3,406 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
Supervised Classification Using Sparse Fisher's LDA
It is well known that in a supervised classification setting when the number
of features is smaller than the number of observations, Fisher's linear
discriminant rule is asymptotically Bayes. However, there are numerous modern
applications where classification is needed in the high-dimensional setting.
Naive implementation of Fisher's rule in this case fails to provide good
results because the sample covariance matrix is singular. Moreover, by
constructing a classifier that relies on all features the interpretation of the
results is challenging. Our goal is to provide robust classification that
relies only on a small subset of important features and accounts for the
underlying correlation structure. We apply a lasso-type penalty to the
discriminant vector to ensure sparsity of the solution and use a shrinkage type
estimator for the covariance matrix. The resulting optimization problem is
solved using an iterative coordinate ascent algorithm. Furthermore, we analyze
the effect of nonconvexity on the sparsity level of the solution and highlight
the difference between the penalized and the constrained versions of the
problem. The simulation results show that the proposed method performs
favorably in comparison to alternatives. The method is used to classify
leukemia patients based on DNA methylation features
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