Low-rank tensor recovery improved with adaptive gradient method
Guaranteed Low-Rank Tensor Recovery from Modewise Measurements via Normalized Block-Weighted Riemannian Gradient Descent
Machine Learning
Summary
The paper addresses how to reconstruct complex data arranged in multiple dimensions, called tensors, when only limited measurements are available. The authors introduce a method that adjusts how parts of the data are weighted during the recovery process to improve speed and reliability. Their approach uses less memory and adapts to different types of measurements, allowing faster recovery with fewer attempts. They tested the method on simulated tensors and found it works particularly well near difficult recovery limits and with specific structured measurements.
What this means in practice
- •For signal processing engineers: Accelerate reconstruction of multidimensional signals from limited or compressed measurements for applications like imaging and communications.
- •For data scientists working with multidimensional data: Improve the efficiency and reliability of inferring low-rank tensor structures from partial or noisy data sets in analytics workflows.
Authors
Yushi Zhou, Feng Zhang
Abstract
We consider the recovery of low-multilinear-rank tensors from linear measurements and propose an adaptive block-weighted modewise Riemannian gradient descent method. The method combines memory-efficient modewise measurements with a normalized adaptive weighting strategy for the core and factor components of the Riemannian gradient. The weighting improves convergence without increasing the multilinear-rank bound of the search direction or the size of the reduced core used for retraction. Under the tensor restricted isometry property and a suitable initialization, we establish local linear convergence and derive sampling guarantees for sub-Gaussian and subsampled orthogonal with random sign (SORS) measurements. Numerical experiments on synthetic low-Tucker-rank tensors show that the proposed method reduces iteration counts and computational time while maintaining reliable recovery performance, especially near the recovery threshold and for structured SORS measurements.