اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFast Sparse Decomposition by Iterative Detection-Estimation
257
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Finding sparse solutions of underdetermined systems of linear equations is a fundamental problem in signal processing and statistics which has become a subject of interest in recent years. In general, these systems have infinitely many solutions. However, it may be shown that sufficiently sparse solutions may be identified uniquely. In other words, the corresponding linear transformation will be invertible if we restrict its domain to sufficiently sparse vectors. This property may be used, for example, to solve the underdetermined Blind Source Separation (BSS) problem, or to find sparse representation of a signal in an `overcomplete dictionary of primitive elements (i.e., the so-called atomic decomposition). The main drawback of current methods of finding sparse solutions is their computational complexity. In this paper, we will show that by detecting `active components of the (potential) solution, i.e., those components having a considerable value, a framework for fast solution of the problem may be devised. The idea leads to a family of algorithms, called `Iterative Detection-Estimation (IDE), which converge to the solution by successive detection and estimation of its active part. Comparing the performance of IDE(s) with one of the most successful method to date, which is based on Linear Programming (LP), an improvement in speed of about two to three orders of magnitude is observed.
قيم البحث
اقرأ أيضاً
We consider online change detection of high dimensional data streams with sparse changes, where only a subset of data streams can be observed at each sensing time point due to limited sensing capacities. On the one hand, the detection scheme should b
e able to deal with partially observable data and meanwhile have efficient detection power for sparse changes. On the other, the scheme should be able to adaptively and actively select the most important variables to observe to maximize the detection power. To address these two points, in this paper, we propose a novel detection scheme called CDSSD. In particular, it describes the structure of high dimensional data with sparse changes by smooth-sparse decomposition, whose parameters can be learned via spike-slab variational Bayesian inference. Then the posterior Bayes factor, which incorporates the learned parameters and sparse change information, is formulated as a detection statistic. Finally, by formulating the statistic as the reward of a combinatorial multi-armed bandit problem, an adaptive sampling strategy based on Thompson sampling is proposed. The efficacy and applicability of our method in practice are demonstrated with numerical studies and a real case study.
Iterative hard thresholding (IHT) is a projected gradient descent algorithm, known to achieve state of the art performance for a wide range of structured estimation problems, such as sparse inference. In this work, we consider IHT as a solution to th
e problem of learning sparse discrete distributions. We study the hardness of using IHT on the space of measures. As a practical alternative, we propose a greedy approximate projection which simultaneously captures appropriate notions of sparsity in distributions, while satisfying the simplex constraint, and investigate the convergence behavior of the resulting procedure in various settings. Our results show, both in theory and practice, that IHT can achieve state of the art results for learning sparse distributions.
We consider the class of convex minimization problems, composed of a self-concordant function, such as the $logdet$ metric, a convex data fidelity term $h(cdot)$ and, a regularizing -- possibly non-smooth -- function $g(cdot)$. This type of problems
have recently attracted a great deal of interest, mainly due to their omnipresence in top-notch applications. Under this emph{locally} Lipschitz continuous gradient setting, we analyze the convergence behavior of proximal Newton schemes with the added twist of a probable presence of inexact evaluations. We prove attractive convergence rate guarantees and enhance state-of-the-art optimization schemes to accommodate such developments. Experimental results on sparse covariance estimation show the merits of our algorithm, both in terms of recovery efficiency and complexity.
We address two shortcomings in online travel time estimation methods for congested urban traffic. The first shortcoming is related to the determination of the number of mixture modes, which can change dynamically, within day and from day to day. The
second shortcoming is the wide-spread use of Gaussian probability densities as mixture components. Gaussian densities fail to capture the positive skew in travel time distributions and, consequently, large numbers of mixture components are needed for reasonable fitting accuracy when applied as mixture components. They also assign positive probabilities to negative travel times. To address these issues, this paper derives a mixture distribution with Gamma component densities, which are asymmetric and supported on the positive numbers. We use sparse estimation techniques to ensure parsimonious models and propose a generalization of Gamma mixture densities using Mittag-Leffler functions, which provides enhanced fitting flexibility and improved parsimony. In order to accommodate within-day variability and allow for online implementation of the proposed methodology (i.e., fast computations on streaming travel time data), we introduce a recursive algorithm which efficiently updates the fitted distribution whenever new data become available. Experimental results using real-world travel time data illustrate the efficacy of the proposed methods.
In this paper, we consider the Graphical Lasso (GL), a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. Recently, we have
shown that the sparsity pattern of the optimal solution of GL is equivalent to the one obtained from simply thresholding the sample covariance matrix, for sparse graphs under different conditions. We have also derived a closed-form solution that is optimal when the thresholded sample covariance matrix has an acyclic structure. As a major generalization of the previous result, in this paper we derive a closed-form solution for the GL for graphs with chordal structures. We show that the GL and thresholding equivalence conditions can significantly be simplified and are expected to hold for high-dimensional problems if the thresholded sample covariance matrix has a chordal structure. We then show that the GL and thresholding equivalence is enough to reduce the GL to a maximum determinant matrix completion problem and drive a recursive closed-form solution for the GL when the thresholded sample covariance matrix has a chordal structure. For large-scale problems with up to 450 million variables, the proposed method can solve the GL problem in less than 2 minutes, while the state-of-the-art methods converge in more than 2 hours.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
