4,304 research outputs found
Generalized power method for sparse principal component analysis
In this paper we develop a new approach to sparse principal component
analysis (sparse PCA). We propose two single-unit and two block optimization
formulations of the sparse PCA problem, aimed at extracting a single sparse
dominant principal component of a data matrix, or more components at once,
respectively. While the initial formulations involve nonconvex functions, and
are therefore computationally intractable, we rewrite them into the form of an
optimization program involving maximization of a convex function on a compact
set. The dimension of the search space is decreased enormously if the data
matrix has many more columns (variables) than rows. We then propose and analyze
a simple gradient method suited for the task. It appears that our algorithm has
best convergence properties in the case when either the objective function or
the feasible set are strongly convex, which is the case with our single-unit
formulations and can be enforced in the block case. Finally, we demonstrate
numerically on a set of random and gene expression test problems that our
approach outperforms existing algorithms both in quality of the obtained
solution and in computational speed.Comment: Submitte
MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation
MADNESS (multiresolution adaptive numerical environment for scientific
simulation) is a high-level software environment for solving integral and
differential equations in many dimensions that uses adaptive and fast harmonic
analysis methods with guaranteed precision based on multiresolution analysis
and separated representations. Underpinning the numerical capabilities is a
powerful petascale parallel programming environment that aims to increase both
programmer productivity and code scalability. This paper describes the features
and capabilities of MADNESS and briefly discusses some current applications in
chemistry and several areas of physics
Generalized power method for sparse principal component analysis
In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.sparse PCA, power method, gradient ascent, strongly convex sets, block algorithms.
Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain
Recovering a sparse signal from its low-pass projections in the Fourier
domain is a problem of broad interest in science and engineering and is
commonly referred to as super-resolution. In many cases, however, Fourier
domain may not be the natural choice. For example, in holography, low-pass
projections of sparse signals are obtained in the Fresnel domain. Similarly,
time-varying system identification relies on low-pass projections on the space
of linear frequency modulated signals. In this paper, we study the recovery of
sparse signals from low-pass projections in the Special Affine Fourier
Transform domain (SAFT). The SAFT parametrically generalizes a number of well
known unitary transformations that are used in signal processing and optics. In
analogy to the Shannon's sampling framework, we specify sampling theorems for
recovery of sparse signals considering three specific cases: (1) sampling with
arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels
and, (3) recovery from Gabor transform measurements linked with the SAFT
domain. Our work offers a unifying perspective on the sparse sampling problem
which is compatible with the Fourier, Fresnel and Fractional Fourier domain
based results. In deriving our results, we introduce the SAFT series (analogous
to the Fourier series) and the short time SAFT, and study convolution theorems
that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie
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