1,108 research outputs found
Sensitivity Analysis for Mirror-Stratifiable Convex Functions
This paper provides a set of sensitivity analysis and activity identification
results for a class of convex functions with a strong geometric structure, that
we coined "mirror-stratifiable". These functions are such that there is a
bijection between a primal and a dual stratification of the space into
partitioning sets, called strata. This pairing is crucial to track the strata
that are identifiable by solutions of parametrized optimization problems or by
iterates of optimization algorithms. This class of functions encompasses all
regularizers routinely used in signal and image processing, machine learning,
and statistics. We show that this "mirror-stratifiable" structure enjoys a nice
sensitivity theory, allowing us to study stability of solutions of optimization
problems to small perturbations, as well as activity identification of
first-order proximal splitting-type algorithms. Existing results in the
literature typically assume that, under a non-degeneracy condition, the active
set associated to a minimizer is stable to small perturbations and is
identified in finite time by optimization schemes. In contrast, our results do
not require any non-degeneracy assumption: in consequence, the optimal active
set is not necessarily stable anymore, but we are able to track precisely the
set of identifiable strata.We show that these results have crucial implications
when solving challenging ill-posed inverse problems via regularization, a
typical scenario where the non-degeneracy condition is not fulfilled. Our
theoretical results, illustrated by numerical simulations, allow to
characterize the instability behaviour of the regularized solutions, by
locating the set of all low-dimensional strata that can be potentially
identified by these solutions
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Activity Identification and Local Linear Convergence of Douglas--Rachford/ADMM under Partial Smoothness
Convex optimization has become ubiquitous in most quantitative disciplines of
science, including variational image processing. Proximal splitting algorithms
are becoming popular to solve such structured convex optimization problems.
Within this class of algorithms, Douglas--Rachford (DR) and alternating
direction method of multipliers (ADMM) are designed to minimize the sum of two
proper lower semi-continuous convex functions whose proximity operators are
easy to compute. The goal of this work is to understand the local convergence
behaviour of DR (resp. ADMM) when the involved functions (resp. their
Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when
both of the two functions (resp. their conjugates) are partly smooth relative
to their respective manifolds, we show that DR (resp. ADMM) identifies these
manifolds in finite time. Moreover, when these manifolds are affine or linear,
we prove that DR/ADMM is locally linearly convergent. When and are
locally polyhedral, we show that the optimal convergence radius is given in
terms of the cosine of the Friedrichs angle between the tangent spaces of the
identified manifolds. This is illustrated by several concrete examples and
supported by numerical experiments.Comment: 17 pages, 1 figure, published in the proceedings of the Fifth
International Conference on Scale Space and Variational Methods in Computer
Visio
Activity Identification and Local Linear Convergence of Forward--Backward-type methods
In this paper, we consider a class of Forward--Backward (FB) splitting
methods that includes several variants (e.g. inertial schemes, FISTA) for
minimizing the sum of two proper convex and lower semi-continuous functions,
one of which has a Lipschitz continuous gradient, and the other is partly
smooth relatively to a smooth active manifold . We propose a
unified framework, under which we show that, this class of FB-type algorithms
(i) correctly identifies the active manifolds in a finite number of iterations
(finite activity identification), and (ii) then enters a local linear
convergence regime, which we characterize precisely in terms of the structure
of the underlying active manifolds. For simpler problems involving polyhedral
functions, we show finite termination. We also establish and explain why FISTA
(with convergent sequences) locally oscillates and can be slower than FB. These
results may have numerous applications including in signal/image processing,
sparse recovery and machine learning. Indeed, the obtained results explain the
typical behaviour that has been observed numerically for many problems in these
fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm
regularization to name only a few.Comment: Full length version of the previous short on
Identifying Active Manifolds
Determining the "active manifold'' for a minimization problem is a large step towards solving the problem. Many researchers have studied under what conditions certain algorithms identify active manifolds in a finite number of iterations. In this work we outline a unifying framework encompassing many earlier results on identification via the Subgradient (Gradient) Projection Method, Newton-like Methods, and the Proximal Point Algorithm. This framework, prox-regular partial smoothness, has the advantage of not requiring convexity for its conclusions, and therefore extends many of these earlier results
Model Consistency of Partly Smooth Regularizers
This paper studies least-square regression penalized with partly smooth
convex regularizers. This class of functions is very large and versatile
allowing to promote solutions conforming to some notion of low-complexity.
Indeed, they force solutions of variational problems to belong to a
low-dimensional manifold (the so-called model) which is stable under small
perturbations of the function. This property is crucial to make the underlying
low-complexity model robust to small noise. We show that a generalized
"irrepresentable condition" implies stable model selection under small noise
perturbations in the observations and the design matrix, when the
regularization parameter is tuned proportionally to the noise level. This
condition is shown to be almost a necessary condition. We then show that this
condition implies model consistency of the regularized estimator. That is, with
a probability tending to one as the number of measurements increases, the
regularized estimator belongs to the correct low-dimensional model manifold.
This work unifies and generalizes several previous ones, where model
consistency is known to hold for sparse, group sparse, total variation and
low-rank regularizations
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