Metric Learning for Semi-Supervised Sparse Source Separation with Spectral Examples

In this article, we investigate how the prior knowledge based on examples of physical spectra can be exploited in sparse Blind Source Separation (sBSS), based on the projection onto the barycentric span of these examples. For that purpose, we investigate different metrics to build such projections, and further introduce a novel machine learning approach to build physically relevant reconstruction. In this context, multi/hyperspectral data are formed of m observations Xi, each of which is made of t samples. The standard mixture model defines each observation as a linear combination of n sources Sj to which Gaussian noise is added: X=AS+N, where X ∈ R^m×t is the data matrix, S ∈ R^n×t the source matrix, A ∈ R^m×n the mixing matrix and N ∈ R^m×t for the noise contribution. BSS aims at recovering both the mixing matrix A and the sources S from the data X. only, which is an unsupervised matrix factorization. Since it is ill-posed, it requires additional assumptions about the sources and/or the mixing matrix, such as statistical independence [1], nonnegativity [2] or sparsity [3] to only name three.