Let M be a random (alpha n) x n matrix of rank r<<n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries with relative root mean square error RMSE <= C(rn |E|)^0.5 . Further, if r=O(1), M can be reconstructed exactly from |E| = O(n log(n)) entries. These results apply beyond random matrices to general low-rank incoherent matrices. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log(n)), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices.
Cai, Cand es and Shen @cite_0 recently proposed a low-complexity procedure to solve the convex program posed in @cite_6 . Our spectral method is akin to a single step of this procedure, with the important novelty of the trimming step that improves significantly its performances. Our analysis techniques might provide a new tool for characterizing the convex relaxation as well.
@cite_0 @cite_6
2121275167 2963537367 2157321686 2166956514
We study learning formulations with non-convex regularizaton that are natural for sparse linear models. There are two approaches to this problem: • Heuristic methods such as gradient descent that only find a local minimum. A drawback of this approach is the lack of theoretical guarantee showing that the local minimum gives a good solution. • Convex relaxation such as L1-regularization that solves the problem under some conditions. However it often leads to sub-optimal sparsity in reality. This paper tries to remedy the above gap between theory and practice. In particular, we investigate a multi-stage convex relaxation scheme for solving problems with non-convex regularization. Theoretically, we analyze the behavior of a resulting two-stage relaxation scheme for the capped-L1 regularization. Our performance bound shows that the procedure is superior to the standard L1 convex relaxation for learning sparse targets. Experiments confirm the effectiveness of this method on some simulation and real data. The problem has found applications in computer vision, computer graphics, and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of MAX-CUT. The contribution of this paper is three-fold: First, we introduce a robust penalty function involving the sum of unsquared deviations and derive a relaxation that leads to a convex optimization problem; Second, we apply the alternating direction method to minimize the penalty function; Finally, under a specific model of the measurement noise and for both complete and random measurement graphs, we prove that the rotations are exactly and stably recovered, exhibiting a phase transition behavior in terms of the proportion of noisy measurements. Numerical simulations confirm the phase transition behavior for our method as well as its improved accuracy compared to existing methods. Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method. We consider learning formulations with non-convex objective functions that often occur in practical applications. There are two approaches to this problem: Heuristic methods such as gradient descent that only find a local minimum. A drawback of this approach is the lack of theoretical guarantee showing that the local minimum gives a good solution. Convex relaxation such as L1-regularization that solves the problem under some conditions. However it often leads to a sub-optimal solution in reality. This paper tries to remedy the above gap between theory and practice. In particular, we present a multi-stage convex relaxation scheme for solving problems with non-convex objective functions. For learning formulations with sparse regularization, we analyze the behavior of a specific multi-stage relaxation scheme. Under appropriate conditions, we show that the local solution obtained by this procedure is superior to the global solution of the standard L1 convex relaxation for learning sparse targets.