Indoor Place Recognition using Online
Independent Support Vector Machines
Francesco Orabona, Claudio Castellini
LIRA-Lab, University of Genova
Genova, Italy{bremen,drwho}@liralab.it

Barbara Caputo, Jie Luo
IDIAP Research Institute / EPFL
Martigny, Switzerland

Giulio Sandini
Italian Institute of Technology
Genova, Italy{bcaputo,jluo}@idiap.ch

giulio.sandini@iit.it

Abstract

In the framework of indoor mobile robotics, place recognition is a challenging task, where it is crucial that self-localization be enforced precisely,
notwithstanding the changing conditions of illumination, objects being shiftedaround and/or people affecting the appearance of the scene. In this scenario
online learning seems the main way out, thanks to the possibility of adapting to changes in a smart and flexible way. Nevertheless, standard machinelearning approaches usually suffer when confronted with massive amounts ofdata and when asked to work online. Online learning requires a high training and testing speed, all the more in place recognition, where a continuous
flow of data comes from one or more cameras. In this paper we follow the
Support Vector Machines-based approach of Pronobis et al. [26], proposing
an improvement that we call Online Independent Support Vector Machines.
This technique exploits linear independence in the image feature space to incrementally keep the size of the learning machine remarkably small while
retaining the accuracy of a standard machine. Since the training and testingtime crucially depend on the size of the machine, this solves the above statedproblems. Our experimental results prove the effectiveness of the approach.

1

Introduction

Place recognition is an open and highly challenging problem in computer vision, especially when applied to mobile robotics in indoor environments. Simply stated, the problem is that of determining what room of a house or office a mobile robot is in, based upon
what the robot sees through one or more cameras. The problem is made very difficult
by at least three factors: (a) the input space is huge, since we deal with images, usuallyat a reasonable resolution and in colour; (b) images of the same place can be quite different as illumination conditions change and moving obstacles get in the way; and (c),recognition must be done on-line in real time, as the robot is moving around. The topic iswidely researched, but incremental learning approaches have been so far mostly used forconstructing the geometrical map, or the environment representation, online [6, 1]. Robustness to illumination changes, and more generally to realistic visual variations in time,
has been addressed in [26], where it was shown that a pure learning approach can be very
effective for tackling the first two issues: indeed it was demonstrated that an approachbased upon Support Vector Machines (SVM, see, e.g., [4]) in batch mode could achieve aremarkable robustness to illumination changes and variability due to the normal use of theenvironments. At the same time the work elicited the problem of the growth of the testingtime when a bigger training set was used, to have better recognition performances. In fact,
as far as the third issue is concerned, it is well known that both the training and testingtime of an SVM crucially depend on the number of samples considered [16]; as well, thenumber of Support Vectors (SVs) found, which determine the complexity of the solution
to the problem, grows proportionally with respect to the number of samples [28]. This
makes the approach unsuitable, at least so far, for on-line learning, where a potentiallyendless flow of data is acquired by the machine. SVMs can be up to 50 times slower thanother specialized approaches with similar performances [8]. Several exact and approximate approaches have been proposed so far for simplifying the SVM decision function:
see for instance [12], based upon linear independence of the SVs in the feature space performed after training, and other after-training simplification methods (e.g. chapter 18.3 in[27] and [23]). The exact solution to online SVM learning was given by Cauwenberghs
and Poggio in 2000 [9], but their algorithm cannot be used to reduce the number of SVs.
In [29] and [25] approximate incremental versions of the SVM are proposed, that alsoachieve a reduction of the number of SVs with small degradation of their performances.
In this paper we propose an improvement to SVMs that we call Online Independent
Support Vector Machines (OISVMs). OISVMs incrementally select “basis vectors” thatare used to build the solution of the SVM training problem, based upon linear independence in the feature space: vectors which are linearly dependent on already stored ones
are rejected, and a smart, incremental minimization algorithm is employed to find the
new minimum of the cost function. This keeps the number of SVs much smaller thanusual, reducing the complexity of the solution and therefore both the training and testingtime. Unsupervised rank reduction methods have been proposed [3] as well as supervised ones [2] that achieve the same goals, but no application of these ideas appears sofar, to the best of our knowledge, in online settings. This is particularly important sincein an online setting the size of a SVM would grow indefinitely, and so would the testing time. Our experiments instead indicate that the number of basis vectors of OISVMs
does not grow linearly with the training set, but reaches a limit and then stops growing.
This result is theoretically confirmed, e.g., in [14], even in the case the feature space isinfinite-dimensional.
Such an approach is actually what is needed to tackle the problem of place recognitionin mobile robotics. To support this claim, we show a set of experimental results obtainedby comparing SVMs and OISVMs on a real-world place recognition problem in an indoor
environment. Data images are acquired continuously by two robot platforms under different weather conditions and across a time span of several months. Our results show that
our method achieves a speed-up of 3.5 − 2.3 times with respect to the time required by
the standard SVM, respectively with χ 2 kernel and matching kernel [30], while retaining
essentially the same accuracy.
The paper is structured as follows: after an overview of background mathematics
proper to SVMs, Section 3 describes OISVMs. Section 4 shows the experimental resultsand lastly, in Section 5, conclusions are drawn and future work is outlined.

2

Background Mathematics

Due to space limitations, this is a very quick account of SVMs — the interested reader
is referred to [7] for a tutorial, and to [11] for a comprehensive introduction to the subject. Assume {xi , yi }li=1 , with xi ∈ Rm and yi ∈ {−1, 1}, is a set of samples and labels
drawn from an unknown probability distribution; we want to find a function f (x) suchthat sign( f (x)) best determines the category of any future sample x. In the most general
setting,lf (x) = ∑ αi yi K(x, xi ) + b

(1)

i=1

where b ∈ R and K(x1 , x2 ) = Φ(x1 ) · Φ(x2 ), the kernel function, evaluates inner products
between images of the samples through a non-linear mapping Φ. The αi s are Lagrangiancoefficients obtained by solving (the dual Lagrangian form of) the problem
min
w,b

p
1
l
2
2 ||w|| +C ∑i=1 ξi(2)subject to yi (w · xi + b) ≥ 1 − ξiξi ≥ 0where w defines a separating hyperplane in the feature space, i.e., the space where Φ
lives, whereas ξi ∈ R are slack variables, C ∈ R+ is an error penalty coefficient and p isusually 1 or 2. In practice, most of the αi are found to be zero after training; the vectors
with an associated αi different from zero are called support vectors. Notice that, from (1),the testing time of a new point is proportional to the number of SVs, hence reducing thenumber of SVs implies reducing the testing time.
In the following, the term kernel dimension will refer, as is customary, to the dimension of the feature space. The kernel dimension is related to the generalization power ofthe machine, and it depends on the choice of the kernel itself. Widely used kernels include
the polynomial one (finite-dimensional) and the Gaussian one (infinite-dimensional).

3

Online Independent Support Vector Machines

Let the kernel matrix K be defined such that Ki j = K(xi , x j ), with i, j = 1, . . . , l. The
possibility to obtain a more compact representation of f (x) follows from the fact that thesolution to a SVM problem (that is, the αi s) is not unique if K does not have full rank [7],which is equivalent to some of the SVs being linearly dependent on some others in thefeature space (this is the core of Downs et al.’s [12] original idea). In an online setting, toapply Downs et al.’s idea, or any other post-training method to reduce the number of SVs,
means to simplify the solution each time a new sample is acquired, which is obviously
infeasible. We need a way to use independent SVs only, that is to decouple the conceptof “basis” vectors, used to build the classification function (1), from the samples used toevaluate the ξi in (2). If the selected basis vectors span the same subspace as the wholesample set, the solution found will be equivalent — that is, we will not lose any precision.
We hereby propose, after having received a new training sample, to incrementally add
it to the basis if it is linearly independent in the feature space from those already present
in the basis itself. The solution found is the same as in the classical SVM formulation;
therefore, no approximation whatsoever is involved, unless one gives it up in order toobtain even fewer support vectors (see Section 4 for a deeper discussion on this point).
Denoting the indexes of the vectors in the current basis, after l training samples, by
B, and the new sample under judgment by xl+1 , the algorithm can then be summed up as
follows:
1. check whether xl+1 is linearly independent from the basis in the feature space; if it
is, add it to B; otherwise, leave B unchanged.
2. incrementally re-train the machine.
Hence the testing time for a new point will be O(|B|), as opposed to O(l) in the standard approach; therefore, keeping B small will improve the testing time without losing
any precision whatsoever.
In the following, the notations AIJ and vI , where A is a matrix, v is a vector and
I, J ⊂ N denote in turn the sub-matrix and the sub-vector obtained from A and v by taking
the indexes in I and J.

3.1 Linear independence
In general, checking whether a matrix has full rank is done via some decomposition, or bylooking at the eigenvalues of the matrix; but here we want to check whether a single vector
is linearly independent from a matrix which is already known to be full-rank. Inspired by
the definition of linear independence, we check how well the vector can be approximatedby a linear combination of the vectors in the set [13]. Let d j ∈ R; then let
2∆ = mind∑ d j φ (x j ) − φ (xl+1 )

(3)

j∈B

If ∆ > 0 then xl+1 is linearly independent with respect to the basis, and {l + 1} is
added to B. In practice, we check whether ∆ ≤ η where η > 0 is a tolerance factor,
and expect that larger values of η lead to worse accuracy, but also to smaller bases. Asa matter of fact, if η is set at machine precision, OISVMs retain the exact accuracy of
SVMs. Notice also that if the feature space has finite dimension n, then no more than nlinearly independent vectors can be found, and B will never contain more than n vectors.
Expanding equation (3) we get∆ = mind!

∑ d j di φ (x j ) · φ (xi ) − 2 ∑ d j φ (x j ) · φ (xl+1 ) + φ (xl+1 ) · φ (xl+1 )i, j∈Bj∈B(4)that is, applying the kernel trick,∆ = min dT KBB d − 2dT k + K(xl+1 , xl+1 )d(5)where ki = K(xi , xl+1 ) with i ∈ B. Solving (5), that is, applying the extremum conditions
with respect to d, we obtain−1kd̃ = KBB(6)∆ = K(xl+1 , xl+1 ) − kT d̃

(7)and, by replacing (6) in (5) once,

Note that KBB can be safely inverted since, by incremental construction, it is fullrank. An efficient way to do it, exploiting the incremental nature of the approach, is that−1of updating it recursively: after the addition of a new sample, the new KBB
then becomes0
..  1  d̃  

−1

. 
KBB
d̃T −1(8)+
0  ∆ −10 ···0 0where d̃ and ∆ are already evaluated during the test (this method matches the one usedin Cauwenberghs and Poggio’s incremental algorithm [9]). Thanks to this incremental
evaluation, the time complexity of the linear independence check is O(|B|2 ), as one caneasily see from Equation (6).
With this method we are approximating the original kernel matrix K with anotherb [2]; the quality of the approximation depends on η . In fact it is possible to show
matrix Kb ≤ η |B| ≤ η l, where l is the number of samples acquired [14]. If wethat trace(K − K)consider a normalized kernel, that is a kernel for which K(x, x) is always equal to 1, web
can write trace(K − K)/trace(K)≤ η . On the other hand a bigger η means of course asmaller number of SVs, hence it controls the trade-off between accuracy and speed of the
OISVM.

3.2 Training the machine
The training method largely follows Keerthi et al. [17, 16], that we have adapted for online
training. The algorithm directly minimizes problem (2) as opposed to the standard wayof minimizing its dual Lagrangian form, allowing to select explicitly the basis vectors touse. We set p = 2 in (2) and transform it to an unconstrained problem. Let D ⊂ {1, . . . , l};then the unconstrained problem is
!
1 l
1 T
2
(9)
min
β KDD β + C ∑ max (0, 1 − yi KiD β )
2
2 i=1
β
where β is the vector of the Lagrangian coefficients involved in f (x), analogously to theαi s in the original formulation. If we set D = B, then the solution to the problem isunique since KBB is full rank by construction. Newton’s method as modified by Keerthi
et al. [17, 16] can then be used to solve (9) after each new sample. When the new sample
xl+1 is received the method goes as follows:
1. let I = {i : 1 − yi oi > 0} where oi = KiB β and β is the vector of optimal coefficients with l training samples; if I has not changed, stop.

Figure 1: Sample images illustrating the variations in the IDOL2 database. Images inthe top row show the variability introduced by changes in illumination as well as people
appearing in the environment. The middle row shows the influence of people’s everyday
activity (first four images) as well as larger variations which happened over a time spanof 6 months. Finally, the bottom row illustrates the changes in viewpoint observed for aseries of images acquired one after another in 1.6 seconds.
T
and g =
2. otherwise, let the new β be β − γ P−1 g, where P = KBB + CKBI KBI
KBB β −CKBI (yI − oI ).

3. go back to Step 1.
In Step 2 above, γ is set to one. In order to speed up the algorithm, we maintain an
updated Cholesky decomposition of P. It turns out that the algorithm converges in very
few iterations, usually 0 to 2; the time complexity of the re-training step is O(|B|l), as
well as its space complexity; hence, keeping B small will speed up the training time aswell as the testing time.

4

Place Recognition via OISVMs

In this section we report the experimental evaluation of OISVMs on the place recognition
scenario, where the aim is to update the model to handle variations in an indoor environment due to human activities over long time spans.
Experiments were conducted on the IDOL2 database (Image Database for rObot Localization 2, [22]), which contains 24 image sequences acquired using a perspective camera mounted on two mobile robot platforms, while moving in an indoor laboratory environment consisting of five different rooms. The sequences were acquired under various
weather and illumination conditions (sunny, cloudy, and night) and across a time span ofsix months. Thus, this data capture natural variability that occurs in real-world environments because of both natural changes in the illumination and human activities. Fig. 1
shows some sample images from the database, illustrating the difficulty of the task. The
image sequences in the database are divided as follows: for each robot platform and foreach type of illumination conditions, there were four sequences recorded. Of these four
sequences, the first two were acquired six months before the last two. This means that, foreach robot and for every illumination condition, there are always two sequences acquired
under similar conditions, and two sequences acquired under very different conditions.

This makes the database suitable for different kinds of evaluation on the adaptability of
an incremental algorithm. For further details about the database see [22].
The evaluation was performed using Composed Receptive Field Histograms (CRFH)[19] as global image features and SIFT descriptors [20] of local features computed using a
Harris-Laplace detector [15]. In the experiments, we consider both exponential χ 2 kernel
for SVM (when use CRFH), and local kernels [30] (SIFT). Note the kernel in [30] is notalways positive semidefinite [5], so this is also a test on non-Mercer kernels that have
proved useful for visual recognition. The kernels used are infinite-dimensional, so forboth kernels we run the OISVM using different values of η .
OISVMs have been implemented in Matlab and tested against LIBSVM v2.82 [10].
The software library has been extended to various families of kernels, and to the fixedpartition incremental SVM [29], an approximate incremental extension of SVM. In thisway we can do a straightforward comparison between exact and approximate methods on
this task. Notice that for the standard SVM the training is not online.
As in the experimental setup of [21], the algorithm was trained incrementally on three
sequences from IDOL2, acquired under similar illumination conditions with the samerobot platform; the fourth sequence was used for testing. In order to test the various
properties of interest of the incremental algorithms, we need a reasonable number of incremental steps. Thus, every sequence was split into 5 subsequences, so that each subset
contained one of the five images acquired by the robot every second (image sequenceswere acquired at a rate of 5fps). Since during acquisition the camera’s viewpoint continuously changes [21], the subsequences could be considered as recorded separately ina static environment but for varying pose. This setup allows us to examine how the algorithms perform on data with less variations. In order to get a feeling of the variationsof the frame images in a sequence, bottom row of Fig. 1 shows some sample images acquired within a time span of 1.6 sec. As a result, training on each sequence was performedin 5 steps, using one subsequence at a time, resulting in 15 steps in total. Overall, we considered 36 different permutations of training and test sequences for both the exponential
χ 2 and matching kernels; here we report average results with standard deviation. Fig. 2,
left, shows the recognition rates of the exponential χ 2 kernel (top) and matching kernel(bottom) experiments obtained at each step using OISVM, the fixed-partition algorithm
and the standard SVM. Fig. 2, right, reports the number of support vectors stored in themodel at each step of the incremental procedure, for both kernel types.
We see that, performance-wise, all methods achieves statistically comparable results;
this is true for both kernel types. As far the machine size is concerned, the OISVM
algorithm shows a considerable advantage with respect to the fixed-partition method. In
the case of the exponential χ 2 kernel this advantage is truly impressive (Fig 2, top right):
for η = 0.017 and 0.025 the size at the final incremental step is 34%/22% of that of the
fixed-partition method and 28%/18% of that of the standard batch method. Even more
important, OISVM, for these two values of η , has found a plateau in memory, while forother methods the trend seems to be of a growth proportional to the number of training
data. Note that the choice of the parameter η is crucial for achieving an optimal trade-off
between compactness of the solution and optimal performance.
It is very interesting to note that, in the case of the matching kernel, the memory
reduction for OISVM is less pronounced, and there is not a clear plateau in memory
growth by any of the algorithms. This behavior might be due to several factors: to begin
with, the matching kernel is not a Mercer kernel [5], which might affect the algorithm.

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Also, the algorithm does not reach a plateau in the SVs growth because, in the induced
space of the matching kernel, there seems to be a high probability that pair of training
points are orthogonal, or almost orthogonal, to each other (notice that, as the kernel is nota Mercer one, the geometric interpretation might not be valid). Anyway, given enoughtraining points, the machine will always reach a maximum size and will stop growing[14]. Other tests on a set of standard databases commonly used in the machine learning
community, as well as more details about OISVM can be found in [24].
It is worth noting that, even if the solution is kept small and the number of support
vectors will be finite in any case, all the received training samples must be stored. Thiscan be a problem in an online setting, but it could be solved using, for example, somekind of forgetting strategy. Another strategy can be the use out-of-core storage of the data
(i.e., storage on the hard disk) in order to be able to deal with big training sets.

5

Discussion and conclusions

In this paper we have shown a promising improvement to Support Vector Machines, that
we call Online Independent Support Vector Machines (OISVM). OISVMs can effectively

solve the problem of place recognition by a mobile robot, at least in the experiment wehave shown. OISVMs were tested on the IDOL2 image database, which consists of image
sequences acquired by two robot platforms under different weather conditions and acrossa time span of several months. OISVMs avoid using in the solution those support vectors
which are linearly dependent of previous ones in the feature space. The optimization
problem is solved via an incremental algorithm which benefits of the small number of thebasis vectors.
As far as we know, this method is different from all analogous procedures presented
so far in literature (e.g., [12, 23, 18, 31, 16]) since it is not an after-training simplificationand it assumes no knowledge whatsoever of the full training set beforehand. Moreoverin case of finite-dimensional kernel and η = 0, the solution is exactly the same of thestandard formulation because no approximation is used.
Our experimental results show that in the case of infinite-dimensional kernels, thenumber of support vectors is dramatically reduced at the price of a negligible degradationin the accuracy. In fact in the case of χ 2 kernel, we get as few as 3.5 times less SVs
with respect to the batch formulation and 3 times less with respect to the fixed-partition
method, while retaining essentially the same accuracy. In the case of the local kernel, thespeed up are respectively 2.3 and 2.1.
Since the training and testing time depend polynomially on the number of support
vectors, reducing them brings an obvious speed up. A careful study of the relationshipbetween η and the degradation in performance is being carried on; in fact, accordingto [14], imposing a value of η strictly larger than zero will eventually result in a finite
number of basis vectors, even in the case the feature space is infinite-dimensional. Further
research about finding a precise relationship between η and this number will allow us toprecisely dimension the machine depending on the required precision.

Acknowledgments
This work was supported by EU projects RobotCub (IST-2004-004370), CONTACT (NEST5010), NEURObotics (FP6-IST-001917) and DIRAC (FP6-0027787).

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