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mass of the clean pointsx which could have given rise to [UNK]. the autoencoder thus learns a vector ﬁeld g ( f ( x ) ) −x indicated by the green arrows. this vector ﬁeld estimates the score ∇xlog pdata ( x ) up to a multiplicative factor that is the average root mean square reconstruction error. 512	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	527	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders 14. 5. 1 estimating the score score matching (, ) is an alternative to maximum likelihood. it hyvarinen 2005 provides a consistent estimator of probability distributions based on encouraging the model to have the same score as the data distribution at every training point x. in this context, the score is a particular gradient ﬁeld : ∇x log ( ) p x. ( 14. 15 ) score matching is discussed further in section. for the present discussion 18. 4 regarding autoencoders, it is [UNK] to understand that learning the gradient ﬁeld of log pdata is one way to learn the structure of pdata itself. a very important property of daes is that their training criterion ( with conditionally gaussian p ( x h \| ) ) makes the autoencoder learn a vector ﬁeld ( g ( f ( x ) ) −x ) that estimates the score of the data distribution. this is illustrated in ﬁgure. 14. 4 denoising training of a speciﬁc kind of autoencoder ( sigmoidal hidden units, linear reconstruction units ) using gaussian noise and mean squared error as the reconstruction cost is equivalent (, ) to training a speci	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	528	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##oising training of a speciﬁc kind of autoencoder ( sigmoidal hidden units, linear reconstruction units ) using gaussian noise and mean squared error as the reconstruction cost is equivalent (, ) to training a speciﬁc kind vincent 2011 of undirected probabilistic model called an rbm with gaussian visible units. this kind of model will be described in detail in section ; for the present 20. 5. 1 discussion it [UNK] to know that it is a model that provides an explicit pmodel ( x ; θ ). when the rbm is trained using denoising score matching (, kingma and lecun 2010 ), its learning algorithm is equivalent to denoising training in the corresponding autoencoder. with a ﬁxed noise level, regularized score matching is not a consistent estimator ; it instead recovers a blurred version of the distribution. however, if the noise level is chosen to approach 0 when the number of examples approaches inﬁnity, then consistency is recovered. denoising score matching is discussed in more detail in section. 18. 5 other connections between autoencoders and rbms exist. score matching applied to rbms yields a cost function that is identical	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	528	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
inﬁnity, then consistency is recovered. denoising score matching is discussed in more detail in section. 18. 5 other connections between autoencoders and rbms exist. score matching applied to rbms yields a cost function that is identical to reconstruction error combined with a regularization term similar to the contractive penalty of the cae ( swersky 2011 bengio and delalleau 2009 et al., ). ( ) showed that an autoen - coder gradient provides an approximation to contrastive divergence training of rbms. for continuous - valued x, the denoising criterion with gaussian corruption and reconstruction distribution yields an estimator of the score that is applicable to general encoder and decoder parametrizations (, ). this alain and bengio 2013 means a generic encoder - decoder architecture may be made to estimate the score 513	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	528	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders by training with the squared error criterion \| \| g f ( ( [UNK] x ) ) − \| \| 2 ( 14. 16 ) and corruption c ( [UNK] = [UNK] x \| ) = ( n [UNK] x ; = µ, σ σ = 2 i ) ( 14. 17 ) with noise variance σ2. see ﬁgure for an illustration of how this works. 14. 5 figure 14. 5 : vector ﬁeld learned by a denoising autoencoder around a 1 - d curved manifold near which the data concentrates in a 2 - d space. each arrow is proportional to the reconstruction minus input vector of the autoencoder and points towards higher probability according to the implicitly estimated probability distribution. the vector ﬁeld has zeros at both maxima of the estimated density function ( on the data manifolds ) and at minima of that density function. for example, the spiral arm forms a one - dimensional manifold of local maxima that are connected to each other. local minima appear near the middle of the gap between two arms. when the norm of reconstruction error ( shown by the length of the arrows ) is large, it means that probability can be signiﬁcantly increased by moving in the direction	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	529	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. local minima appear near the middle of the gap between two arms. when the norm of reconstruction error ( shown by the length of the arrows ) is large, it means that probability can be signiﬁcantly increased by moving in the direction of the arrow, and that is mostly the case in places of low probability. the autoencoder maps these low probability points to higher probability reconstructions. where probability is maximal, the arrows shrink because the reconstruction becomes more accurate. figure reproduced with permission from ( ). alain and bengio 2013 in general, there is no guarantee that the reconstruction g ( f ( x ) ) minus the input x corresponds to the gradient of any function, let alone to the score. that is 514	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	529	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders why the early results (, ) are specialized to particular parametrizations vincent 2011 where g ( f ( x ) ) −x may be obtained by taking the derivative of another function. kamyshanska and memisevic 2015 vincent 2011 ( ) generalized the results of ( ) by identifying a family of shallow autoencoders such that g ( f ( x ) ) −x corresponds to a score for all members of the family. so far we have described only how the denoising autoencoder learns to represent a probability distribution. more generally, one may want to use the autoencoder as a generative model and draw samples from this distribution. this will be described later, in section. 20. 11 14. 5. 1. 1 historical perspective the idea of using mlps for denoising dates back to the work of ( ) lecun 1987 and ( ). ( ) also used recurrent networks to denoise gallinari et al. 1987 behnke 2001 images. denoising autoencoders are, in some sense, just mlps trained to denoise. however, the name “ denoising autoencoder ” refers to a model that is intended not merely	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	530	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
1987 behnke 2001 images. denoising autoencoders are, in some sense, just mlps trained to denoise. however, the name “ denoising autoencoder ” refers to a model that is intended not merely to learn to denoise its input but to learn a good internal representation as a side [UNK] of learning to denoise. this idea came much later ( vincent et al.,, ). the learned representation may then be used to pretrain a 2008 2010 deeper unsupervised network or a supervised network. like sparse autoencoders, sparse coding, contractive autoencoders and other regularized autoencoders, the motivation for daes was to allow the learning of a very high - capacity encoder while preventing the encoder and decoder from learning a useless identity function. prior to the introduction of the modern dae, inayoshi and kurita 2005 ( ) explored some of the same goals with some of the same methods. their approach minimizes reconstruction error in addition to a supervised objective while injecting noise in the hidden layer of a supervised mlp, with the objective to improve generalization by introducing the reconstruction error and the injected noise. however, their method was based on a	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	530	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
minimizes reconstruction error in addition to a supervised objective while injecting noise in the hidden layer of a supervised mlp, with the objective to improve generalization by introducing the reconstruction error and the injected noise. however, their method was based on a linear encoder and could not learn function families as powerful as can the modern dae. 14. 6 learning manifolds with autoencoders like many other machine learning algorithms, autoencoders exploit the idea that data concentrates around a low - dimensional manifold or a small set of such manifolds, as described in section. some machine learning algorithms exploit 5. 11. 3 this idea only insofar as that they learn a function that behaves correctly on the manifold but may have unusual behavior if given an input that is [UNK] manifold. 515	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	530	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders autoencoders take this idea further and aim to learn the structure of the manifold. to understand how autoencoders do this, we must present some important characteristics of manifolds. an important characterization of a manifold is the set of its tangent planes. at a point x on a d - dimensional manifold, the tangent plane is given by d basis vectors that span the local directions of variation allowed on the manifold. as illustrated in ﬁgure, these local directions specify how one can change 14. 6 x inﬁnitesimally while staying on the manifold. all autoencoder training procedures involve a compromise between two forces : 1. learning a representation h of a training example x such that x can be approximately recovered from h through a decoder. the fact that x is drawn from the training data is crucial, because it means the autoencoder need not successfully reconstruct inputs that are not probable under the data generating distribution. 2. satisfying the constraint or regularization penalty. this can be an architec - tural constraint that limits the capacity of the autoencoder, or it can be a regularization term added to the reconstruction cost. these techniques generally prefer solutions that are less sensitive to the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	531	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##ization penalty. this can be an architec - tural constraint that limits the capacity of the autoencoder, or it can be a regularization term added to the reconstruction cost. these techniques generally prefer solutions that are less sensitive to the input. clearly, neither force alone would be useful — copying the input to the output is not useful on its own, nor is ignoring the input. instead, the two forces together are useful because they force the hidden representation to capture information about the structure of the data generating distribution. the important principle is that the autoencoder can [UNK] to represent only the variations that are needed to reconstruct training examples. if the data generating distribution concentrates near a low - dimensional manifold, this yields representations that implicitly capture a local coordinate system for this manifold : only the variations tangent to the manifold around x need to correspond to changes in h = f ( x ). hence the encoder learns a mapping from the input space x to a representation space, a mapping that is only sensitive to changes along the manifold directions, but that is insensitive to changes orthogonal to the manifold. a one - dimensional example is illustrated in ﬁgure, showing that, by making 14. 7 the reconstruction function insensitive to per	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	531	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
only sensitive to changes along the manifold directions, but that is insensitive to changes orthogonal to the manifold. a one - dimensional example is illustrated in ﬁgure, showing that, by making 14. 7 the reconstruction function insensitive to perturbations of the input around the data points, we cause the autoencoder to recover the manifold structure. to understand why autoencoders are useful for manifold learning, it is in - structive to compare them to other approaches. what is most commonly learned to characterize a manifold is a representation of the data points on ( or near ) 516	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	531	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders figure 14. 6 : an illustration of the concept of a tangent hyperplane. here we create a one - dimensional manifold in 784 - dimensional space. we take an mnist image with 784 pixels and transform it by translating it vertically. the amount of vertical translation deﬁnes a coordinate along a one - dimensional manifold that traces out a curved path through image space. this plot shows a few points along this manifold. for visualization, we have projected the manifold into two dimensional space using pca. an n - dimensional manifold has an n - dimensional tangent plane at every point. this tangent plane touches the manifold exactly at that point and is oriented parallel to the surface at that point. it deﬁnes the space of directions in which it is possible to move while remaining on the manifold. this one - dimensional manifold has a single tangent line. we indicate an example tangent line at one point, with an image showing how this tangent direction appears in image space. gray pixels indicate pixels that do not change as we move along the tangent line, white pixels indicate pixels that brighten, and black pixels indicate pixels that darken. 517	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	532	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders x0 x 1 x2 x 0 0. 0 2. 0 4. 0 6. 0 8. 1 0. r x ( ) identity optimal reconstruction figure 14. 7 : if the autoencoder learns a reconstruction function that is invariant to small perturbations near the data points, it captures the manifold structure of the data. here the manifold structure is a collection of - dimensional manifolds. the dashed diagonal 0 line indicates the identity function target for reconstruction. the optimal reconstruction function crosses the identity function wherever there is a data point. the horizontal arrows at the bottom of the plot indicate the r ( x ) −x reconstruction direction vector at the base of the arrow, in input space, always pointing towards the nearest “ manifold ” ( a single datapoint, in the 1 - d case ). the denoising autoencoder explicitly tries to make the derivative of the reconstruction function r ( x ) small around the data points. the contractive autoencoder does the same for the encoder. although the derivative ofr ( x ) is asked to be small around the data points, it can be large between the data points. the space between the data points corresponds to the region between the manifolds	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	533	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
does the same for the encoder. although the derivative ofr ( x ) is asked to be small around the data points, it can be large between the data points. the space between the data points corresponds to the region between the manifolds, where the reconstruction function must have a large derivative in order to map corrupted points back onto the manifold. the manifold. such a representation for a particular example is also called its embedding. it is typically given by a low - dimensional vector, with less dimensions than the “ ambient ” space of which the manifold is a low - dimensional subset. some algorithms ( non - parametric manifold learning algorithms, discussed below ) directly learn an embedding for each training example, while others learn a more general mapping, sometimes called an encoder, or representation function, that maps any point in the ambient space ( the input space ) to its embedding. manifold learning has mostly focused on unsupervised learning procedures that attempt to capture these manifolds. most of the initial machine learning research on learning nonlinear manifolds has focused on non - parametric methods based on the nearest - neighbor graph. this graph has one node per training example and edges connecting near neighbors to each other. these methods ( scholkopf	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	533	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
the initial machine learning research on learning nonlinear manifolds has focused on non - parametric methods based on the nearest - neighbor graph. this graph has one node per training example and edges connecting near neighbors to each other. these methods ( scholkopf et al., ; 1998 roweis and saul 2000 tenenbaum 2000 brand 2003 belkin, ; et al., ;, ; 518	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	533	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders figure 14. 8 : non - parametric manifold learning procedures build a nearest neighbor graph in which nodes represent training examples a directed edges indicate nearest neighbor relationships. various procedures can thus obtain the tangent plane associated with a neighborhood of the graph as well as a coordinate system that associates each training example with a real - valued vector position, or embedding. it is possible to generalize such a representation to new examples by a form of interpolation. so long as the number of examples is large enough to cover the curvature and twists of the manifold, these approaches work well. images from the qmul multiview face dataset (, gong et al. 2000 ). and niyogi 2003 donoho and grimes 2003 weinberger and saul 2004 hinton, ;, ;, ; and roweis 2003 van der maaten and hinton 2008, ;, ) associate each of nodes with a tangent plane that spans the directions of variations associated with the [UNK] vectors between the example and its neighbors, as illustrated in ﬁgure. 14. 8 a global coordinate system can then be obtained through an optimization or solving a linear system. figure illustrates how a manifold can be tiled by a 14. 9 large number of locally linear gauss	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	534	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
its neighbors, as illustrated in ﬁgure. 14. 8 a global coordinate system can then be obtained through an optimization or solving a linear system. figure illustrates how a manifold can be tiled by a 14. 9 large number of locally linear gaussian - like patches ( or “ pancakes, ” because the gaussians are ﬂat in the tangent directions ). however, there is a fundamental [UNK] with such local non - parametric approaches to manifold learning, raised in ( ) : if the bengio and monperrus 2005 manifolds are not very smooth ( they have many peaks and troughs and twists ), one may need a very large number of training examples to cover each one of 519	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	534	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders figure 14. 9 : if the tangent planes ( see ﬁgure ) at each location are known, then they 14. 6 can be tiled to form a global coordinate system or a density function. each local patch can be thought of as a local euclidean coordinate system or as a locally ﬂat gaussian, or “ pancake, ” with a very small variance in the directions orthogonal to the pancake and a very large variance in the directions deﬁning the coordinate system on the pancake. a mixture of these gaussians provides an estimated density function, as in the manifold parzen window algorithm (, ) or its non - local neural - net based vincent and bengio 2003 variant (, ). bengio et al. 2006c these variations, with no chance to generalize to unseen variations. indeed, these methods can only generalize the shape of the manifold by interpolating between neighboring examples. unfortunately, the manifolds involved in ai problems can have very complicated structure that can be [UNK] to capture from only local interpolation. consider for example the manifold resulting from translation shown in ﬁgure. if we watch just one coordinate within the input vector, 14. 6 xi, as the image is translated, we	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	535	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
that can be [UNK] to capture from only local interpolation. consider for example the manifold resulting from translation shown in ﬁgure. if we watch just one coordinate within the input vector, 14. 6 xi, as the image is translated, we will observe that one coordinate encounters a peak or a trough in its value once for every peak or trough in brightness in the image. in other words, the complexity of the patterns of brightness in an underlying image template drives the complexity of the manifolds that are generated by performing simple image transformations. this motivates the use of distributed representations and deep learning for capturing manifold structure. 520	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	535	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders 14. 7 contractive autoencoders the contractive autoencoder (,, ) introduces an explicit regularizer rifai et al. 2011a b on the code h = f ( x ), encouraging the derivatives of f to be as small as possible : ω ( ) = h λ ∂f ( ) x ∂x 2 f. ( 14. 18 ) the penalty ω ( h ) is the squared frobenius norm ( sum of squared elements ) of the jacobian matrix of partial derivatives associated with the encoder function. there is a connection between the denoising autoencoder and the contractive autoencoder : ( ) showed that in the limit of small gaussian alain and bengio 2013 input noise, the denoising reconstruction error is equivalent to a contractive penalty on the reconstruction function that maps x to r = g ( f ( x ) ). in other words, denoising autoencoders make the reconstruction function resist small but ﬁnite - sized perturbations of the input, while contractive autoencoders make the feature extraction function resist inﬁnitesimal perturbations of the input. when using the jacobian -	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	536	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
function resist small but ﬁnite - sized perturbations of the input, while contractive autoencoders make the feature extraction function resist inﬁnitesimal perturbations of the input. when using the jacobian - based contractive penalty to pretrain features f ( x ) for use with a classiﬁer, the best classiﬁcation accuracy usually results from applying the contractive penalty to f ( x ) rather than to g ( f ( x ) ). a contractive penalty on f ( x ) also has close connections to score matching, as discussed in section. 14. 5. 1 the name contractive arises from the way that the cae warps space. speciﬁ - cally, because the cae is trained to resist perturbations of its input, it is encouraged to map a neighborhood of input points to a smaller neighborhood of output points. we can think of this as contracting the input neighborhood to a smaller output neighborhood. to clarify, the cae is contractive only locally — all perturbations of a training point x are mapped near to f ( x ). globally, two [UNK] points x and xmay be mapped to f ( x ) and f ( x ) points that	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	536	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
cae is contractive only locally — all perturbations of a training point x are mapped near to f ( x ). globally, two [UNK] points x and xmay be mapped to f ( x ) and f ( x ) points that are farther apart than the original points. it is plausible that f be expanding in - between or far from the data manifolds ( see for example what happens in the 1 - d toy example of ﬁgure ). when the 14. 7 ω ( h ) penalty is applied to sigmoidal units, one easy way to shrink the jacobian is to make the sigmoid units saturate to or. this encourages the cae to encode 0 1 input points with extreme values of the sigmoid that may be interpreted as a binary code. it also ensures that the cae will spread its code values throughout most of the hypercube that its sigmoidal hidden units can span. we can think of the jacobian matrix j at a point x as approximating the nonlinear encoder f ( x ) as being a linear operator. this allows us to use the word “ contractive ” more formally. in the theory of linear operators, a linear operator 521	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	536	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders is said to be contractive if the norm of jx remains less than or equal to for 1 all unit - norm x. in other words, j is contractive if it shrinks the unit sphere. we can think of the cae as penalizing the frobenius norm of the local linear approximation of f ( x ) at every training point x in order to encourage each of these local linear operator to become a contraction. as described in section, regularized autoencoders learn manifolds by 14. 6 balancing two opposing forces. in the case of the cae, these two forces are reconstruction error and the contractive penalty ω ( h ). reconstruction error alone would encourage the cae to learn an identity function. the contractive penalty alone would encourage the cae to learn features that are constant with respect tox. the compromise between these two forces yields an autoencoder whose derivatives ∂f ( ) x ∂x are mostly tiny. only a small number of hidden units, corresponding to a small number of directions in the input, may have signiﬁcant derivatives. the goal of the cae is to learn the manifold structure of the data. directions x with large jx rapidly change h, so	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	537	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
hidden units, corresponding to a small number of directions in the input, may have signiﬁcant derivatives. the goal of the cae is to learn the manifold structure of the data. directions x with large jx rapidly change h, so these are likely to be directions which approximate the tangent planes of the manifold. experiments by ( ) rifai et al. 2011a and ( ) show that training the cae results in most singular values rifai et al. 2011b of j dropping below in magnitude and therefore becoming contractive. however, 1 some singular values remain above, because the reconstruction error penalty 1 encourages the cae to encode the directions with the most local variance. the directions corresponding to the largest singular values are interpreted as the tangent directions that the contractive autoencoder has learned. ideally, these tangent directions should correspond to real variations in the data. for example, a cae applied to images should learn tangent vectors that show how the image changes as objects in the image gradually change pose, as shown in ﬁgure. visualizations 14. 6 of the experimentally obtained singular vectors do seem to correspond to meaningful transformations of the input image, as shown in ﬁgure. 14. 10 one practical issue with the cae	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	537	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
pose, as shown in ﬁgure. visualizations 14. 6 of the experimentally obtained singular vectors do seem to correspond to meaningful transformations of the input image, as shown in ﬁgure. 14. 10 one practical issue with the cae regularization criterion is that although it is cheap to compute in the case of a single hidden layer autoencoder, it becomes much more expensive in the case of deeper autoencoders. the strategy followed by rifai 2011a et al. ( ) is to separately train a series of single - layer autoencoders, each trained to reconstruct the previous autoencoder ’ s hidden layer. the composition of these autoencoders then forms a deep autoencoder. because each layer was separately trained to be locally contractive, the deep autoencoder is contractive as well. the result is not the same as what would be obtained by jointly training the entire architecture with a penalty on the jacobian of the deep model, but it captures many of the desirable qualitative characteristics. another practical issue is that the contraction penalty can obtain useless results 522	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	537	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders input point tangent vectors local pca ( no sharing across regions ) contractive autoencoder figure 14. 10 : illustration of tangent vectors of the manifold estimated by local pca and by a contractive autoencoder. the location on the manifold is deﬁned by the input image of a dog drawn from the cifar - 10 dataset. the tangent vectors are estimated by the leading singular vectors of the jacobian matrix ∂h ∂x of the input - to - code mapping. although both local pca and the cae can capture local tangents, the cae is able to form more accurate estimates from limited training data because it exploits parameter sharing across [UNK] locations that share a subset of active hidden units. the cae tangent directions typically correspond to moving or changing parts of the object ( such as the head or legs ). images reproduced with permission from ( ). rifai et al. 2011c if we do not impose some sort of scale on the decoder. for example, the encoder could consist of multiplying the input by a small constant and the decoder could consist of dividing the code by. as approaches, the encoder drives the 0 contractive penalty ω ( h ) to	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	538	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. for example, the encoder could consist of multiplying the input by a small constant and the decoder could consist of dividing the code by. as approaches, the encoder drives the 0 contractive penalty ω ( h ) to approach without having learned anything about the 0 distribution. meanwhile, the decoder maintains perfect reconstruction. in rifai et al. ( ), this is prevented by tying the weights of 2011a f and g. both f and g are standard neural network layers consisting of an [UNK] transformation followed by an element - wise nonlinearity, so it is straightforward to set the weight matrix of g to be the transpose of the weight matrix of. f 14. 8 predictive sparse decomposition predictive sparse decomposition ( psd ) is a model that is a hybrid of sparse coding and parametric autoencoders ( kavukcuoglu 2008 et al., ). a parametric encoder is trained to predict the output of iterative inference. psd has been applied to unsupervised feature learning for object recognition in images and video ( kavukcuoglu 2009 2010 jarrett 2009 farabet 2011 et al.,, ; et al., ; et al., ), as well	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	538	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
applied to unsupervised feature learning for object recognition in images and video ( kavukcuoglu 2009 2010 jarrett 2009 farabet 2011 et al.,, ; et al., ; et al., ), as well as for audio (, ). the model consists of an encoder [UNK] al. 2011 f ( x ) and a decoder g ( h ) that are both parametric. during training, h is controlled by the 523	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	538	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders optimization algorithm. training proceeds by minimizing \| \| − \| \| x g ( ) h 2 + λ \| \| h 1 + ( ) γ f \| \| − h x \| \| 2. ( 14. 19 ) like in sparse coding, the training algorithm alternates between minimization with respect to h and minimization with respect to the model parameters. minimization with respect to h is fast because f ( x ) provides a good initial value of h and the cost function constrains h to remain near f ( x ) anyway. simple gradient descent can obtain reasonable values of in as few as ten steps. h the training procedure used by psd is [UNK] from ﬁrst training a sparse coding model and then training f ( x ) to predict the values of the sparse coding features. the psd training procedure regularizes the decoder to use parameters for which can infer good code values. f ( ) x predictive sparse coding is an example of learned approximate inference. in section, this topic is developed further. the tools presented in chapter 19. 5 19 make it clear that psd can be interpreted as training a directed sparse coding probabilistic model by maximizing a lower bound on the log - likelihood	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	539	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. in section, this topic is developed further. the tools presented in chapter 19. 5 19 make it clear that psd can be interpreted as training a directed sparse coding probabilistic model by maximizing a lower bound on the log - likelihood of the model. in practical applications of psd, the iterative optimization is only used during training. the parametric encoder f is used to compute the learned features when the model is deployed. evaluating f is computationally inexpensive compared to inferring h via gradient descent. because f is a [UNK] parametric function, psd models may be stacked and used to initialize a deep network to be trained with another criterion. 14. 9 applications of autoencoders autoencoders have been successfully applied to dimensionality reduction and infor - mation retrieval tasks. dimensionality reduction was one of the ﬁrst applications of representation learning and deep learning. it was one of the early motivations for studying autoencoders. for example, hinton and salakhutdinov 2006 ( ) trained a stack of rbms and then used their weights to initialize a deep autoencoder with gradually smaller hidden layers, culminating in a bottleneck of 30 units. the resulting code yielded less reconstruction error than	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	539	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##akhutdinov 2006 ( ) trained a stack of rbms and then used their weights to initialize a deep autoencoder with gradually smaller hidden layers, culminating in a bottleneck of 30 units. the resulting code yielded less reconstruction error than pca into 30 dimensions and the learned representation was qualitatively easier to interpret and relate to the underlying categories, with these categories manifesting as well - separated clusters. lower - dimensional representations can improve performance on many tasks, such as classiﬁcation. models of smaller spaces consume less memory and runtime. 524	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	539	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 14. autoencoders many forms of dimensionality reduction place semantically related examples near each other, as observed by salakhutdinov and hinton 2007b torralba ( ) and et al. ( ). the hints provided by the mapping to the lower - dimensional space aid 2008 generalization. one task that beneﬁts even more than usual from dimensionality reduction is information retrieval, the task of ﬁnding entries in a database that resemble a query entry. this task derives the usual beneﬁts from dimensionality reduction that other tasks do, but also derives the additional beneﬁt that search can become extremely [UNK] in certain kinds of low dimensional spaces. speciﬁcally, if we train the dimensionality reduction algorithm to produce a code that is low - dimensional and, then we can store all database entries in a hash table binary mapping binary code vectors to entries. this hash table allows us to perform information retrieval by returning all database entries that have the same binary code as the query. we can also search over slightly less similar entries very [UNK], just by ﬂipping individual bits from the encoding of the query. this approach to information retrieval via dimensionality reduction and binarization is called semantic hashing ( salakhutdin	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	540	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. we can also search over slightly less similar entries very [UNK], just by ﬂipping individual bits from the encoding of the query. this approach to information retrieval via dimensionality reduction and binarization is called semantic hashing ( salakhutdinov and hinton 2007b 2009b,, ), and has been applied to both textual input ( salakhutdinov and hinton 2007b 2009b,, ) and images ( torralba 2008 weiss 2008 krizhevsky and hinton 2011 et al., ; et al., ;, ). to produce binary codes for semantic hashing, one typically uses an encoding function with sigmoids on the ﬁnal layer. the sigmoid units must be trained to be saturated to nearly 0 or nearly 1 for all input values. one trick that can accomplish this is simply to inject additive noise just before the sigmoid nonlinearity during training. the magnitude of the noise should increase over time. to ﬁght that noise and preserve as much information as possible, the network must increase the magnitude of the inputs to the sigmoid function, until saturation occurs. the idea of learning a hashing function has been further explored in several directions, including the idea of training the representations so as	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	540	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
information as possible, the network must increase the magnitude of the inputs to the sigmoid function, until saturation occurs. the idea of learning a hashing function has been further explored in several directions, including the idea of training the representations so as to optimize a loss more directly linked to the task of ﬁnding nearby examples in the hash table (, ). norouzi and fleet 2011 525	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	540	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15 representation learning in this chapter, we ﬁrst discuss what it means to learn representations and how the notion of representation can be useful to design deep architectures. we discuss how learning algorithms share statistical strength across [UNK] tasks, including using information from unsupervised tasks to perform supervised tasks. shared representations are useful to handle multiple modalities or domains, or to transfer learned knowledge to tasks for which few or no examples are given but a task representation exists. finally, we step back and argue about the reasons for the success of representation learning, starting with the theoretical advantages of distributed representations ( hinton 1986 et al., ) and deep representations and ending with the more general idea of underlying assumptions about the data generating process, in particular about underlying causes of the observed data. many information processing tasks can be very easy or very [UNK] depending on how the information is represented. this is a general principle applicable to daily life, computer science in general, and to machine learning. for example, it is straightforward for a person to divide 210 by 6 using long division. the task becomes considerably less straightforward if it is instead posed using the roman numeral representation of the numbers. most modern people asked to divide ccx by vi would begin by converting the numbers to the arabic numeral	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	541	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
divide 210 by 6 using long division. the task becomes considerably less straightforward if it is instead posed using the roman numeral representation of the numbers. most modern people asked to divide ccx by vi would begin by converting the numbers to the arabic numeral representation, permitting long division procedures that make use of the place value system. more concretely, we can quantify the asymptotic runtime of various operations using appropriate or inappropriate representations. for example, inserting a number into the correct position in a sorted list of numbers is an o ( n ) operation if the list is represented as a linked list, but only o ( log n ) if the list is represented as a red - black tree. in the context of machine learning, what makes one representation better than 526	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	541	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning another? generally speaking, a good representation is one that makes a subsequent learning task easier. the choice of representation will usually depend on the choice of the subsequent learning task. we can think of feedforward networks trained by supervised learning as per - forming a kind of representation learning. speciﬁcally, the last layer of the network is typically a linear classiﬁer, such as a softmax regression classiﬁer. the rest of the network learns to provide a representation to this classiﬁer. training with a supervised criterion naturally leads to the representation at every hidden layer ( but more so near the top hidden layer ) taking on properties that make the classiﬁcation task easier. for example, classes that were not linearly separable in the input features may become linearly separable in the last hidden layer. in principle, the last layer could be another kind of model, such as a nearest neighbor classiﬁer ( salakhutdinov and hinton 2007a, ). the features in the penultimate layer should learn [UNK] properties depending on the type of the last layer. supervised training of feedforward networks does not involve explicitly imposing any condition on the learned intermediate features. other kinds of representation learning algorithms are	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	542	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##a, ). the features in the penultimate layer should learn [UNK] properties depending on the type of the last layer. supervised training of feedforward networks does not involve explicitly imposing any condition on the learned intermediate features. other kinds of representation learning algorithms are often explicitly designed to shape the representation in some particular way. for example, suppose we want to learn a representation that makes density estimation easier. distributions with more independences are easier to model, so we could design an objective function that encourages the elements of the representation vector h to be independent. just like supervised networks, unsupervised deep learning algorithms have a main training objective but also learn a representation as a side [UNK]. regardless of how a representation was obtained, it can be used for another task. alternatively, multiple tasks ( some supervised, some unsupervised ) can be learned together with some shared internal representation. most representation learning problems face a [UNK] preserving as much information about the input as possible and attaining nice properties ( such as independence ). representation learning is particularly interesting because it provides one way to perform unsupervised and semi - supervised learning. we often have very large amounts of unlabeled training data and relatively little labeled training data. training with supervised learning techniques on the labeled subset often	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	542	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
particularly interesting because it provides one way to perform unsupervised and semi - supervised learning. we often have very large amounts of unlabeled training data and relatively little labeled training data. training with supervised learning techniques on the labeled subset often results in severe overﬁtting. semi - supervised learning [UNK] the chance to resolve this overﬁtting problem by also learning from the unlabeled data. speciﬁcally, we can learn good representations for the unlabeled data, and then use these representations to solve the supervised learning task. humans and animals are able to learn from very few labeled examples. we do 527	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	542	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)