Efficient Tensor Completion Algorithms for Highly Oscillatory Operators

2025-10-20

AI Summary

This paper addresses the challenge of efficiently reconstructing highly oscillatory operators—such as $n imes n$ oscillatory matrices arising in seismic wave propagation. To this end, we propose a low-complexity tensor completion algorithm. Our method reformulates the butterfly decomposition into a logarithmic-order tensor representation and leverages tensor reshaping coupled with low-rank matrix completion to generate high-fidelity initial estimates. Subsequently, it integrates alternating least squares with gradient-based optimization for rapid convergence. The algorithm requires only $O(n log n)$ measurements, accelerates each iteration by several orders of magnitude, and achieves an overall computational complexity of $O(n log^3 n)$. Experimental results demonstrate that our approach reduces reconstruction error by one order of magnitude compared to state-of-the-art methods, significantly improving both accuracy and efficiency in structural recovery on synthetic seismic data.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Algorithm Configuration
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

This paper presents low-complexity tensor completion algorithms and their efficient implementation to reconstruct highly oscillatory operators discretized as $n imes n$ matrices. The underlying tensor decomposition is based on the reshaping of the input matrix and its butterfly decomposition into an order $mathcal{O} (log n)$ 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 $mathcal {O} (n log n)$ observed entries in the input matrix and demonstrate an $mathcal{O}(nlog^3 n)$ 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.

Problem

• Reconstruct highly oscillatory operators using tensor completion algorithms
• Achieve efficient computation with O(n log^3 n) complexity for large matrices
• Improve reconstruction accuracy with novel initialization strategy

Innovation

• Reshapes matrix into logarithmic order tensor
• Uses butterfly decomposition for tensor completion
• Applies novel initial guess via low-rank completion