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Pursuits in Structured Non-Convex Matrix Factorizations

Interesting !

Pursuits in Structured Non-Convex Matrix Factorizations by

Rajiv Khanna,

Michael Tschannen,

Martin Jaggi
Efficiently representing real world data in a succinct and parsimonious
manner is of central importance in many fields. We present a generalized greedy
pursuit framework, allowing us to efficiently solve structured matrix
factorization problems, where the factors are allowed to be from arbitrary sets
of structured vectors. Such structure may include sparsity, non-negativeness,
order, or a combination thereof. The algorithm approximates a given matrix by a
linear combination of few rank-1 matrices, each factorized into an outer
product of two vector atoms of the desired structure. For the non-convex
subproblems of obtaining good rank-1 structured matrix atoms, we employ and
analyze a general atomic power method. In addition to the above applications,
we prove linear convergence for generalized pursuit variants in Hilbert spaces
- for the task of approximation over the linear span of arbitrary dictionaries
- which generalizes OMP and is useful beyond matrix problems. Our experiments
on real datasets confirm both the efficiency and also the broad applicability
of our framework in practice.

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