Tensor Matched Subspace Detection

TL;DR

This paper introduces a novel tensor matched subspace detection method using transform-based tensor models, estimators, and detectors, applicable to high-dimensional tensor data, with theoretical analysis and simulation validation.

Contribution

It develops tensor matched subspace detection techniques based on tubal-sampling and elementwise-sampling, extending vector methods to tensor data with theoretical guarantees.

Findings

Estimators effectively estimate signal energy outside a subspace with sufficient samples.

Detectors perform well for both noiseless and noisy tensor data.

Simulation results confirm the approach's effectiveness.

Abstract

The problem of testing whether a signal lies within a given subspace, also named matched subspace detection, has been well studied when the signal is represented as a vector. However, the matched subspace detection methods based on vectors can not be applied to the situations that signals are naturally represented as multi-dimensional data arrays or tensors. Considering that tensor subspaces and orthogonal projections onto these subspaces are well defined in the recently proposed transform-based tensor model, which motivates us to investigate the problem of matched subspace detection in high dimensional case. In this paper, we propose an approach for tensor matched subspace detection based on the transform-based tensor model with tubal-sampling and elementwise-sampling, respectively. First, we construct estimators based on tubal-sampling and elementwise-sampling to estimate the energy of a signal outside a given subspace of a third-order tensor and then give the probability bounds of our estimators, which show that our estimators work effectively when the sample size is greater than a constant. Secondly, the detectors both for noiseless data and noisy data are given, and the corresponding detection performance analyses are also provided. Finally, based on discrete Fourier transform (DFT) and discrete cosine transform (DCT), the performance of our estimators and detectors are evaluated by several simulations, and simulation results verify the effectiveness of our approach.