4 research outputs found
On a class of optimization-based robust estimators
We consider in this paper the problem of estimating a parameter matrix from
observations which are affected by two types of noise components: (i) a sparse
noise sequence which, whenever nonzero can have arbitrarily large amplitude
(ii) and a dense and bounded noise sequence of "moderate" amount. This is
termed a robust regression problem. To tackle it, a quite general
optimization-based framework is proposed and analyzed. When only the sparse
noise is present, a sufficient bound is derived on the number of nonzero
elements in the sparse noise sequence that can be accommodated by the estimator
while still returning the true parameter matrix. While almost all the
restricted isometry-based bounds from the literature are not verifiable, our
bound can be easily computed through solving a convex optimization problem.
Moreover, empirical evidence tends to suggest that it is generally tight. If in
addition to the sparse noise sequence, the training data are affected by a
bounded dense noise, we derive an upper bound on the estimation error.Comment: To appear in IEEE Transactions on Automatic Contro
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
Analysis of A Nonsmooth Optimization Approach to Robust Estimation
In this paper, we consider the problem of identifying a linear map from
measurements which are subject to intermittent and arbitarily large errors.
This is a fundamental problem in many estimation-related applications such as
fault detection, state estimation in lossy networks, hybrid system
identification, robust estimation, etc. The problem is hard because it exhibits
some intrinsic combinatorial features. Therefore, obtaining an effective
solution necessitates relaxations that are both solvable at a reasonable cost
and effective in the sense that they can return the true parameter vector. The
current paper discusses a nonsmooth convex optimization approach and provides a
new analysis of its behavior. In particular, it is shown that under appropriate
conditions on the data, an exact estimate can be recovered from data corrupted
by a large (even infinite) number of gross errors.Comment: 17 pages, 9 figure
Subspace clustering through parametric representation and sparse optimization
International audienceWe consider the problem of recovering a finite number of linear subspaces from a collection of unlabeled data points that lie in the union of the subspaces. The data are such that it is not known which data point originates from which subspace. To address this challenge, we show that the clustering problem is amenable to a sparse optimization problem. Considering a candidate subspace and the distances of the data points to that subspace, the foundation of the proposed method lies in the maximization of the number of zero distances. This can be relaxed into a convex optimization. Efficiency of the relaxation can be significantly increased by solving a sequence of reweighted convex optimization problems