80 research outputs found
Error Bounds for Piecewise Smooth and Switching Regression
The paper deals with regression problems, in which the nonsmooth target is
assumed to switch between different operating modes. Specifically, piecewise
smooth (PWS) regression considers target functions switching deterministically
via a partition of the input space, while switching regression considers
arbitrary switching laws. The paper derives generalization error bounds in
these two settings by following the approach based on Rademacher complexities.
For PWS regression, our derivation involves a chaining argument and a
decomposition of the covering numbers of PWS classes in terms of the ones of
their component functions and the capacity of the classifier partitioning the
input space. This yields error bounds with a radical dependency on the number
of modes. For switching regression, the decomposition can be performed directly
at the level of the Rademacher complexities, which yields bounds with a linear
dependency on the number of modes. By using once more chaining and a
decomposition at the level of covering numbers, we show how to recover a
radical dependency. Examples of applications are given in particular for PWS
and swichting regression with linear and kernel-based component functions.Comment: This work has been submitted to the IEEE for possible publication.
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Global optimization for low-dimensional switching linear regression and bounded-error estimation
The paper provides global optimization algorithms for two particularly
difficult nonconvex problems raised by hybrid system identification: switching
linear regression and bounded-error estimation. While most works focus on local
optimization heuristics without global optimality guarantees or with guarantees
valid only under restrictive conditions, the proposed approach always yields a
solution with a certificate of global optimality. This approach relies on a
branch-and-bound strategy for which we devise lower bounds that can be
efficiently computed. In order to obtain scalable algorithms with respect to
the number of data, we directly optimize the model parameters in a continuous
optimization setting without involving integer variables. Numerical experiments
show that the proposed algorithms offer a higher accuracy than convex
relaxations with a reasonable computational burden for hybrid system
identification. In addition, we discuss how bounded-error estimation is related
to robust estimation in the presence of outliers and exact recovery under
sparse noise, for which we also obtain promising numerical results
On the complexity of switching linear regression
This technical note extends recent results on the computational complexity of
globally minimizing the error of piecewise-affine models to the related problem
of minimizing the error of switching linear regression models. In particular,
we show that, on the one hand the problem is NP-hard, but on the other hand, it
admits a polynomial-time algorithm with respect to the number of data points
for any fixed data dimension and number of modes.Comment: Automatica, Elsevier, 201
Risk Bounds for Learning Multiple Components with Permutation-Invariant Losses
This paper proposes a simple approach to derive efficient error bounds for
learning multiple components with sparsity-inducing regularization. We show
that for such regularization schemes, known decompositions of the Rademacher
complexity over the components can be used in a more efficient manner to result
in tighter bounds without too much effort. We give examples of application to
switching regression and center-based clustering/vector quantization. Then, the
complete workflow is illustrated on the problem of subspace clustering, for
which decomposition results were not previously available. For all these
problems, the proposed approach yields risk bounds with mild dependencies on
the number of components and completely removes this dependence for nonconvex
regularization schemes that could not be handled by previous methods
Finding sparse solutions of systems of polynomial equations via group-sparsity optimization
The paper deals with the problem of finding sparse solutions to systems of
polynomial equations possibly perturbed by noise. In particular, we show how
these solutions can be recovered from group-sparse solutions of a derived
system of linear equations. Then, two approaches are considered to find these
group-sparse solutions. The first one is based on a convex relaxation resulting
in a second-order cone programming formulation which can benefit from efficient
reweighting techniques for sparsity enhancement. For this approach, sufficient
conditions for the exact recovery of the sparsest solution to the polynomial
system are derived in the noiseless setting, while stable recovery results are
obtained for the noisy case. Though lacking a similar analysis, the second
approach provides a more computationally efficient algorithm based on a greedy
strategy adding the groups one-by-one. With respect to previous work, the
proposed methods recover the sparsest solution in a very short computing time
while remaining at least as accurate in terms of the probability of success.
This probability is empirically analyzed to emphasize the relationship between
the ability of the methods to solve the polynomial system and the sparsity of
the solution.Comment: Journal of Global Optimization (2014) to appea
Sparse phase retrieval via group-sparse optimization
This paper deals with sparse phase retrieval, i.e., the problem of estimating
a vector from quadratic measurements under the assumption that few components
are nonzero. In particular, we consider the problem of finding the sparsest
vector consistent with the measurements and reformulate it as a group-sparse
optimization problem with linear constraints. Then, we analyze the convex
relaxation of the latter based on the minimization of a block l1-norm and show
various exact recovery and stability results in the real and complex cases.
Invariance to circular shifts and reflections are also discussed for real
vectors measured via complex matrices
On the exact minimization of saturated loss functions for robust regression and subspace estimation
This paper deals with robust regression and subspace estimation and more
precisely with the problem of minimizing a saturated loss function. In
particular, we focus on computational complexity issues and show that an exact
algorithm with polynomial time-complexity with respect to the number of data
can be devised for robust regression and subspace estimation. This result is
obtained by adopting a classification point of view and relating the problems
to the search for a linear model that can approximate the maximal number of
points with a given error. Approximate variants of the algorithms based on
ramdom sampling are also discussed and experiments show that it offers an
accuracy gain over the traditional RANSAC for a similar algorithmic simplicity.Comment: Pattern Recognition Letters, Elsevier, 201
Error bounds with almost radical dependence on the number of components for multi-category classification, vector quantization and switching regression
National audienceThis paper presents a simple approach for obtaining efficient error bounds in learning problems involving multiple components. In particular, we obtain error bounds with a close to radical dependency on the number of components for multi-category classification, vector quantification and switching regression when the regularization scheme relies on a sum of norms over the components. These results are obtained thanks to the combination of a number of structural results on Rademacher complexities and a suitable handling of the structure of the regularizer
Estimating the probability of success of a simple algorithm for switched linear regression
International audienceThis paper deals with the switched linear regression problem inherent in hybrid system identification. In particular, we discuss k-LinReg, a straightforward and easy to implement algorithm in the spirit of k-means for the nonconvex optimization problem at the core of switched linear regression, and focus on the question of its accuracy on large data sets and its ability to reach global optimality. To this end, we emphasize the relationship between the sample size and the probability of obtaining a local minimum close to the global one with a random initialization. This is achieved through the estimation of a model of the behavior of this probability with respect to the problem dimensions. This model can then be used to tune the number of restarts required to obtain a global solution with high probability. Experiments show that the model can accurately predict the probability of success and that, despite its simplicity, the resulting algorithm can outperform more complicated approaches in both speed and accuracy
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