In this letter, we solve the problem of identifying matrices S is an element of R(n x N) and A is an element of R(m x 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.
It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C 1 functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C 1 functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex.
In many practical problems for data mining the data X under consideration (given as (m x A^)-matrix) is of the form X = AS, where the matrices A and S with dimensions mxn and nx N respectively (often called mixing matrix or dictionary and source matrix) are unknown (m < n < N). We formulate conditions (SCA-conditions) under which we can recover A and S uniquely (up to scaling and permutation), such that S is sparse in the sense that each column of S has at least one zero element. We call this the Sparse Component Analysis problem (SCA). We present new algorithms for identification of the mixing matrix (under SCAconditions), and for source recovery (under identifiability conditions). The methods are illustrated with examples showing good performance of the algorithms. Typical examples are EEC and fMRI data sets, in which the SCA algorithm allows us to detect some features of the brain signals. Special attention is given to the application of our method to the transposed system X^ = S^ A^ utilizing the sparseness of the mixing matrix A in appropriate situations. We note that the sparseness conditions could be obtained with some preprocessing methods and no independence conditions for the source signals are imposed (in contrast to Independent Component Analysis). We applied our method to fMRI data sets with dimension (128 x 128 x 98) and to EEC data sets from a 256-channels EEC machine.
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