84,431 research outputs found
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
Recovering Structured Probability Matrices
We consider the problem of accurately recovering a matrix B of size M by M ,
which represents a probability distribution over M2 outcomes, given access to
an observed matrix of "counts" generated by taking independent samples from the
distribution B. How can structural properties of the underlying matrix B be
leveraged to yield computationally efficient and information theoretically
optimal reconstruction algorithms? When can accurate reconstruction be
accomplished in the sparse data regime? This basic problem lies at the core of
a number of questions that are currently being considered by different
communities, including building recommendation systems and collaborative
filtering in the sparse data regime, community detection in sparse random
graphs, learning structured models such as topic models or hidden Markov
models, and the efforts from the natural language processing community to
compute "word embeddings".
Our results apply to the setting where B has a low rank structure. For this
setting, we propose an efficient algorithm that accurately recovers the
underlying M by M matrix using Theta(M) samples. This result easily translates
to Theta(M) sample algorithms for learning topic models and learning hidden
Markov Models. These linear sample complexities are optimal, up to constant
factors, in an extremely strong sense: even testing basic properties of the
underlying matrix (such as whether it has rank 1 or 2) requires Omega(M)
samples. We provide an even stronger lower bound where distinguishing whether a
sequence of observations were drawn from the uniform distribution over M
observations versus being generated by an HMM with two hidden states requires
Omega(M) observations. This precludes sublinear-sample hypothesis tests for
basic properties, such as identity or uniformity, as well as sublinear sample
estimators for quantities such as the entropy rate of HMMs
Proximal Methods for Hierarchical Sparse Coding
Sparse coding consists in representing signals as sparse linear combinations
of atoms selected from a dictionary. We consider an extension of this framework
where the atoms are further assumed to be embedded in a tree. This is achieved
using a recently introduced tree-structured sparse regularization norm, which
has proven useful in several applications. This norm leads to regularized
problems that are difficult to optimize, and we propose in this paper efficient
algorithms for solving them. More precisely, we show that the proximal operator
associated with this norm is computable exactly via a dual approach that can be
viewed as the composition of elementary proximal operators. Our procedure has a
complexity linear, or close to linear, in the number of atoms, and allows the
use of accelerated gradient techniques to solve the tree-structured sparse
approximation problem at the same computational cost as traditional ones using
the L1-norm. Our method is efficient and scales gracefully to millions of
variables, which we illustrate in two types of applications: first, we consider
fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we
apply our optimization tools in the context of dictionary learning, where
learned dictionary elements naturally organize in a prespecified arborescent
structure, leading to a better performance in reconstruction of natural image
patches. When applied to text documents, our method learns hierarchies of
topics, thus providing a competitive alternative to probabilistic topic models
Efficient Multi-Template Learning for Structured Prediction
Conditional random field (CRF) and Structural Support Vector Machine
(Structural SVM) are two state-of-the-art methods for structured prediction
which captures the interdependencies among output variables. The success of
these methods is attributed to the fact that their discriminative models are
able to account for overlapping features on the whole input observations. These
features are usually generated by applying a given set of templates on labeled
data, but improper templates may lead to degraded performance. To alleviate
this issue, in this paper, we propose a novel multiple template learning
paradigm to learn structured prediction and the importance of each template
simultaneously, so that hundreds of arbitrary templates could be added into the
learning model without caution. This paradigm can be formulated as a special
multiple kernel learning problem with exponential number of constraints. Then
we introduce an efficient cutting plane algorithm to solve this problem in the
primal, and its convergence is presented. We also evaluate the proposed
learning paradigm on two widely-studied structured prediction tasks,
\emph{i.e.} sequence labeling and dependency parsing. Extensive experimental
results show that the proposed method outperforms CRFs and Structural SVMs due
to exploiting the importance of each template. Our complexity analysis and
empirical results also show that our proposed method is more efficient than
OnlineMKL on very sparse and high-dimensional data. We further extend this
paradigm for structured prediction using generalized -block norm
regularization with , and experiments show competitive performances when
Gaussian Multiresolution Models: Exploiting Sparse Markov and Covariance Structure
In this paper, we consider the problem of learning Gaussian multiresolution (MR) models in which data are only available at the finest scale, and the coarser, hidden variables serve to capture long-distance dependencies. Tree-structured MR models have limited modeling capabilities, as variables at one scale are forced to be uncorrelated with each other conditioned on other scales. We propose a new class of Gaussian MR models in which variables at each scale have sparse conditional covariance structure conditioned on other scales. Our goal is to learn a tree-structured graphical model connecting variables across scales (which translates into sparsity in inverse covariance), while at the same time learning sparse structure for the conditional covariance (not its inverse) within each scale conditioned on other scales. This model leads to an efficient, new inference algorithm that is similar to multipole methods in computational physics. We demonstrate the modeling and inference advantages of our approach over methods that use MR tree models and single-scale approximation methods that do not use hidden variables
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