Joint Majorization-Minimization for Nonnegative CP and Tucker Decompositions under $\beta$-Divergences: Unfolding-Free Updates

TL;DR

This paper introduces unfolding-free majorization-minimization algorithms for nonnegative tensor decompositions under $eta$-divergences, achieving faster computations and convergence guarantees by using tensor contractions instead of mode unfoldings.

Contribution

It proposes novel separable surrogates and joint MM strategies for nonnegative CP and Tucker tensor decompositions that avoid explicit mode unfoldings and large auxiliary matrices.

Findings

Significant speedups over unfolding-based methods.

Convergence to critical points under standard assumptions.

Competitive runtime with recent einsum-based frameworks.

Abstract

We study majorization-minimization methods for nonnegative tensor decompositions under the $\beta$-divergence family, focusing on nonnegative CP and Tucker models. Our aim is to avoid explicit mode unfoldings and large auxiliary matrices by deriving separable surrogates whose multiplicative updates can be implemented using only tensor contractions (einsum-style operations). We present both classical block-MM updates in contraction-only form and a joint majorization strategy, inspired by joint MM for matrix $\beta$-NMF, that reuses cached reference quantities across inexpensive inner updates. We prove tightness of the proposed majorizers, establish monotonic decrease of the objective, and show convergence of the sequence of objective values. For block-MM, we discuss how BSUM theory applies to the analysis of stationary accumulation points. For J-CoMM, we further establish, under a set of standard regularity assumptions and for one inner sweep per outer iteration, convergence of the iterates to a critical point through a KL-based analysis. Finally, experiments on synthetic tensors and the Uber spatiotemporal count tensor demonstrate substantial speedups over unfolding-based baselines and competitive runtime relative to a recent einsum-factorization framework.