Identifiability of Low-Rank Sparse Component Analysis
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Sparse component analysis (SCA) is the following problem: Given an input matrix M and an integer r, find a dictionary D with r columns and a sparse matrix B with r rows such that M ≈ DB. A key issue in SCA is identifiability, that is, characterizing the conditions under which D and B are essentially unique (that is, they are unique up to permutation and scaling of the columns of D and rows of B). Although SCA has been vastly investigated in the last two decades, only a few works have tackled this issue in the deterministic scenario, and no work provides reasonable bounds in the minimum number of data points (that is, columns of M) that leads to identifiability. In this work, we provide new results in the deterministic scenario when the data has a low-rank structure, that is, when D has rank r, drastically improving with respect to previous results. In particular, we show that if each column of B contains at least s zeros then O(r^3/s^2) data points are sufficient to obtain an essentially unique decomposition, as long as these data points are well spread among the subspaces spanned by r-1 columns of D. This implies for example that for a fixed proportion of zeros (constant and independent of r, e.g., 10% of zero entries in B), one only requires O(r) data points to guarantee identifiability.
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