We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.
There are several algorithm suggested and tested for dictionary learning, see e.g. @cite_18 @cite_17 @cite_8 @cite_4 . The algorithm we derive in this paper is closely related but different to the bilinear AMP proposed by @cite_2 as explained in Sec. .
@cite_18 @cite_4 @cite_8 @cite_2 @cite_17
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In this paper, we extend the generalized approximate message passing (G-AMP) approach, originally proposed for high-dimensional generalized-linear regression in the context of compressive sensing, to the generalized-bilinear case, which enables its application to matrix completion, robust PCA, dictionary learning, and related matrix-factorization problems. Here, in Part I of a two-part paper, we derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the sum-product belief propagation algorithm in the high-dimensional limit, where central-limit theorem arguments and Taylor-series approximations apply, and under the assumption of statistically independent matrix entries with known priors. In addition, we propose an adaptive damping mechanism that aids convergence under finite problem sizes, an expectation-maximization (EM)-based method to automatically tune the parameters of the assumed priors, and two rank-selection strategies. In Part II of the paper, we will discuss the specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA, and dictionary learning, and we will present the results of an extensive empirical study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem. Algorithms for data-driven learning of domain-specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of Bayesian models with concave Schur-concave (CSC) negative log priors. Such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen (environmentally matched) dictionary. The elements of the dictionary can be interpreted as concepts, features, or words capable of succinct expression of events encountered in the environment (the source of the measured signals). This is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries (the proverbial "25 words or less"), but not necessarily as succinct as one entry. To learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment, we develop algorithms that iterate between a representative set of sparse representations found by variants of FOCUSS and an update of the dictionary using these sparse representations.Experiments were performed using synthetic data and natural images. For complete dictionaries, we demonstrate that our algorithms have improved performance over other independent component analysis (ICA) methods, measured in terms of signal-to-noise ratios of separated sources. In the overcomplete case, we show that the true underlying dictionary and sparse sources can be accurately recovered. In tests with natural images, learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries; that is, images encoded with an overcomplete dictionary have both higher compression (fewer bits per pixel) and higher accuracy (lower mean square error). We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^ Omega(n) size. 2. Any multilinear depth three circuit computing IMM_ n,d (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^ Omega(d) size. IMM_ n,d can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^ Omega(d) . This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field. 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.