146 research outputs found
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
An Efficient Primal-Dual Prox Method for Non-Smooth Optimization
We study the non-smooth optimization problems in machine learning, where both
the loss function and the regularizer are non-smooth functions. Previous
studies on efficient empirical loss minimization assume either a smooth loss
function or a strongly convex regularizer, making them unsuitable for
non-smooth optimization. We develop a simple yet efficient method for a family
of non-smooth optimization problems where the dual form of the loss function is
bilinear in primal and dual variables. We cast a non-smooth optimization
problem into a minimax optimization problem, and develop a primal dual prox
method that solves the minimax optimization problem at a rate of
{assuming that the proximal step can be efficiently solved}, significantly
faster than a standard subgradient descent method that has an
convergence rate. Our empirical study verifies the efficiency of the proposed
method for various non-smooth optimization problems that arise ubiquitously in
machine learning by comparing it to the state-of-the-art first order methods
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
Blind Source Separation with Optimal Transport Non-negative Matrix Factorization
Optimal transport as a loss for machine learning optimization problems has
recently gained a lot of attention. Building upon recent advances in
computational optimal transport, we develop an optimal transport non-negative
matrix factorization (NMF) algorithm for supervised speech blind source
separation (BSS). Optimal transport allows us to design and leverage a cost
between short-time Fourier transform (STFT) spectrogram frequencies, which
takes into account how humans perceive sound. We give empirical evidence that
using our proposed optimal transport NMF leads to perceptually better results
than Euclidean NMF, for both isolated voice reconstruction and BSS tasks.
Finally, we demonstrate how to use optimal transport for cross domain sound
processing tasks, where frequencies represented in the input spectrograms may
be different from one spectrogram to another.Comment: 22 pages, 7 figures, 2 additional file
A Primal-dual Framework For Mixtures Of Regularisers
Effectively solving many inverse problems in engineering requires to leverage all possible prior information about the structure of the signal to be estimated. This often leads to tackling constrained optimization problems with mixtures of regularizers. Providing a general purpose optimization algorithm for these cases, with both guaranteed convergence rate as well as fast implementation remains an important challenge. In this paper, we describe how a recent primaldual algorithm for non-smooth constrained optimization can be successfully used to tackle these problems. Its simple iterations can be easily parallelized, allowing very efficient computations. Furthermore, the algorithm is guaranteed to achieve an optimal convergence rate for this class of problems. We illustrate its performance on two problems, a compressive magnetic resonance imaging application and an approach for improving the quality of analog-to-digital conversion of amplitude-modulated signals
Generating structured non-smooth priors and associated primal-dual methods
The purpose of the present chapter is to bind together and extend some recent developments regarding data-driven non-smooth regularization techniques in image processing through the means of a bilevel minimization scheme. The scheme, considered in function space, takes advantage of a dualization framework and it is designed to produce spatially varying regularization parameters adapted to the data for well-known regularizers, e.g. Total Variation and Total Generalized variation, leading to automated (monolithic), image reconstruction workflows. An inclusion of the theory of bilevel optimization and the theoretical background of the dualization framework, as well as a brief review of the aforementioned regularizers and their parameterization, makes this chapter a self-contained one. Aspects of the numerical implementation of the scheme are discussed and numerical examples are provided
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