41
 \usepackage{todonotes}

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