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We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range…
A classification algorithm, called the Linear Centralization Classifier (LCC), is introduced. The algorithm seeks to find a transformation that best maps instances from the feature space to a space where they concentrate towards the center…
We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of…
Reduced-rank linear discriminant analysis (RRLDA) is a foundational method of dimension reduction for classification that has been useful in a wide range of applications. The goal is to identify an optimal subspace to project the…
Sparse auto-encoders are useful for extracting low-dimensional representations from high-dimensional data. However, their performance degrades sharply when the input noise at test time differs from the noise employed during training. This…
This paper addresses the challenge of efficient principal component analysis (PCA) in high-dimensional spaces by analyzing a compressively sampled variant of Oja's algorithm with adaptive sensing. Traditional PCA methods incur substantial…
In this paper, we propose an efficient semidefinite programming (SDP) approach to worst-case linear discriminant analysis (WLDA). Compared with the traditional LDA, WLDA considers the dimensionality reduction problem from the worst-case…
This paper develops a convex approach for sparse one-dimensional deconvolution that improves upon L1-norm regularization, the standard convex approach. We propose a sparsity-inducing non-separable non-convex bivariate penalty function for…
Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components…
High-dimensional simulation optimization is notoriously challenging. We propose a new sampling algorithm that converges to a global optimal solution and suffers minimally from the curse of dimensionality. The algorithm consists of two…
L1-norm Principal-Component Analysis (L1-PCA) of real-valued data has attracted significant research interest over the past decade. However, L1-PCA of complex-valued data remains to date unexplored despite the many possible applications…
We propose the first general and scalable framework to design certifiable algorithms for robust geometric perception in the presence of outliers. Our first contribution is to show that estimation using common robust costs, such as truncated…
Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting a sequence in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric…
In high-dimensional model selection problems, penalized simple least-square approaches have been extensively used. This paper addresses the question of both robustness and efficiency of penalized model selection methods, and proposes a…
We present and analyze a simple, two-step algorithm to approximate the optimal solution of the sparse PCA problem. Our approach first solves a L1 penalized version of the NP-hard sparse PCA optimization problem and then uses a randomized…
This work proposes a research problem of finding sparse solution of undetermined Linear system with some applications. Two approaches how to solve the compressive sensing problem: using l_1 approach , the l_q approach with 0 < q < 1.…
In this paper we revisit random linear under-determined systems with sparse solutions. We consider $\ell_1$ optimization heuristic known to work very well when used to solve these systems. A collection of fundamental results that relate to…
Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the…
We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to…
Robust model fitting is a core algorithm in a large number of computer vision applications. Solving this problem efficiently for datasets highly contaminated with outliers is, however, still challenging due to the underlying computational…