Fast and Cheap Krylov-Based Covariance Smoothing

TL;DR

The paper presents TReK, a fast and memory-efficient Krylov-based algorithm for large-scale covariance tensor estimation, significantly improving computational speed and flexibility over existing methods.

Contribution

Introduction of TReK, a novel tensorized Krylov method that extends conjugate gradient with range restrictions for efficient large-scale covariance smoothing.

Findings

Achieves an order of magnitude speedup over existing methods

Ensures finite-step convergence without rounding errors

Supports a wide range of tensor operations and restrictions

Abstract

We introduce the Tensorized-and-Restricted Krylov (TReK) method, a simple and efficient algorithm for estimating covariance tensors with large observational sizes. TReK extends the conjugate gradient method to incorporate range restrictions, enabling its use in a variety of covariance smoothing applications. By leveraging matrix-level operations, it achieves significant improvements in both computational speed and memory cost, improving over existing methods by an order of magnitude. TReK ensures finite-step convergence in the absence of rounding errors and converges fast in practice, making it well-suited for large-scale problems. The algorithm is also highly flexible, supporting a wide range of forward and projection tensors.