![]() RANK OF A MATRIX FULLWe do note that in some cases, the matrices A U and A V will not have full column rank, meaning that the LLS solution is nonunique for example, if V is initialized to be identically zero, then A V is also identically zero. However, LLS problems are classical, and a number of efficient algorithms exist to compute solutions. The main computation in the PF procedure is solving the LLS problems in ( 8) and ( 9). In this incremented-rank version of PF (IRPF), we initialize the new components of U and V using a rank-1 PF fit to the current residual. Otherwise, repeat steps 2)–4).įor matrix recovery, we have obtained best results by starting PF with r = 1, and gradually incrementing until the desired rank constraint is achieved (or until the point where the relative error is smaller than ε in the case where the true rank is unknown). If q exceeds a maximum number of iterations, if the iterations stagnate, or if the relative error ∥□(U ( q)V ( q))−b∥ 2 ∕ ∥b∥ 2 is smaller than a desired threshold ε, then terminate the iterative procedure. In contrast to NNM methods, PF optimizes U and V in alternation to find a local solution to This type of low-rank parameterization has been used in previous NNM algorithms, to improve computational efficiency at the expense of introducing nonconvexity. PF seeks to find a matrix that can be factored as X = UV, with U ∈ ℂ m× r and V ∈ ℂ r× n so that rank( X) ≤ r. In this work, we propose to use the PowerFactorization (PF) algorithm to solve a rank-constrained analogue of ( 1). As a result, various fast algorithms for NNM have appeared,. However, for large problem sizes, these methods are limited by large computation and storage requirements. Nuclear norm minimization (NNM) can be cast as a semidefinite programming problem (SDP), and can be solved using off-the-shelf interior point solvers like SDPT3 or SeDuMi. Similar to results in compressed sensing, theoretical conditions for the equivalence of ( 1) and ( 2) have been derived -, and rely on □ having special restricted isometry or incoherence properties. The nuclear norm is convex, and its use as a surrogate for matrix rank is analogous to the way that the l 1 norm is used as a surrogate for vector sparsity in the emerging field of compressed sensing. Where ∥X∥ ∗ is the nuclear norm of X, and is defined as the sum of its singular values. ![]()
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