Author manuscript, published in "Neurocomputing 71, 7-9 (2008) 1578-1594"
DOI : 10.1016/j.neucom.2007.04.010
Incorporating Prior Knowledge in Support
Vector Machines for Classification: a Review
Fabien Lauer ∗ and Gérard Bloch
Centre de Recherche en Automatique de Nancy (CRAN - UMR 7039),
Nancy–University, CNRS, CRAN–ESSTIN, Rue Jean Lamour, 54519 Vandœuvre
Cedex, France
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Abstract
For classification, support vector machines (SVMs) have recently been introduced
and quickly became the state of the art. Now, the incorporation of prior knowledge
into SVMs is the key element that allows to increase the performance in many applications.
This paper gives a review of the current state of research regarding the
incorporation of two general types of prior knowledge into SVMs for classification.
The particular forms of prior knowledge considered here are presented in two main
groups: class-invariance and knowledge on the data. The first one includes invariances
to transformations, to permutations and in domains of input space, whereas
the second one contains knowledge on unlabeled data, the imbalance of the training
set or the quality of the data. The methods are then described and classified in
the three categories that have been used in literature: sample methods based on
the modification of the training data, kernel methods based on the modification
of the kernel and optimization methods based on the modification of the problem
formulation. A recent method, developed for support vector regression, considers
prior knowledge on arbitrary regions of the input space. It is exposed here when
applied to the classification case. A discussion is then conducted to regroup sample
and optimization methods under a regularization framework.
Key words: pattern recognition, classification, support vector machines (SVM),
prior knowledge, invariance
∗ Corresponding author.
Email addresses: fabien.lauer@esstin.uhp-nancy.fr (Fabien Lauer),
gerard.bloch@esstin.uhp-nancy.fr (Gérard Bloch).
Preprint submitted to Elsevier Science 13 April 20071 Introduction
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Pattern recognition is a very active field of research intimately bound to machine
learning. As part of this area, classification aims at building classifiers
that can determine the class of an input pattern. An extensive amount of
work has been done in the past decades to develop classifiers that can learn
from data to perform recognition tasks. Typical fields of application include
image recognition such as character recognition, text categorization, speech
recognition, biometric applications, bioinformatics, fault detection, diagnostic
applications, decision support, network intrusion detection and so on. The
classification problems can be divided into binary problems and multi-class
problems. Many classifiers like support vector machines (SVMs) [63] consider
the first case where the patterns can be of two classes. For multi-class applications,
there also exists learning machines that can tackle directly multi-class
applications such as neural networks [4,33] or even multi-class support vector
machines (MSVM) [69,21,8]. Nonetheless, a very common approach consists
in building a set of binary classifiers, each one either trained to separate one
class from the others (the one-against-all method) or only to distinguish between
two classes (the one-against-one method). Thus, this paper will focus
on binary SVMs without restricting the area of applications.
Support vector machines aim at learning an unknown decision function based
only on a set of N input-output pairs (xi, yi). Nonetheless, in real world applications,
a certain amount of information on the problem is usually known
beforehand. For instance, in character recognition, if an image is slightly translated
or rotated it still represents the same character. This prior knowledge
indicates that one should incorporate invariance to translations and rotations
into the classifier.
This paper gives a review of the current state of research regarding the incorporation
of two main types of prior knowledge into SVMs for classification.
The different forms of prior knowledge considered here are presented hierarchically
and divided into two main groups: class-invariance and knowledge on the
data. The first one includes invariances to transformations, to permutations
and in domains of input space, whereas the second one contains knowledge on
unlabeled data, the imbalance of the training set or the quality of the data.
This review chooses to present general methods that can be used for different
applications rather than to attempt to provide an exhaustive list of application
specific prior knowledge with its practical implementation into SVMs.
However, some interesting methods derived from an application specific point
of view can still be used in other fields and thus deserve to be presented. This
paper focuses on the methods and reuses a categorization from the literature
based on the component of the problem (the samples, the kernel or the optimization
program) which is modified to include the prior knowledge rather
2than on the prior knowledge itself. A regularization framework is then used to
regroup the sample and optimization based methods.
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The present paper aims at giving an up-to-date review that synthesizes the
existing methods. In the last decade, authors considered the introduction of
prior knowledge into SVMs and some reviews can be found in the literature.
A chapter of the well-known book [53] is dedicated to the incorporation of invariances,
but deals only with transformation-invariances. Thus, the authors
do not present methods to include knowledge on the data or class-invariance
in a domain (which were not available at the time). Nonetheless, they expose
the three different ways of exploiting prior knowledge on which relies
our categorization of the methods into three groups: sample methods, kernel
methods and optimization methods. In [22], an overview of the works in the
field is also given. However, this overview focuses on invariant kernel methods
for pattern recognition in general. Here, we are interested in different types of
prior knowledge (not only invariance) that can be included in the particular
learning machine known as SVM, either in the kernel or not. In particular,
this review focuses on and is limited to two main types of prior knowledge:
class-invariance and knowledge on the data.
The paper is organized as follows. The SVM principles and the different problem
formulations (QP and LP) are first introduced in Section 2, before giving
a definition and categorization of the types of prior knowledge that we consider
in Section 3. A review of the literature for the incorporation of prior
knowledge into SVMs is then presented in Section 4 where the methods are
classified in 3 categories: sample methods (Sect. 4.1) based on the modification
of the training data, kernel methods (Sect. 4.2) based on the modification of
the kernel and optimization methods (Sect. 4.3) based on the modification of
the problem formulation. This last subsection includes the presentation of a
recently developed approach considering prior knowledge on arbitrary regions
of the input space for support vector regression [38] to propose its application
to the classification case. The links and differences between the methods are
discussed in Section 5 with the unifying framework before the exposition of
some perspectives regarding the combination of methods. Finally, the conclusion
is given in Section 6.
Notations: all vectors are column vectors written in boldface and lowercase
letters whereas matrices are boldface and uppercase, except for the ith column
of a matrix A that is denoted Ai. The vectors 0 and 1 are vectors of
appropriate dimensions with all their components respectively equal to 0 and
1. For A ∈ R d×m and B ∈ R d×n containing d-dimensional sample vectors, the
“kernel” K(A, B) maps R d×m ×R d×n in R m×n with K(A, B)i,j = k(Ai, Bj),
where k : R d×d → R is the kernel function. In particular, if x ∈ R d is a column
vector then K(x, B) is a row vector in R 1×n . The matrix X ∈ R N×d
contains all the training samples xi, i = 1, . . ., N, as rows. The kernel matrix
3K(X T , X T ) will be written K for short. The symbol 〈., .〉 stands for the inner
product (or dot product).
2 Support vector machines for classification
The classifier is built from a training set of N samples
(x1, y1), . . .,(xi, yi), . . ., (xN, yN) (1)
where xi ∈ R d is the input vector corresponding to the ith sample labelled
by yi ∈ {−1, +1} depending on its class (only binary problems are considered
here). For the linear case, the machine implements the decision function
f(x) = sign (〈x, w〉 + b) (2)
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of parameters w ∈ R d and b ∈ R. This function determines on which side of
the separating hyperplane (〈x, w〉 + b = 0) the sample x lies.
Support vector machines (SVMs) were first introduced as large margin classifiers
[63]. For a separable training set, the margin is defined as the minimum
distance between the points of the two classes, measured perpendicularly to
the separating hyperplane. Maximizing this margin is a way for a learning
algorithm to control the capacity and the complexity of the machine, and to
select the optimal separating hyperplane amongst all the hyperplanes that separate
the two classes of the training set. The control of the capacity allows to
bound the generalization error [63] which is the probability of misclassification
for new test samples [46].
In its original form, the SVM learning leads to a quadratic program which is
a convex constrained optimization problem and thus has a unique solution.
This is a large advantage in comparison to other learning algorithms such as
the back-propagation for neural networks [48,4]. The SVM problem can be
formulated as follows: find the parameters w (also called the weights) and b
that maximize the margin while ensuring that the training samples are well
classified. This can be written as the QP optimization problem [9]
min 1
2 ‖w‖2
(3)
s.t. yi(〈xi, w〉 + b) ≥ 1, i = 1, . . .,N (4)
whose solution corresponds to the saddle point of the primal Lagrangian
LP = 1
2 ‖w‖2 −
N∑
αi[yi(〈xi, w〉 + b) − 1] (5)
i=1
4where the αi ≥ 0 are the Lagrange multipliers. The problem is equivalently
solved by maximizing the dual Lagrangian with respect to αi as
max LD =
N∑
i=1
αi − 1
2
N∑
i,j=1
s.t. αi ≥ 0, i = 1, . . ., N
∑ Ni=1
αiyi = 0
αiαjyiyj〈xi, xj〉
(6)
The resulting decision function is given [9] by
⎛
f(x) = sign ⎝ ∑
⎞
yiαi〈x, xi〉 + b⎠ (7)
αi>0
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where the xi are the support vectors (SVs), i.e. with non-zero corresponding
Lagrange multipliers αi. The SVs are the training patterns that lie on the
margin boundaries. An advantage of this algorithm is its sparsity since only a
small subset of the training samples is finally retained for the classifier.
In order to deal with non-separable data, the soft margin hyperplane is used.
A set of slack variables ξi is introduced to allow errors or points inside the
margin and a hyperparameter C is used to tune the trade-off between the
amount of accepted errors and the maximization of the margin
min 1
2 ‖w‖2 N∑
+ C ξi
i=1
s.t. yi(〈xi, w〉 + b) ≥ 1 − ξi,
ξi ≥ 0
i = 1, . . ., N
This new formulation leads to the same dual problem (6) but with the addition
of an upper bound on the Lagrange multipliers [9]
(8)
0 ≤ αi ≤ C, i = 1, . . .,N (9)
For non-linear classification problems, the data are first mapped into a higher
dimensional feature space F by
R d ∋ x Φ
↦→ Φ(x) ∈ F (10)
in which a separating hyperplane is built. This leads to the decision function
f(x) = sign (〈Φ(x), w〉 + b), where w is now a vector of F. To avoid the
curse of dimensionality [9], the ”kernel trick” is used, which leads, for (7),
⎛
f(x) = sign ⎝ ∑
⎞
yiαik(x, xi) + b⎠ (11)
αi>0
5where k(x, xi) = 〈Φ(x),Φ(xi)〉 stands for the kernel function, or, in matrix
form
f(x) = sign (
K(x, X T )Dα + b )
(12)
where D = diag(y1, . . .,yi, . . .,yN) and α = [α1 . . . αi . . . αN] T .
For the training of a non-linear SVM, one only has to replace the inner product
〈xi, xj〉 in (6) by the kernel function k(xi, xj) that corresponds to the inner
product in the feature space F. To be an admissible kernel, this function must
satisfy Mercer’s conditions (positive semidefiniteness of the kernel matrix K)
[9]. Typical kernel functions used for classification are the linear, Gaussian
RBF or polynomial kernels.
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The training of SVMs can also result in a linear program. This is now presented,
since some of the methods exposed in the next section for the incorporation
of prior knowledge use this form of SVMs. Following the approach
of [37], the ℓ1-norm of the parameters α in (12) is minimized instead of the
ℓ2-norm of the weights w as in (8). In practice, to yield a linear program,
a new set of variables a bounding the ℓ1-norm of the parameters is used. In
matrix form, the linear program corresponding to the soft margin SVM is
s.t.
min
(α,b,ξ,a)
1T a + C1 T ξ
D(KDα + b1) ≥ 1 − ξ
−a ≤ α ≤ a
ξ ≥ 0
(13)
In this formulation, no assumption on the symmetry or positive definiteness of
the kernel matrix K is needed [37]. The form of the resulting output function
(12) remains unchanged. Here, the sparsity is enforced by the minimization
of the ℓ1-norm of the parameters α which makes some αi vanish to zero. It
has also been noticed [3] that, compared to the original QP formulation, this
approach offers an increased sparsity of support vectors.
One advantage of the SVMs is the form of the learning problems, since many
general optimization softwares such as CPLEX, LOQO, Matlab linprog and
quadprog are capable of solving the linear and quadratic programs derived
from SVMs. Nonetheless, the scale of the problems led to develop specific
methods such as chunking [29], decomposition [41] or its extreme case known
as the Sequential Minimal Optimization (SMO) algorithm [42], of which modifications
have been proposed [30]. Many softwares, usually available on the
Internet, have been developed in the last years to speed up the training time or
to deal with large data sets such as SVM light [25], libSVM [6,15] and libsvmTL
[47], HeroSVM [12,13] or Core Vector Machine (CVM) [61].
63 Prior knowledge for classification
This Section starts by giving a definition of prior knowledge as considered in
this paper. Different types of prior knowledge encountered in pattern recognition
are then regrouped under two main categories: class-invariance and
knowledge on the data. The tree-like diagram of Figure 1 categorizes the types
of prior knowledge from the most general at the top to the particular at the
bottom where arrows point to suited methods. In particular, this review focuses
on two main types of prior knowledge: class-invariance and knowledge
on the data. The next Section provides the description and categorization of
the suited methods.
3.1 Definition
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In this review, prior knowledge is defined as in [53] and refers to all information
about the problem available in addition to the training data. Determining
a model from a finite set of samples without prior knowledge is an ill-posed
problem, in the sense that, for instance, a unique model may not exist. Many
classifiers incorporate the general smoothness assumption that a test pattern
similar to one of the training samples tends to be assigned to the same class.
Also, choosing the soft margin version of SVMs can be seen as a use of prior
knowledge on the non-separability of the data or the presence of outliers and
noise in the training set. However, in both cases, these assumptions are intrinsically
made by the support vector learning and are thus excluded from
the definition of prior knowledge in the remainder of the paper. In machine
learning, the importance of prior knowledge can be seen from the No Free
Lunch theorem [71] which states that all the algorithms perform the same
when averaged over the different problems and thus implies that to gain in
performance one must use a specialized algorithm that includes some prior
knowledge about the problem at hand.
3.2 Class-invariance
A very common type of prior knowledge in pattern recognition is the invariance
of the class (or the output of the classifier) to a transformation of the input
pattern. Throughout this paper this type of knowledge will be referred to as
transformation-invariance. Incorporating the invariance to a transformation
Tθ : x ↦→ Tθx, parametrized in θ, into a classifier of output f(x) for an input
pattern x corresponds to enforce the equality
f(x) = f(Tθx), ∀x, θ (14)
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Fig. 1. Two main types of prior knowledge that can be incorporated into SVM with the corresponding methods expressed by their
number given in the companion diagram of Figure 2.However, local invariance is sometimes considered instead. In this case, the
invariance is only imposed around a fixed value of θ. For a transformation
centered at θ = 0, so that T0x = x, local invariance can be enforced by the
constraint
∂
∣ f(Tθx) = 0 (15)
∂θ
∣
θ=0
thus limiting the variation of f for a variation of θ. It must be noted that f
in (15) is considered to be the real-valued output of the classifier, i.e. without
the sign function in (7) or (11).
Some methods are based on another approach, which is to consider the classinvariance
with respect to a domain of the input space instead of a transformation.
In this case, the problem becomes finding f so that
f(x) = yP, ∀x ∈ P (16)
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where yP is the class label of the region P of the input space. In practice, this
approach is particularly useful to provide prior knowledge in regions of input
space that lack training samples.
Another type of class-invariance found in pattern recognition is the
permutation-invariance, i.e. invariance of the class to a permutation of elements
in a structured input. A typical application is a classifier invariant to
permutations of rows in matrix inputs. Since permutations are no more than
particular transformations, permutation-invariance can also be considered as
transformation-invariance.
3.3 Knowledge on the data
Other forms of prior knowledge than class-invariance concern the data more
specifically and are thus of particular interest for real-world applications. In
this review, the three particular cases that most often occur when gathering
data are studied.
• Unlabeled samples are available with supposed class-memberships.
• Imbalance of the training set is encountered when a high proportion of
samples is of the same class.
• Quality of the data may vary from a sample to another.
Prior knowledge in relationship with these cases can enhance the quality of
the recognition if included in the learning. Moreover, not taking into account
the poor quality of some data or a large imbalance between the classes can
mislead the decision of a classifier.
94 Incorporating prior knowledge into SVM: a survey
In this Section, a review of the methods for incorporating prior knowledge
into SVMs, however restricted to class-invariance and knowledge on the data,
is given. These methods are classified in three categories, as defined by [53],
depending on the means used to include the prior knowledge and the component
of the problem that is modified. As presented in Figure 2, these three
groups of methods are:
• sample methods that incorporate the prior knowledge either by generating
new data or by modifying the way they are taken into account;
• kernel methods that incorporate the prior knowledge in the kernel function
either by selecting the most appropriate kernel or by creating a new kernel;
• optimization methods that incorporate the prior knowledge in the problem
formulation either by adding constraints to the original problem or by defining
a new formulation which includes intrinsically the prior knowledge.
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The methods are detailed in the following. Further discussions and links between
the methods can be found in the next Section.
4.1 Sample methods
Two different approaches will be exposed in this section. The first one is based
on the generation of virtual samples, while the second one aims at weighting
the influence of different samples. Whereas the virtual samples methods focus
on the incorporation of transformation-invariance, the weighting of samples
allows to include knowledge on the data.
4.1.1 Virtual samples
In machine learning, the generalization ability of the obtained model depends
on the number of data at hand. The more representative samples we have, the
better we learn. Based on this simple fact, the idea of creating new samples
to enlarge the training set was first introduced by [43] as virtual samples and
in [1,2] as “hints”. In [40], learning on an extended training set by virtual
samples was linked to regularization and it thus showed a justification for the
method.
The basic idea of the virtual samples approach [40] is to incorporate a known
transformation-invariance as defined by (14). The new samples are generated
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C Ci
+, C− ��℄ �
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Fig. 2. Hierarchy of the methods for the incorporation of prior knowledge into SVM from the component of the problem which is modified
at the top to the methods at the bottom. For each method are given the main references, the number of the corresponding section and
a reference number in a circle.from the training data as follows
(xi, yi) ↦→ (Txi, yi), i = 1, . . ., N (17)
This method can be easily implemented in the context of pattern recognition.
For instance, in image recognition, invariances to translations or rotations are
often considered.
In the neural networks framework, hints introduced in [1] are specific properties
the output function f of the network must satisfy such as oddness or
invariance to a transformation. An error measure em is defined to introduce
the mth hint Hm in the learning process. For instance, for the invariance hint
that implies that the samples xp and xi are of the same class, we have
em(xp) = (f(xp) − f(xi)) 2
(18)
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which is zero when the two estimated classes are the same. A set Xm of Nm new
samples is generated and the overall error Em for the hint Hm is estimated
∑
by Em = 1/Nm xp∈Xm em(xp) to test the accordance of f with the hint
Hm. It must be noticed that the training set itself can be considered as a
hint H0 for which e0(xi) = (f(xi) − yi) 2 and the estimated overall error
is E0 = 1/N ∑N i=1(f(xi) − yi) 2 . Originally applied to a neural network that
minimizes a least squares criterion, the learning process can be summarized
as
min ∑
λmEm = ∑ λm ∑
em(xp) (19)
m
m
Nm xp∈Xm
where λm is a weighting factor for the hint Hm.
In addition, in [49], the Virtual SVM (VSVM) is introduced to incorporate
transformation-invariance into SVMs. The idea is to generate the virtual samples
only from the support vectors (SVs) since they contain all the information
about the problem. The virtual samples are thus called “virtual SVs” (VSVs).
The proposed procedure requires two SVM trainings. The first one extracts
the SVs from the training set while the second one is performed on a dataset
composed of the SVs and the VSVs.
In character recognition, the generation of virtual samples became very popular
and almost necessary to achieve first-class performances. It is clear that
an image representing a character will still represent the same character if,
for instance, translated by one pixel. Thus, one often looks for classifiers that
can incorporate some translation-invariance of the output as prior knowledge.
Whereas simple distortions such as translations, rotations and scaling are generated
by applying affine displacement fields to images, elastic distortion, introduced
in [58] to imitate the variations of the handwriting, uses random
displacement fields. Other transformations, such as morphing [28], were specifically
developed and it appears that the best results (at least on the MNIST
12database [34]) are obtained by elastic distortions [58,32] even if it is based
on random displacement of pixels in the image. This highlights the fact that
more samples help to learn better even if they are not absolutely accurate.
4.1.2 Weighting of samples
The weighting of samples allows to include other forms of prior knowledge
than transformation-invariance. It is typically used to express knowledge on
the data such as an imbalance between classes, the relative quality of the
samples or prior knowledge on unlabeled samples. In practice this amounts to
weight the errors or to choose a different trade-off parameter C for different
samples.
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Originally, different misclassification costs Ci were used to deal with unbalanced
data, i.e. providing much more samples of a class than of the other
[5,27]. Ci is set to a higher value for the less represented class, thus penalizing
more the errors on this class. Besides, in [9], an equivalence is shown between
the soft margin SVM using the ℓ2-norm of the errors and a hard margin SVM
trained with a modified kernel matrix
K ′ = K + 1
I (20)
C
This idea is extended in [65] to deal with unbalanced data when different
misclassification costs C + and C − are assigned for the positive and negative
classes. The kernel matrix is then given by
K ′ = K + D (21)
where D is a diagonal matrix with components Dii = 1/C + for yi = +1
and Dii = 1/C− for yi = −1. This method amounts to define an asymmetric
margin keeping further away from the decision boundary the class with a
higher C. Heuristics are proposed in [5] to tune Dii based on the knowledge
of unbalanced data: set Dii = λn +/N for yi = +1 and Dii = λn− /N for
yi = −1 , where n + and n− are respectively the numbers of positive and
negative samples in the training set of size N and where λ is a scaling factor.
Another approach is developed in [72] in order to incorporate prior knowledge
on the quality of the training data which may vary from a sample to another.
The method, known as Weighted-SVM (WSVM), proposes to set a different
cost Ci for each sample with respect to a confidence value based on some
knowledge of the data acquisition or labeling procedure.
In [68], prior knowledge on unlabeled samples is considered. Based on supposed
class-memberships, pseudo-labels are assigned to these samples which are then
added to the training set with a different weight Ci in the cost function. The
13incorporation of test samples as unlabeled data in the training is introduced
in [64] as ”transductive learning”. A transductive learning for SVMs, in which
the prior knowledge takes the form of the number num+ of positive samples in
the test set, has also been proposed for text classification [26]. In this scheme,
the test samples are assigned a misclassification cost C ∗ which is different of
the one used for the training samples and is further refined for each class as
C ∗ + and C∗ − in accordance with the number num+ to deal with unbalanced
data.
4.2 Kernel methods
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The following presents five methods based on the direct modification of the kernel
function: the jittering kernels, the tangent distance, the Haar-integration
kernels, the kernels between sets and the knowledge-driven kernel design. The
first three methods aim at building invariant kernels k that can provide the
same value for a sample x and its transformed Tx:
k(x, z) = k(Tx, z) (22)
thus leading to the transformation-invariance (14). Besides, the kernels between
sets introduce permutation-invariance into the learning, which is another
form of class-invariance. The last method considers the problem of selecting
the kernel amongst admissible kernels with respect to prior knowledge
on the imbalance of the training set.
4.2.1 Jittering kernels
Jittering kernels were first developed for kernel k-Nearest-Neighbors [10] and
then presented for the incorporation of transformation-invariance into SVMs
[11]. This approach is related to the virtual SV (VSV) method (see Sect. 4.1.1).
Instead of considering an extended training set with all the jittered forms
(translated, rotated...) of the training samples, these forms are considered in
the kernel itself. Using the notation kij = k(xi, xj), the jittered form k J ij of
the kernel kij is computed in two steps [11]:
(1) consider the sample xi and all its jittered forms, and select the one, xq,
closest to xj by minimizing the distance between xq and xj in the space
induced by the kernel: √
kqq − 2kqj + kjj (23)
(2) let k J ij
= kqj
Using such a jittering kernel may provide an output invariant to transformations.
For instance, for a sample x and a sample issued from a translation of
14this sample Tx, the jittering kernel function can yield k(Tx, x) = k(x, x).
The computation of such a kernel can be time consuming. However, for a RBF
kernel, only kqj needs to be considered for the minimization since kqq and kjj
are constants and equal 1. Moreover it is argued that, compared to the VSVM
method, the jittering kernel can still be faster for the training by making use
of kernel caching. Nonetheless, in testing phase, this kernel might be slower
since it requires to repeat the steps (1) and (2) for each new test sample.
4.2.2 Tangent distance
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Another approach to incorporate knowledge of transformation-invariance is via
the distance measurement, not in the space induced by the kernel but in the
input space. In order to do so, one can implement a different distance ρ(x, z)
instead of the Euclidean distance commonly used in radial basis kernels. For
the Gaussian kernel, this yields
(
2
−ρ(x, z)
k(x, z) = exp
2σ2 )
(24)
The use of another distance for transformation-invariance has been extensively
studied in [57] for neural networks under the name of tangent distance (TD)
and originally incorporated in SVMs as TD kernels by [23]. The main idea is
to measure the distance not between the samples x and z but between the sets
Px and Pz. These sets contain all the patterns generated by the transformation
T of the samples x and z that leaves the label unchanged. Thus, the distance
between a sample and its translation can be made so that it equals zero.
As simple image transformations can correspond to highly non-linear transformations
in the input space, the tangent distance uses linear approximations.
Considering the transformation Tx as a combination of nL local and simple
transformations Lαk
of parameter αk in [αmin k
approximated by using the tangent vectors ℓαk
(x) as
, α max
k
], it can be linearly
Tx ≈ x +
nL∑
k=1
αkℓαk
(x) (25)
The tangent distance is thus defined as the minimal Euclidean distance between
the linearly approximated sets of all transformed samples
ρ(Px, Pz) 2 =
min
α1,...,αn ,β1,...,βn L L
s.t. αk ∈ [α min
k
(
x − z +
, α max
k
nL∑
(αkℓαk
k=1
], βk ∈ [β min
k
(x) − βkℓβk
(z))
) 2
, β max
], k = 1, . . .,nL
k
(26)
where βk and ℓβk
(z) correspond to the parameter and the tangent vector for
the kth local transformation of z.
15A similar approach was originally taken in [16] where a joint manifold distance
was defined by minimizing a distance between sets of transformed samples.
When the transformation is approximated by a Taylor expansion, this method
is analogous to the tangent distance method of [57]. In [45], these concepts
are considered under the name of object-to-object distance, where an object
corresponds to the set of all transformed samples Px or Pz. Sample-to-object
distance is also considered, in which case the transformation of only one of
the two samples is allowed. This corresponds to a one-sided tangent distance
which is computed for a sample x and an object Pz by
ρ(x, Pz) 2 ≈ min
β1,...,βn L
(
x − z −
s.t. βk ∈ [β min
k
nL∑
k=1
βkℓβk (z)
) 2
, β max
], k = 1, . . ., nL
k
(27)
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The sample-to-object (or one-sided tangent distance) method can be related to
the jittering kernel method. They both amount to compute the kernel k(x, z)
between the sample x and the closest pattern generated around the center z by
a transformation that does not change the class label. The main difference lies
in the implementation where the sample-to-object distance can be considered
as an analytical form of the one used in jittering kernels, these latter requiring
to test every jittered form of the center. Therefore, the sample-to-object
method can be faster but introduces restrictions on the class of admissible
transformations [45].
Objects can also be coded by local distributions centered at the samples.
In this case they are called soft-objects. One has then to define a similarity
measure between a sample and such an object. Here, tangent vectors can be
used to locally approximate the transformations [45] and lead to the Tangent
Vector Kernels (TVK) introduced in [44].
4.2.3 Haar-integration kernels
Haar-integration has been introduced in [54] for the construction of invariant
features. In a similar approach, Haar-integration has been used to generate
invariant kernels known as Haar-integration kernels (HI-kernels) [24]. Consider
a standard kernel k0 and a transformation group T which contains the
admissible transformations (see [54] for a complete definition). The idea is to
compute the average of the kernel output k0 (Tx, T ′ z) over all pairwise combinations
of the transformed samples (Tx, T ′ z), ∀T, T ′ ∈ T . The HI-kernel k
of k0 with respect to T is thus
∫
k(x, z) =
T
∫
k0 (Tx, T
T
′ z)dTdT ′
(28)
16under the condition of existence and finiteness of the integral (which can be
satisfied for instance by discretization of T ). An interpretation of this kernel
in the feature space F can be given thanks to the following equality [24]
∫
〈
T
∫
Φ(Tx)dT,
T
Φ(T ′ z)dT ′ ∫ ∫
〉 = 〈Φ(Tx) ,Φ(T
T T
′ z)〉 dTdT ′ = k(x, z)
(29)
where the mapping Φ is defined by (10). In other words, averaging over
k0 (Tx, T ′ z) is equivalent to computing the inner product between the averages
Φ(x) and Φ(z) of the sets of transformed samples {Φ(Tx)| T ∈ T }
and {Φ(T ′ z)| T ′ ∈ T }.
4.2.4 Kernels between sets
In [31], a kernel between sets of vectors is proposed. The idea is to classify the
samples defined as sets of d-dimensional vectors xi and now written
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χ = {x1, . . .,xi, . . .,xn} (30)
where n is the size of the set. This representation allows to intrinsically incorporate
invariance to permutations of vectors xi in the set. For instance, in
image recognition, a vector xi = [x, y, γ] T represents a point of the image identified
by its coordinates (x, y) and a gray-level γ. A sample χ is then composed
of all the points corresponding to an image. It is clear that the ordering of the
vectors inside this sample is irrelevant for the image classification. Thus, the
recognition algorithm must include an invariance to permutations of vectors
inside a sample, which is included here in the kernel.
The kernel between two sets χ and χ ′ is defined as the Bhattacharyya’s affinity
between the distributions p and p ′ fitted to the sets χ and χ ′
k(χ, χ ′ ) = k(p, p ′ ) =
∫ √ √
p(x) p ′(x) dx (31)
The approach here is to consider χ and χ ′ as i.i.d. samples from unknown
distributions p and p ′ from a parametric family P. The kernel requires to fit
the distributions p and p ′ to the sets as an intermediate step, which ensures
permutation-invariance [31]. When P is chosen as the family of multivariate
normal distributions N(µ,Σ), p and p ′ are fitted by setting µ and Σ to their
maximum likelihood estimates given by the sample mean and the empirical
covariance matrix.
So far, the distributions are fitted in the input space R d , which might be
limited (for instance, d = 3 for image recognition). However, the method is
extended in [31] with an additional kernel κ : R d × R d → R defined between
the vectors x in order to consider p and p ′ as distributions on the feature
17space induced by κ. In this case, a regularized estimate of the covariance
matrix, involving the computation of eigenvectors in the feature space by
Kernel Principal Component Analysis (KPCA) [52], is used.
Another similar approach, independently developed at the same time, can
be found in [70], where a positive definite kernel is defined over sets of
vectors represented as matrices. Considering two matrices of identical sizes
A = [Φ(a1), . . .,Φ(an)] and B = [Φ(b1), . . .,Φ(bn)], where ai and bi are
vectors of R d and Φ(.) is defined by (10), the proposed kernel between these
matrices with column order invariance is given by
n∏
k(A, B) = cos(θi) (32)
i=1
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where the θi stand for the principal angles in the feature space. In [70], a
method is proposed to compute the principal angles in feature space using
only inner products between columns of the input matrices. This extension
allows to introduce an additional kernel as in the method of [31] in order to
deal with non-linear cases without requiring to explicitly compute the feature
map.
4.2.5 Knowledge-driven kernel selection
All previously described methods involving a modification of the kernel aim
at building invariant kernels. The following method applies to unbalanced
training sets and the cases where prior knowledge indicates that the negative
class includes a wide variety of samples with only a few available for training.
For instance, in face recognition, when the positive class includes the training
images of a particular man, the training set cannot contain all the possible
faces of the other men for the negative class.
In image retrieval, this problem has been tackled in [67] by a knowledgedriven
kernel design (KDKD) procedure. The authors highlight the fact that,
when the training set is small, the data cannot effectively represent the true
distributions of the positive and negative classes, especially the negative one.
Based on this prior knowledge, the kernel is designed so that, in the feature
space, the positive samples are tightly clustered while the negative samples
are pushed away from the positive ones, anywhere, but scattered.
In practice, the kernel k is designed by maximizing the ratio between the
scatter tr(S Φ
) of
np ) of the negative samples (xi ∈ Dn) and the scatter tr(S Φ
p
the positive ones (xi ∈ Dp), with respect to the mean mΦ p of the positive ones
J (k, θ) = tr(SΦ np )
tr(S Φ
p )
(33)
18where the matrices S Φ
np and SΦp are defined by
S Φ
∑
np =
S Φ
p
xi∈Dn
=
∑
xi∈Dp
(
Φ(xi) − m Φ ) (
p Φ(xi) − m Φ ) T
p
(
Φ(xi) − m Φ ) (
p Φ(xi) − m Φ ) T
p
(34)
(35)
where Φ(xi) is defined by (10). The traces of the matrices can be computed
from the kernel function as described in [67] and the criterion has continuous
first and second order derivatives with respect to the kernel parameters as
long as the kernel function has. However, non-linear optimization techniques
are required to solve the problem.
The maximization of the ratio (33) can be used to find the optimal kernel
function amongst a set of admissible kernels, but also to tune the parameters
of a previously chosen kernel.
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4.3 Optimization methods
This Section presents methods that incorporate prior knowledge directly in
the problem formulation: invariant kernels, Semidefinite Programming Machines
(SDPM), Invariant SimpleSVM (ISSVM), Knowledge-Based Linear
Programming (KBLP), Non-Linear Knowledge-Based Linear Programming
(NLKBLP) and π-SVM. Though it may be argued that the first one belongs
to the category of kernel methods, it is derived from a regularization
approach minimizing a composite criterion. It is thus categorized as an optimization
based method. The first three methods of the list aim at incorporating
transformation-invariance, whereas the KBLP method considers classinvariance
in polyhedral regions of the input space. From a method originally
formulated for support vector regression, an extension of KBLP to arbitrary
non-linear domains, NLKBLP, is proposed in Sect. 4.3.5 for classification. The
last method exposed in this review, π-SVM, concerns permutation-invariance
for SVMs that classify sets of elements instead of vectors.
4.3.1 Invariant kernels
Regularization of the cost function has been extensively used for neural networks
[19,4], allowing to incorporate prior knowledge on a property of the
function to estimate (usually the smoothness). But the implicit inclusion of
regularization in SVMs, equivalent to regularization networks [59,14], might
explain why few articles studied the application of regularization techniques
for the incorporation of other forms of prior knowledge into SVMs. However,
in the case of classification, the addition of a term to be minimized in the
19cost function for this purpose has been considered. Nonetheless, it results in
a modification of the kernel rather than a modification of the optimization
problem as exposed in the following.
The authors of [51] incorporated local invariance in the sense of (15) and
proposed invariant kernels. Defining the tangent vectors by
dxi = ∂
∣
∂θ
∣
θ=0
Tθxi (36)
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allows to include the local invariance (15) in the learning of a linear SVM by
minimizing
1
N∑ (
w
N
T (dxi) ) 2
(37)
i=1
and thus making the weight vector w as orthogonal as possible to the tangent
vectors. For the original QP formulation and a linear kernel, the regularized
cost becomes
J(w) = (1 − γ)‖w‖ 2 N∑ (
+ γ w T (dxi) )
(38)
where γ ∈ [0, 1] controls the trade-off between the standard SVM (γ = 0) and
a full enforcement of the orthogonality between the hyperplane and the invariance
directions (γ → 1). Let define Cγ as the square root of the regularized
covariance matrix of the tangent vectors
i=1
(
N∑
Cγ = (1 − γ)I + γ dxidx
i=1
T i
) 1
2
(39)
Then, minimizing the regularized cost (38) under the original constraints (4)
leads to a standard SVM problem [51], yielding the output function
N∑
f(x) = αiyi〈C
i=1
−1
γ xi, C −1
γ x〉 + b (40)
Thus a linear invariant SVM is equivalent to a standard SVM where the input
is first transformed via the linear mapping x ↦→ C −1
γ x.
In order to extend directly the approach to non-linear kernels [7], one would
have to compute the matrix Cγ in the feature space F by
(
N∑
Cγ = (1 − γ)I + γ dΦ(xi)dΦ(xi)
i=1
T
) 1
2
(41)
where Φ(xi) is defined by (10), and use the new kernel
k(xi, x) = 〈C −1
γ Φ(xi), C −1
γ Φ(x)〉 = Φ(xi) T C −2
γ
Φ(x) (42)
20which is impossible to do directly because of the high dimensionality of F
and the implicit nature of Φ. Nonetheless, two methods were proposed in [7]
but still suffer from computational problems when applied to large datasets
or when more than one invariances are considered.
This approach for the incorporation of invariance can be related to the virtual
samples method that simply adds the transformed samples to the training
set (see Section 4.1.1) and some equivalence between the two methods can
be shown [35]. However, the invariant SVM does not only make the class
invariant to the transformation but also the real value of the output (which
can be considered as a class-conditional probability) [7].
4.3.2 Semidefinite Programming Machines (SDPM)
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In [20], another formulation for the large margin classifier is developed for
the incorporation of transformation-invariance. The aim is to find an optimal
separating hyperplane between trajectories rather than between points. In
practice, the trajectories, defined as sets of the type {Tθxi : θ ∈ R}, are based
on training samples xi and a differentiable transformation T of parameter θ
to which the class is known to be invariant. The problem can be solved by
approximating T by a transformation ˜ T polynomial in θ that can be a Taylor
expansion of the form
Tθxi ≈ ˜ r∑
Tθxi =
j=0
θ j
(
1
j!
d j Tθxi
dθ j
)
∣
θ=0
= ˜ X T
i θ (43)
where the (r + 1) × d-dimensional matrix ˜ Xi contains the derivative components
and θ = [1 θ θ 2 . . . θ r ] T . For this type of transformations, the problem
of finding the optimal separating hyperplane between trajectories can be formulated
as
min 1
2 ‖w‖2 , s.t. yiw T T
X˜ i
θ ≤ 0, ∀θ ∈ R, i = 1, . . .,N (44)
For this problem, the authors of [20] propose an equivalent semidefinite program
(SDP) [62], for which efficient algorithms exist. They also show that the
resulting expansion of the optimal weight vector w∗ in terms of ˜ Xi is sparse.
Moreover, not only the examples ˜ Xi are determined but also their corresponding
optimal transformation parameter θ∗ i . Thus, the so-called Semidefinite Programming
Machines (SDPM) extends the idea of virtual samples (Sect. 4.1.1),
since truly virtual samples that are not in the training set are used as SVs.
However, practical issues regarding the application of SDPM to non-linear
classification with kernels remain open.
214.3.3 Invariant SimpleSVM
The authors of [36] propose a general framework for the incorporation of
transformation-invariance into the learning based on a modification of the
problem formulation. For the hard-margin SVM, the problem reads
1
min
g,b 2 ‖g‖2H (45)
s.t. yi (g(Tθxi) + b) ≥ 1, i = 1, . . .,N, θ ∈ Θ
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where the function g lies in the Hilbert space H of admissible functions. This
setting allows to ensure that not only the training sample xi = T0xi, but also
the admissible transformations of this sample Tθxi, are classified as belonging
to class the yi. To solve the problem, the method requires a discretization of
the parameter space Θ based on the assumption that only a finite number of
values for θ will yield support vectors. This is reasonable since for the hardmargin
SVM, only samples that lie exactly on the margin borders are SVs.
In this case, a dual form of the problem with a finite number of Lagrange
multipliers αi(θ) can lead to the solution. However, for the soft-margin case,
any sample lying inside the margin corresponds to a SV. Thus the trajectory of
a transformation that goes through the margin would yield an infinite number
of SVs.
As highlighted in [36], this method can include other methods that incorporate
invariance as follows:
• the virtual samples approach (see Sect. 4.1.1) is recovered after a discretization
of the parameter space Θ;
• the tangent distance based methods (Sect. 4.3.1 and 4.2.2) are recovered by
approximating the transformation Tθxi by a first order polynomial;
• the semidefinite programming machine (SDPM, Sect. 4.3.2) is recovered if
the transformation Tθxi is approximated by a transformation polynomial in
θ, such as a Taylor expansion.
In this framework, the authors proposed an efficient algorithm called Invariant
SimpleSVM (ISSVM) based on the SimpleSVM algorithm developed in [66].
However, this algorithm requires the discretization of the parameter space Θ
for the non-separable case, which is not necessary for SDPM which considers
all θ in R.
4.3.4 Knowledge-Based Linear Programming (KBLP)
The following methods consider the incorporation of prior knowledge into
support vector learning by the addition of constraints to the optimization
problem. In this framework, class-invariance inside polyhedral regions has been
22introduced for linear classification in [17] and was then extended to the nonlinear
case via a reformulation of the kernel in [18]. These two methods are
regrouped under the name Knowledge-Based Linear Programming (KBLP).
The learning machine considered here uses the linear programming formulation
(13). Assuming as prior knowledge that all the points x on a polyhedral
domain P = {x| B+x ≤ d+} are positive samples (y = +1) can be written
as the implication
B+x ≤ d+ ⇒ w T x + b ≥ 1 (46)
with, for a domain of dimension n, B+ ∈ R n×d , x ∈ R d and d+ ∈ R n . The
implication (46) can be transformed for a linear SVM in an equivalent system
of linear inequalities having the solution u ∈ R n [17]
B T +u + w = 0,
d T
+u − b + 1 ≤ 0, u ≥ 0 (47)
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For prior knowledge on negative samples (y = −1), we have
which is equivalent to
B−x ≤ d− ⇒ w T x + b ≤ −1 (48)
B T −u − w = 0,
d T
−u + b + 1 ≤ 0, u ≥ 0 (49)
This result is then extended for SVMs with non-linear kernels by assuming
that x = X T t is a linear combination of the training samples. For the positive
class, the “kernelized” prior knowledge becomes
K(B T
+ , XT )t ≤ d+ ⇒ α T DKt + b ≥ 1 (50)
with the diagonal matrix D = diag(y1, . . .,yi, . . .,yN) and α defined as in
(12), the following system of linear inequalities is equivalent
K(X T , B T
+
)u + KDα = 0, dT u − b + 1 ≤ 0, u ≥ 0 (51)
+
These inequalities can then be easily incorporated to the linear program (13).
With the introduction of N + 1 slack variables z = [z1, . . ., zi, . . ., zN] T and
ζ, this leads to the addition of N + 1 linear constraints (not counting u ≥ 0
23and ζ ≥ 0) as
min
(α,b,ξ≥0,a,u≥0,z,ζ≥0) 1Ta + C1T ξ + µ11T z + µ2ζ
s.t. D(KDα + b1) ≥ 1 − ξ
−a ≤ α ≤ a
(52)
−z ≤
K(X T , B T
+)u + KDα ≤ z
d T
+ u − b + 1 ≤ ζ
where µ1 and µ2 are two trade-off parameters between the training on the
data and the learning of the prior knowledge.
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Similar constraints can be derived for prior knowledge on negative samples
and added to the problem [18]. As prior knowledge on a polyhedral set only
requires the addition of a set of linear constraints, knowledge on many regions
for the two classes can be easily combined and included to the problem.
It must be noticed that the three kernels appearing in (52) could be distinct
kernels and do not need to be positive semidefinite.
4.3.5 Non-Linear Knowledge Based Linear Programming
In the framework of regression and kernel approximation of functions, Mangasarian
and his coworkers proposed a new approach based on a non-linear
formulation of the knowledge [38] to overcome the lacks of the previously described
KBLP method. It is presented here to allow its extension to classification
and thus to class-invariance inside an input domain. The prior knowledge
can now be considered on any non-linear region of the input space and takes
the general form
g(x) ≤ 0 ⇒ f(x) ≥ h(x) (53)
which is equivalent to
f(x) − h(x) + v T g(x) ≥ 0, ∀x (54)
for v ≥ 0 [38]. In this formulation, f, g, h are arbitrary non-linear functions.
Indeed the demonstration of the equivalence requires f, g, −h to be convex but
only for the implication (53) ⇒ (54). The implication (54) ⇒ (53), in which
we are interested, holds for any f, g, h. Nonetheless, the inequality (54) must
be verified ∀x and thus cannot directly be included in the linear program as
a finite set of constraints. To overcome this, a discretization of the knowledge
is performed over a set of points {xp} ⊂ {x|g(x) ≤ 0}, yielding a finite set of
constraints.
24For the classification case, we can set f(x) to the real value of the output
of the classifier (the bracketed expression in (12) without the sign function)
f(x) = K(x, X T )Dα + b and h(x) = +1. Of course, if the region considered
is known to belong to the negative class instead of the positive class, the
setting becomes f(x) = −K(x, X T )Dα − b and h(x) = −1. Associating
slack variables zp to the Np points of discretization {xp} gives the Nonlinear
Knowledge-Based Linear Program (NLKBLP)
min
(α,b,ξ≥0,a,v≥0,zp≥0)
1 T a + C1 T ξ + µ ∑ Np
p
s.t. D(KDα + 1b) ≥ 1 − ξ
−a ≤ α ≤ a
zp
f(xp) − h(xp) + v T g(xp) + zp ≥ 0, p = 1, . . .,Np
(55)
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A difference between this method and the polyhedral method is that the constraints
are applied on an arbitrary discretized domain. With a polyhedral
domain, the constraints hold on the whole domain (without discretization).
It is thus possible to include prior knowledge such as the class-membership
of an arbitrary region into support vector learning. For each region a set of
constraints is added to the linear program. The method is thus only limited
by computational power.
4.3.6 Permutation-invariant SVM (π-SVM)
The paper [56] focuses on the issue of Section 4.2.4, i.e. to build a classifier
separating sets of vectors (here in matrix form) that incorporates permutationinvariance
(here between rows of matrices). But the approach is rather different
from the ones of [31] or [70]. Here, the permutation invariance is not incorporated
in the kernel but in the learning machine itself. To do so, a SVM with
matrix inputs Z i ∈ R m×d instead of vectors xi is considered by defining a
function π : R m×d × R m×d → R as
d∑
d∑ m∑
π(A, B) = 〈Aj, Bj〉 = AjkBjk (56)
j=1
j=1 k=1
and the norm of a matrix as ‖A‖ = √ ∑ dj=1
‖Aj‖ 2 . The training of such a SVM
can still be written as (8) by replacing the vector w by a matrix W in R m×d
and replacing the inner product 〈xi, w〉 in the constraints by π(Z i , W). Similarly,
the class of a sample Z is given by f(Z) = sign(π(Z, W)+b). The margin
M of this classifier can be defined as the minimal value for yi(π(Z i , W)+b)
over the training set, since maximizing M would lead to a better separation
25of the training data.
The main steps of the proposed procedure to train a permutation-invariant
SVM (π-SVM) [56] are as follows:
(1) compute the radius R and centroid of the smallest hypersphere enclosing
all training data;
(2) solve the SVM on the training data composed of matrices;
(3) find the permutation of rows for each training sample that both minimizes
the radius R and maximizes the margin (with a chosen trade-off
parameter);
(4) permute the rows of the training matrices accordingly;
(5) repeat from step 1 until some criterion is met.
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For all these steps, efficient algorithms, of which the description can be found
in [56], exist. The idea here is to permute the rows of training matrices so as
to minimize the bound on the generalization error based on the ratio between
the radius of the data and the margin. The class of a test sample Z is then
determined by f(Z) computed for the permutation of rows of Z that yields
the larger margin.
5 Discussions
This Section starts with some general remarks on the main directions for the
incorporation of prior knowledge into SVM. Then, a technical discussion on
particular aspects of the methods is proposed to highlight the links between
the methods. Notably, a unifying framework from an implementation point of
view is given in Section 5.2.1.
5.1 General issues
In this overview of the methods, the approaches to incorporating prior knowledge
into SVM are classified in the three categories defined in [53]: sample
methods, kernel methods and optimization methods.
Sample methods are often chosen in practice for their simplicity of use. On
the contrary, kernel methods may suffer from computational issues. However,
most of the current research aim at building invariant kernels. One reason is
that these kernels may also be directly applied to other kernel-based classifiers
such as Kernel Principal Component Analysis (KPCA) [52] or Kernel Fisher
Discriminant Analysis (KFDA) [39].
26Also, most of the work focus on one particular type of prior knowledge: classinvariance
to a transformation of the input. This can be explained by the
availability of this type of prior knowledge in many of the pattern recognition
problems such as image recognition and all its various applications. However,
on a smaller scale, other forms of invariances are also studied when, for instance,
considering structured inputs such as matrices. In this setting invariance
to permutations of rows as proposed by [31], [70] or [56] can be crucial for
the problem. The methods proposed by [17], [18] and their extension based on
[38] allow to include some class-invariance knowledge on regions of the input
space, which might be interesting if, for instance, these regions lack training
samples.
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Methods that incorporate knowledge on the data, such as the Weighted-SVM
[72], are also interesting since they allow to include knowledge on the experimental
setup used to provide the training data. From a practical point of view,
it is clear that, in real-world applications, the process of gathering data may
suffer from a weak or variable accuracy. Being able to track this and learn
in accordance is an important issue. As an example, the labeling of images
in handwriting recognition is not always exact and can sometimes differ with
respect to the person who is asked to label a pattern. As highlighted in [60], a
certain amount of errors made by the classifiers are due to images that humans
have difficulty in identifying them because of cursive writing, degradation and
distortion due to the quality of the scanner or the width of the tip of the
writing instrument. These problems are present and identifiable in most of
the common databases (see [60] for an analysis on the MNIST, CENPARMI,
USPS and NIST SD 19 databases) and should be taken into account to be
able to improve the performances of the recognition systems.
5.2 Technical discussions
In the following, a regularization framework is used to regroup both the sample
and optimization methods. A comparison of the invariant kernel methods is
then proposed to show the different spaces in which the distance measures are
considered by the different methods. The end of this Section is dedicated to
the perspective of combining the methods.
5.2.1 Sample and constrained methods in a regularization framework
Though stemming from different approaches, many of the presented methods
can be seen in a regularization framework from the way they are implemented.
Actually both sample and constrained methods amount to minimize a com-
27posite criterion JC with additional constraints
min JC = JSV M + JPK (57)
s.t. cSV M ≤ 0
cPK ≤ 0
where JSV M and cSV M correspond to the usual SVM criterion and constraints
(8), whereas JPK and cPK are added to the problem to include the prior
knowledge. Their setting for the different methods are derived in the following
and summarized in Table 1.
All virtual sample methods (Sect. 4.1.1) basically add new samples to the
training set, which simply corresponds to augmenting the size of the training
set N by the number of virtual samples NV in the optimization problem (8).
Separating the training samples xi from the virtual samples ˜xi in the writing
yields the setting of Table 1 for (57).
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Though originally developed for different purposes and from different points
of view, all the methods based on weighting (Sect. 4.1.2) can be written as a
standard SVM problem with the parameter C set to a different value Ci for
each sample. For asymmetric margins, Ci can only take two values C + and C −
depending on the class of the sample xi: positive (for i in P = {i : yi = +1})
or negative (for i in N = {i : yi = −1}). Considering C as an average
weight allows to write the problem as in (57) with the settings of Table 1 for
asymmetric margin methods and the Weighted-SVM. Besides, the method of
[68] for the incorporation of unlabeled samples can be seen as a mixture of
virtual samples and weighting by considering the NU unlabeled samples as
extra samples ˜xi weighted by Ci.
The KBLP and NLKBLP problems (Sect. 4.3.4 and 4.3.5) are directly formulated
as in (57), except for JSV M and cSV M, which correspond now to the
criterion and constraints used by the linear programming form of SVM (13).
Because of the required discretization of the domain of knowledge, the NLK-
BLP method can be seen as adding virtual samples to the training set. In this
case, the samples are not generated by transformations of other samples as
usual, but chosen in the input space by discretization of a region that might
actually contain no sample. An interesting point is that though basically corresponding
to the same problem from an optimization viewpoint, the VSVM
(Sect. 4.1.1) and NLKBLP methods are radically different. The first one aims
at improving the boundary between two regions with training samples that
are close but that belong to different classes. The second one becomes of particular
interest when enhancing the decision function in a region of input space
that lacks training samples but on which prior knowledge is available.
The methods of Section 4.3.1 for the incorporation of invariances conform
obviously to this framework since they are initially formulated as the mini-
28hal-00021555, version 2 - 24 Apr 2007
Table 1
Setting for the regularization framework with respect to the different methods.
Method JPK cPK ≤ 0
Virtual
samples
Asymmetric
margin
WSVM
Unlabeled
samples
KBLP µ1
C
NV
∑
i=1
˜ξi
(C + − C) ∑
i∈P
+(C − − C) ∑
i∈N
ξi
N∑
(Ci − C)ξi
i=1
NU
∑
i=1
N∑
i=1
Ci ˜ ξi
ξi
˜yi(〈˜xi,w〉+b) ≥ 1− ˜ ξi, i = 1,... ,NV
˜yi(〈˜xi,w〉 + b) ≥ 1 − ˜ ξi
zi + µ2ζ −z ≤ K(X T ,B T )u + KDα ≤ z,
d T u − b + 1 ≤ ζ
Np
∑
NLKBLP µ zp f(xp) − h(xp) + v
p
Tg(xp) + zp ≥ 0,
p = 1,... ,Np
mization of a composite criterion. However, as they correspond in practice to
a modification of the kernel, they do not appear in Table 1, which focuses on
the practical implementation of the methods.
5.2.2 Kernel methods: invariance via distance measure, but in which space?
Most of the kernel methods presented in Section 4.2 implement transformation
invariance by modifying the computation of the distance between the
patterns. These methods are summarized in Table 2. It can be noticed that
while the jittering kernel minimizes a distance in the feature space F, the
tangent distance (TD) and the sample-to-object or object-to-object distances
are considered in the input space. On the other hand, the Haar-integration
kernel does not look for a minimal distance, but rather computes the distance
in F between averages over transformed samples.
5.3 Combinations of methods
Combining methods from different categories with respect to our classification
seems possible since they act on different parts of the problem. For example,
nothing prevents us from using a jittering kernel on an extended training set
with prior knowledge on polyhedral regions of input space. But this does not
29Table 2
Invariant kernel functions. Z contains all the jittered forms of z.
Method kernel function specificity
Jittering kernels k(x,z) = k(x, ˆz) ˆz = arg min‖Φ(x) − Φ(z)‖F
z∈Z
One-sided TD
(sample-to-object) k(x,z) = k(x, ˆz) ˆz = arg min‖x − z‖
z∈Pz
Two-sided TD
(object-to-object) k(x,z) = k(ˆx, ˆz) (ˆx, ˆz) = arg min ‖x − z‖
(x,z)∈Px×Pz
Haar-integration k(x,z) = k0(x,z) k0(x,z) = 〈Φ(z),Φ(x)〉
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exclude combinations of methods of the same category. Actually all the methods
of Table 1 can be combined since they all amount to the addition of a
term in the criterion and optionally some constraints. Such combinations become
interesting when two methods are used for different purposes. Consider
the example of character recognition for which we have seen that generating
virtual samples by a known transformation such as translation helps the classifier
to be invariant to translations. Also, random transformations such as
elastic distortions lead to even better results [58] by incorporating invariance
to small random variations of the image. But elastic distortion may sometimes
yield very distorted images that become unrecognizable [32] and can mislead
the classifier. This drawback could be diminished by combining elastic distortions
with a Weighted-SVM and associating a smaller confidence to the virtual
samples than to the original training ones. Thus, the combination of virtual
samples and weighting of samples can be used to incorporate invariance to an
approximately known transformation. The weights Ci associated to the virtual
samples are then a confidence measure of the transformation that generated
these samples.
However, the combination of methods leads to an increase of the algorithm
complexity. The methods must be chosen with care by looking at their complementarity
in order to yield a respectable improvment.
6 Conclusion
The fundamentals of the Support Vector Machines (SVMs) for classification
have been presented together with the different formulations of the optimization
problem resulting from the training of such machines. A review of the literature
concerning the incorporation of prior knowledge into SVMs has been
exposed. The methods are classified, with respect to the categorization of [53],
30in three categories depending on the implementation approach (via samples, in
the kernel or in the problem formulation). Two main types of prior knowledge
that can be included by these methods have been considered: class-invariance
and knowledge on the data. Most of the work in this field has been focused
so far on transformation-invariance, either via the kernel function or via extended
training sets. Nonetheless, a recent approach considers prior knowledge
on a polyhedral domain of the input space for which the class is known. It
has been extended here for arbitrary regions and results in the addition of
linear constraints to the optimization problem, thus providing an easy mean
to code knowledge on multiple regions for the two classes. Finally, a regularization
framework has been used to regroup both the sample and constrained
methods from an implementation point of of view.
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Being able to include expert knowledge in the learning will be a key element
in the future for the increase of classifiers performance on benchmark datasets
and practical applications. Further research might explore other forms of prior
knowledge together with optimized algorithms for their implementations. Also,
the combination of different types of knowledge might be explored for practical
applications. In addition, an interesting problem not covered in this review,
though it may be regarded as the inclusion of prior knowledge, is the classification
of structured inputs such as graphs or strings. Regarding this issue,
the reader can refer to the book [55] and the references therein that provide a
framework for building kernels for structured data.
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