|
585
|
1 \documentclass{article} % For LaTeX2e
|
|
|
2 \usepackage{times}
|
|
|
3 \usepackage{wrapfig}
|
|
|
4 \usepackage{amsthm,amsmath,bbm}
|
|
|
5 \usepackage[psamsfonts]{amssymb}
|
|
|
6 \usepackage{algorithm,algorithmic}
|
|
|
7 \usepackage[utf8]{inputenc}
|
|
|
8 \usepackage{graphicx,subfigure}
|
|
|
9 \usepackage[numbers]{natbib}
|
|
|
10
|
|
|
11 \addtolength{\textwidth}{10mm}
|
|
|
12 \addtolength{\evensidemargin}{-5mm}
|
|
|
13 \addtolength{\oddsidemargin}{-5mm}
|
|
|
14
|
|
|
15 %\setlength\parindent{0mm}
|
|
|
16
|
|
|
17 \title{Deep Self-Taught Learning for Handwritten Character Recognition}
|
|
|
18 \author{
|
|
|
19 Frédéric Bastien \and
|
|
|
20 Yoshua Bengio \and
|
|
|
21 Arnaud Bergeron \and
|
|
|
22 Nicolas Boulanger-Lewandowski \and
|
|
|
23 Thomas Breuel \and
|
|
|
24 Youssouf Chherawala \and
|
|
|
25 Moustapha Cisse \and
|
|
|
26 Myriam Côté \and
|
|
|
27 Dumitru Erhan \and
|
|
|
28 Jeremy Eustache \and
|
|
|
29 Xavier Glorot \and
|
|
|
30 Xavier Muller \and
|
|
|
31 Sylvain Pannetier Lebeuf \and
|
|
|
32 Razvan Pascanu \and
|
|
|
33 Salah Rifai \and
|
|
|
34 Francois Savard \and
|
|
|
35 Guillaume Sicard
|
|
|
36 }
|
|
|
37 \date{September 30th, submission to MLJ special issue on learning from multi-label data}
|
|
|
38
|
|
|
39 \begin{document}
|
|
|
40
|
|
|
41 %\makeanontitle
|
|
|
42 \maketitle
|
|
|
43
|
|
|
44 %\vspace*{-2mm}
|
|
|
45 \begin{abstract}
|
|
|
46 Recent theoretical and empirical work in statistical machine learning has demonstrated the importance of learning algorithms for deep architectures, i.e., function classes obtained by composing multiple non-linear transformations. Self-taught learning (exploiting unlabeled examples or examples from other distributions) has already been applied to deep learners, but mostly to show the advantage of unlabeled examples. Here we explore the advantage brought by {\em out-of-distribution examples}. For this purpose we developed a powerful generator of stochastic variations and noise processes for character images, including not only affine transformations but also slant, local elastic deformations, changes in thickness, background images, grey level changes, contrast, occlusion, and various types of noise. The out-of-distribution examples are obtained from these highly distorted images or by including examples of object classes different from those in the target test set. We show that {\em deep learners benefit more from out-of-distribution examples than a corresponding shallow learner}, at least in the area of handwritten character recognition. In fact, we show that they beat previously published results and reach human-level performance on both handwritten digit classification and 62-class handwritten character recognition.
|
|
|
47 \end{abstract}
|
|
|
48 %\vspace*{-3mm}
|
|
|
49
|
|
|
50 Keywords: self-taught learning, multi-task learning, out-of-distribution examples, deep learning, handwriting recognition.
|
|
|
51
|
|
|
52 \section{Introduction}
|
|
|
53 %\vspace*{-1mm}
|
|
|
54
|
|
|
55 {\bf Deep Learning} has emerged as a promising new area of research in
|
|
|
56 statistical machine learning (see~\citet{Bengio-2009} for a review).
|
|
|
57 Learning algorithms for deep architectures are centered on the learning
|
|
|
58 of useful representations of data, which are better suited to the task at hand,
|
|
|
59 and are organized in a hierarchy with multiple levels.
|
|
|
60 This is in part inspired by observations of the mammalian visual cortex,
|
|
|
61 which consists of a chain of processing elements, each of which is associated with a
|
|
|
62 different representation of the raw visual input. In fact,
|
|
|
63 it was found recently that the features learnt in deep architectures resemble
|
|
|
64 those observed in the first two of these stages (in areas V1 and V2
|
|
|
65 of visual cortex)~\citep{HonglakL2008}, and that they become more and
|
|
|
66 more invariant to factors of variation (such as camera movement) in
|
|
|
67 higher layers~\citep{Goodfellow2009}.
|
|
|
68 Learning a hierarchy of features increases the
|
|
|
69 ease and practicality of developing representations that are at once
|
|
|
70 tailored to specific tasks, yet are able to borrow statistical strength
|
|
|
71 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
|
|
|
72 feature representation can lead to higher-level (more abstract, more
|
|
|
73 general) features that are more robust to unanticipated sources of
|
|
|
74 variance extant in real data.
|
|
|
75
|
|
|
76 {\bf Self-taught learning}~\citep{RainaR2007} is a paradigm that combines principles
|
|
|
77 of semi-supervised and multi-task learning: the learner can exploit examples
|
|
|
78 that are unlabeled and possibly come from a distribution different from the target
|
|
|
79 distribution, e.g., from other classes than those of interest.
|
|
|
80 It has already been shown that deep learners can clearly take advantage of
|
|
|
81 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small},
|
|
|
82 but more needs to be done to explore the impact
|
|
|
83 of {\em out-of-distribution} examples and of the {\em multi-task} setting
|
|
|
84 (one exception is~\citep{CollobertR2008}, which uses a different kind
|
|
|
85 of learning algorithm). In particular the {\em relative
|
|
|
86 advantage of deep learning} for these settings has not been evaluated.
|
|
|
87 The hypothesis discussed in the conclusion is that in the context of
|
|
|
88 multi-task learning and the availability of out-of-distribution training examples,
|
|
|
89 a deep hierarchy of features
|
|
|
90 may be better able to provide sharing of statistical strength
|
|
|
91 between different regions in input space or different tasks, compared to
|
|
|
92 a shallow learner.
|
|
|
93
|
|
|
94 Whereas a deep architecture can in principle be more powerful than a
|
|
|
95 shallow one in terms of representation, depth appears to render the
|
|
|
96 training problem more difficult in terms of optimization and local minima.
|
|
|
97 It is also only recently that successful algorithms were proposed to
|
|
|
98 overcome some of these difficulties. All are based on unsupervised
|
|
|
99 learning, often in an greedy layer-wise ``unsupervised pre-training''
|
|
|
100 stage~\citep{Bengio-2009}. One of these layer initialization techniques,
|
|
|
101 applied here, is the Denoising
|
|
|
102 Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see Figure~\ref{fig:da}),
|
|
|
103 which
|
|
|
104 performed similarly or better than previously proposed Restricted Boltzmann
|
|
|
105 Machines in terms of unsupervised extraction of a hierarchy of features
|
|
|
106 useful for classification. Each layer is trained to denoise its
|
|
|
107 input, creating a layer of features that can be used as input for the next layer.
|
|
|
108
|
|
|
109 %The principle is that each layer starting from
|
|
|
110 %the bottom is trained to encode its input (the output of the previous
|
|
|
111 %layer) and to reconstruct it from a corrupted version. After this
|
|
|
112 %unsupervised initialization, the stack of DAs can be
|
|
|
113 %converted into a deep supervised feedforward neural network and fine-tuned by
|
|
|
114 %stochastic gradient descent.
|
|
|
115
|
|
|
116 %
|
|
|
117 In this paper we ask the following questions:
|
|
|
118
|
|
|
119 %\begin{enumerate}
|
|
|
120 $\bullet$ %\item
|
|
|
121 Do the good results previously obtained with deep architectures on the
|
|
|
122 MNIST digit images generalize to the setting of a much larger and richer (but similar)
|
|
|
123 dataset, the NIST special database 19, with 62 classes and around 800k examples?
|
|
|
124
|
|
|
125 $\bullet$ %\item
|
|
|
126 To what extent does the perturbation of input images (e.g. adding
|
|
|
127 noise, affine transformations, background images) make the resulting
|
|
|
128 classifiers better not only on similarly perturbed images but also on
|
|
|
129 the {\em original clean examples}? We study this question in the
|
|
|
130 context of the 62-class and 10-class tasks of the NIST special database 19.
|
|
|
131
|
|
|
132 $\bullet$ %\item
|
|
|
133 Do deep architectures {\em benefit {\bf more} from such out-of-distribution}
|
|
|
134 examples, i.e. do they benefit more from the self-taught learning~\citep{RainaR2007} framework?
|
|
|
135 We use highly perturbed examples to generate out-of-distribution examples.
|
|
|
136
|
|
|
137 $\bullet$ %\item
|
|
|
138 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more}
|
|
|
139 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than
|
|
|
140 a corresponding shallow and purely supervised architecture?
|
|
|
141 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case)
|
|
|
142 to answer this question.
|
|
|
143 %\end{enumerate}
|
|
|
144
|
|
|
145 Our experimental results provide positive evidence towards all of these questions,
|
|
|
146 as well as classifiers that reach human-level performance on 62-class isolated character
|
|
|
147 recognition and beat previously published results on the NIST dataset (special database 19).
|
|
|
148 To achieve these results, we introduce in the next section a sophisticated system
|
|
|
149 for stochastically transforming character images and then explain the methodology,
|
|
|
150 which is based on training with or without these transformed images and testing on
|
|
|
151 clean ones. We measure the relative advantage of out-of-distribution examples
|
|
|
152 (perturbed or out-of-class)
|
|
|
153 for a deep learner vs a supervised shallow one.
|
|
|
154 Code for generating these transformations as well as for the deep learning
|
|
|
155 algorithms are made available at {\tt http://hg.assembla.com/ift6266}.
|
|
|
156 We estimate the relative advantage for deep learners of training with
|
|
|
157 other classes than those of interest, by comparing learners trained with
|
|
|
158 62 classes with learners trained with only a subset (on which they
|
|
|
159 are then tested).
|
|
|
160 The conclusion discusses
|
|
|
161 the more general question of why deep learners may benefit so much from
|
|
|
162 the self-taught learning framework. Since out-of-distribution data
|
|
|
163 (perturbed or from other related classes) is very common, this conclusion
|
|
|
164 is of practical importance.
|
|
|
165
|
|
|
166 %\vspace*{-3mm}
|
|
|
167 %\newpage
|
|
|
168 \section{Perturbed and Transformed Character Images}
|
|
|
169 \label{s:perturbations}
|
|
|
170 %\vspace*{-2mm}
|
|
|
171
|
|
|
172 \begin{wrapfigure}[8]{l}{0.15\textwidth}
|
|
|
173 %\begin{minipage}[b]{0.14\linewidth}
|
|
|
174 %\vspace*{-5mm}
|
|
|
175 \begin{center}
|
|
|
176 \includegraphics[scale=.4]{images/Original.png}\\
|
|
|
177 {\bf Original}
|
|
|
178 \end{center}
|
|
|
179 \end{wrapfigure}
|
|
|
180 %%\vspace{0.7cm}
|
|
|
181 %\end{minipage}%
|
|
|
182 %\hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth}
|
|
|
183 This section describes the different transformations we used to stochastically
|
|
|
184 transform $32 \times 32$ source images (such as the one on the left)
|
|
|
185 in order to obtain data from a larger distribution which
|
|
|
186 covers a domain substantially larger than the clean characters distribution from
|
|
|
187 which we start.
|
|
|
188 Although character transformations have been used before to
|
|
|
189 improve character recognizers, this effort is on a large scale both
|
|
|
190 in number of classes and in the complexity of the transformations, hence
|
|
|
191 in the complexity of the learning task.
|
|
|
192 The code for these transformations (mostly python) is available at
|
|
|
193 {\tt http://hg.assembla.com/ift6266}. All the modules in the pipeline share
|
|
|
194 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
|
|
|
195 amount of deformation or noise introduced.
|
|
|
196 There are two main parts in the pipeline. The first one,
|
|
|
197 from slant to pinch below, performs transformations. The second
|
|
|
198 part, from blur to contrast, adds different kinds of noise.
|
|
|
199 %\end{minipage}
|
|
|
200
|
|
|
201 %\vspace*{1mm}
|
|
|
202 \subsection{Transformations}
|
|
|
203 %{\large\bf 2.1 Transformations}
|
|
|
204 %\vspace*{1mm}
|
|
|
205
|
|
|
206 \subsubsection*{Thickness}
|
|
|
207
|
|
|
208 %\begin{wrapfigure}[7]{l}{0.15\textwidth}
|
|
|
209 \begin{minipage}[b]{0.14\linewidth}
|
|
|
210 %\centering
|
|
|
211 \begin{center}
|
|
|
212 \vspace*{-5mm}
|
|
|
213 \includegraphics[scale=.4]{images/Thick_only.png}\\
|
|
|
214 %{\bf Thickness}
|
|
|
215 \end{center}
|
|
|
216 \vspace{.6cm}
|
|
|
217 \end{minipage}%
|
|
|
218 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth}
|
|
|
219 %\end{wrapfigure}
|
|
|
220 To change character {\bf thickness}, morphological operators of dilation and erosion~\citep{Haralick87,Serra82}
|
|
|
221 are applied. The neighborhood of each pixel is multiplied
|
|
|
222 element-wise with a {\em structuring element} matrix.
|
|
|
223 The pixel value is replaced by the maximum or the minimum of the resulting
|
|
|
224 matrix, respectively for dilation or erosion. Ten different structural elements with
|
|
|
225 increasing dimensions (largest is $5\times5$) were used. For each image,
|
|
|
226 randomly sample the operator type (dilation or erosion) with equal probability and one structural
|
|
|
227 element from a subset of the $n=round(m \times complexity)$ smallest structuring elements
|
|
|
228 where $m=10$ for dilation and $m=6$ for erosion (to avoid completely erasing thin characters).
|
|
|
229 A neutral element (no transformation)
|
|
|
230 is always present in the set.
|
|
|
231 %%\vspace{.4cm}
|
|
|
232 \end{minipage}
|
|
|
233
|
|
|
234 \vspace{2mm}
|
|
|
235
|
|
|
236 \subsubsection*{Slant}
|
|
|
237 \vspace*{2mm}
|
|
|
238
|
|
|
239 \begin{minipage}[b]{0.14\linewidth}
|
|
|
240 \centering
|
|
|
241 \includegraphics[scale=.4]{images/Slant_only.png}\\
|
|
|
242 %{\bf Slant}
|
|
|
243 \end{minipage}%
|
|
|
244 \hspace{0.3cm}
|
|
|
245 \begin{minipage}[b]{0.83\linewidth}
|
|
|
246 %\centering
|
|
|
247 To produce {\bf slant}, each row of the image is shifted
|
|
|
248 proportionally to its height: $shift = round(slant \times height)$.
|
|
|
249 $slant \sim U[-complexity,complexity]$.
|
|
|
250 The shift is randomly chosen to be either to the left or to the right.
|
|
|
251 \vspace{5mm}
|
|
|
252 \end{minipage}
|
|
|
253 %\vspace*{-4mm}
|
|
|
254
|
|
|
255 %\newpage
|
|
|
256
|
|
|
257 \subsubsection*{Affine Transformations}
|
|
|
258
|
|
|
259 \begin{minipage}[b]{0.14\linewidth}
|
|
|
260 %\centering
|
|
|
261 %\begin{wrapfigure}[8]{l}{0.15\textwidth}
|
|
|
262 \begin{center}
|
|
|
263 \includegraphics[scale=.4]{images/Affine_only.png}
|
|
|
264 \vspace*{6mm}
|
|
|
265 %{\small {\bf Affine \mbox{Transformation}}}
|
|
|
266 \end{center}
|
|
|
267 %\end{wrapfigure}
|
|
|
268 \end{minipage}%
|
|
|
269 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth}
|
|
|
270 \noindent A $2 \times 3$ {\bf affine transform} matrix (with
|
|
|
271 parameters $(a,b,c,d,e,f)$) is sampled according to the $complexity$.
|
|
|
272 Output pixel $(x,y)$ takes the value of input pixel
|
|
|
273 nearest to $(ax+by+c,dx+ey+f)$,
|
|
|
274 producing scaling, translation, rotation and shearing.
|
|
|
275 Marginal distributions of $(a,b,c,d,e,f)$ have been tuned to
|
|
|
276 forbid large rotations (to avoid confusing classes) but to give good
|
|
|
277 variability of the transformation: $a$ and $d$ $\sim U[1-3
|
|
|
278 complexity,1+3\,complexity]$, $b$ and $e$ $\sim U[-3 \,complexity,3\,
|
|
|
279 complexity]$, and $c$ and $f \sim U[-4 \,complexity, 4 \,
|
|
|
280 complexity]$.\\
|
|
|
281 \end{minipage}
|
|
|
282
|
|
|
283 %\vspace*{-4.5mm}
|
|
|
284 \subsubsection*{Local Elastic Deformations}
|
|
|
285
|
|
|
286 %\begin{minipage}[t]{\linewidth}
|
|
|
287 %\begin{wrapfigure}[7]{l}{0.15\textwidth}
|
|
|
288 %\hspace*{-8mm}
|
|
|
289 \begin{minipage}[b]{0.14\linewidth}
|
|
|
290 %\centering
|
|
|
291 \begin{center}
|
|
|
292 \vspace*{5mm}
|
|
|
293 \includegraphics[scale=.4]{images/Localelasticdistorsions_only.png}
|
|
|
294 %{\bf Local Elastic Deformation}
|
|
|
295 \end{center}
|
|
|
296 %\end{wrapfigure}
|
|
|
297 \end{minipage}%
|
|
|
298 \hspace{3mm}
|
|
|
299 \begin{minipage}[b]{0.85\linewidth}
|
|
|
300 %%\vspace*{-20mm}
|
|
|
301 The {\bf local elastic deformation}
|
|
|
302 module induces a ``wiggly'' effect in the image, following~\citet{SimardSP03-short},
|
|
|
303 which provides more details.
|
|
|
304 The intensity of the displacement fields is given by
|
|
|
305 $\alpha = \sqrt[3]{complexity} \times 10.0$, which are
|
|
|
306 convolved with a Gaussian 2D kernel (resulting in a blur) of
|
|
|
307 standard deviation $\sigma = 10 - 7 \times\sqrt[3]{complexity}$.
|
|
|
308 \vspace{2mm}
|
|
|
309 \end{minipage}
|
|
|
310
|
|
|
311 \vspace*{4mm}
|
|
|
312
|
|
|
313 \subsubsection*{Pinch}
|
|
|
314
|
|
|
315 \begin{minipage}[b]{0.14\linewidth}
|
|
|
316 %\centering
|
|
|
317 %\begin{wrapfigure}[7]{l}{0.15\textwidth}
|
|
|
318 %\vspace*{-5mm}
|
|
|
319 \begin{center}
|
|
|
320 \includegraphics[scale=.4]{images/Pinch_only.png}\\
|
|
|
321 \vspace*{15mm}
|
|
|
322 %{\bf Pinch}
|
|
|
323 \end{center}
|
|
|
324 %\end{wrapfigure}
|
|
|
325 %%\vspace{.6cm}
|
|
|
326 \end{minipage}%
|
|
|
327 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth}
|
|
|
328 The {\bf pinch} module applies the ``Whirl and pinch'' GIMP filter with whirl set to 0.
|
|
|
329 A pinch is ``similar to projecting the image onto an elastic
|
|
|
330 surface and pressing or pulling on the center of the surface'' (GIMP documentation manual).
|
|
|
331 For a square input image, draw a radius-$r$ disk
|
|
|
332 around its center $C$. Any pixel $P$ belonging to
|
|
|
333 that disk has its value replaced by
|
|
|
334 the value of a ``source'' pixel in the original image,
|
|
|
335 on the line that goes through $C$ and $P$, but
|
|
|
336 at some other distance $d_2$. Define $d_1=distance(P,C)$
|
|
|
337 and $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times
|
|
|
338 d_1$, where $pinch$ is a parameter of the filter.
|
|
|
339 The actual value is given by bilinear interpolation considering the pixels
|
|
|
340 around the (non-integer) source position thus found.
|
|
|
341 Here $pinch \sim U[-complexity, 0.7 \times complexity]$.
|
|
|
342 %%\vspace{1.5cm}
|
|
|
343 \end{minipage}
|
|
|
344
|
|
|
345 %\vspace{1mm}
|
|
|
346
|
|
|
347 %{\large\bf 2.2 Injecting Noise}
|
|
|
348 \subsection{Injecting Noise}
|
|
|
349 %\vspace{2mm}
|
|
|
350
|
|
|
351 \subsubsection*{Motion Blur}
|
|
|
352
|
|
|
353 %%\vspace*{-.2cm}
|
|
|
354 \begin{minipage}[t]{0.14\linewidth}
|
|
|
355 \centering
|
|
|
356 \vspace*{0mm}
|
|
|
357 \includegraphics[scale=.4]{images/Motionblur_only.png}
|
|
|
358 %{\bf Motion Blur}
|
|
|
359 \end{minipage}%
|
|
|
360 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth}
|
|
|
361 %%\vspace*{.5mm}
|
|
|
362 \vspace*{2mm}
|
|
|
363 The {\bf motion blur} module is GIMP's ``linear motion blur'', which
|
|
|
364 has parameters $length$ and $angle$. The value of
|
|
|
365 a pixel in the final image is approximately the mean of the first $length$ pixels
|
|
|
366 found by moving in the $angle$ direction,
|
|
|
367 $angle \sim U[0,360]$ degrees, and $length \sim {\rm Normal}(0,(3 \times complexity)^2)$.
|
|
|
368 %\vspace{5mm}
|
|
|
369 \end{minipage}
|
|
|
370
|
|
|
371 %\vspace*{1mm}
|
|
|
372
|
|
|
373 \subsubsection*{Occlusion}
|
|
|
374
|
|
|
375 \begin{minipage}[t]{0.14\linewidth}
|
|
|
376 \centering
|
|
|
377 \vspace*{3mm}
|
|
|
378 \includegraphics[scale=.4]{images/occlusion_only.png}\\
|
|
|
379 %{\bf Occlusion}
|
|
|
380 %%\vspace{.5cm}
|
|
|
381 \end{minipage}%
|
|
|
382 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth}
|
|
|
383 %\vspace*{-18mm}
|
|
|
384 The {\bf occlusion} module selects a random rectangle from an {\em occluder} character
|
|
|
385 image and places it over the original {\em occluded}
|
|
|
386 image. Pixels are combined by taking the max(occluder, occluded),
|
|
|
387 i.e. keeping the lighter ones.
|
|
|
388 The rectangle corners
|
|
|
389 are sampled so that larger complexity gives larger rectangles.
|
|
|
390 The destination position in the occluded image are also sampled
|
|
|
391 according to a normal distribution.
|
|
|
392 This module is skipped with probability 60\%.
|
|
|
393 %%\vspace{7mm}
|
|
|
394 \end{minipage}
|
|
|
395
|
|
|
396 %\vspace*{1mm}
|
|
|
397 \subsubsection*{Gaussian Smoothing}
|
|
|
398
|
|
|
399 %\begin{wrapfigure}[8]{l}{0.15\textwidth}
|
|
|
400 %\vspace*{-6mm}
|
|
|
401 \begin{minipage}[t]{0.14\linewidth}
|
|
|
402 \begin{center}
|
|
|
403 %\centering
|
|
|
404 \vspace*{6mm}
|
|
|
405 \includegraphics[scale=.4]{images/Bruitgauss_only.png}
|
|
|
406 %{\bf Gaussian Smoothing}
|
|
|
407 \end{center}
|
|
|
408 %\end{wrapfigure}
|
|
|
409 %%\vspace{.5cm}
|
|
|
410 \end{minipage}%
|
|
|
411 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth}
|
|
|
412 With the {\bf Gaussian smoothing} module,
|
|
|
413 different regions of the image are spatially smoothed.
|
|
|
414 This is achieved by first convolving
|
|
|
415 the image with an isotropic Gaussian kernel of
|
|
|
416 size and variance chosen uniformly in the ranges $[12,12 + 20 \times
|
|
|
417 complexity]$ and $[2,2 + 6 \times complexity]$. This filtered image is normalized
|
|
|
418 between $0$ and $1$. We also create an isotropic weighted averaging window, of the
|
|
|
419 kernel size, with maximum value at the center. For each image we sample
|
|
|
420 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be
|
|
|
421 averaging centers between the original image and the filtered one. We
|
|
|
422 initialize to zero a mask matrix of the image size. For each selected pixel
|
|
|
423 we add to the mask the averaging window centered on it. The final image is
|
|
|
424 computed from the following element-wise operation: $\frac{image + filtered\_image
|
|
|
425 \times mask}{mask+1}$.
|
|
|
426 This module is skipped with probability 75\%.
|
|
|
427 \end{minipage}
|
|
|
428
|
|
|
429 %\newpage
|
|
|
430
|
|
|
431 %\vspace*{-9mm}
|
|
|
432 \subsubsection*{Permute Pixels}
|
|
|
433
|
|
|
434 %\hspace*{-3mm}\begin{minipage}[t]{0.18\linewidth}
|
|
|
435 %\centering
|
|
|
436 \begin{minipage}[t]{0.14\textwidth}
|
|
|
437 %\begin{wrapfigure}[7]{l}{
|
|
|
438 %\vspace*{-5mm}
|
|
|
439 \begin{center}
|
|
|
440 \vspace*{1mm}
|
|
|
441 \includegraphics[scale=.4]{images/Permutpixel_only.png}
|
|
|
442 %{\small\bf Permute Pixels}
|
|
|
443 \end{center}
|
|
|
444 %\end{wrapfigure}
|
|
|
445 \end{minipage}%
|
|
|
446 \hspace{3mm}\begin{minipage}[t]{0.86\linewidth}
|
|
|
447 \vspace*{1mm}
|
|
|
448 %%\vspace*{-20mm}
|
|
|
449 This module {\bf permutes neighbouring pixels}. It first selects a
|
|
|
450 fraction $\frac{complexity}{3}$ of pixels randomly in the image. Each
|
|
|
451 of these pixels is then sequentially exchanged with a random pixel
|
|
|
452 among its four nearest neighbors (on its left, right, top or bottom).
|
|
|
453 This module is skipped with probability 80\%.\\
|
|
|
454 %\vspace*{1mm}
|
|
|
455 \end{minipage}
|
|
|
456
|
|
|
457 %\vspace{-3mm}
|
|
|
458
|
|
|
459 \subsubsection*{Gaussian Noise}
|
|
|
460
|
|
|
461 \begin{minipage}[t]{0.14\textwidth}
|
|
|
462 %\begin{wrapfigure}[7]{l}{
|
|
|
463 %%\vspace*{-3mm}
|
|
|
464 \begin{center}
|
|
|
465 %\hspace*{-3mm}\begin{minipage}[t]{0.18\linewidth}
|
|
|
466 %\centering
|
|
|
467 \vspace*{0mm}
|
|
|
468 \includegraphics[scale=.4]{images/Distorsiongauss_only.png}
|
|
|
469 %{\small \bf Gauss. Noise}
|
|
|
470 \end{center}
|
|
|
471 %\end{wrapfigure}
|
|
|
472 \end{minipage}%
|
|
|
473 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth}
|
|
|
474 \vspace*{1mm}
|
|
|
475 %\vspace*{12mm}
|
|
|
476 The {\bf Gaussian noise} module simply adds, to each pixel of the image independently, a
|
|
|
477 noise $\sim Normal(0,(\frac{complexity}{10})^2)$.
|
|
|
478 This module is skipped with probability 70\%.
|
|
|
479 %%\vspace{1.1cm}
|
|
|
480 \end{minipage}
|
|
|
481
|
|
|
482 %\vspace*{1.2cm}
|
|
|
483
|
|
|
484 \subsubsection*{Background Image Addition}
|
|
|
485
|
|
|
486 \begin{minipage}[t]{\linewidth}
|
|
|
487 \begin{minipage}[t]{0.14\linewidth}
|
|
|
488 \centering
|
|
|
489 \vspace*{0mm}
|
|
|
490 \includegraphics[scale=.4]{images/background_other_only.png}
|
|
|
491 %{\small \bf Bg Image}
|
|
|
492 \end{minipage}%
|
|
|
493 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth}
|
|
|
494 \vspace*{1mm}
|
|
|
495 Following~\citet{Larochelle-jmlr-2009}, the {\bf background image} module adds a random
|
|
|
496 background image behind the letter, from a randomly chosen natural image,
|
|
|
497 with contrast adjustments depending on $complexity$, to preserve
|
|
|
498 more or less of the original character image.
|
|
|
499 %%\vspace{.8cm}
|
|
|
500 \end{minipage}
|
|
|
501 \end{minipage}
|
|
|
502 %%\vspace{-.7cm}
|
|
|
503
|
|
|
504 \subsubsection*{Salt and Pepper Noise}
|
|
|
505
|
|
|
506 \begin{minipage}[t]{0.14\linewidth}
|
|
|
507 \centering
|
|
|
508 \vspace*{0mm}
|
|
|
509 \includegraphics[scale=.4]{images/Poivresel_only.png}
|
|
|
510 %{\small \bf Salt \& Pepper}
|
|
|
511 \end{minipage}%
|
|
|
512 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth}
|
|
|
513 \vspace*{1mm}
|
|
|
514 The {\bf salt and pepper noise} module adds noise $\sim U[0,1]$ to random subsets of pixels.
|
|
|
515 The number of selected pixels is $0.2 \times complexity$.
|
|
|
516 This module is skipped with probability 75\%.
|
|
|
517 %%\vspace{.9cm}
|
|
|
518 \end{minipage}
|
|
|
519 %%\vspace{-.7cm}
|
|
|
520
|
|
|
521 %\vspace{1mm}
|
|
|
522 \subsubsection*{Scratches}
|
|
|
523
|
|
|
524 \begin{minipage}[t]{0.14\textwidth}
|
|
|
525 %\begin{wrapfigure}[7]{l}{
|
|
|
526 %\begin{minipage}[t]{0.14\linewidth}
|
|
|
527 %\centering
|
|
|
528 \begin{center}
|
|
|
529 \vspace*{4mm}
|
|
|
530 %\hspace*{-1mm}
|
|
|
531 \includegraphics[scale=.4]{images/Rature_only.png}\\
|
|
|
532 %{\bf Scratches}
|
|
|
533 \end{center}
|
|
|
534 \end{minipage}%
|
|
|
535 %\end{wrapfigure}
|
|
|
536 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth}
|
|
|
537 %%\vspace{.4cm}
|
|
|
538 The {\bf scratches} module places line-like white patches on the image. The
|
|
|
539 lines are heavily transformed images of the digit ``1'' (one), chosen
|
|
|
540 at random among 500 such 1 images,
|
|
|
541 randomly cropped and rotated by an angle $\sim Normal(0,(100 \times
|
|
|
542 complexity)^2$ (in degrees), using bi-cubic interpolation.
|
|
|
543 Two passes of a grey-scale morphological erosion filter
|
|
|
544 are applied, reducing the width of the line
|
|
|
545 by an amount controlled by $complexity$.
|
|
|
546 This module is skipped with probability 85\%. The probabilities
|
|
|
547 of applying 1, 2, or 3 patches are (50\%,30\%,20\%).
|
|
|
548 \end{minipage}
|
|
|
549
|
|
|
550 %\vspace*{1mm}
|
|
|
551
|
|
|
552 \subsubsection*{Grey Level and Contrast Changes}
|
|
|
553
|
|
|
554 \begin{minipage}[t]{0.15\linewidth}
|
|
|
555 \centering
|
|
|
556 \vspace*{0mm}
|
|
|
557 \includegraphics[scale=.4]{images/Contrast_only.png}
|
|
|
558 %{\bf Grey Level \& Contrast}
|
|
|
559 \end{minipage}%
|
|
|
560 \hspace{3mm}\begin{minipage}[t]{0.85\linewidth}
|
|
|
561 \vspace*{1mm}
|
|
|
562 The {\bf grey level and contrast} module changes the contrast by changing grey levels, and may invert the image polarity (white
|
|
|
563 to black and black to white). The contrast is $C \sim U[1-0.85 \times complexity,1]$
|
|
|
564 so the image is normalized into $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The
|
|
|
565 polarity is inverted with probability 50\%.
|
|
|
566 %%\vspace{.7cm}
|
|
|
567 \end{minipage}
|
|
|
568 %\vspace{2mm}
|
|
|
569
|
|
|
570
|
|
|
571 \iffalse
|
|
|
572 \begin{figure}[ht]
|
|
|
573 \centerline{\resizebox{.9\textwidth}{!}{\includegraphics{example_t.png}}}\\
|
|
|
574 \caption{Illustration of the pipeline of stochastic
|
|
|
575 transformations applied to the image of a lower-case \emph{t}
|
|
|
576 (the upper left image). Each image in the pipeline (going from
|
|
|
577 left to right, first top line, then bottom line) shows the result
|
|
|
578 of applying one of the modules in the pipeline. The last image
|
|
|
579 (bottom right) is used as training example.}
|
|
|
580 \label{fig:pipeline}
|
|
|
581 \end{figure}
|
|
|
582 \fi
|
|
|
583
|
|
|
584 %\vspace*{-3mm}
|
|
|
585 \section{Experimental Setup}
|
|
|
586 %\vspace*{-1mm}
|
|
|
587
|
|
|
588 Much previous work on deep learning had been performed on
|
|
|
589 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009},
|
|
|
590 with 60~000 examples, and variants involving 10~000
|
|
|
591 examples~\citep{Larochelle-jmlr-toappear-2008,VincentPLarochelleH2008}.
|
|
|
592 The focus here is on much larger training sets, from 10 times to
|
|
|
593 to 1000 times larger, and 62 classes.
|
|
|
594
|
|
|
595 The first step in constructing the larger datasets (called NISTP and P07) is to sample from
|
|
|
596 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas},
|
|
|
597 and {\bf OCR data} (scanned machine printed characters). Once a character
|
|
|
598 is sampled from one of these sources (chosen randomly), the second step is to
|
|
|
599 apply a pipeline of transformations and/or noise processes described in section \ref{s:perturbations}.
|
|
|
600
|
|
|
601 To provide a baseline of error rate comparison we also estimate human performance
|
|
|
602 on both the 62-class task and the 10-class digits task.
|
|
|
603 We compare the best Multi-Layer Perceptrons (MLP) against
|
|
|
604 the best Stacked Denoising Auto-encoders (SDA), when
|
|
|
605 both models' hyper-parameters are selected to minimize the validation set error.
|
|
|
606 We also provide a comparison against a precise estimate
|
|
|
607 of human performance obtained via Amazon's Mechanical Turk (AMT)
|
|
|
608 service (http://mturk.com).
|
|
|
609 AMT users are paid small amounts
|
|
|
610 of money to perform tasks for which human intelligence is required.
|
|
|
611 Mechanical Turk has been used extensively in natural language processing and vision.
|
|
|
612 %processing \citep{SnowEtAl2008} and vision
|
|
|
613 %\citep{SorokinAndForsyth2008,whitehill09}.
|
|
|
614 AMT users were presented
|
|
|
615 with 10 character images (from a test set) and asked to choose 10 corresponding ASCII
|
|
|
616 characters. They were forced to choose a single character class (either among the
|
|
|
617 62 or 10 character classes) for each image.
|
|
|
618 80 subjects classified 2500 images per (dataset,task) pair.
|
|
|
619 Different humans labelers sometimes provided a different label for the same
|
|
|
620 example, and we were able to estimate the error variance due to this effect
|
|
|
621 because each image was classified by 3 different persons.
|
|
|
622 The average error of humans on the 62-class task NIST test set
|
|
|
623 is 18.2\%, with a standard error of 0.1\%.
|
|
|
624
|
|
|
625 %\vspace*{-3mm}
|
|
|
626 \subsection{Data Sources}
|
|
|
627 %\vspace*{-2mm}
|
|
|
628
|
|
|
629 %\begin{itemize}
|
|
|
630 %\item
|
|
|
631 {\bf NIST.}
|
|
|
632 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995},
|
|
|
633 widely used for training and testing character
|
|
|
634 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}.
|
|
|
635 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications,
|
|
|
636 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes
|
|
|
637 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity.
|
|
|
638 The fourth partition (called $hsf_4$, 82587 examples),
|
|
|
639 experimentally recognized to be the most difficult one, is the one recommended
|
|
|
640 by NIST as a testing set and is used in our work as well as some previous work~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}
|
|
|
641 for that purpose. We randomly split the remainder (731668 examples) into a training set and a validation set for
|
|
|
642 model selection.
|
|
|
643 The performances reported by previous work on that dataset mostly use only the digits.
|
|
|
644 Here we use all the classes both in the training and testing phase. This is especially
|
|
|
645 useful to estimate the effect of a multi-task setting.
|
|
|
646 The distribution of the classes in the NIST training and test sets differs
|
|
|
647 substantially, with relatively many more digits in the test set, and a more uniform distribution
|
|
|
648 of letters in the test set (whereas in the training set they are distributed
|
|
|
649 more like in natural text).
|
|
|
650 %\vspace*{-1mm}
|
|
|
651
|
|
|
652 %\item
|
|
|
653 {\bf Fonts.}
|
|
|
654 In order to have a good variety of sources we downloaded an important number of free fonts from:
|
|
|
655 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}.
|
|
|
656 % TODO: pointless to anonymize, it's not pointing to our work
|
|
|
657 Including the operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from.
|
|
|
658 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image,
|
|
|
659 directly as input to our models.
|
|
|
660 %\vspace*{-1mm}
|
|
|
661
|
|
|
662 %\item
|
|
|
663 {\bf Captchas.}
|
|
|
664 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for
|
|
|
665 generating characters of the same format as the NIST dataset. This software is based on
|
|
|
666 a random character class generator and various kinds of transformations similar to those described in the previous sections.
|
|
|
667 In order to increase the variability of the data generated, many different fonts are used for generating the characters.
|
|
|
668 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity
|
|
|
669 depending on the value of the complexity parameter provided by the user of the data source.
|
|
|
670 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class?
|
|
|
671 %\vspace*{-1mm}
|
|
|
672
|
|
|
673 %\item
|
|
|
674 {\bf OCR data.}
|
|
|
675 A large set (2 million) of scanned, OCRed and manually verified machine-printed
|
|
|
676 characters where included as an
|
|
|
677 additional source. This set is part of a larger corpus being collected by the Image Understanding
|
|
|
678 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern
|
|
|
679 ({\tt http://www.iupr.com}), and which will be publicly released.
|
|
|
680 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this
|
|
|
681 %\end{itemize}
|
|
|
682
|
|
|
683 %\vspace*{-3mm}
|
|
|
684 \subsection{Data Sets}
|
|
|
685 %\vspace*{-2mm}
|
|
|
686
|
|
|
687 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label
|
|
|
688 from one of the 62 character classes.
|
|
|
689 %\begin{itemize}
|
|
|
690 %\vspace*{-1mm}
|
|
|
691
|
|
|
692 %\item
|
|
|
693 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has
|
|
|
694 \{651668 / 80000 / 82587\} \{training / validation / test\} examples.
|
|
|
695 %\vspace*{-1mm}
|
|
|
696
|
|
|
697 %\item
|
|
|
698 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources
|
|
|
699 and sending them through the transformation pipeline described in section \ref{s:perturbations}.
|
|
|
700 For each new example to generate, a data source is selected with probability $10\%$ from the fonts,
|
|
|
701 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the
|
|
|
702 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$.
|
|
|
703 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples.
|
|
|
704 %\vspace*{-1mm}
|
|
|
705
|
|
|
706 %\item
|
|
|
707 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources)
|
|
|
708 except that we only apply
|
|
|
709 transformations from slant to pinch. Therefore, the character is
|
|
|
710 transformed but no additional noise is added to the image, giving images
|
|
|
711 closer to the NIST dataset.
|
|
|
712 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples.
|
|
|
713 %\end{itemize}
|
|
|
714
|
|
|
715 %\vspace*{-3mm}
|
|
|
716 \subsection{Models and their Hyperparameters}
|
|
|
717 %\vspace*{-2mm}
|
|
|
718
|
|
|
719 The experiments are performed using MLPs (with a single
|
|
|
720 hidden layer) and SDAs.
|
|
|
721 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.}
|
|
|
722
|
|
|
723 {\bf Multi-Layer Perceptrons (MLP).}
|
|
|
724 Whereas previous work had compared deep architectures to both shallow MLPs and
|
|
|
725 SVMs, we only compared to MLPs here because of the very large datasets used
|
|
|
726 (making the use of SVMs computationally challenging because of their quadratic
|
|
|
727 scaling behavior). Preliminary experiments on training SVMs (libSVM) with subsets of the training
|
|
|
728 set allowing the program to fit in memory yielded substantially worse results
|
|
|
729 than those obtained with MLPs. For training on nearly a billion examples
|
|
|
730 (with the perturbed data), the MLPs and SDA are much more convenient than
|
|
|
731 classifiers based on kernel methods.
|
|
|
732 The MLP has a single hidden layer with $\tanh$ activation functions, and softmax (normalized
|
|
|
733 exponentials) on the output layer for estimating $P(class | image)$.
|
|
|
734 The number of hidden units is taken in $\{300,500,800,1000,1500\}$.
|
|
|
735 Training examples are presented in minibatches of size 20. A constant learning
|
|
|
736 rate was chosen among $\{0.001, 0.01, 0.025, 0.075, 0.1, 0.5\}$.
|
|
|
737 %through preliminary experiments (measuring performance on a validation set),
|
|
|
738 %and $0.1$ (which was found to work best) was then selected for optimizing on
|
|
|
739 %the whole training sets.
|
|
|
740 %\vspace*{-1mm}
|
|
|
741
|
|
|
742
|
|
|
743 {\bf Stacked Denoising Auto-Encoders (SDA).}
|
|
|
744 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs)
|
|
|
745 can be used to initialize the weights of each layer of a deep MLP (with many hidden
|
|
|
746 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006},
|
|
|
747 apparently setting parameters in the
|
|
|
748 basin of attraction of supervised gradient descent yielding better
|
|
|
749 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised
|
|
|
750 pre-training phase} uses all of the training images but not the training labels.
|
|
|
751 Each layer is trained in turn to produce a new representation of its input
|
|
|
752 (starting from the raw pixels).
|
|
|
753 It is hypothesized that the
|
|
|
754 advantage brought by this procedure stems from a better prior,
|
|
|
755 on the one hand taking advantage of the link between the input
|
|
|
756 distribution $P(x)$ and the conditional distribution of interest
|
|
|
757 $P(y|x)$ (like in semi-supervised learning), and on the other hand
|
|
|
758 taking advantage of the expressive power and bias implicit in the
|
|
|
759 deep architecture (whereby complex concepts are expressed as
|
|
|
760 compositions of simpler ones through a deep hierarchy).
|
|
|
761
|
|
|
762 \begin{figure}[ht]
|
|
|
763 %\vspace*{-2mm}
|
|
|
764 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}}
|
|
|
765 %\vspace*{-2mm}
|
|
|
766 \caption{Illustration of the computations and training criterion for the denoising
|
|
|
767 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of
|
|
|
768 the layer (i.e. raw input or output of previous layer)
|
|
|
769 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$.
|
|
|
770 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which
|
|
|
771 is compared to the uncorrupted input $x$ through the loss function
|
|
|
772 $L_H(x,z)$, whose expected value is approximately minimized during training
|
|
|
773 by tuning $\theta$ and $\theta'$.}
|
|
|
774 \label{fig:da}
|
|
|
775 %\vspace*{-2mm}
|
|
|
776 \end{figure}
|
|
|
777
|
|
|
778 Here we chose to use the Denoising
|
|
|
779 Auto-encoder~\citep{VincentPLarochelleH2008} as the building block for
|
|
|
780 these deep hierarchies of features, as it is simple to train and
|
|
|
781 explain (see Figure~\ref{fig:da}, as well as
|
|
|
782 tutorial and code there: {\tt http://deeplearning.net/tutorial}),
|
|
|
783 provides efficient inference, and yielded results
|
|
|
784 comparable or better than RBMs in series of experiments
|
|
|
785 \citep{VincentPLarochelleH2008}. During training, a Denoising
|
|
|
786 Auto-encoder is presented with a stochastically corrupted version
|
|
|
787 of the input and trained to reconstruct the uncorrupted input,
|
|
|
788 forcing the hidden units to represent the leading regularities in
|
|
|
789 the data. Here we use the random binary masking corruption
|
|
|
790 (which sets to 0 a random subset of the inputs).
|
|
|
791 Once it is trained, in a purely unsupervised way,
|
|
|
792 its hidden units' activations can
|
|
|
793 be used as inputs for training a second one, etc.
|
|
|
794 After this unsupervised pre-training stage, the parameters
|
|
|
795 are used to initialize a deep MLP, which is fine-tuned by
|
|
|
796 the same standard procedure used to train them (see previous section).
|
|
|
797 The SDA hyper-parameters are the same as for the MLP, with the addition of the
|
|
|
798 amount of corruption noise (we used the masking noise process, whereby a
|
|
|
799 fixed proportion of the input values, randomly selected, are zeroed), and a
|
|
|
800 separate learning rate for the unsupervised pre-training stage (selected
|
|
|
801 from the same above set). The fraction of inputs corrupted was selected
|
|
|
802 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number
|
|
|
803 of hidden layers but it was fixed to 3 based on previous work with
|
|
|
804 SDAs on MNIST~\citep{VincentPLarochelleH2008}. The size of the hidden
|
|
|
805 layers was kept constant across hidden layers, and the best results
|
|
|
806 were obtained with the largest values that we could experiment
|
|
|
807 with given our patience, with 1000 hidden units.
|
|
|
808
|
|
|
809 %\vspace*{-1mm}
|
|
|
810
|
|
|
811 \begin{figure}[ht]
|
|
|
812 %\vspace*{-2mm}
|
|
|
813 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}}
|
|
|
814 %\vspace*{-3mm}
|
|
|
815 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained
|
|
|
816 on NIST, 1 on NISTP, and 2 on P07. Left: overall results
|
|
|
817 of all models, on NIST and NISTP test sets.
|
|
|
818 Right: error rates on NIST test digits only, along with the previous results from
|
|
|
819 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}
|
|
|
820 respectively based on ART, nearest neighbors, MLPs, and SVMs.}
|
|
|
821 \label{fig:error-rates-charts}
|
|
|
822 %\vspace*{-2mm}
|
|
|
823 \end{figure}
|
|
|
824
|
|
|
825
|
|
|
826 \begin{figure}[ht]
|
|
|
827 %\vspace*{-3mm}
|
|
|
828 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}}
|
|
|
829 %\vspace*{-3mm}
|
|
|
830 \caption{Relative improvement in error rate due to self-taught learning.
|
|
|
831 Left: Improvement (or loss, when negative)
|
|
|
832 induced by out-of-distribution examples (perturbed data).
|
|
|
833 Right: Improvement (or loss, when negative) induced by multi-task
|
|
|
834 learning (training on all classes and testing only on either digits,
|
|
|
835 upper case, or lower-case). The deep learner (SDA) benefits more from
|
|
|
836 both self-taught learning scenarios, compared to the shallow MLP.}
|
|
|
837 \label{fig:improvements-charts}
|
|
|
838 %\vspace*{-2mm}
|
|
|
839 \end{figure}
|
|
|
840
|
|
|
841 \section{Experimental Results}
|
|
|
842 %\vspace*{-2mm}
|
|
|
843
|
|
|
844 %%\vspace*{-1mm}
|
|
|
845 %\subsection{SDA vs MLP vs Humans}
|
|
|
846 %%\vspace*{-1mm}
|
|
|
847 The models are either trained on NIST (MLP0 and SDA0),
|
|
|
848 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested
|
|
|
849 on either NIST, NISTP or P07, either on the 62-class task
|
|
|
850 or on the 10-digits task. Training (including about half
|
|
|
851 for unsupervised pre-training, for DAs) on the larger
|
|
|
852 datasets takes around one day on a GPU-285.
|
|
|
853 Figure~\ref{fig:error-rates-charts} summarizes the results obtained,
|
|
|
854 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1,
|
|
|
855 SDA2), along with the previous results on the digits NIST special database
|
|
|
856 19 test set from the literature, respectively based on ARTMAP neural
|
|
|
857 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search
|
|
|
858 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs
|
|
|
859 ~\citep{Milgram+al-2005}. More detailed and complete numerical results
|
|
|
860 (figures and tables, including standard errors on the error rates) can be
|
|
|
861 found in Appendix.
|
|
|
862 The deep learner not only outperformed the shallow ones and
|
|
|
863 previously published performance (in a statistically and qualitatively
|
|
|
864 significant way) but when trained with perturbed data
|
|
|
865 reaches human performance on both the 62-class task
|
|
|
866 and the 10-class (digits) task.
|
|
|
867 17\% error (SDA1) or 18\% error (humans) may seem large but a large
|
|
|
868 majority of the errors from humans and from SDA1 are from out-of-context
|
|
|
869 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a
|
|
|
870 ``c'' and a ``C'' are often indistinguishible).
|
|
|
871
|
|
|
872 In addition, as shown in the left of
|
|
|
873 Figure~\ref{fig:improvements-charts}, the relative improvement in error
|
|
|
874 rate brought by self-taught learning is greater for the SDA, and these
|
|
|
875 differences with the MLP are statistically and qualitatively
|
|
|
876 significant.
|
|
|
877 The left side of the figure shows the improvement to the clean
|
|
|
878 NIST test set error brought by the use of out-of-distribution examples
|
|
|
879 (i.e. the perturbed examples examples from NISTP or P07).
|
|
|
880 Relative percent change is measured by taking
|
|
|
881 $100 \% \times$ (original model's error / perturbed-data model's error - 1).
|
|
|
882 The right side of
|
|
|
883 Figure~\ref{fig:improvements-charts} shows the relative improvement
|
|
|
884 brought by the use of a multi-task setting, in which the same model is
|
|
|
885 trained for more classes than the target classes of interest (i.e. training
|
|
|
886 with all 62 classes when the target classes are respectively the digits,
|
|
|
887 lower-case, or upper-case characters). Again, whereas the gain from the
|
|
|
888 multi-task setting is marginal or negative for the MLP, it is substantial
|
|
|
889 for the SDA. Note that to simplify these multi-task experiments, only the original
|
|
|
890 NIST dataset is used. For example, the MLP-digits bar shows the relative
|
|
|
891 percent improvement in MLP error rate on the NIST digits test set
|
|
|
892 is $100\% \times$ (single-task
|
|
|
893 model's error / multi-task model's error - 1). The single-task model is
|
|
|
894 trained with only 10 outputs (one per digit), seeing only digit examples,
|
|
|
895 whereas the multi-task model is trained with 62 outputs, with all 62
|
|
|
896 character classes as examples. Hence the hidden units are shared across
|
|
|
897 all tasks. For the multi-task model, the digit error rate is measured by
|
|
|
898 comparing the correct digit class with the output class associated with the
|
|
|
899 maximum conditional probability among only the digit classes outputs. The
|
|
|
900 setting is similar for the other two target classes (lower case characters
|
|
|
901 and upper case characters).
|
|
|
902 %%\vspace*{-1mm}
|
|
|
903 %\subsection{Perturbed Training Data More Helpful for SDA}
|
|
|
904 %%\vspace*{-1mm}
|
|
|
905
|
|
|
906 %%\vspace*{-1mm}
|
|
|
907 %\subsection{Multi-Task Learning Effects}
|
|
|
908 %%\vspace*{-1mm}
|
|
|
909
|
|
|
910 \iffalse
|
|
|
911 As previously seen, the SDA is better able to benefit from the
|
|
|
912 transformations applied to the data than the MLP. In this experiment we
|
|
|
913 define three tasks: recognizing digits (knowing that the input is a digit),
|
|
|
914 recognizing upper case characters (knowing that the input is one), and
|
|
|
915 recognizing lower case characters (knowing that the input is one). We
|
|
|
916 consider the digit classification task as the target task and we want to
|
|
|
917 evaluate whether training with the other tasks can help or hurt, and
|
|
|
918 whether the effect is different for MLPs versus SDAs. The goal is to find
|
|
|
919 out if deep learning can benefit more (or less) from multiple related tasks
|
|
|
920 (i.e. the multi-task setting) compared to a corresponding purely supervised
|
|
|
921 shallow learner.
|
|
|
922
|
|
|
923 We use a single hidden layer MLP with 1000 hidden units, and a SDA
|
|
|
924 with 3 hidden layers (1000 hidden units per layer), pre-trained and
|
|
|
925 fine-tuned on NIST.
|
|
|
926
|
|
|
927 Our results show that the MLP benefits marginally from the multi-task setting
|
|
|
928 in the case of digits (5\% relative improvement) but is actually hurt in the case
|
|
|
929 of characters (respectively 3\% and 4\% worse for lower and upper class characters).
|
|
|
930 On the other hand the SDA benefited from the multi-task setting, with relative
|
|
|
931 error rate improvements of 27\%, 15\% and 13\% respectively for digits,
|
|
|
932 lower and upper case characters, as shown in Table~\ref{tab:multi-task}.
|
|
|
933 \fi
|
|
|
934
|
|
|
935
|
|
|
936 %\vspace*{-2mm}
|
|
|
937 \section{Conclusions and Discussion}
|
|
|
938 %\vspace*{-2mm}
|
|
|
939
|
|
|
940 We have found that the self-taught learning framework is more beneficial
|
|
|
941 to a deep learner than to a traditional shallow and purely
|
|
|
942 supervised learner. More precisely,
|
|
|
943 the answers are positive for all the questions asked in the introduction.
|
|
|
944 %\begin{itemize}
|
|
|
945
|
|
|
946 $\bullet$ %\item
|
|
|
947 {\bf Do the good results previously obtained with deep architectures on the
|
|
|
948 MNIST digits generalize to a much larger and richer (but similar)
|
|
|
949 dataset, the NIST special database 19, with 62 classes and around 800k examples}?
|
|
|
950 Yes, the SDA {\em systematically outperformed the MLP and all the previously
|
|
|
951 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level
|
|
|
952 performance} at around 17\% error on the 62-class task and 1.4\% on the digits,
|
|
|
953 and beating previously published results on the same data.
|
|
|
954
|
|
|
955 $\bullet$ %\item
|
|
|
956 {\bf To what extent do self-taught learning scenarios help deep learners,
|
|
|
957 and do they help them more than shallow supervised ones}?
|
|
|
958 We found that distorted training examples not only made the resulting
|
|
|
959 classifier better on similarly perturbed images but also on
|
|
|
960 the {\em original clean examples}, and more importantly and more novel,
|
|
|
961 that deep architectures benefit more from such {\em out-of-distribution}
|
|
|
962 examples. MLPs were helped by perturbed training examples when tested on perturbed input
|
|
|
963 images (65\% relative improvement on NISTP)
|
|
|
964 but only marginally helped (5\% relative improvement on all classes)
|
|
|
965 or even hurt (10\% relative loss on digits)
|
|
|
966 with respect to clean examples . On the other hand, the deep SDAs
|
|
|
967 were significantly boosted by these out-of-distribution examples.
|
|
|
968 Similarly, whereas the improvement due to the multi-task setting was marginal or
|
|
|
969 negative for the MLP (from +5.6\% to -3.6\% relative change),
|
|
|
970 it was quite significant for the SDA (from +13\% to +27\% relative change),
|
|
|
971 which may be explained by the arguments below.
|
|
|
972 %\end{itemize}
|
|
|
973
|
|
|
974 In the original self-taught learning framework~\citep{RainaR2007}, the
|
|
|
975 out-of-sample examples were used as a source of unsupervised data, and
|
|
|
976 experiments showed its positive effects in a \emph{limited labeled data}
|
|
|
977 scenario. However, many of the results by \citet{RainaR2007} (who used a
|
|
|
978 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught
|
|
|
979 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases.
|
|
|
980 We note instead that, for deep
|
|
|
981 architectures, our experiments show that such a positive effect is accomplished
|
|
|
982 even in a scenario with a \emph{large number of labeled examples},
|
|
|
983 i.e., here, the relative gain of self-taught learning is probably preserved
|
|
|
984 in the asymptotic regime.
|
|
|
985
|
|
|
986 {\bf Why would deep learners benefit more from the self-taught learning framework}?
|
|
|
987 The key idea is that the lower layers of the predictor compute a hierarchy
|
|
|
988 of features that can be shared across tasks or across variants of the
|
|
|
989 input distribution. A theoretical analysis of generalization improvements
|
|
|
990 due to sharing of intermediate features across tasks already points
|
|
|
991 towards that explanation~\cite{baxter95a}.
|
|
|
992 Intermediate features that can be used in different
|
|
|
993 contexts can be estimated in a way that allows to share statistical
|
|
|
994 strength. Features extracted through many levels are more likely to
|
|
|
995 be more abstract (as the experiments in~\citet{Goodfellow2009} suggest),
|
|
|
996 increasing the likelihood that they would be useful for a larger array
|
|
|
997 of tasks and input conditions.
|
|
|
998 Therefore, we hypothesize that both depth and unsupervised
|
|
|
999 pre-training play a part in explaining the advantages observed here, and future
|
|
|
1000 experiments could attempt at teasing apart these factors.
|
|
|
1001 And why would deep learners benefit from the self-taught learning
|
|
|
1002 scenarios even when the number of labeled examples is very large?
|
|
|
1003 We hypothesize that this is related to the hypotheses studied
|
|
|
1004 in~\citet{Erhan+al-2010}. Whereas in~\citet{Erhan+al-2010}
|
|
|
1005 it was found that online learning on a huge dataset did not make the
|
|
|
1006 advantage of the deep learning bias vanish, a similar phenomenon
|
|
|
1007 may be happening here. We hypothesize that unsupervised pre-training
|
|
|
1008 of a deep hierarchy with self-taught learning initializes the
|
|
|
1009 model in the basin of attraction of supervised gradient descent
|
|
|
1010 that corresponds to better generalization. Furthermore, such good
|
|
|
1011 basins of attraction are not discovered by pure supervised learning
|
|
|
1012 (with or without self-taught settings), and more labeled examples
|
|
|
1013 does not allow the model to go from the poorer basins of attraction discovered
|
|
|
1014 by the purely supervised shallow models to the kind of better basins associated
|
|
|
1015 with deep learning and self-taught learning.
|
|
|
1016
|
|
|
1017 A Flash demo of the recognizer (where both the MLP and the SDA can be compared)
|
|
|
1018 can be executed on-line at {\tt http://deep.host22.com}.
|
|
|
1019
|
|
|
1020
|
|
|
1021 \section*{Appendix I: Detailed Numerical Results}
|
|
|
1022
|
|
|
1023 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered.
|
|
|
1024 They also contain additional data such as test errors on P07 and standard errors.
|
|
|
1025
|
|
|
1026 \begin{table}[ht]
|
|
|
1027 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits +
|
|
|
1028 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training
|
|
|
1029 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture
|
|
|
1030 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07)
|
|
|
1031 and using a validation set to select hyper-parameters and other training choices.
|
|
|
1032 \{SDA,MLP\}0 are trained on NIST,
|
|
|
1033 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07.
|
|
|
1034 The human error rate on digits is a lower bound because it does not count digits that were
|
|
|
1035 recognized as letters. For comparison, the results found in the literature
|
|
|
1036 on NIST digits classification using the same test set are included.}
|
|
|
1037 \label{tab:sda-vs-mlp-vs-humans}
|
|
|
1038 \begin{center}
|
|
|
1039 \begin{tabular}{|l|r|r|r|r|} \hline
|
|
|
1040 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
|
|
|
1041 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline
|
|
|
1042 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline
|
|
|
1043 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline
|
|
|
1044 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline
|
|
|
1045 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline
|
|
|
1046 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline
|
|
|
1047 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline
|
|
|
1048 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline
|
|
|
1049 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline
|
|
|
1050 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline
|
|
|
1051 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline
|
|
|
1052 \end{tabular}
|
|
|
1053 \end{center}
|
|
|
1054 \end{table}
|
|
|
1055
|
|
|
1056 \begin{table}[ht]
|
|
|
1057 \caption{Relative change in error rates due to the use of perturbed training data,
|
|
|
1058 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models.
|
|
|
1059 A positive value indicates that training on the perturbed data helped for the
|
|
|
1060 given test set (the first 3 columns on the 62-class tasks and the last one is
|
|
|
1061 on the clean 10-class digits). Clearly, the deep learning models did benefit more
|
|
|
1062 from perturbed training data, even when testing on clean data, whereas the MLP
|
|
|
1063 trained on perturbed data performed worse on the clean digits and about the same
|
|
|
1064 on the clean characters. }
|
|
|
1065 \label{tab:perturbation-effect}
|
|
|
1066 \begin{center}
|
|
|
1067 \begin{tabular}{|l|r|r|r|r|} \hline
|
|
|
1068 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
|
|
|
1069 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline
|
|
|
1070 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline
|
|
|
1071 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline
|
|
|
1072 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline
|
|
|
1073 \end{tabular}
|
|
|
1074 \end{center}
|
|
|
1075 \end{table}
|
|
|
1076
|
|
|
1077 \begin{table}[ht]
|
|
|
1078 \caption{Test error rates and relative change in error rates due to the use of
|
|
|
1079 a multi-task setting, i.e., training on each task in isolation vs training
|
|
|
1080 for all three tasks together, for MLPs vs SDAs. The SDA benefits much
|
|
|
1081 more from the multi-task setting. All experiments on only on the
|
|
|
1082 unperturbed NIST data, using validation error for model selection.
|
|
|
1083 Relative improvement is 1 - single-task error / multi-task error.}
|
|
|
1084 \label{tab:multi-task}
|
|
|
1085 \begin{center}
|
|
|
1086 \begin{tabular}{|l|r|r|r|} \hline
|
|
|
1087 & single-task & multi-task & relative \\
|
|
|
1088 & setting & setting & improvement \\ \hline
|
|
|
1089 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline
|
|
|
1090 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline
|
|
|
1091 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline
|
|
|
1092 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline
|
|
|
1093 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline
|
|
|
1094 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline
|
|
|
1095 \end{tabular}
|
|
|
1096 \end{center}
|
|
|
1097 \end{table}
|
|
|
1098
|
|
|
1099 %\afterpage{\clearpage}
|
|
|
1100 \clearpage
|
|
|
1101 {
|
|
|
1102 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter}
|
|
|
1103 %\bibliographystyle{plainnat}
|
|
|
1104 \bibliographystyle{unsrtnat}
|
|
|
1105 %\bibliographystyle{apalike}
|
|
|
1106 }
|
|
|
1107
|
|
|
1108
|
|
|
1109 \end{document}
|
