  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.
