We analyze the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications, such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis, or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics—the cavity and replica methods—to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random-independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable, in principle, in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.
The blind source separation problem is to extract the underlying source signals from a set of linear mixtures, where the mixing matrix is unknown. This situation is common in acoustics, radio, medical signal and image processing, hyperspectral imaging, and other areas. We suggest a two-stage separation process: a priori selection of a possibly overcomplete signal dictionary (for instance, a wavelet frame or a learned dictionary) in which the sources are assumed to be sparsely representable, followed by unmixing the sources by exploiting the their sparse representability. We consider the general case of more sources than mixtures, but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures. Experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques. Empirical results were obtained for the blind source separation of more sources than mixtures using a previously proposed framework for learning overcomplete representations. This technique assumes a linear mixing model with additive noise and involves two steps: (1) learning an overcomplete representation for the observed data and (2) inferring sources given a sparse prior on the coefficients. We demonstrate that three speech signals can be separated with good fidelity given only two mixtures of the three signals. Similar results were obtained with mixtures of two speech signals and one music signal. In this survey, we highlight the appealing features and challenges of Sparse Component Analysis (SCA) for blind source separation (BSS). SCA is a simple yet powerful framework to separate several sources from few sensors, even when the independence assumption is dropped. So far, SCA has been most successfully applied when the sources can be represented sparsely in a given basis, but many other potential uses of SCA remain unexplored. Among other challenging perspectives, we discuss how SCA could be used to exploit both the spatial diversity corresponding to the mixing process and the morphological diversity between sources to unmix even underdetermined convolutive mixtures. This raises several challenges, including the design of both provably good and numerically efficient algorithms for large-scale sparse approximation with overcomplete signal dictionaries. Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis (PCA), and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming-based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse PCA problem. The method has complexity @math , where @math is the size of the underlying covariance matrix and @math is the desired absolute accuracy on the optimal value of the problem. The scope of this work is the separation of N sources from M linear mixtures when the underlying system is underdetermined, that is, when Mi N. If the input distribution is sparse the mixing matrix can be estimated either by external optimization or by clustering and, given the mixing matrix, a minimal l1 norm representation of the sources can be obtained by solving a low-dimensional linear programming problem for each of the data points. Yet, when the signals per se do not satisfy this assumption, sparsity can still be achieved by realizing the separation in a sparser transformed domain. The approach is illustrated here for M = 2. In this case we estimate both the number of sources and the mixing matrix by the maxima of a potential function along the circle of unit length, and we obtain the minimal l1 norm representation of each data point by a linear combination of the pair of basis vectors that enclose it. Several experiments with music and speech signals show that their time-domain representation is not sparse enough. Yet, excellent results were obtained using their short-time Fourier transform, including the separation of up to six sources from two mixtures. ? 2001 Elsevier Science B.V. All rights reserved. In this letter, we solve the problem of identifying matrices S spl isin spl Ropf sup n spl times N and A spl isin spl Ropf sup m spl times n knowing only their multiplication X = AS, under some conditions, expressed either in terms of A and sparsity of S (identifiability conditions), or in terms of X (sparse component analysis (SCA) conditions). We present algorithms for such identification and illustrate them by examples.