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Blind Separation of QuasiStationary Sources: Exploiting Convex Geometry in Covariance Domain
"... Abstract—This paper revisits blind source separation of instantaneously mixed quasistationary sources (BSSQSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity ..."
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Abstract—This paper revisits blind source separation of instantaneously mixed quasistationary sources (BSSQSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity of source powers, this endows the problem with a nice convex geometry that enables elegant and efficient BSS solutions. Local dominance is tantamount to the socalled pure pixel/separability assumption in hyperspectral unmixing/nonnegative matrix factorization, respectively. Building on this link, a very simple algorithm called successive projection algorithm (SPA) is considered for estimating the mixing system in closed form. To complement SPA in the specific BSSQSS context, an algebraic preprocessing procedure is proposed to suppress shortterm source crosscorrelation interference. The proposed procedure is simple, effective, and supported by theoretical analysis. Solutions based on volume minimization (VolMin) are also considered. By theoretical analysis, it is shown that VolMin guarantees perfect mixing system identifiability under an assumption more relaxed than (exact) local dominance—which means wider applicability in practice. Exploiting the specific structure of BSSQSS, a fast VolMin algorithm is proposed for the overdetermined case. Careful simulations using real speech sources showcase the simplicity, efficiency, and accuracy of the proposed algorithms. Index Terms—Blind source separation, local dominance, purepixel, separability, volume minimization, identifiability, speech, audio. I.
Sampling Versus Unambiguous Nondeterminism in Communication Complexity
, 2014
"... In this note, we investigate the relationship between the following two communication complexity measures for a 2party function f: X × Y → {0, 1}: on one hand, sampling a uniformly random 1entry of the matrix of f such that Alice learns the row index and Bob learns the column index, and on the ot ..."
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In this note, we investigate the relationship between the following two communication complexity measures for a 2party function f: X × Y → {0, 1}: on one hand, sampling a uniformly random 1entry of the matrix of f such that Alice learns the row index and Bob learns the column index, and on the other hand, unambiguous nondeterminism, which corresponds to partitioning the 1’s of the matrix into combinatorial rectangles. The former complexity measure equals the log of the nonnegative rank of the matrix, and the latter equals the log of the binary rank of the matrix (which is always at least the nonnegative rank). Thus we consider the relationship between these two ranks of 01 matrices. We prove that if the nonnegative rank is at most 3 then the two ranks are equal. We also show a separation by exhibiting a matrix with nonnegative rank 4 and binary rank 5, as well as a family of matrices for which the binary rank is 4/3 times the nonnegative rank. 1
Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation
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A rank estimation criterion using an NMF algorithm under an inner dimension condition
, 2014
"... We introduce a rank selection criterion for nonnegative factorization algorithms, for the cases where the rank of the matrix coincides with the inner dimension of the matrix. The criteria is motivated by noting that provided that a unique factorization exists, the factorization is a solution to a f ..."
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We introduce a rank selection criterion for nonnegative factorization algorithms, for the cases where the rank of the matrix coincides with the inner dimension of the matrix. The criteria is motivated by noting that provided that a unique factorization exists, the factorization is a solution to a fixed point iteration formula that can be obtained by rewriting nonnegative factorization together with singular value decomposition. We characterize the asymptotic error rate for our fixed point formula when the nonnegative matrix is observed with noise generated according to the socalled random dot product model for graphs. 1