2,540 research outputs found

    Multiple pattern classification by sparse subspace decomposition

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    A robust classification method is developed on the basis of sparse subspace decomposition. This method tries to decompose a mixture of subspaces of unlabeled data (queries) into class subspaces as few as possible. Each query is classified into the class whose subspace significantly contributes to the decomposed subspace. Multiple queries from different classes can be simultaneously classified into their respective classes. A practical greedy algorithm of the sparse subspace decomposition is designed for the classification. The present method achieves high recognition rate and robust performance exploiting joint sparsity.Comment: 8 pages, 3 figures, 2nd IEEE International Workshop on Subspace Methods, Workshop Proceedings of ICCV 200

    A Nonconvex Projection Method for Robust PCA

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    Robust principal component analysis (RPCA) is a well-studied problem with the goal of decomposing a matrix into the sum of low-rank and sparse components. In this paper, we propose a nonconvex feasibility reformulation of RPCA problem and apply an alternating projection method to solve it. To the best of our knowledge, we are the first to propose a method that solves RPCA problem without considering any objective function, convex relaxation, or surrogate convex constraints. We demonstrate through extensive numerical experiments on a variety of applications, including shadow removal, background estimation, face detection, and galaxy evolution, that our approach matches and often significantly outperforms current state-of-the-art in various ways.Comment: In the proceedings of Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19

    Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion

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    A spectrally sparse signal of order rr is a mixture of rr damped or undamped complex sinusoids. This paper investigates the problem of reconstructing spectrally sparse signals from a random subset of nn regular time domain samples, which can be reformulated as a low rank Hankel matrix completion problem. We introduce an iterative hard thresholding (IHT) algorithm and a fast iterative hard thresholding (FIHT) algorithm for efficient reconstruction of spectrally sparse signals via low rank Hankel matrix completion. Theoretical recovery guarantees have been established for FIHT, showing that O(r2log2(n))O(r^2\log^2(n)) number of samples are sufficient for exact recovery with high probability. Empirical performance comparisons establish significant computational advantages for IHT and FIHT. In particular, numerical simulations on 33D arrays demonstrate the capability of FIHT on handling large and high-dimensional real data

    Geometric approach to error correcting codes and reconstruction of signals

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    We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes.Comment: 17 pages, 3 figure
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