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Low rank approximation is a commonly occurring problem in many computervision and machine learning applications. There are two common ways ofoptimizing the resulting models. Either the set of matrices with a given rankcan be explicitly parametrized using a bilinear factorization, or low rank canbe implicitly enforced using regularization terms penalizing non-zero singularvalues. While the former approach results in differentiable problems that canbe efficiently optimized using local quadratic approximation, the latter istypically not differentiable (sometimes even discontinuous) and requires firstorder subgradient or splitting methods. It is well known that gradient basedmethods exhibit slow convergence for ill-conditioned problems.  In this paper we show how many non-differentiable regularization methods canbe reformulated into smooth objectives using bilinear parameterization. Thisallows us to use standard second order methods, such as Levenberg--Marquardt(LM) and Variable Projection (VarPro), to achieve accurate solutions forill-conditioned cases. We show on several real and synthetic experiments thatour second order formulation converges to substantially more accurate solutionsthan competing state-of-the-art methods.

Bilinear Parameterization For Differentiable Rank-Regularization