Mercurial > ift6266

annotate writeup/techreport.tex @ 393:4c840798d290

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
added examples of figure and table of results
author Yoshua Bengio <bengioy@iro.umontreal.ca>
date Tue, 27 Apr 2010 09:59:40 -0400
parents 5f8fffd7347f
children fe2e2964e7a3
rev   line source
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
1 \documentclass[12pt,letterpaper]{article}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
2 \usepackage[utf8]{inputenc}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
3 \usepackage{graphicx}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
4 \usepackage{times}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
5 \usepackage{mlapa}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
6
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
7 \begin{document}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
8 \title{Generating and Exploiting Perturbed Training Data for Deep Architectures}
381
0a91fc69ff90 authors
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 379
diff changeset
9 \author{The IFT6266 Gang}
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
10 \date{April 2010, Technical Report, Dept. IRO, U. Montreal}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
11
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
12 \maketitle
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
13
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
14 \begin{abstract}
392
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
15 Recent theoretical and empirical work in statistical machine learning has
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
16 demonstrated the importance of learning algorithms for deep
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
17 architectures, i.e., function classes obtained by composing multiple
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
18 non-linear transformations. In the area of handwriting recognition,
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
19 deep learning algorithms
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
20 had been evaluated on rather small datasets with a few tens of thousands
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
21 of examples. Here we propose a powerful generator of variations
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
22 of examples for character images based on a pipeline of stochastic
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
23 transformations that include not only the usual affine transformations
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
24 but also the addition of slant, local elastic deformations, changes
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
25 in thickness, background images, color, contrast, occlusion, and
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
26 various types of pixel and spatially correlated noise.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
27 We evaluate a deep learning algorithm (Stacked Denoising Autoencoders)
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
28 on the task of learning to classify digits and letters transformed
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
29 with this pipeline, using the hundreds of millions of generated examples
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
30 and testing on the full NIST test set.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
31 We find that the SDA outperforms its
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
32 shallow counterpart, an ordinary Multi-Layer Perceptron,
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
33 and that it is better able to take advantage of the additional
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
34 generated data.
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
35 \end{abstract}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
36
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
37 \section{Introduction}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
38
392
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
39 Deep Learning has emerged as a promising new area of research in
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
40 statistical machine learning (see~\emcite{Bengio-2009} for a review).
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
41 Learning algorithms for deep architectures are centered on the learning
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
42 of useful representations of data, which are better suited to the task at hand.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
43 This is in great part inspired by observations of the mammalian visual cortex,
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
44 which consists of a chain of processing elements, each of which is associated with a
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
45 different representation. In fact,
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
46 it was found recently that the features learnt in deep architectures resemble
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
47 those observed in the first two of these stages (in areas V1 and V2
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
48 of visual cortex)~\cite{HonglakL2008}.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
49 Processing images typically involves transforming the raw pixel data into
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
50 new {\bf representations} that can be used for analysis or classification.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
51 For example, a principal component analysis representation linearly projects
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
52 the input image into a lower-dimensional feature space.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
53 Why learn a representation? Current practice in the computer vision
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
54 literature converts the raw pixels into a hand-crafted representation
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
55 (e.g.\ SIFT features~\cite{Lowe04}), but deep learning algorithms
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
56 tend to discover similar features in their first few
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
57 levels~\cite{HonglakL2008,ranzato-08,Koray-08,VincentPLarochelleH2008-very-small}.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
58 Learning increases the
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
59 ease and practicality of developing representations that are at once
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
60 tailored to specific tasks, yet are able to borrow statistical strength
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
61 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
62 feature representation can lead to higher-level (more abstract, more
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
63 general) features that are more robust to unanticipated sources of
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
64 variance extant in real data.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
65
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
66 Whereas a deep architecture can in principle be more powerful than a shallow
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
67 one in terms of representation, depth appears to render the training problem
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
68 more difficult in terms of optimization and local minima.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
69 It is also only recently that
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
70 successful algorithms were proposed to overcome some of these
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
71 difficulties.
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
72
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
73 \section{Perturbation and Transformation of Character Images}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
74
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
75 \subsection{Affine Transformations}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
76 \subsection{Adding Slant}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
77 \subsection{Local Elastic Deformations}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
78 \subsection{Changing Thickness}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
79 \subsection{Occlusion}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
80 \subsection{Background Images}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
81 \subsection{Salt and Pepper Noise}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
82 \subsection{Spatially Gaussian Noise}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
83 \subsection{Color and Contrast Changes}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
84
393
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
85 \begin{figure}[h]
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
86 \resizebox{.99\textwidth}{!}{\includegraphics{images/example_t.png}}\\
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
87 \caption{Illustration of the pipeline of stochastic
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
88 transformations applied to the image of a lower-case t
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
89 (the upper left image). Each image in the pipeline (going from
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
90 left to right, first top line, then bottom line) shows the result
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
91 of applying one of the modules in the pipeline. The last image
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
92 (bottom right) is used as training example.}
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
93 \label{fig:pipeline}
4c840798d290 added examples of figure and table of results
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 392
diff changeset
94 \end{figure}
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
95
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
96 \section{Learning Algorithms for Deep Architectures}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
97
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
98 \section{Experimental Setup}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
99
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
100 \subsection{Training Datasets}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
101
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
102 \subsubsection{Data Sources}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
103
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
104 \begin{itemize}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
105 \item {\bf NIST}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
106 \item {\bf Fonts}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
107 \item {\bf Captchas}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
108 \item {\bf OCR data}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
109 \end{itemize}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
110
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
111 \subsubsection{Data Sets}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
112 \begin{itemize}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
113 \item {\bf NIST}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
114 \item {\bf P07}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
115 \item {\bf NISTP} {\em ne pas utiliser PNIST mais NISTP, pour rester politically correct...}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
116 \end{itemize}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
117
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
118 \subsection{Models and their Hyperparameters}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
119
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
120 \subsubsection{Multi-Layer Perceptrons (MLP)}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
121
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
122 \subsubsection{Stacked Denoising Auto-Encoders (SDAE)}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
123
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
124 \section{Experimental Results}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
125
392
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
126 \subsection{SDA vs MLP}
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
127
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
128 \begin{center}
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
129 \begin{tabular}{lcc}
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
130 & train w/ & train w/ \\
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
131 & NIST & P07 + NIST \\ \hline
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
132 SDA & & \\ \hline
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
133 MLP & & \\ \hline
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
134 \end{tabular}
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
135 \end{center}
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
136
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
137 \subsection{Perturbed Training Data More Helpful for SDAE}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
138
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
139 \subsection{Training with More Classes than Necessary}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
140
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
141 \section{Conclusions}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
142
392
5f8fffd7347f possible image for illustrating perturbations
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 381
diff changeset
143 \bibliography{strings,ml,aigaion}
379
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
144 \bibliographystyle{mlapa}
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
145
a21a174c1c18 added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
146 \end{document}