In sparse recovery we are given a matrix @math (the dictionary) and a vector of the form @math where @math is sparse, and the goal is to recover @math . This is a central notion in signal processing, statistics and machine learning. But in applications such as sparse coding, edge detection, compression and super resolution, the dictionary @math is unknown and has to be learned from random examples of the form @math where @math is drawn from an appropriate distribution --- this is the dictionary learning problem. In most settings, @math is overcomplete: it has more columns than rows. This paper presents a polynomial-time algorithm for learning overcomplete dictionaries; the only previously known algorithm with provable guarantees is the recent work of Spielman, Wang and Wright who gave an algorithm for the full-rank case, which is rarely the case in applications. Our algorithm applies to incoherent dictionaries which have been a central object of study since they were introduced in seminal work of Donoho and Huo. In particular, a dictionary is @math -incoherent if each pair of columns has inner product at most @math . The algorithm makes natural stochastic assumptions about the unknown sparse vector @math , which can contain @math non-zero entries (for any @math ). This is close to the best @math allowable by the best sparse recovery algorithms even if one knows the dictionary @math exactly. Moreover, both the running time and sample complexity depend on @math , where @math is the target accuracy, and so our algorithms converge very quickly to the true dictionary. Our algorithm can also tolerate substantial amounts of noise provided it is incoherent with respect to the dictionary (e.g., Gaussian). In the noisy setting, our running time and sample complexity depend polynomially on @math , and this is necessary.
After this work, @cite_20 give an quasi-polynomial time algorithm for dictionary learning using sum-of-squares SDP hierarchy. The algorithm can output an approximate dictionary even when sparsity is almost linear in the dimensions with weaker assumptions.
@cite_20
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We propose an algorithm for approximately maximizing a concave function over the bounded semi-definite cone, which produces sparse solutions. Sparsity for SDP corresponds to low rank matrices, and is a important property for both computational as well as learning theoretic reasons. As an application, building on Aaronson's recent work, we derive a linear time algorithm for Quantum State Tomography. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7 8-approximation for max-sat. By solving a linear inverse problem under a sparsity constraint, one can successfully recover the coefficients, if there exists such a sparse approximation for the proposed class of signals. In this framework the dictionary can be adapted to a given set of signals using dictionary learning methods. The learned dictionary often does not have useful structures for a fast implementation, i.e. fast matrix-vector multiplication. This prevents such a dictionary being used for the real applications or large scale problems. The structure can be induced on the dictionary throughout the learning progress. Examples of such structures are shift-invariance and being multi-scale. These dictionaries can be efficiently implemented using a filter bank. In this paper a well-known structure, called compressibility, is adapted to be used in the dictionary learning problem. As a result, the complexity of the implementation of a compressible dictionary can be reduced by wisely choosing a generative model. By some simulations, it has been shown that the learned dictionary provides sparser approximations, while it does not increase the computational complexity of the algorithms, with respect to the pre-designed fast structured dictionaries.