nips nips2000 nips2000-54 knowledge-graph by maker-knowledge-mining
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Author: Jason Weston, Sayan Mukherjee, Olivier Chapelle, Massimiliano Pontil, Tomaso Poggio, Vladimir Vapnik
Abstract: We introduce a method of feature selection for Support Vector Machines. The method is based upon finding those features which minimize bounds on the leave-one-out error. This search can be efficiently performed via gradient descent. The resulting algorithms are shown to be superior to some standard feature selection algorithms on both toy data and real-life problems of face recognition, pedestrian detection and analyzing DNA micro array data.
Reference: text
sentIndex sentText sentNum sentScore
1 Abstract We introduce a method of feature selection for Support Vector Machines. [sent-12, score-0.655]
2 The method is based upon finding those features which minimize bounds on the leave-one-out error. [sent-13, score-0.292]
3 This search can be efficiently performed via gradient descent. [sent-14, score-0.071]
4 The resulting algorithms are shown to be superior to some standard feature selection algorithms on both toy data and real-life problems of face recognition, pedestrian detection and analyzing DNA micro array data. [sent-15, score-0.833]
5 1 Introduction In many supervised learning problems feature selection is important for a variety of reasons: generalization performance, running time requirements, and constraints and interpretational issues imposed by the problem itself. [sent-16, score-0.856]
6 In classification problems we are given f data points Xi E ~n labeled Y E ±1 drawn i. [sent-17, score-0.047]
7 We would like to select a subset of features while preserving or improving the discriminative ability of a classifier. [sent-20, score-0.167]
8 As a brute force search of all possible features is a combinatorial problem one needs to take into account both the quality of solution and the computational expense of any given algorithm. [sent-21, score-0.319]
9 Support vector machines (SVMs) have been extensively used as a classification tool with a great deal of success from object recognition [5, 11] to classification of cancer morphologies [10] and a variety of other areas, see e. [sent-22, score-0.362]
10 In this article we introduce feature selection algorithms for SVMs. [sent-24, score-0.797]
11 The methods are based on minimizing generalization bounds via gradient descent and are feasible to compute. [sent-25, score-0.376]
12 g object recognition) and one can perform feature discovery (e. [sent-27, score-0.271]
13 We also show how SVMs can perform badly in the situation of many irrelevant features, a problem which is remedied by using our feature selection approach. [sent-29, score-0.73]
14 In section 2 we describe the feature selection problem, in section 3 we review SVMs and some of their generalization bounds and in section 4 we introduce the new SVM feature selection method. [sent-31, score-1.43]
15 Section 5 then describes results on toy and real life data indicating the usefulness of our approach. [sent-32, score-0.135]
16 2 The Feature Selection problem The feature selection problem can be addressed in the following two ways: (1) given a fixed m « n , find the m features that give the smallest expected generalization error; or (2) given a maximum allowable generalization error "(, find the smallest m. [sent-33, score-1.214]
17 In both of these problems the expected generalization error is of course unknown, and thus must be estimated. [sent-34, score-0.216]
18 Note that choices of m in problem (1) can usually can be reparameterized as choices of"( in problem (2). [sent-36, score-0.268]
19 In the literature one distinguishes between two types of method to solve this problem: the so-called filter and wrapper methods [2]. [sent-44, score-0.515]
20 Filter methods are defined as a preprocessing step to induction that can remove irrelevant attributes before induction occurs, and thus wish to be valid for any set of functions f(x, a). [sent-45, score-0.714]
21 For example one popular filter method is to use Pearson correlation coefficients. [sent-46, score-0.17]
22 The wrapper method, on the other hand, is defined as a search through the space of feature subsets using the estimated accuracy from an induction algorithm as a measure of goodness of a particular feature subset. [sent-47, score-1.093]
23 Thus, one approximates T(O', a) by minimizing Twrap(O', a) = min Talg(O') (2) IT subject to 0' E {a, l}n where Talg is a learning algorithm trained on data preprocessed with fixed 0'. [sent-48, score-0.192]
24 In this article we introduce a feature selection algorithm for SVMs that takes advantage of the performance increase of wrapper methods whilst avoiding their computational complexity. [sent-50, score-1.192]
25 Note, some previous work on feature selection for SVMs does exist, however results have been limited to linear kernels [3, 7] or linear probabilistic models [8]. [sent-51, score-0.532]
26 In order to describe this algorithm, we first review the SVM method and some of its properties. [sent-53, score-0.099]
27 3 Support Vector Learning Support Vector Machines [13] realize the following idea: they map x E IRn into a high (possibly infinite) dimensional space and construct an optimal hyperplane in this space. [sent-54, score-0.13]
28 (3) The optimal hyperplane is the one with the maximal distance (in 1l space) to the closest image ~(Xi) from the training data (called the maximal margin). [sent-62, score-0.35]
29 For the non-separable case one can quadratically penalize errors with the modified kernel K +- K + I where I is the identity matrix and A a constant penalizing the training errors (see [4] for reasons for this choice). [sent-67, score-0.467]
30 t Suppose that the size of the maximal margin is M and the images (Xl), . [sent-68, score-0.257]
31 , (Xl) of the training vectors are within a sphere of radius R. [sent-71, score-0.043]
32 Theorem 1 lfimages of training data of size £ belonging to a . [sent-73, score-0.106]
33 Iphere of size R are separable with the corresponding margin M, then the expectation of the error probability has the bound 1 1 O} EPerr ~ £E {R2} = £E { R 2 W 2 (0:) , M2 (5) where expectation is taken over sets of training data of size £. [sent-74, score-0.459]
34 This theorem justifies the idea that the performance depends on the ratio E{ R2 / M2} and not simply on the large margin M, where R is controlled by the mapping function <1>(. [sent-75, score-0.216]
35 Other bounds also exist, in particular Vapnik and Chapelle [4] derived an estimate using the concept of the span of support vectors. [sent-77, score-0.281]
36 Theorem 2 Under the assumption that the set of support vectors does not change when removing the example p Ep l - err 1 < ! [sent-78, score-0.184]
37 E ~ \II ( (K- 1 ) o:~ - £ ~ p=1 sv pp -1) (6) where \II is the step function , Ksv is the matrix of dot products between support vectors, p~;:-; is the probability of test error for the machine trained on a sample of size £ - 1 and the expectations are taken over the random choice of the sample. [sent-79, score-0.316]
38 4 Feature Selection for SVMs In the problem of feature selection we wish to minimize equation (1) over u and 0:. [sent-80, score-0.75]
39 The support vector method attempts to find the function from the set f(x, w, b) = w . [sent-81, score-0.213]
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