Efficient coordinate-descent for orthogonal matrices through Givens rotations
This addresses computational bottlenecks in machine learning problems like sparse-PCA and tensor decomposition, though it appears incremental as it adapts coordinate-descent to orthogonal matrices.
The authors tackled the problem of optimizing over orthogonal matrices, which is computationally expensive due to orthogonality constraints, by proposing a coordinate-descent framework using Givens rotations that preserves orthogonality efficiently. They demonstrated applications in tensor decomposition and sparse-PCA, showing faster convergence and superior performance on a genome-wide brain-wide mRNA expression dataset.
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of 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 number of entries in the learned matrix, and preserves orthogonality. We show two applications of this approach: an algorithm for tensor decomposition that is used in learning mixture models, and an algorithm for sparse-PCA. We study the parameter regime where a Givens rotation approach converges faster and achieves a superior model on a genome-wide brain-wide mRNA expression dataset.