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\begin{document}
\title{Generating and Exploiting Perturbed and Multi-Task Handwritten Training Data for Deep Architectures}
\author{The IFT6266 Gang}
\date{April 2010, Technical Report, Dept. IRO, U. Montreal}

\maketitle

\begin{abstract}
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. In the area of handwriting recognition,
deep learning algorithms
had been evaluated on rather small datasets with a few tens of thousands
of examples. Here we propose a powerful generator of variations
of examples for character images based on a pipeline of stochastic
transformations that include not only the usual affine transformations
but also the addition of slant, local elastic deformations, changes
in thickness, background images, color, contrast, occlusion, and
various types of pixel and spatially correlated noise.
We evaluate a deep learning algorithm (Stacked Denoising Autoencoders)
on the task of learning to classify digits and letters transformed
with this pipeline, using the hundreds of millions of generated examples
and testing on the full 62-class NIST test set.
We find that the SDA outperforms its
shallow counterpart, an ordinary Multi-Layer Perceptron,
and that it is better able to take advantage of the additional
generated data, as well as better able to take advantage of
the multi-task setting, i.e., 
training from more classes than those of interest in the end.
In fact, we find that the SDA reaches human performance as
estimated by the Amazon Mechanical Turk on the 62-class NIST test characters.
\end{abstract}

\section{Introduction}

Deep Learning has emerged as a promising new area of research in
statistical machine learning (see~\emcite{Bengio-2009} for a review).
Learning algorithms for deep architectures are centered on the learning
of useful representations of data, which are better suited to the task at hand.
This is in great part inspired by observations of the mammalian visual cortex, 
which consists of a chain of processing elements, each of which is associated with a
different representation. In fact,
it was found recently that the features learnt in deep architectures resemble
those observed in the first two of these stages (in areas V1 and V2
of visual cortex)~\cite{HonglakL2008}.
Processing images typically involves transforming the raw pixel data into
new {\bf representations} that can be used for analysis or classification.
For example, a principal component analysis representation linearly projects 
the input image into a lower-dimensional feature space.
Why learn a representation?  Current practice in the computer vision
literature converts the raw pixels into a hand-crafted representation
(e.g.\ SIFT features~\cite{Lowe04}), but deep learning algorithms
tend to discover similar features in their first few 
levels~\cite{HonglakL2008,ranzato-08,Koray-08,VincentPLarochelleH2008-very-small}.
Learning increases the
ease and practicality of developing representations that are at once
tailored to specific tasks, yet are able to borrow statistical strength
from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
feature representation can lead to higher-level (more abstract, more
general) features that are more robust to unanticipated sources of
variance extant in real data.

Whereas a deep architecture can in principle be more powerful than a
shallow one in terms of representation, depth appears to render the
training problem more difficult in terms of optimization and local minima.
It is also only recently that successful algorithms were proposed to
overcome some of these difficulties.  All are based on unsupervised
learning, often in an greedy layer-wise ``unsupervised pre-training''
stage~\cite{Bengio-2009}.  One of these layer initialization techniques,
applied here, is the Denoising
Auto-Encoder~(DEA)~\cite{VincentPLarochelleH2008-very-small}, which
performed similarly or better than previously proposed Restricted Boltzmann
Machines in terms of unsupervised extraction of a hierarchy of features
useful for classification.  The principle is that each layer starting from
the bottom is trained to encode their input (the output of the previous
layer) and try to reconstruct it from a corrupted version of it. After this
unsupervised initialization, the stack of denoising auto-encoders can be
converted into a deep supervised feedforward neural network and trained by
stochastic gradient descent.


\section{Perturbation and Transformation of Character Images}

This section describes the different transformations we used to generate data, in their order.
The code for these transformations (mostly python) is available at 
{\tt http://anonymous.url.net}. All the modules in the pipeline share
a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
amount of deformation or noise introduced.

We can differentiate two important parts in the pipeline. The first one,
from slant to pinch, performs transformations of the character. The second
part, from blur to contrast, adds noise to the image.

\subsection{Slant}

In order to mimic a slant effect, we simply shift each row of the image
proportionnaly to its height: $shift = round(slant \times height)$.  We
round the shift in order to have a discret displacement. We do not use a
filter to smooth the result in order to save computing time and also
because latter transformations have similar effects.

The $slant$ coefficient can be negative or positive with equal probability
and its value is randomly sampled according to the complexity level.  In
our case we take uniformly a number in the range $[0,complexity]$, so the
maximum displacement for the lowest or highest pixel line is of
$round(complexity \times 32)$.


\subsection{Thickness}

To change the thickness of the characters we used morpholigical operators:
dilation and erosion~\cite{Haralick87,Serra82}.

The basic idea of such transform is, for each pixel, to multiply in the
element-wise manner its neighbourhood with a matrix called the structuring
element.  Then for dilation we remplace the pixel value by the maximum of
the result, or the minimum for erosion.  This will dilate or erode objects
in the image and the strength of the transform only depends on the
structuring element.

We used ten different structural elements with increasing dimensions (the
biggest is $5\times5$).  for each image, we radomly sample the operator
type (dilation or erosion) with equal probability and one structural
element from a subset of the $n$ smallest structuring elements where $n$ is
$round(10 \times complexity)$ for dilation and $round(6 \times complexity)$
for erosion.  A neutral element is always present in the set, if it is
chosen the transformation is not applied.  Erosion allows only the six
smallest structural elements because when the character is too thin it may
erase it completly.

\subsection{Affine Transformations}

We generate an affine transform matrix according to the complexity level,
then we apply it directly to the image.  The matrix is of size $2 \times
3$, so we can represent it by six parameters $(a,b,c,d,e,f)$.  Formally,
for each pixel $(x,y)$ of the output image, we give the value of the pixel
nearest to : $(ax+by+c,dx+ey+f)$, in the input image.  This allows to
produce scaling, translation, rotation and shearing variances.

The sampling of the parameters $(a,b,c,d,e,f)$ have been tuned by hand to
forbid important rotations (not to confuse classes) but to give good
variability of the transformation. For each image we sample uniformly the
parameters in the following ranges: $a$ and $d$ in $[1-3 \times
complexity,1+3 \times complexity]$, $b$ and $e$ in $[-3 \times complexity,3
\times complexity]$ and $c$ and $f$ in $[-4 \times complexity, 4 \times
complexity]$.


\subsection{Local Elastic Deformations}

This filter induces a "wiggly" effect in the image. The description here
will be brief, as the algorithm follows precisely what is described in
\cite{SimardSP03}.

The general idea is to generate two "displacements" fields, for horizontal
and vertical displacements of pixels. Each of these fields has the same
size as the original image.

When generating the transformed image, we'll loop over the x and y
positions in the fields and select, as a value, the value of the pixel in
the original image at the (relative) position given by the displacement
fields for this x and y. If the position we'd retrieve is outside the
borders of the image, we use a 0 value instead.

To generate a pixel in either field, first a value between -1 and 1 is
chosen from a uniform distribution. Then all the pixels, in both fields, is
multiplied by a constant $\alpha$ which controls the intensity of the
displacements (bigger $\alpha$ translates into larger wiggles).

As a final step, each field is convoluted with a Gaussian 2D kernel of
standard deviation $\sigma$. Visually, this results in a "blur"
filter. This has the effect of making values next to each other in the
displacement fields similar. In effect, this makes the wiggles more
coherent, less noisy.

As displacement fields were long to compute, 50 pairs of fields were
generated per complexity in increments of 0.1 (50 pairs for 0.1, 50 pairs
for 0.2, etc.), and afterwards, given a complexity, we selected randomly
among the 50 corresponding pairs.

$\sigma$ and $\alpha$ were linked to complexity through the formulas
$\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times
\sqrt[3]{complexity}$.


\subsection{Pinch}

This is another GIMP filter we used. The filter is in fact named "Whirl and
pinch", but we don't use the "whirl" part (whirl is set to 0). As described
in GIMP, a pinch is "similar to projecting the image onto an elastic
surface and pressing or pulling on the center of the surface".

Mathematically, for a square input image, think of drawing a circle of
radius $r$ around a center point $C$. Any point (pixel) $P$ belonging to
that disk (region inside circle) will have its value recalculated by taking
the value of another "source" pixel in the original image. The position of
that source pixel is found on the line thats goes through $C$ and $P$, but
at some other distance $d_2$. Define $d_1$ to be the distance between $P$
and $C$. $d_2$ is given by $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times
d_1$, where $pinch$ is a parameter to the filter.

If the region considered is not square then, before computing $d_2$, the
smallest dimension (x or y) is stretched such that we may consider the
region as if it was square. Then, after $d_2$ has been computed and
corresponding components $d_2\_x$ and $d_2\_y$ have been found, the
component corresponding to the stretched dimension is compressed back by an
inverse ratio.

The actual value is given by bilinear interpolation considering the pixels
around the (non-integer) source position thus found.

The value for $pinch$ in our case was given by sampling from an uniform
distribution over the range $[-complexity, 0.7 \times complexity]$.

\subsection{Motion Blur}

This is a GIMP filter we applied, a "linear motion blur" in GIMP
terminology. The description will be brief as it is a well-known filter.

This algorithm has two input parameters, $length$ and $angle$. The value of
a pixel in the final image is the mean value of the $length$ first pixels
found by moving in the $angle$ direction. An approximation of this idea is
used, as we won't fall onto precise pixels by following that
direction. This is done using the Bresenham line algorithm.

The angle, in our case, is chosen from a uniform distribution over
$[0,360]$ degrees. The length, though, depends on the complexity; it's
sampled from a Gaussian distribution of mean 0 and standard deviation
$\sigma = 3 \times complexity$.

\subsection{Occlusion}

This filter selects random parts of other (hereafter "occlusive") letter
images and places them over the original letter (hereafter "occluded")
image. To be more precise, having selected a subregion of the occlusive
image and a desination position in the occluded image, to determine the
final value for a given overlapping pixel, it selects whichever pixel is
the lightest. As a reminder, the background value is 0, black, so the value
nearest to 1 is selected.

To select a subpart of the occlusive image, four numbers are generated. For
compability with the code, we'll call them "haut", "bas", "gauche" and
"droite" (respectively meaning top, bottom, left and right). Each of these
numbers is selected according to a Gaussian distribution of mean $8 \times
complexity$ and standard deviation $2$. This means the largest the
complexity is, the biggest the occlusion will be. The absolute value is
taken, as the numbers must be positive, and the maximum value is capped at
15.

These four sizes collectively define a window centered on the middle pixel
of the occlusive image. This is the part that will be extracted as the
occlusion.

The next step is to select a destination position in the occluded
image. Vertical and horizontal displacements $y\_arrivee$ and $x\_arrivee$
are selected according to Gaussian distributions of mean 0 and of standard
deviations of, respectively, 3 and 2. Then an horizontal placement mode,
$place$, is selected to be of three values meaning
left, middle or right.

If $place$ is "middle", the occlusion will be horizontally centered
around the horizontal middle of the occluded image, then shifted according
to $x\_arrivee$. If $place$ is "left", it will be placed on the left of
the occluded image, then displaced right according to $x\_arrivee$. The
contrary happens if $place$ is $right$.

In both the horizontal and vertical positionning, the maximum position in
either direction is such that the selected occlusion won't go beyond the
borders of the occluded image.

This filter has a probability of not being applied, at all, of 60\%.


\subsection{Pixel Permutation}

This filter permuts neighbouring pixels. It selects first
$\frac{complexity}{3}$ pixels randomly in the image. Each of them are then
sequentially exchanged to one other pixel in its $V4$ neighbourhood. Number
of exchanges to the left, right, top, bottom are equal or does not differ
from more than 1 if the number of selected pixels is not a multiple of 4.

It has has a probability of not being applied, at all, of 80\%.


\subsection{Gaussian Noise}

This filter simply adds, to each pixel of the image independently, a
Gaussian noise of mean $0$ and standard deviation $\frac{complexity}{10}$.

It has has a probability of not being applied, at all, of 70\%.


\subsection{Background Images}

Following~\cite{Larochelle-jmlr-2009}, this transformation adds a random
background behind the letter. The background is chosen by first selecting,
at random, an image from a set of images. Then we choose a 32x32 subregion
of that image as the background image (by sampling x and y positions
uniformly while making sure not to cross image borders).

To combine the original letter image and the background image, contrast
adjustments are made. We first get the maximal values (i.e. maximal
intensity) for both the original image and the background image, $maximage$
and $maxbg$. We also have a parameter, $contrast$, given by sampling from a
uniform distribution over $[complexity, 1]$.

Once we have all these numbers, we first adjust the values for the
background image. Each pixel value is multiplied by $\frac{max(maximage -
  contrast, 0)}{maxbg}$. Therefore the higher the contrast, the darkest the
background will be.

The final image is found by taking the brightest (i.e. value nearest to 1)
pixel from either the background image or the corresponding pixel in the
original image.

\subsection{Salt and Pepper Noise}

This filter adds noise to the image by randomly selecting a certain number
of them and, for those selected pixels, assign a random value according to
a uniform distribution over the $[0,1]$ ranges. This last distribution does
not change according to complexity. Instead, the number of selected pixels
does: the proportion of changed pixels corresponds to $complexity / 5$,
which means, as a maximum, 20\% of the pixels will be randomized. On the
lowest extreme, no pixel is changed.

This filter also has a probability of not being applied, at all, of 75\%.

\subsection{Spatially Gaussian Noise}

The aim of this transformation is to filter, with a gaussian kernel,
different regions of the image. In order to save computing time we decided
to convolve the whole image only once with a symmetric gaussian kernel of
size and variance choosen uniformly in the ranges: $[12,12 + 20 \times
complexity]$ and $[2,2 + 6 \times complexity]$. The result is normalized
between $0$ and $1$.  We also create a symmetric averaging window, of the
kernel size, with maximum value at the center.  For each image we sample
uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be
averaging centers between the original image and the filtered one.  We
initialize to zero a mask matrix of the image size. For each selected pixel
we add to the mask the averaging window centered to it.  The final image is
computed from the following element-wise operation: $\frac{image + filtered
  image \times mask}{mask+1}$.

This filter has a probability of not being applied, at all, of 75\%.

\subsection{Scratches}

The scratches module places line-like white patches on the image.  The
lines are in fact heavily transformed images of the digit "1" (one), chosen
at random among five thousands such start images of this digit.

Once the image is selected, the transformation begins by finding the first
$top$, $bottom$, $right$ and $left$ non-zero pixels in the image. It is
then cropped to the region thus delimited, then this cropped version is
expanded to $32\times32$ again. It is then rotated by a random angle having a
Gaussian distribution of mean 90 and standard deviation $100 \times
complexity$ (in degrees). The rotation is done with bicubic interpolation.

The rotated image is then resized to $50\times50$, with anti-aliasing. In
that image, we crop the image again by selecting a region delimited
horizontally to $left$ to $left+32$ and vertically by $top$ to $top+32$.

Once this is done, two passes of a greyscale morphological erosion filter
are applied. Put briefly, this erosion filter reduces the width of the line
by a certain $smoothing$ amount. For small complexities (< 0.5),
$smoothing$ is 6, so the line is very small. For complexities ranging from
0.25 to 0.5, $smoothing$ is 5. It is 4 for complexities 0.5 to 0.75, and 3
for higher complexities.

To compensate for border effects, the image is then cropped to 28x28 by
removing two pixels everywhere on the borders, then expanded to 32x32
again. The pixel values are then linearly expanded such that the minimum
value is 0 and the maximal one is 1. Then, 50\% of the time, the image is
vertically flipped.

This filter is only applied only 15\% of the time. When it is applied, 50\%
of the time, only one patch image is generated and applied. In 30\% of
cases, two patches are generated, and otherwise three patches are
generated. The patch is applied by taking the maximal value on any given
patch or the original image, for each of the 32x32 pixel locations.

\subsection{Color and Contrast Changes}

This filter changes the constrast and may invert the image polarity (white
on black to black on white). The contrast $C$ is defined here as the
difference between the maximum and the minimum pixel value of the image. A
contrast value is sampled uniformly between $1$ and $1-0.85 \times
complexity$ (this insure a minimum constrast of $0.15$). We then simply
normalize the image to the range $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The
polarity is inverted with $0.5$ probability.


\begin{figure}[h]
\resizebox{.99\textwidth}{!}{\includegraphics{images/example_t.png}}\\
\caption{Illustration of the pipeline of stochastic 
transformations applied to the image of a lower-case t
(the upper left image). Each image in the pipeline (going from
left to right, first top line, then bottom line) shows the result
of applying one of the modules in the pipeline. The last image
(bottom right) is used as training example.}
\label{fig:pipeline}
\end{figure}


\section{Experimental Setup}

\subsection{Training Datasets}

\subsubsection{Data Sources}

\begin{itemize}
\item {\bf NIST}
The NIST Special Database 19 (NIST19) is a very widely used dataset for training and testing OCR systems. 
The dataset is composed with over 800 000 digits and characters (upper and lower cases), with hand checked classifications,
extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes 
corresponding to "0"-"9","A"-"Z" and "a"-"z". The dataset contains 8 series of different complexity. 
The fourth series, $hsf_4$, experimentally recognized to be the most difficult one for classification task is recommended 
by NIST as testing set and is used in our work for that purpose.
The performances reported by previous work on that dataset mostly use only the digits.
Here we use the whole classes both in the training and testing phase.   


\item {\bf Fonts}
\item {\bf Captchas}
The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for 
generating characters of the same format as the NIST dataset. The core of this data source is composed with a random character
generator and various kinds of tranformations similar to those described in the previous sections. 
In order to increase the variability of the data generated, different fonts are used for generating the characters. 
Transformations (slant, distorsions, rotation, translation) are applied to each randomly generated character with a complexity
depending on the value of the complexity parameter provided by the user of the data source. Two levels of complexity are 
allowed and can be controlled via an easy to use facade class.    
\item {\bf OCR data}
\end{itemize}

\subsubsection{Data Sets}
\begin{itemize}
\item {\bf NIST}
\item {\bf P07}
The dataset P07 is sampled with our transformation pipeline with a complexity parameter of $0.7$. 
For each new exemple to generate, we choose one source with the following probability: $0.1$ for the fonts,
$0.25$ for the captchas, $0.25$ for OCR data and $0.4$ for NIST. We apply all the transformations in their order
and for each of them we sample uniformly a complexity in the range $[0,0.7]$.
\item {\bf NISTP} {\em ne pas utiliser PNIST mais NISTP, pour rester politically correct...}
NISTP is equivalent to P07 except that we only apply transformations from slant to pinch. Therefore, the character is transformed
but no additionnal noise is added to the image, this gives images closer to the NIST dataset.
\end{itemize}

We noticed that the distribution of the training sets and the test sets differ.
Since our validation sets are sampled from the training set, they have approximately the same distribution, but the test set has a completely different distribution as illustrated in figure \ref {setsdata}.

\begin{figure}
\subfigure[NIST training]{\includegraphics[width=0.5\textwidth]{images/nisttrainstats}}
\subfigure[NIST validation]{\includegraphics[width=0.5\textwidth]{images/nistvalidstats}}
\subfigure[NIST test]{\includegraphics[width=0.5\textwidth]{images/nistteststats}}
\subfigure[NISTP validation]{\includegraphics[width=0.5\textwidth]{images/nistpvalidstats}}
\caption{Proportion of each class in some of the data sets}
\label{setsdata}
\end{figure}

\subsection{Models and their Hyperparameters}

\subsubsection{Multi-Layer Perceptrons (MLP)}

An MLP is a family of functions that are described by stacking layers of of a function similar to
$$g(x) = \tanh(b+Wx)$$
The input, $x$, is a $d$-dimension vector.  
The output, $g(x)$, is a $m$-dimension vector.
The parameter $W$ is a $m\times d$ matrix and is called the weight matrix.
The parameter  $b$ is a $m$-vector and is called the bias vector.
The non-linearity (here $\tanh$) is applied element-wise to the output vector.
Usually the input is referred to a input layer and similarly for the output.
You can of course chain several such functions to obtain a more complex one.
Here is a common example
$$f(x) = c + V\tanh(b+Wx)$$
In this case the intermediate layer corresponding to $\tanh(b+Wx)$ is called a hidden layer.
Here the output layer does not have the same non-linearity as the hidden layer.
This is a common case where some specialized non-linearity is applied to the output layer only depending on the task at hand.

If you put 3 or more hidden layers in such a network you obtain what is called a deep MLP.
The parameters to adapt are the weight matrix and the bias vector for each layer.

\subsubsection{Stacked Denoising Auto-Encoders (SDAE)}
\label{SdA}

Auto-encoders are essentially a way to initialize the weights of the network to enable better generalization.
This is essentially unsupervised training where the layer is made to reconstruct its input through and encoding and decoding phase.
Denoising auto-encoders are a variant where the input is corrupted with random noise but the target is the uncorrupted input.
The principle behind these initialization methods is that the network will learn the inherent relation between portions of the data and be able to represent them thus helping with whatever task we want to perform.

An auto-encoder unit is formed of two MLP layers with the bottom one called the encoding layer and the top one the decoding layer.
Usually the top and bottom weight matrices are the transpose of each other and are fixed this way.
The network is trained as such and, when sufficiently trained, the MLP layer is initialized with the parameters of the encoding layer.
The other parameters are discarded.

The stacked version is an adaptation to deep MLPs where you initialize each layer with a denoising auto-encoder  starting from the bottom.
During the initialization, which is usually called pre-training, the bottom layer is treated as if it were an isolated auto-encoder.
The second and following layers receive the same treatment except that they take as input the encoded version of the data that has gone through the layers before it.
For additional details see \cite{vincent:icml08}.

\section{Experimental Results}

\subsection{SDA vs MLP vs Humans}

We compare here the best MLP (according to validation set error) that we found against
the best SDA (again according to validation set error), along with a precise estimate
of human performance obtained via Amazon's Mechanical Turk (AMT)
service\footnote{http://mturk.com}. AMT users are paid small amounts
of money to perform tasks for which human intelligence is required.
Mechanical Turk has been used extensively in natural language
processing \cite{SnowEtAl2008} and vision
\cite{SorokinAndForsyth2008,whitehill09}. AMT users where presented
with 10 character images and asked to type 10 corresponding ascii
characters. Hence they were forced to make a hard choice among the
62 character classes. Three users classified each image, allowing
to estimate inter-human variability (shown as +/- in parenthesis below).

\begin{table}
\caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits +
26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training
(SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture 
(MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07)
and using a validation set to select hyper-parameters and other training choices. 
\{SDA,MLP\}0 are trained on NIST,
\{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07.
The human error rate on digits is a lower bound because it does not count digits that were
recognized as letters. For comparison, the results found in the literature
on NIST digits classification using the same test set are included.}
\label{tab:sda-vs-mlp-vs-humans}
\begin{center}
\begin{tabular}{|l|r|r|r|r|} \hline
      & NIST test          & NISTP test       & P07 test       & NIST test digits   \\ \hline
Humans&   18.2\% $\pm$.1\%   &  39.4\%$\pm$.1\%   &  46.9\%$\pm$.1\%  &  $>1.1\%$ \\ \hline 
SDA0   &  23.7\% $\pm$.14\%  &  65.2\%$\pm$.34\%  & 97.45\%$\pm$.06\%  & 2.7\% $\pm$.14\%\\ \hline 
SDA1   &  17.1\% $\pm$.13\%  &  29.7\%$\pm$.3\%  & 29.7\%$\pm$.3\%  & 1.4\% $\pm$.1\%\\ \hline 
SDA2   &  18.7\% $\pm$.13\%  &  33.6\%$\pm$.3\%  & 39.9\%$\pm$.17\%  & 1.7\% $\pm$.1\%\\ \hline 
MLP0   &  24.2\% $\pm$.15\%  & 68.8\%$\pm$.33\%  & 78.70\%$\pm$.14\%  & 3.45\% $\pm$.15\% \\ \hline 
MLP1   &  23.0\% $\pm$.15\%  &  41.8\%$\pm$.35\%  & 90.4\%$\pm$.1\%  & 3.85\% $\pm$.16\% \\ \hline 
MLP2   &  24.3\% $\pm$.15\%  &  46.0\%$\pm$.35\%  & 54.7\%$\pm$.17\%  & 4.85\% $\pm$.18\% \\ \hline 
[5]    &                     &                    &                   & 4.95\% $\pm$.18\% \\ \hline
[2]    &                     &                    &                   & 3.71\% $\pm$.16\% \\ \hline
[3]    &                     &                    &                   & 2.4\% $\pm$.13\% \\ \hline
[4]    &                     &                    &                   & 2.1\% $\pm$.12\% \\ \hline
\end{tabular}
\end{center}
\end{table}

\subsection{Perturbed Training Data More Helpful for SDAE}

\begin{table}
\caption{Relative change in error rates due to the use of perturbed training data,
either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models.
A positive value indicates that training on the perturbed data helped for the
given test set (the first 3 columns on the 62-class tasks and the last one is
on the clean 10-class digits). Clearly, the deep learning models did benefit more
from perturbed training data, even when testing on clean data, whereas the MLP
trained on perturbed data performed worse on the clean digits and about the same
on the clean characters. }
\label{tab:sda-vs-mlp-vs-humans}
\begin{center}
\begin{tabular}{|l|r|r|r|r|} \hline
      & NIST test          & NISTP test      & P07 test       & NIST test digits   \\ \hline
SDA0/SDA1-1   &  38\%      &  84\%           & 228\%          &  93\% \\ \hline 
SDA0/SDA2-1   &  27\%      &  94\%           & 144\%          &  59\% \\ \hline 
MLP0/MLP1-1   &  5.2\%     &  65\%           & -13\%          & -10\%  \\ \hline 
MLP0/MLP2-1   &  -0.4\%    &  49\%           & 44\%           & -29\% \\ \hline 
\end{tabular}
\end{center}
\end{table}


\subsection{Multi-Task Learning Effects}

As previously seen, the SDA is better able to benefit from the
transformations applied to the data than the MLP. In this experiment we
define three tasks: recognizing digits (knowing that the input is a digit),
recognizing upper case characters (knowing that the input is one), and
recognizing lower case characters (knowing that the input is one).  We
consider the digit classification task as the target task and we want to
evaluate whether training with the other tasks can help or hurt, and
whether the effect is different for MLPs versus SDAs.  The goal is to find
out if deep learning can benefit more (or less) from multiple related tasks
(i.e. the multi-task setting) compared to a corresponding purely supervised
shallow learner.

We use a single hidden layer MLP with 1000 hidden units, and a SDA
with 3 hidden layers (1000 hidden units per layer), pre-trained and
fine-tuned on NIST.

Our results show that the MLP benefits marginally from the multi-task setting
in the case of digits (5\% relative improvement) but is actually hurt in the case
of characters (respectively 3\% and 4\% worse for lower and upper class characters).
On the other hand the SDA benefitted from the multi-task setting, with relative
error rate improvements of 27\%, 15\% and 13\% respectively for digits,
lower and upper case characters, as shown in Table~\ref{tab:multi-task}.

\begin{table}
\caption{Test error rates and relative change in error rates due to the use of
a multi-task setting, i.e., training on each task in isolation vs training
for all three tasks together, for MLPs vs SDAs. The SDA benefits much
more from the multi-task setting. All experiments on only on the
unperturbed NIST data, using validation error for model selection.
Relative improvement is 1 - single-task error / multi-task error.}
\label{tab:multi-task}
\begin{center}
\begin{tabular}{|l|r|r|r|} \hline
             & single-task  & multi-task  & relative \\ 
             & setting      & setting     & improvement \\ \hline
MLP-digits   &  3.77\%      &  3.99\%     & 5.6\%   \\ \hline 
MLP-lower   &  17.4\%      &  16.8\%     &  -4.1\%    \\ \hline 
MLP-upper   &  7.84\%     &  7.54\%      & -3.6\%    \\ \hline 
SDA-digits   &  2.6\%      &  3.56\%     & 27\%    \\ \hline 
SDA-lower   &  12.3\%      &  14.4\%    & 15\%    \\ \hline 
SDA-upper   &  5.93\%     &  6.78\%      & 13\%    \\ \hline 
\end{tabular}
\end{center}
\end{table}

\section{Conclusions}

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