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In this paper, we study the following nonlinear matrix decomposition (NMD)problem: given a sparse nonnegative matrix $X$, find a low-rank matrix $\Theta$such that $X \approx f(\Theta)$, where $f$ is an element-wise nonlinearfunction. We focus on the case where $f(\cdot) = \max(0, \cdot)$, the rectifiedunit (ReLU) non-linear activation. We refer to the corresponding problem asReLU-NMD. We first provide a brief overview of the existing approaches thatwere developed to tackle ReLU-NMD. Then we introduce two new algorithms: (1)aggressive accelerated NMD (A-NMD) which uses an adaptive Nesterovextrapolation to accelerate an existing algorithm, and (2) three-block NMD(3B-NMD) which parametrizes $\Theta = WH$ and leads to a significant reductionin the computational cost. We also propose an effective initialization strategybased on the nuclear norm as a proxy for the rank function. We illustrate theeffectiveness of the proposed algorithms (available on gitlab) on synthetic andreal-world data sets.

Accelerated Algorithms for Nonlinear Matrix Decomposition with the ReLU function