17,484 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
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