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+\documentclass{article} % For LaTeX2e
+\usepackage{nips10submit_e,times}
+
+\usepackage{amsthm,amsmath,amssymb,bbold,bbm} 
+\usepackage{algorithm,algorithmic}
+\usepackage[utf8]{inputenc}
+\usepackage{graphicx,subfigure}
+\usepackage{mlapa}
+
+\title{Generating and Exploiting Perturbed and Multi-Task Handwritten Training Data for Deep Architectures}
+\author{The IFT6266 Gang}
+
+\begin{document}
+
+%\makeanontitle
+\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 8????? 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. It contains 82600 examples,
+while the training and validation sets (which have the same distribution) contain XXXXX and
+XXXXX examples respectively.
+The performances reported by previous work on that dataset mostly use only the digits.
+Here we use all the classes both in the training and testing phase. This is especially
+useful to estimate the effect of a multi-task setting.
+Note that the distribution of the classes in the NIST training and test sets differs
+substantially, with relatively many more digits in the test set, and uniform distribution
+of letters in the test set, not in the training set (more like the natural distribution
+of letters in text).
+
+\item {\bf Fonts} TODO!!!
+
+\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} This is the raw NIST special database 19.
+\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} NISTP is equivalent to P07 (complexity parameter of $0.7$ with the same sources proportion)
+  except that we only apply
+  transformations from slant to pinch. Therefore, the character is
+  transformed but no additionnal noise is added to the image, giving images
+  closer to the NIST dataset.
+\end{itemize}
+
+\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}
+
+\bibliography{strings,ml,aigaion,specials}
+\bibliographystyle{mlapa}
+%\bibliographystyle{apalike}
+
+\end{document}