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Optimizing over the set of orthogonal matrices is a central component inproblems like sparse-PCA or tensor decomposition. Unfortunately, suchoptimization is hard since simple operations on orthogonal matrices easilybreak orthogonality, and correcting orthogonality usually costs a large amountof computation. Here we propose a framework for optimizing orthogonal matrices,that is the parallel of coordinate-descent in Euclidean spaces. It is based on{\em Givens-rotations}, a fast-to-compute operation that affects a small numberof entries in the learned matrix, and preserves orthogonality. We show twoapplications of this approach: an algorithm for tensor decomposition that isused in learning mixture models, and an algorithm for sparse-PCA. We study theparameter regime where a Givens rotation approach converges faster and achievesa superior model on a genome-wide brain-wide mRNA expression dataset.

Efficient coordinate-descent for orthogonal matrices through Givens rotations