55 research outputs found
Collaborative Filtering via Group-Structured Dictionary Learning
Structured sparse coding and the related structured dictionary learning
problems are novel research areas in machine learning. In this paper we present
a new application of structured dictionary learning for collaborative filtering
based recommender systems. Our extensive numerical experiments demonstrate that
the presented technique outperforms its state-of-the-art competitors and has
several advantages over approaches that do not put structured constraints on
the dictionary elements.Comment: A compressed version of the paper has been accepted for publication
at the 10th International Conference on Latent Variable Analysis and Source
Separation (LVA/ICA 2012
Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
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
Deep Learning of Representations: Looking Forward
Deep learning research aims at discovering learning algorithms that discover
multiple levels of distributed representations, with higher levels representing
more abstract concepts. Although the study of deep learning has already led to
impressive theoretical results, learning algorithms and breakthrough
experiments, several challenges lie ahead. This paper proposes to examine some
of these challenges, centering on the questions of scaling deep learning
algorithms to much larger models and datasets, reducing optimization
difficulties due to ill-conditioning or local minima, designing more efficient
and powerful inference and sampling procedures, and learning to disentangle the
factors of variation underlying the observed data. It also proposes a few
forward-looking research directions aimed at overcoming these challenges
A flexible framework for sparse simultaneous component based data integration
<p>Abstract</p> <p>1 Background</p> <p>High throughput data are complex and methods that reveal structure underlying the data are most useful. Principal component analysis, frequently implemented as a singular value decomposition, is a popular technique in this respect. Nowadays often the challenge is to reveal structure in several sources of information (e.g., transcriptomics, proteomics) that are available for the same biological entities under study. Simultaneous component methods are most promising in this respect. However, the interpretation of the principal and simultaneous components is often daunting because contributions of each of the biomolecules (transcripts, proteins) have to be taken into account.</p> <p>2 Results</p> <p>We propose a sparse simultaneous component method that makes many of the parameters redundant by shrinking them to zero. It includes principal component analysis, sparse principal component analysis, and ordinary simultaneous component analysis as special cases. Several penalties can be tuned that account in different ways for the block structure present in the integrated data. This yields known sparse approaches as the lasso, the ridge penalty, the elastic net, the group lasso, sparse group lasso, and elitist lasso. In addition, the algorithmic results can be easily transposed to the context of regression. Metabolomics data obtained with two measurement platforms for the same set of <it>Escherichia coli </it>samples are used to illustrate the proposed methodology and the properties of different penalties with respect to sparseness across and within data blocks.</p> <p>3 Conclusion</p> <p>Sparse simultaneous component analysis is a useful method for data integration: First, simultaneous analyses of multiple blocks offer advantages over sequential and separate analyses and second, interpretation of the results is highly facilitated by their sparseness. The approach offered is flexible and allows to take the block structure in different ways into account. As such, structures can be found that are exclusively tied to one data platform (group lasso approach) as well as structures that involve all data platforms (Elitist lasso approach).</p> <p>4 Availability</p> <p>The additional file contains a MATLAB implementation of the sparse simultaneous component method.</p
New inexact explicit thresholding/shrinkage formulas for inverse problems with overlapping group sparsity
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