Efficient Tensor Completion Algorithms for Highly Oscillatory Operators

TL;DR

This paper introduces efficient tensor completion algorithms tailored for highly oscillatory operators, leveraging butterfly tensor decompositions to achieve significant speedups and improved accuracy over existing methods.

Contribution

The paper develops novel tensor completion algorithms using butterfly decompositions and low-rank matrix initialization, optimized for reconstructing oscillatory operators with reduced computational complexity.

Findings

Achieves $O(n ext{log}^3 n)$ computational cost for large matrices.

Demonstrates orders of magnitude speedup per iteration compared to existing methods.

Attains reconstruction errors an order of magnitude smaller than state-of-the-art algorithms.

Abstract

This paper presents low-complexity tensor completion algorithms and their efficient implementation to reconstruct highly oscillatory operators discretized as $n\times n$ matrices. The underlying tensor decomposition is based on the reshaping of the input matrix and its butterfly decomposition into an order $O (\log n)$ tensor. The reshaping of the input matrix into a tensor allows for representation of the butterfly decomposition as a tensor decomposition with dense tensors. This leads to efficient utilization of the existing software infrastructure for dense and sparse tensor computations. We propose two tensor completion algorithms in the butterfly format, using alternating least squares and gradient-based optimization, as well as a novel strategy that uses low-rank matrix completion to efficiently generate an initial guess for the proposed algorithms. To demonstrate the efficiency and applicability of our proposed algorithms, we perform three numerical experiments using simulated oscillatory operators in seismic applications. In these experiments, we use $O (n \log n)$ observed entries in the input matrix and demonstrate an $O(n\log^3 n)$ computational cost of the proposed algorithms, leading to a speedup of orders of magnitudes per iteration for large matrices compared to the low-rank matrix and quantized tensor-train completion. Moreover, the proposed butterfly completion algorithms, equipped with the novel initial guess generation strategy, achieve reconstruction errors that are smaller by an order of magnitude, enabling accurate recovery of the underlying structure compared to the state-of-the-art completion algorithms.