An Efficient Alternative to SVM Based
Recursive Feature Elimination with Applications
in Natural Language Processing and
Bioinformatics
Justin Bedo1,2, Conrad Sanderson1,2, and Adam Kowalczyk1,2,3
1 Australian National University, ACT 0200, Australia
2 National ICT Australia (NICTA), Locked Bag 8001, ACT 2601, Australia
3 Dept. Electrical & Electronic Eng., University of Melbourne, VIC 3010, Australia
Abstract. The SVM based Recursive Feature Elimination (RFE-SVM)
algorithm is a popular technique for feature selection, used in natural
language processing and bioinformatics. Recently it was demonstrated
that a small regularisation constant C can considerably improve the per-
formance of RFE-SVM on microarray datasets. In this paper we show
that further improvements are possible if the explicitly computable limit
C → 0 is used. We prove that in this limit most forms of SVM and ridge
regression classifiers scaled by the factor 1C converge to a centroid clas-
sifier. As this classifier can be used directly for feature ranking, in the
limit we can avoid the computationally demanding recursion and convex
optimisation in RFE-SVM. Comparisons on two text based author verifi-
cation tasks and on three genomic microarray classification tasks indicate
that this straightforward method can surprisingly obtain comparable (at
times superior) performance and is about an order of magnitude faster.
1 Introduction
The Support Vector Machine based Recursive Feature Elimination (RFE-SVM)
approach [1] is a popular technique for feature selection and subsequent classi-
fication, especially in the bioinformatics area. At each iteration a linear SVM is
trained, followed by removing one or more “bad” features from further considera-
tion. The goodness of the features is determined by the absolute value of the cor-
responding weights used in the SVM. The features remaining after a number of
iterations are deemed to be the most useful for discrimination, and can be used to
provide insights into the given data. A similar feature selection strategy was used
in the author unmasking approach, proposed for the task of authorship verifica-
tion [2] (a sub-area within the natural language processing field). However, rather
than removing the worst features, the best features were iteratively dropped.
Recently it has been observed experimentally on two microarray datasets that
using very low values for the regularisation constant C can improve the perfor-
mance of RFE-SVM [3]. In this paper we take the next step and rather than pursu-
ing the elaborate scheme of recursively generating SVMs, we use the limit C → 0.
We show that this limit can be explicitly calculated and results in a centroid based
classifier. Furthermore, unlike RFE-SVM, in this limit the removal of one or more
A. Sattar and B.H. Kang (Eds.): AI 2006, LNAI 4304, pp. 170–180, 2006.
c
© Springer-Verlag Berlin Heidelberg 2006

An Efficient Alternative to RFE-SVM 171
features does not affect the solution of the classifier for the remaining features.
The need for multiple recursion is hence obviated, resulting in considerable com-
putational savings.
This paper is structured as follows. In Section 2 we calculate the limit C → 0.
In Sections 3 and 4 we provide empirical evaluations on two authorship attri-
bution tasks and on three bioinformatics tasks, respectively, showing that the
centroid based approach can obtain comparable (at times superior) performance.
A concluding discussion is given in Section 5.
2 Theory
Empirical risk minimisation is a popular machine learning technique in which
one optimises the performance of an estimator (often called a predictor or hy-
pothesis) on a “training set” subject to constraints on the hypothesis class. The
predictor f , which is constrained to belong to a Hilbert space H, is selected as the
minimiser of a regularised risk which is the sum of two terms: a regulariser that
is typically the squared norm of the hypothesis, and a loss function capturing
its error on the training set, weighted by a factor C > 0.
The well studied limit of C → ∞ results in a hard margin Support Vector
Machine (SVM). In this paper, we focus on the other extreme, the limit when
C → 0. This is a heavy regularisation scheme as the regularised risk is dominated
by the regulariser. We formally prove that if the classifier is scaled by the factor
1/C, then it converges in most cases to a well defined and explicitly calculable
limit: the centroid classifier.
We start with an introduction of the notation necessary to precisely state our
formal results. Given a training set Z = {(xi, yi)}i=1,...,m ∈ Rn × {±1} a Mercer
kernel k : Rn × Rn → R with the associated reproducing kernel Hilbert space H,
and with a feature map Φ : Rn → H, x → k(x, .) [4,5], the regularised risk on the
training set is defined as
Rreg[f, b] := ‖f‖2H +

y=±1
Cy

i,yi=y
l(1 − yi(f(xi) + b)) (1)
for any (f, b) ∈ H × R. Here C+1, C−1 ≥ 0 are two class dependent regularisation
constants. These constants can be always written in the form
Cy = C
1 + yB
2my
, (2)
where C > 0, 0 ≤ B ≤ 1 and my := |{i ; yi = y}| is the number of instances with
label y = ±1. Further, l(ξ) is the loss function of one of the following three forms:
l(ξ) := ξ2 for the ridge regression (RRegr), the hinge loss l(ξ) := max(0, ξ) for
the support vector machine (SVM) with the linear loss, and l(ξ) := max(0, ξ)2
for the SVM2 with quadratic loss [4,5,6].
We are concerned with kernel machines defined as
f := fH := arg min
f∈H
Rreg[f, 0], (3)
in the homogeneous case (no bias b), or, in general, as the sum
f = fH + b, (fH , b) := arg min
(f,b)∈H×R
Rreg[f, b]. (4)

172 J. Bedo, C. Sanderson, and A. Kowalczyk
2.1 Centroid Based Classification and Feature Selection
The centroid classifier, is defined for every −1 ≤ B ≤ 1 as
gB(x) =
1 + B
2

i,yi=+1
k(x, xi)
m+1
− 1 − B
2

i,yi=−1
k(x, xi)
m−1
+ b, (5)
where b is a bias term. In terms of the feature space, it is the projection onto
the vector connecting (weighted) arithmetic means of samples from both class
labels, degenerating to the mean of a single class for B = ±1, respectively. For a
suitable kernel k it implements the difference between Parzen window estimates
of probability densities for both labels. For the linear kernel it simplifies to
gB(x) = wB · x + b :=

1 + B
2
x¯+1 −
1 − B
2
x¯−1

· x + b (6)
where x¯y is the centroid of the samples with label y = ±1 and wB ∈ Rn is the cen-
troid weight vector. In this work we have used B = 0 and b = wB · (x¯+1 + x¯−1)/2,
such that the decision hyperplane is exactly halfway between x¯+1 and x¯−1.
The weight vector can be used for feature selection in the same manner as in
RFE-SVM, i.e. the features are selected based on a ranking given by ri = |w(i)B |,
where w(i)B is i-th dimension of wB. However, unlike RFE-SVM, the removal of a
feature does not affect the solution for the remaining features, thus the need for
recursion is obviated. This form of combined feature selection and subsequent
classification will be referred to as the centroid approach.
For the case of the linear kernel, we note that the centroid classifier can be
interpreted as a form of Linear Discriminant Analysis (LDA) [7], where the two
classes share a diagonal covariance matrix with equal diagonal entries.
2.2 Formal Results
In the following theorem we prove that using either form of quadratic loss spec-
ified above, the machine converges to the centroid classifier as C → 0; the case
of linear loss is covered by Theorem 2.
Theorem 1. Let |B| < 1, the loss be either l(ξ) = ξ2 or l(ξ) = max(0, ξ)2 and
fC := fH,C + bC and (fH,C , bC) := arg min
f,b
Rreg[f, b]
for every C > 0. Then limC→0+ | fCC | = ∞ if B = 0 but limC→0+
fH,C
C =
1−B2
2 g0.
Thus although the whole predictor fC scaled by C−1 diverges with C → 0, its
homogeneous part fH,C converges to an explicitly calculable limit.
Proof outline. The kernel trick reduces the proof to the linear kernel k(x,x′) :=
x · x′ on Rn and H isomorphic to Rn. In such a case f(x) = fH(x) + b = x · w + b,
where w ∈ Rn and ‖fH‖2H = ‖w‖2. The regularised risk (1-2) takes the form
Rreg[w, b] = ‖w‖2 + C

i
yi + B
2myi
l(1 − yi(w · xi + b))

An Efficient Alternative to RFE-SVM 173
and fC(x) = wC · x + bC for x ∈ Rn, where (wC , bC) = arg minw,b Rreg[w, b]. This
implies
‖wC‖2 ≤ min
w,b
Rreg[w, b] ≤ Rreg[0, 0] = C (7)
and due to continuous differentiability of the loss functions in our case:
∂Rreg
∂b
[wC , bC ] = 0 and
∂Rreg
∂w
[wC , bC ] = 0. (8)
Solving the first of these equations we obtain
bC = B −

i∈SV
1 + yiB
2myi
wC · xi
where SV := {i : 1 − yixiwC − yibC > 0} is the set of the support vectors in
the case of the hinge loss and the whole training set in the case of regression.
Furthermore,





i∈SV
1 + yiB
2myi
wC · xi




≤ ‖wC‖max
i
‖xi‖ ≤
√
C max
i
‖xi‖,
by (7), hence, denoting by O(ξ) a term such that |O(ξ)/ξ| is bounded on for ξ > 0,
we get
bC = B + O(
√
C). (9)
Case l(ξ) = ξ2. The first equation in (8) takes the form
2wC − 2C

i
yi + B
2myi
(1 − yi(wC · xi + bC))xi = 0
which upon consideration of (7) and (9) implies
fH,C(x)
C
=

i
yi
(1 + yiB)(1 − yiB + O(
√
C))
2myi
xi · x =
1 − B2
2
g0(x) + O(
√
C),
fC(x)
C
=
fH,C(x) + bC
C
=
1 − B2
2
g0(x) + O(
√
C) +
B + O(
√
C)
C
.
This immediately proves the current case of the theorem.
Case l(ξ) = max(0, ξ)2. This case reduces to the previous one, once we
observe that every training sample becomes a support vector for sufficiently
small C. Indeed, Eqns. (7) and (9) imply
1 − yiwC · xi − yibC = 1 − yiB + O(
√
C) > 0
if C is sufficiently small, since |B| < 1. 
Note that the above result does not cover the most popular case of SVM, namely
the linear loss l(ξ) := max(0, ξ). The example in Figure 1 shows that in such a
case the limit depends on the particular data configuration in a complicated way
and cannot be simply expressed in terms of the centroid classifier.
The extension of Theorem 1 to the case of homogeneous machines follows.
A proof, similar to the above one, is not included.

174 J. Bedo, C. Sanderson, and A. Kowalczyk
+
+
wc
wB

2

1
-
wB
+
+
wc
<
/2
2

1 ,
-

2

1
(a) (b)
1
2
3
1
2
3
<
/2
2

1 ,
-
<
/2
1
< 
2(c)
+
+
wc
wB
1
2
3 1
2
-
<
/2
2
< 
1(d)
+
+
wc
wB
1
2
3 1
2
Fig. 1. Four subsets of R2 that share the same centroid classifier (the blue line), but
which have different vectors wC (the red line) for the SVM1 solutions x → wC ·x+ bC
Theorem 2. Let |B| ≤ 1, the loss be either l(ξ) = ξ2 or l(ξ) = max(0, ξ)p for p = 1, 2
and fC := arg minf Rreg[f, 0] for every C > 0. Then limC→0+ fC/C = gB(x).
Using the kernel k(x,x′) + 1 rather than k(x,x′) in the last theorem we obtain:
Corollary 1. Let fC := fH,C + bC for every C > 0, where (fH,C , bC) := arg minf,b
Rreg[f, 0] + b2. Then limC→0+ fC/C = gB(x) + B.
Note the differences between the above of results. Theorem 2 and Corollary 1
hold for B = ±1, which is virtually the case of one-class-SVM, while in Theorem 1
this is excluded. The limit limC→0 fC/C exists in the case of Corollary 1 but does
not exist in the case of Theorem 1. Finally, in Theorem 1 the convergence is to
a scaled version of the balanced centroid g0 for all |B| < 1, while in the latter
two results to gB or gB + B, respectively.
3 Authorship Verification Experiments
Recently an RFE-SVM based technique was proposed for the task of authorship
verification [2], a particular problem that spans the fields of natural language
processing and humanities [8,9]. Given a reference text from a purported author
and a text of unknown authorship, an author unmasking curve is built as follows.
Both texts are divided into chunks, with each chunk represented by counts of pre-
selected words. The chunks are partitioned into training and test sets. Each point
in the author unmasking curve is the accuracy of discriminating (using a linear
SVM) between the test chunks from the two texts. At the end of each iteration
several of the most discriminative words are removed from further consideration.
A vector representing the essential features of the curve is then classified as
representing either the “same author” or a “different author” [2].
The underlying hypothesis is that if the two given texts have been written by
the same author, the differences between them will be reflected in a relatively

An Efficient Alternative to RFE-SVM 175
0 1 2 3 4 5 6 7 8 9
60
65
70
75
80
85
90
95
100
ITERATION
A
CC
UR
AC
Y
(%
)


WILDE vs WILDE (CENT)
WILDE vs WILDE (SVM)
WILDE vs SHAW (CENT)
WILDE vs SHAW (SVM)
WILDE vs HAWTHORNE (CENT)
WILDE vs HAWTHORNE (SVM)
Fig. 2. Demonstration of the unmask-
ing effect for Wilde’s An Ideal Hus-
band using Wilde’s Woman of No Im-
portance as well as the works of other
authors as reference texts.
Table 1. Classification accuracies for same-
author (sa) and different-author (da) curves
on the Gutenberg dataset, using centroid and
rfe-svm (r/s) based unmasking. The second
term in the method indicates the secondary
classifier type.
Method SA Acc. DA Acc. Time
Cent. + Cent. 92.3% 95.4% 14 mins
Cent. + svm 92.3% 95.2% 14 mins
r/s + Cent. 69.2% 97.2% 124 mins
r/s + svm 92.3% 94.9% 124 mins
Table 2. As per Table 1, but for the
Columnists dataset
Method SA Acc. DA Acc. Time
Cent. + Cent. 82% 89.6% 99 mins
Cent. + svm 86% 90.3% 104 mins
r/s + Cent. 84% 88% 709 mins
r/s + svm 84% 88.5% 714 mins
small number of features. In [2] it was observed that for texts authored by the
same person, the extent of the accuracy degradation is much larger than for
texts written by different authors.
We used two datasets in our experiments: Gutenberg and Columnists. The
former is comprised of 21 books1 from 10 authors (as used in [2]), while the
latter (used in [10]) is comprised of 50 journalists (each covering several topics)
with two texts per journalist; each text has approximately 5000 words.
The setup of experiments was similar to that in [2], with the main difference
being that books by the same author were not combined into one large text. For
the Gutenberg dataset we used chunks with a size of 500 words, 10 iterations of
removing 6 features, and using 250 words with the highest average frequency in
both texts as the set of pre-selected words. For each pair of texts, 10 fold cross-
validation was used to obtain 10 curves, which were then represented by a mean
curve prior to further processing. For the Columnist dataset we used a chunk size
of 200 words and 100 pre-selected words (based on preliminary experiments).
A leave-one-text-out approach was used, where the remaining texts were used
for generation of same-author (SA) and different-author (DA) curves, which in
turn were used for training a secondary classifier. The left-out text was then
unmasked against the remaining texts and the resulting curves classified. For
the Gutenberg dataset, there was a total of 24 SA and 394 DA classifications,
while for Columnists there were 100 SA and 9800 DA classifications. Both the
SVM and the centroid were also used as secondary classifiers.
1 Available from Project Gutenberg (www.gutenberg.org).

176 J. Bedo, C. Sanderson, and A. Kowalczyk
As can be observed in Figure 2 (which closely resembles Fig. 2 in [2]) the
unmasking effect is also well present for the centroid based approach. Tables 1
and 2 indicate that the RFE-SVM and the centroid based unmasking approaches
are comparable in terms of performance, but differ greatly in terms of wall clock
time2. The centroid based approach is about seven to nine times faster even if
we neglect the significant time required for searching for the optimal C for SVM.
4 Experiments with Classification of Microarray Datasets
In this section we perform experiments on three publicly available microarray
datasets: colon cancer [1,3,11], lymphoma3 cancer [3,12,13], and Cancer of Un-
known Primary (CUP) [14]. All these datasets are characterised by a relatively
small number of high dimensional samples. The colon dataset contains 62 sam-
ples (22 normal and 40 cancerous), with each sample containing the expression
values for 2000 genes. The lymphoma dataset contains three subtypes, with a to-
tal of 62 samples. Each sample contains measurements of 4026 expressed genes.
We used a subset of the CUP dataset, containing samples for 226 primary and
metastatic tumours, representing 12 tumour types4, with each sample contain-
ing expressions for approximately 10500 genes measured on a cDNA microarray
platform. The number of samples per class widely varied, form 8 to 50.
In addition to comparing the performance of the RFE-SVM and centroid ap-
proaches, we also evaluated the multi-class Shrunken Centroid (SC) approach [15],
proposed specifically for processing microarray datasets. Briefly, in SC the class
centroids are shrunk towards the overall centroid, involving a form of t-statistic
which standardises each feature for each class by its within-class standard devi-
ation. The degree of shrinking for all features is controlled by single tunable pa-
rameter. A feature is not used for prediction of a class when it has been shrunk
“past” the overall centroid. We note that one of the main differences between SC
and the centroid approach (described in Section 2.1) is that in the latter standard
deviation estimates are not used.
For classification of multiclass datasets (lymphoma and CUP) with the cen-
troid and SVM approaches we have used the popular one-vs-all architecture [16].
Feature selection for these datasets was done follows. Each class had its own rank-
ing of features, of which the top T were selected. The final set of features was the
union of the selected features from all classes. Note that since the class-specific
feature subsets can intersect, the final set can have less than KT features, where
K is the number of classes.
The experiment setup for the colon and lymphoma datasets followed [3]. Test-
ing was based on 50 random splits of data into disjoint training and test sets.
In each split approximately 66% of the samples from each class were assigned to
2 Using LibSVM 2.71 (www.csie.ntu.edu.tw/˜cjlin/libsvm) on a Pentium 4, 3.2 GHz.
3 Available from http://llmpp.nih.gov/lymphoma/data.shtml
4 The full CUP dataset contains 13 tumour types, however we have omitted the small-
est class (‘testicular’) which was represented by only three samples.

An Efficient Alternative to RFE-SVM 177
2 5 10 20 50 100 200 500 2000
15
20
25
30
N. Features
Er
ro
r R
at
e
(%
)
Centroid min error = 11.43%
RFE−SVM C=.01 min error = 14.76%
RFE−SVM C=1000 min error = 17.14%
Shrunken Centroid min error = 13.14%
Fig. 3. Results for the colon dataset
5 10 20 50 100 200 500 2000 5000
0
10
20
30
40
N. Features
Er
ro
r R
at
e
(%
)
Centroid min error = 0%
RFE−SVM C=1000 min error = 2.19%
RFE−SVM C=.01 min error = 1.81%
Shrunken Centroid min error = 0.76%
Fig. 4. Results for the lymphoma dataset
the training set, with the remainder assigned to the test set. Results are shown5
in Figures 3 and 4.
As the value of C decreases, the performance of RFE-SVM improves, achieving
a minimum error rate of 14.76% and 1.82% with C = 0.01 on the colon and
lymphoma datasets, respectively. This agrees with results in [3]. When using the
C → 0 limit (i.e. the centroid approach), the minimum error rate drops to 11.43%
and 0% on the colon and lymphoma datasets, respectively. The SC approach
obtained a comparable minimum of 13.14% and 0.76% on the respective datasets.
However, on the lymphoma dataset its error rate increased rapidly when fewer
features were used. Specifically, the centroid approach selects about 200 genes
that achieve near perfect discrimination, while the SC approach requires the full
set of 4026 genes to achieve its minimum error rate.
The CUP dataset was compiled for building a clinical test for determination
of the origin of a cancer from the tissue of a secondary tumour. In [14] it was
demonstrated (for a subset of five cancers), that in order for CUP classification
to be practical for use in pathology tests, it is necessary to select a relatively
small subset of marker genes which are then used to build a considerably cheaper
and also more robust test, i.e., via a Polymerase Chain Reaction based micro-
fluidic card, where the maximum number of gene probes is 384 [14]. Results in
Figure 5 indicate that the centroid method outperforms both RFE-SVM and SC
in the practically important region of a few hundred genes.
Finally, wall clock timing results shown in Table 3 indicate that the cen-
troid approach is between four to 17 times faster than RFE-SVM. For for the
two-class dataset (colon), the SC and centroid approaches have comparable time
requirements. For multi-class datasets (lymphoma and CUP), the SC approach
is roughly twice as fast as the centroid approach, due to the use of the one-vs-all
architecture for the latter.
5 We have evaluated a large set of C values for RFE-SVM, however for purposes of
clarity only a subset of the results is shown in Figures 3, 4 and 5.

178 J. Bedo, C. Sanderson, and A. Kowalczyk
20 50 100 200 500 1000 2000 5000
10
20
30
40
50
N. Features
Er
ro
r R
at
e
(%
)
Centroid
RFE−SVM C=.01
Shrunken Centroid
Fig. 5. Results for the CUP dataset
Table 3. Timing results for the three
bioinformatics datasets for a single 50-
permutation test (1.83 GHz Intel Core
Duo machine)
Dataset Algorithm Time
Colon
RFE-SVM 146 sec.
Centroid 20 sec.
Shrunken Cent. 28 sec.
Lymp.
RFE-SVM 505 sec.
Centroid 113 sec.
Shrunken Cent. 66 sec.
CUP
RFE-SVM 19844 sec.
Centroid 1135 sec.
Shrunken Cent. 478 sec.
5 Discussion and Conclusions
In this paper we have shown that the limit C → 0 of most forms of SVMs
is explicitly computable and converges to a centroid based classifier. As this
classifier can be used directly for feature ranking, in the limit we can avoid the
computationally demanding recursion and convex optimisation in RFE-SVM.
We have also shown that on some real life datasets the use of the centroid
approach can be advantageous.
The results may at first be surprising, raising the question: Why is the centroid
approach a competitor to its more sophisticated counterparts? One explanation
is that its relative simplicity makes it well matched for classification and fea-
ture selection on datasets comprised of a small number of samples from a noisy
distribution. On such datasets, even the simple estimates of variance which are
employed by, say, the Shrunken Centroid approach [15], are so unreliable that
their usage brings more harm than good. Indeed, the results presented in Sec-
tion 4 support this view.
Another independent argument can be built on the theoretical link established
in Section 2, relating the centroid classifier to the extreme case of regularisation
of SVMs. Once we accept the well supported perspective of the SVM commu-
nity that regularisation is the way to generate robust empirical estimators and
classifiers from noisy data [4,6], it naturally follows that we should get robust
classifiers in the limit of extreme regularisation. It is worth noting that the same
argument can be extended to the theoretical and empirical comparison with
LDA [7], which due to space restrictions is not included here.
We note that the centroid approach has been applied and used sporadically
in the past: we have already linked it to the “ancient” Parzen window; its kernel
form can be found in [4] and its application to classification of breast cancer
microarrays is given in a well cited paper [17]. However, what this paper brings
is a formal link to SVMs and RFE-SVM in particular. Furthermore, to our

An Efficient Alternative to RFE-SVM 179
knowledge, this type of centroid based feature selection and subsequent classifi-
cation has not been explored previously and its application to genomic microar-
ray classification as well as natural language processing (author unmasking) is
novel.
To conclude, we have shown that the centroid approach is a good baseline
alternative to RFE-SVM, especially when dealing with datasets comprised of a
low number of samples from a very high dimensional space. It it very fast and
straightforward to implement, especially in the linear case. It has a nice theoreti-
cal link to SVMs and regularisation networks, and given the right circumstances
it can deliver surprisingly good performance. We also recommend it as a base-
line classifier, facilitating quick sanity cross checks of more involved supervised
learning techniques.
Acknowledgements
We thank Simon Guenter for useful discussions. National ICT Australia (NICTA)
is funded by the Australian Government’s Backing Australia’s Ability initiative,
in part through the Australian Research Council.
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