53 research outputs found
Regularization-free estimation in trace regression with symmetric positive semidefinite matrices
Over the past few years, trace regression models have received considerable
attention in the context of matrix completion, quantum state tomography, and
compressed sensing. Estimation of the underlying matrix from
regularization-based approaches promoting low-rankedness, notably nuclear norm
regularization, have enjoyed great popularity. In the present paper, we argue
that such regularization may no longer be necessary if the underlying matrix is
symmetric positive semidefinite (\textsf{spd}) and the design satisfies certain
conditions. In this situation, simple least squares estimation subject to an
\textsf{spd} constraint may perform as well as regularization-based approaches
with a proper choice of the regularization parameter, which entails knowledge
of the noise level and/or tuning. By contrast, constrained least squares
estimation comes without any tuning parameter and may hence be preferred due to
its simplicity
Matrix factorization with Binary Components
Motivated by an application in computational biology, we consider low-rank
matrix factorization with -constraints on one of the factors and
optionally convex constraints on the second one. In addition to the
non-convexity shared with other matrix factorization schemes, our problem is
further complicated by a combinatorial constraint set of size ,
where is the dimension of the data points and the rank of the
factorization. Despite apparent intractability, we provide - in the line of
recent work on non-negative matrix factorization by Arora et al. (2012) - an
algorithm that provably recovers the underlying factorization in the exact case
with operations for datapoints. To obtain this
result, we use theory around the Littlewood-Offord lemma from combinatorics.Comment: appeared in NIPS 201
Feature selection guided by structural information
In generalized linear regression problems with an abundant number of
features, lasso-type regularization which imposes an -constraint on the
regression coefficients has become a widely established technique. Deficiencies
of the lasso in certain scenarios, notably strongly correlated design, were
unmasked when Zou and Hastie [J. Roy. Statist. Soc. Ser. B 67 (2005) 301--320]
introduced the elastic net. In this paper we propose to extend the elastic net
by admitting general nonnegative quadratic constraints as a second form of
regularization. The generalized ridge-type constraint will typically make use
of the known association structure of features, for example, by using temporal-
or spatial closeness. We study properties of the resulting "structured elastic
net" regression estimation procedure, including basic asymptotics and the issue
of model selection consistency. In this vein, we provide an analog to the
so-called "irrepresentable condition" which holds for the lasso. Moreover, we
outline algorithmic solutions for the structured elastic net within the
generalized linear model family. The rationale and the performance of our
approach is illustrated by means of simulated and real world data, with a focus
on signal regression.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS302 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Topics in learning sparse and low-rank models of non-negative data
Advances in information and measurement technology have led to a surge in prevalence of high-dimensional data. Sparse and low-rank modeling can both be seen as techniques of dimensionality reduction, which is essential for obtaining compact and interpretable representations of such data. In this thesis, we investigate aspects of sparse and low-rank modeling in conjunction with non-negative data or non-negativity constraints. The first part is devoted to the problem of learning sparse non-negative representations, with a focus on how non-negativity can be taken advantage of. We work out a detailed analysis of non-negative least squares regression, showing that under certain conditions sparsity-promoting regularization, the approach advocated paradigmatically over the past years, is not required. Our results have implications for problems in signal processing such as compressed sensing and spike train deconvolution. In the second part, we consider the problem of factorizing a given matrix into two factors of low rank, out of which one is binary. We devise a provably correct algorithm computing such factorization whose running time is exponential only in the rank of the factorization, but linear in the dimensions of the input matrix. Our approach is extended to noisy settings and applied to an unmixing problem in DNA methylation array analysis. On the theoretical side, we relate the uniqueness of the factorization to Littlewood-Offord theory in combinatorics.Fortschritte in Informations- und Messtechnologie fĂŒhren zu erhöhtem Vorkommen hochdimensionaler Daten. ModellierungsansĂ€tze basierend auf Sparsity oder niedrigem Rang können als Dimensionsreduktion betrachtet werden, die notwendig ist, um kompakte und interpretierbare Darstellungen solcher Daten zu erhalten. In dieser Arbeit untersuchen wir Aspekte dieser AnsĂ€tze in Verbindung mit nichtnegativen Daten oder NichtnegativitĂ€tsbeschrĂ€nkungen. Der erste Teil handelt vom Lernen nichtnegativer sparsamer Darstellungen, mit einem Schwerpunkt darauf, wie NichtnegativitĂ€t ausgenutzt werden kann. Wir analysieren nichtnegative kleinste Quadrate im Detail und zeigen, dass unter gewissen Bedingungen Sparsity-fördernde Regularisierung - der in den letzten Jahren paradigmatisch enpfohlene Ansatz - nicht notwendig ist. Unsere Resultate haben Auswirkungen auf Probleme in der Signalverarbeitung wie Compressed Sensing und die Entfaltung von Pulsfolgen. Im zweiten Teil betrachten wir das Problem, eine Matrix in zwei Faktoren mit niedrigem Rang, von denen einer binĂ€r ist, zu zerlegen. Wir entwickeln dafĂŒr einen Algorithmus, dessen Laufzeit nur exponentiell in dem Rang der Faktorisierung, aber linear in den Dimensionen der gegebenen Matrix ist. Wir erweitern unseren Ansatz fĂŒr verrauschte Szenarien und wenden ihn zur Analyse von DNA-Methylierungsdaten an. Auf theoretischer Ebene setzen wir die Eindeutigkeit der Faktorisierung in Beziehung zur Littlewood-Offord-Theorie aus der Kombinatorik
Topics in learning sparse and low-rank models of non-negative data
Advances in information and measurement technology have led to a surge in prevalence of high-dimensional data. Sparse and low-rank modeling can both be seen as techniques of dimensionality reduction, which is essential for obtaining compact and interpretable representations of such data. In this thesis, we investigate aspects of sparse and low-rank modeling in conjunction with non-negative data or non-negativity constraints. The first part is devoted to the problem of learning sparse non-negative representations, with a focus on how non-negativity can be taken advantage of. We work out a detailed analysis of non-negative least squares regression, showing that under certain conditions sparsity-promoting regularization, the approach advocated paradigmatically over the past years, is not required. Our results have implications for problems in signal processing such as compressed sensing and spike train deconvolution. In the second part, we consider the problem of factorizing a given matrix into two factors of low rank, out of which one is binary. We devise a provably correct algorithm computing such factorization whose running time is exponential only in the rank of the factorization, but linear in the dimensions of the input matrix. Our approach is extended to noisy settings and applied to an unmixing problem in DNA methylation array analysis. On the theoretical side, we relate the uniqueness of the factorization to Littlewood-Offord theory in combinatorics.Fortschritte in Informations- und Messtechnologie fĂŒhren zu erhöhtem Vorkommen hochdimensionaler Daten. ModellierungsansĂ€tze basierend auf Sparsity oder niedrigem Rang können als Dimensionsreduktion betrachtet werden, die notwendig ist, um kompakte und interpretierbare Darstellungen solcher Daten zu erhalten. In dieser Arbeit untersuchen wir Aspekte dieser AnsĂ€tze in Verbindung mit nichtnegativen Daten oder NichtnegativitĂ€tsbeschrĂ€nkungen. Der erste Teil handelt vom Lernen nichtnegativer sparsamer Darstellungen, mit einem Schwerpunkt darauf, wie NichtnegativitĂ€t ausgenutzt werden kann. Wir analysieren nichtnegative kleinste Quadrate im Detail und zeigen, dass unter gewissen Bedingungen Sparsity-fördernde Regularisierung - der in den letzten Jahren paradigmatisch enpfohlene Ansatz - nicht notwendig ist. Unsere Resultate haben Auswirkungen auf Probleme in der Signalverarbeitung wie Compressed Sensing und die Entfaltung von Pulsfolgen. Im zweiten Teil betrachten wir das Problem, eine Matrix in zwei Faktoren mit niedrigem Rang, von denen einer binĂ€r ist, zu zerlegen. Wir entwickeln dafĂŒr einen Algorithmus, dessen Laufzeit nur exponentiell in dem Rang der Faktorisierung, aber linear in den Dimensionen der gegebenen Matrix ist. Wir erweitern unseren Ansatz fĂŒr verrauschte Szenarien und wenden ihn zur Analyse von DNA-Methylierungsdaten an. Auf theoretischer Ebene setzen wir die Eindeutigkeit der Faktorisierung in Beziehung zur Littlewood-Offord-Theorie aus der Kombinatorik
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