Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from phase retrieval and from blind deconvolution, which are designed to yield rank-1 solutions. An algorithm is described that is based on solving a certain constrained eigenvalue optimization problem that corresponds to the gauge dual which, unlike the more typical Lagrange dual, has an especially simple constraint. The dominant cost at each iteration is the computation of rightmost eigenpairs of a Hermitian operator. A range of numerical examples illustrate the scalability of the approach.
There are at least two issues that need to be addressed when comparing the FW algorithm to the approach we take here. First, as @cite_8 make clear, even in cases where low-rank solutions are expected, it is not possible to anticipate the rank of early iterates @math generated by the FW method. In particular, they observe that the rank of @math quickly increases during early iterations, and only slowly starts to decrease as the solution is approached. This motivates their development of algorithmic devices that attenuate rank growth in intermediate iterates. Any implementation, however, must be prepared to increase storage for the factors of @math during intermediate iterates. In contrast, a subgradient method applied to the gauge dual problem can be implemented with constant storage. Second, although in principle there exists a parameter @math that causes the optimization problems and to share the same solution, this parameter is not generally known in advance. One way around this is to solve a sequence of problems for varying parameters @math using, for example, a level-set procedure described by * 2016arXiv160201506A .
@cite_8
2952852844 1983068660 2189538311 2949254743
The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank- @math matrix. We propose the learning problem of a Bernoulli rank- @math bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank- @math bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards @math is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as @math . In this paper we propose Rank1ElimKL which simply replaces the (crude) confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences, and with the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of @math . Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim, while experiments with models derived from real data confirm that the improvements are significant across the board, regardless of whether the data is benign or not. We study the problem of determining if an input matrix A eRm x n can be well-approximated by a low rank matrix. Specifically, we study the problem of quickly estimating the rank or stable rank of A, the latter often providing a more robust measure of the rank. Since we seek significantly sublinear time algorithms, we cast these problems in the property testing framework. In this framework, A either has low rank or stable rank, or is far from having this property. The algorithm should read only a small number of entries or rows of A and decide which case A is in with high probability. If neither case occurs, the output is allowed to be arbitrary. We consider two notions of being far: (1) A requires changing at least an e-fraction of its entries, or (2) A requires changing at least an e-fraction of its rows. We call the former the "entry model" and the latter the "row model". We show: For testing if a matrix has rank at most d in the entry model, we improve the previous number of entries of A that need to be read from O(d2 e2) (Krauthgamer and Sasson, SODA 2003) to O(d2 e). Our algorithm is the first to adaptively query the entries of A, which for constant d we show is necessary to achieve O(1 e) queries. For the important case of d = 1 we also give a new non-adaptive algorithm, improving the previous O(1 e2) queries to O(log2(1 e) e). For testing if a matrix has rank at most d in the row model, we prove an Ω(d e) lower bound on the number of rows that need to be read, even for adaptive algorithms. Our lower bound matches a non-adaptive upper bound of Krauthgamer and Sasson. For testing if a matrix has stable rank at most d in the row model or requires changing an e d-fraction of its rows in order to have stable rank at most d, we prove that reading θ(d e2) rows is necessary and sufficient. We also give an empirical evaluation of our rank and stable rank algorithms on real and synthetic datasets. In this paper we propose a global optimization-based approach to jointly matching a set of images. The estimated correspondences simultaneously maximize pairwise feature affinities and cycle consistency across multiple images. Unlike previous convex methods relying on semidefinite programming, we formulate the problem as a low-rank matrix recovery problem and show that the desired semidefiniteness of a solution can be spontaneously fulfilled. The low-rank formulation enables us to derive a fast alternating minimization algorithm in order to handle practical problems with thousands of features. Both simulation and real experiments demonstrate that the proposed algorithm can achieve a competitive performance with an order of magnitude speedup compared to the state-of-the-art algorithm. In the end, we demonstrate the applicability of the proposed method to match the images of different object instances and as a result the potential to reconstruct category-specific object models from those images. In this paper we propose a global optimization-based approach to jointly matching a set of images. The estimated correspondences simultaneously maximize pairwise feature affinities and cycle consistency across multiple images. Unlike previous convex methods relying on semidefinite programming, we formulate the problem as a low-rank matrix recovery problem and show that the desired semidefiniteness of a solution can be spontaneously fulfilled. The low-rank formulation enables us to derive a fast alternating minimization algorithm in order to handle practical problems with thousands of features. Both simulation and real experiments demonstrate that the proposed algorithm can achieve a competitive performance with an order of magnitude speedup compared to the state-of-the-art algorithm. In the end, we demonstrate the applicability of the proposed method to match the images of different object instances and as a result the potential to reconstruct category-specific object models from those images.