Chapter 17

The Manifold Perspective on

Representation Learning

Manifold learning is an approach to machine learning that is capitalizing on the

manifold hypothesis (Cayton, 2005; Narayanan and Mitter, 2010): the data gen-

erating distribution is assumed to concentrate near regions of low dimensionality.

The notion of manifold in mathematics refers to continuous spaces that locally

resemble Euclidean space, and the term we should be using is really submanifold,

which corresponds to a subset which has a manifold structure. The use of the

term manifold in machine learning is much looser than its use in mathematics,

though:

• the data may not be strictly on the manifold, but only near it,

• the dimensionality may not be the same everywhere,

• the notion actually referred to in machine learning naturally extends to

discrete spaces.

Indeed, although the very notions of a manifold or submanifold are deﬁned

for continuous spaces, the more general notion of probability concentration ap-

plies equally well to discrete data. It is a kind of informal prior assumption about

the data generating distribution that seems particularly well-suited for AI tasks

such as those involving images, video, speech, music, text, etc. In all of these

cases the natural data has the property that randomly choosing conﬁgurations of

the observed variables according to a factored distribution (e.g. uniformly) are

very unlikely to generate the kind of observations we want to model. What is the

probability of generating a natural looking image by choosing pixel intensities

independently of each other? What is the probability of generating a meaning-

ful natural language paragraph by independently choosing each character in a

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⇓

Figure 17.1: Top: data sampled from a distribution in a high-dimensional space (one 2

dimensions shown for illustration) that is actually concentrated near a one-dimensional

manifold, which here is like a twisted string. Bottom: the underlying manifold that the

learner should infer.

string? Doing a thought experiment should give a clear answer: an exponentially

tiny probability. This is because the probability distribution of interest concen-

trates in a tiny volume of the total space of conﬁgurations. That means that to

the ﬁrst degree, the problem of characterizing the data generating distribution

can be reduced to a binary classiﬁcation problem: is this conﬁguration probable

or not?. Is this a grammatically and semantically plausible sentence in English?

Is this a natural-looking image? Answering these questions tells us much more

about the nature of natural language or text than the additional information one

would have by being able to assign a precise probability to each possible sequence

of characters or set of pixels. Hence, simply characterizing where probability

concentrates is a fundamental importance, and this is what manifold learning

algorithms attempt to do. Because it is a where question, it is more about ge-

ometry than about probability distributions, although we ﬁnd both views useful

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

when designing learning algorithms for AI tasks.

tangent directions

tangent plane

Data on a curved manifold

Figure 17.2: A two-dimensional manifold near which training examples are concentrated,

along with a tangent plane and its associated tangent directions, forming a basis that

specify the directions of small moves one can make to stay on the manifold.

Figure 17.3: Illustration of tangent vectors of the manifold estimated by a contractive

auto-encoder (CAE), at some input point (top left, image of a zero). Each image on the

top right corresponds to a tangent vector. They are obtained by picking the dominant

singular vectors (with largest singular value) of the Jacobian

∂f (x)

∂x

(see Section 15.10).

Taking the original image plus a small quantity of any of these tangent vectors yields

another plausible image, as illustrated in the bottom. The leading tangent vectors seem

to correspond to small deformations, such as translation, or shifting ink around locally in

the original image. Reproduced with permission from the authors of Rifai et al. (2011a).

In addition to the property of probability concentration, there is another one

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that characterizes the manifold hypothesis: when a conﬁguration is probable it

is generally surrounded (at least in some directions) by other probable conﬁgura-

tions. If a conﬁguration of pixels looks like a natural image, then there are tiny

changes one can make to the image (like translating everything by 0.1 pixel to the

left) which yield another natural-looking image. The number of independent ways

(each characterized by a number indicating how much or whether we do it) by

which a probable conﬁguration can be locally transformed into another probable

conﬁguration indicates the local dimension of the manifold. Whereas maximum

likelihood procedures tend to concentrate probability mass on the training ex-

amples (which can each become a local maximum of probability when the model

overﬁts), the manifold hypothesis suggests that good solutions instead concen-

trate probability along ridges of high probability (or their high-dimensional gen-

eralization) that connect nearby examples to each other. This is illustrated in

Figure 17.1.

What is most commonly learned to characterize a manifold is a representation

of the data points on (or near, i.e. projected on) the manifold. Such a representa-

tion 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 man-

ifold 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.

Another 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 17.2, these local directions specify how one can change x

inﬁnitesimally while staying on the manifold.

Manifold learning has mostly focused on unsupervised learning procedures

that attempt to capture these manifolds. Most of the initial machine learning re-

search on learning non-linear manifolds has focused on non-parametric methods

based on the nearest-neighbor graph. This graph has one node per training ex-

ample and edges connecting near neighbors. Basically, these methods (Sch¨olkopf

et al., 1998; Roweis and Saul, 2000; Tenenbaum et al., 2000; Brand, 2003; Belkin

and Niyogi, 2003; Donoho and Grimes, 2003; Weinberger and Saul, 2004; Hinton

and Roweis, 2003; van der Maaten and Hinton, 2008a) associate each of these

nodes with a tangent plane that spans the directions of variations associated with

the diﬀerence vectors between the example and its neighbors, as illustrated in

Figure 17.4.

A global coordinate system can then be obtained through an optimization or

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

Figure 17.4: Non-parametric manifold learning procedures build a nearest neighbor graph

whose nodes are training examples and arcs connect nearest neighbors. Various proce-

dures can thus obtain the tangent plane associated with a neighborhood of the graph,

and 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).

solving a linear system. Figure 17.5 illustrates how a manifold can be tiled by a

large number of locally linear Gaussian-like patches (or “pancakes”, because the

Gaussians are ﬂat in the tangent directions).

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

Figure 17.5: If the tangent plane at each location is known, then they can be tiled to

form a global coordinate system or a density function. In the ﬁgure, 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.

The average of all these Gaussians would provide an estimated density function, as in the

Manifold Parzen algorithm (Vincent and Bengio, 2003) or its non-local neural-net based

variant (Bengio et al., 2006c).

tangent directions

tangent image

tangent directions

tangent image

shifted

image

high−contrast image

Figure 17.6: When the data are images, the tangent vectors can also be visualized like

images. Here we show the tangent vector associated with translation: it corresponds to

the diﬀerence between an image and a slightly translated version. This basically extracts

edges, with the sign (shown as red or green color) indicating whether ink is being added

or subtracted when moving to the right. The ﬁgure illustrates the fact that the tangent

vectors along the manifolds associated with translation, rotation and such apparently

benign transformations of images, can be highly curved: the tangent vector at a position

x is very diﬀerent from the tangent vector at a nearby position (slightly translated image).

Note how the two tangent vectors shown have almost no overlap, i.e., their dot product

is nearly 0, and they are as far as two such images can be. If the tangent vectors of a

manifold change quickly, it means that the manifold is highly curved. This is going to

make it very diﬃcult to capture such manifolds (with a huge numbers of twists, ups and

downs) with local non-parametric methods such as graph-based non-parametric manifold

learning. This point was made in Bengio and Monperrus (2005).

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

However, there is a fundamental diﬃculty with such non-parametric neighborhood-

based approaches to manifold learning, raised in Bengio and Monperrus (2005):

if the manifolds are not very smooth (they have many ups and downs and twists),

one may need a very large number of training examples to cover each one of

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 of interest in AI have many

ups and downs and twists and strong curvature, as illustrated in Figure 17.6. This

motivates the use of distributed representations and deep learning for capturing

manifold structure, which is the subject of this chapter.

Figure 17.7: Training examples of a face dataset – the QMUL Multiview Face

Dataset (Gong et al., 2000) – for which the subjects were asked to move in such a way as

to cover the two-dimensional manifold corresponding to two angles of rotation. We would

like learning algorithms to be able to discover and disentangle such factors. Figure 17.8

illustrates such a feat.

The hope of many manifold learning algorithms, including those based on deep

learning and auto-encoders, is that one learns an explicit or implicit coordinate

system for the leading factors of variation that explain most of the structure in

the unknown data generating distribution. An example of explicit coordinate

system is one where the dimensions of the representation (e.g., the outputs of the

encoder, i.e., of the hidden units that compute the “code” associated with the

input) are directly the coordinates that map the unknown manifold. Training

examples of a face dataset in which the images have been arranged visually on a

2-D manifold are shown in Figure 17.7, with the images laid down so that each

of the two axes corresponds to one of the two angles of rotation of the face.

However, the objective is to discover such manifolds, and Figure 17.8 illus-

trates the images generated by a variational auto-encoder (Kingma and Welling,

2014a) when the two-dimensional auto-encoder code (representation) is varied

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

on the 2-D plane. Note how the algorithm actually discovered two independent

factors of variation: angle of rotation and emotional expression.

Another kind of interesting illustration of manifold learning involves the dis-

covery of distributed representations for words. Neural language models were

initiated with the work of Bengio et al. (2001c, 2003b), in which a neural network

is trained to predict the next word in a sequence of natural language text, given

the previous words, and where each word is represented by a real-valued vector,

called embedding or neural word embedding.

Figure 17.8: Two-dimensional representation space (for easier visualization), i.e., a Eu-

clidean coordinate system for Frey faces (left) and MNIST digits (right), learned by a

variational auto-encoder (Kingma and Welling, 2014a). Figures reproduced with permis-

sion from the authors. The images shown are not examples from the training set but

images x actually generated by the model P(x | h), simply by changing the 2-D “code”

h (each image corresponds to a diﬀerent choice of “code” h on a 2-D uniform grid). On

the left, one dimension that has been discovered (horizontal) mostly corresponds to a

rotation of the face, while the other (vertical) corresponds to the emotional expression.

The decoder deterministically maps codes (here two numbers) to images. The encoder

maps images to codes (and adds noise, during training).

Figure 17.9 shows such neural word embeddings reduced to two dimensions

(originally 50 or 100) using the t-SNE non-linear dimensionality reduction algo-

rithm (van der Maaten and Hinton, 2008a). The ﬁgures zooms into diﬀerent areas

of the word-space and illustrates that words that are semantically and syntacti-

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

cally close end up having nearby embeddings.

Figure 17.9: Two-dimensional representation space (for easier visualization), of English

words, learned by a neural language model as in Bengio et al. (2001c, 2003b), with t-SNE

for the non-linear dimensionality reduction from 100 to 2. Diﬀerent regions are zoomed

to better see the details. At the global level one can identify big clusters correspond-

ing to part-of-speech, while locally one sees mostly semantic similarity explaining the

neighborhood structure.

17.1 Manifold Interpretation of PCA and Linear Auto-

Encoders

The above view of probabilistic PCA as a thin “pancake” of high probability

is related to the manifold interpretation of PCA and linear auto-encoders, in

which we are looking for projections of x into a subspace that preserves as much

information as possible about x. This is illustrated in Figure 17.10. Let the

encoder be

h = f(x) = W

(x −µ)

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

computing such a projection, a low-dimensional representation of h. With the

auto-encoder view, we have a decoder computing the reconstruction

x = g(h) = b + V h.

Figure 17.10: Flat Gaussian capturing probability concentration near a low-dimensional

manifold. The ﬁgure shows the upper half of the “pancake” above the “manifold plane”

which goes through its middle. The variance in the direction orthogonal to the manifold

is very small (upward red arrow) and can be considered like “noise”, where the other

variances are large (larger red arrows) and correspond to “signal”, and a coordinate

system for the reduced-dimension data.

It turns out that the choices of linear encoder and decoder that minimize

reconstruction error

E[||x −

x||

]

correspond to V = W , µ = b = E[x] and the rows of W form an orthonormal

basis which spans the same subspace as the principal eigenvectors of the covariance

matrix

C = E[(x − µ)(x −µ)

In the case of PCA, the rows of W are these eigenvectors, ordered by the magni-

tude of the corresponding eigenvalues (which are all real and non-negative). This

is illustrated in Figure 17.11.

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

!"#$%&'!(#)$%*"!!$!*+"#'$!*

!"#$%&'!(#)$%,x-*

Figure 17.11: Manifold view of PCA and linear auto-encoders. The data distribution is

concentrated near a manifold aligned with the leading eigenvectors (here, this is just v

)

of the data covariance matrix. The other eigenvectors (here, just v

) are orthogonal to the

manifold. A data point (in red, x) is encoded into a lower-dimensional representation or

code h (here the scalar which indicates the position on the manifold, starting from h = 0).

The decoder (transpose of the encoder) maps h to the data space, and corresponds to a

point lying exactly on the manifold (green cross), the orthogonal projection of x on the

manifold. The optimal encoder and decoder minimize the sum of reconstruction errors

(diﬀerence vector between x and its reconstruction).

One can also show that eigenvalue λ

of C corresponds to the variance of x

in the direction of eigenvector v

. If x ∈ R

and h ∈ R

with d < D, then the

optimal reconstruction error (choosing µ, b, V and W as above) is

min E[||x −

x||

] =

i=d+1

Hence, if the covariance has rank d, the eigenvalues λ

d+1

to λ

are 0 and recon-

struction error is 0.

Furthermore, one can also show that the above solution can be obtained by

maximizing the variances of the elements of h, under orthonormal W , instead of

minimizing reconstruction error.

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

17.2 Manifold Interpretation of Sparse Coding

Sparse coding was introduced in Section 15.6.2 a linear factors generative model.

It also has an interesting manifold learning interpretation. The codes h inferred

with the above equation do not ﬁll the space in which h lives. Instead, probability

mass is concentrated on axis-aligned subspaces: sets of values of h for which most

of the axes are set at 0. We can thus decompose h into two pieces of information:

• A binary pattern β which speciﬁes which h

are non-zero, with N

the number of “active” (non-zero) dimensions.

• A variable-length real-valued vector α ∈ R

which speciﬁes the coordinates

for each of the active dimensions.

The pattern β can be viewed as specifying an N

-dimensional region in input

space (the set of x = W h + b where h

= 0 if b

= 0). That region is actually a

linear manifold, an N

-dimensional hyperplane. All those hyperplanes go through

a “center” x = b. The vector α then speciﬁes a Euclidean coordinate on that

hyperplane.

Because the prior P (h) is concentrated around 0, the probability mass of P (x)

is concentrated on the regions of these hyperplanes near x = b. Depending on

the amount of reconstruction error (output variance for P(x | g(h))), there is also

probability mass bleeding around these hyperplanes and making them look more

like pancakes. Each of these hyperplane-aligned manifolds and the associated

distribution is just like the ones we associate to probabilistic PCA and factor

analysis. The crucial diﬀerence is that instead of one hyperplane, we have 2

hyperplanes if h ∈ R

. Due to the sparsity prior, however, most of these ﬂat

Gaussians are unlikely: only the ones corresponding to a small N

(with only

a few of the axes being active) are likely. For example, if we were to restrict

ourselves to only those values of b for which N

= k, then one would have





Gaussians. With this exponentially large number of Gaussians, the interesting

thing to observe is that the sparse coding model only has a number of parameters

linear in the number of dimensions of h. This property is shared with other

distributed representation learning algorithms described in this chapter, such as

the regularized auto-encoders.

17.3 The Entropy Bias from Maximum Likelihood

TODO: how the log-likelihood criterion forces a learner that is not able to gen-

eralize perfectly to yield an estimator that is much smoother than the target

distribution. Phrase it in terms of entropy, not smoothness.

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17.4 Manifold Learning via Regularized Auto-Encoders

Auto-encoders have been described in Section 15. What is their connection to

manifold learning? This is what we discuss here.

We denote f the encoder function, with h = f(x) the representation of x,

and g the decoding function, with

x = g(h) the reconstruction of x, although in

some cases the encoder is a conditional distribution q(h | x) and the decoder is a

conditional distribution P (x | h).

What all auto-encoders have in common, when they are prevented from sim-

ply learning the identity function for all possible input x, is that training them

involves a compromise between two “forces”:

1. Learning a representation h of training examples x such that x can be

approximately recovered from h through a decoder. Note that this needs

not be true for any x, only for those that are probable under the data

generating distribution.

2. Some constraint or regularization is imposed, either on the code h or on the

composition of the encoder/decoder, so as to make the transformed data

somehow simpler or to prevent the auto-encoder from achieving perfect

reconstruction everywhere. We can think of these constraints or regulariza-

tion as a preference for solutions in which the representation is as simple

as possible, e.g., factorized or as constant as possible, in as many directions

as possible. In the case of the bottleneck auto-encoder a ﬁxed number of

representation dimensions is allowed, that is smaller than the dimension of

x. In the case of sparse auto-encoders (Section 15.8) the representation

elements h

are pushed towards 0. In the case of denoising auto-encoders

(Section 15.9), the encoder/decoder function is encouraged to be contrac-

tive (have small derivatives). In the case of the contractive auto-encoder

(Section 15.10), the encoder function alone is encouraged to be contractive,

while the decoder function is tied (by symmetric weights) to the encoder

function. In the case of the variational auto-encoder (Section 20.9.3), a

prior log P (h) is imposed on h to make its distribution factorize and con-

centrate as much as possible. Note how in the limit, for all of these cases,

the regularization prefers representations that are insensitive to the input.

Clearly, the second type of force alone would not make any sense (as would

any regularizer, in general). How can these two forces (reconstruction error on

one hand, and “simplicity” of the representation on the other hand) be recon-

ciled? The solution of the optimization problem is that only the variations that

are needed to distinguish training examples need to be represented. If the data

generating distribution concentrates near a low-dimensional manifold, this yields

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

Figure 17.12: A regularized auto-encoder or a bottleneck auto-encoder has to reconcile

two forces: reconstruction error (which forces it to keep enough information to distinguish

training examples from each other), and a regularizer or constraint that aims at reducing

its representational ability, to make it as insensitive as possible to the input in as many

directions as possible. The solution is for the learned representation to be sensitive to

changes along the manifold (green arrow going to the right, tangent to the manifold) but

invariant to changes orthogonal to the manifold (blue arrow going down). This yields to

contraction of the representation in the directions orthogonal to the manifold.

representations that implicitly capture a local coordinate 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 embedding 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 man-

ifold. This idea is illustrated in Figure 17.12. A one-dimensional example is

illustrated in Figure 17.13, showing that by making the auto-encoder contractive

around the data points (and the reconstruction point towards the nearest data

point), we recover the manifold structure (of a set of 0-dimensional manifolds in

a 1-dimensional embedding space, in the ﬁgure).

17.5 Tangent Distance, Tangent-Prop, and Manifold

Tangent Classiﬁer

One of the early attempts to take advantage of the manifold hypothesis is the

Tangent Distance algorithm (Simard et al., 1993, 1998). It is a non-parametric

nearest-neighbor algorithm in which the metric used is not the generic Euclidean

distance but one that is derived from knowledge of the manifolds near which

probability concentrates. It is assumed that we are trying to classify examples

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

r(x)"

Figure 17.13: If the auto-encoder learns to be contractive around the data points, with

the reconstruction pointing towards the nearest data points, it captures the manifold

structure of the data. This is a 1-dimensional version of Figure 17.12. The denoising

auto-encoder explicitly tries to make the derivative of the reconstruction function r(x)

small around the data points. The contractive auto-encoder does the same thing for the

encoder. Although the derivative of r(x) is asked to be small around the data points, it

can be large between the data points (e.g. in the regions between manifolds), and it has

to be large there so as to reconcile reconstruction error (r(x) ≈ x for data points x) and

contraction (small derivatives of r(x) near data points).

and that examples on the same manifold share the same category. Since the

classiﬁer should be invariant to the local factors of variation that correspond

to movement on the manifold, it would make sense to use as nearest-neighbor

distance between points x

and x

the distance between the manifolds M

and

to which they respectively belong. Although that may be computationally

diﬃcult (it would require an optimization, to ﬁnd the nearest pair of points on

and M

), a cheap alternative that makes sense locally is to approximate M

by its tangent plane at x

and measure the distance between the two tangents,

or between a tangent plane and a point. That can be achieved by solving a low-

dimensional linear system (in the dimension of the manifolds). Of course, this

algorithm requires one to specify the tangent vectors at any point

In a related spirit, the Tangent-Prop algorithm (Simard et al., 1992) proposes

to train a neural net classiﬁer with an extra penalty to make the output f(x) of

the neural net locally invariant to known factors of variation. These factors of

variation correspond to movement of the manifold near which examples of the

same class concentrate. Local invariance is achieved by requiring

∂f (x)

∂x

to be

orthogonal to the known manifold tangent vectors v

at x, or equivalently that

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

the directional derivative of f at x in the directions v

be small:

regularizer = λ



∂f(x)

∂x

· v



. (17.1)

Like for tangent distance, the tangent vectors are derived a priori, e.g., from the

formal knowledge of the eﬀect of transformations such as translation, rotation,

and scaling in images. Tangent-Prop has been used not just for supervised learn-

ing (Simard et al., 1992) but also in the context of reinforcement learning (Thrun,

1995).

∂h

∂x

∂f

∂x

Figure 17.14: Illustration of the main idea of the tangent-prop algorithm (Simard et al.,

1992) and manifold tangent classiﬁer (Rifai et al., 2011d), which both regularize the

classiﬁer output function f(x) (e.g. estimating conditional class probabilities given the

input) so as to make it invariant to the local directions of variations

∂h

∂x

(manifold tangent

directions). This can be achieved by penalizing the magnitude of the dot product of

all the rows of

∂h

∂x

(the tangent directions) with all the rows of

∂f

∂x

(the directions of

sensitivity of each output to the input). In the case of the tangent-prop algorithm, the

tangent directions are given a priori, whereas in the case of the manifold tangent classiﬁer,

they are learned, with h(x) being the learned representation of the input x. The ﬁgure

illustrates two manifolds, one per class, and we see that the classiﬁer output increases

the most as we move from one manifold to the other, in input space.

A more recent paper introduces the Manifold Tangent Classiﬁer (Rifai et al.,

2011d), which eliminates the need to know the tangent vectors a priori, and

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CHAPTER 17. THE MANIFOLD PERSPECTIVE ON REPRESENTATION LEARNING

instead uses a contractive auto-encoder to estimate them at any point. As we

have seen in the previous section and Figure 15.9, auto-encoders in general, and

contractive auto-encoders especially well, learn a representation h that is most

sensitive to the factors of variation present in the data x, so that the leading

singular vectors of

∂h

∂x

correspond to the estimated tangent vectors. As illustrated

in Figure 15.10, these estimated tangent vectors go beyond the classical invariants

that arise out of the geometry of images (such as translation, rotation and scaling)

and include factors that must be learned because they are object-speciﬁc (such

as adding or moving body parts). The algorithm proposed with the manifold

tangent classiﬁer is therefore simple: (1) use a regularized auto-encoder such

as the contractive auto-encoder to learn the manifold structure by unsupervised

learning (2) use these tangents to regularize a neural net classiﬁer as in Tangent-

Prop (Eq. 17.1). TODO Tangent Prop or Tangent-Prop?

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