3 years ago

# Tensor Matched Subspace Detection.

Yue Sun, Cuiping Li, Xiao-Yang Liu

The problem of testing whether an incomplete tensor lies in a given tensor subspace is significant when it is impossible to observe all entries of a tensor. We focus on this problem under tubal-sampling and elementwise-sampling. Different from the matrix case, our problem is much more challenging due to the curse of dimensionality and different definitions for tensor operators. In this paper, the problem of matched subspace detections is discussed based on the tensor product (t-product) and tensor-tensor product with invertible linear transformation ($\mathcal{L}$-product). Based on t-product and $\mathcal{L}$-product, a tensor subspace can be defined, and the energy of a tensor outside the given subspace (also called residual energy in statistics) is bounded with high probability based on samples. For a tensor in $\mathbb{R}^{n_1\times 1\times n_3}$, the reliable detection is possible when the number of its elements we obtained is slightly greater than $r\times n_3$ both with t-product and $\mathcal{L}$-product, where $r$ is the dimension of the given tensor subspace.

Publisher URL: http://arxiv.org/abs/1710.08308

DOI: arXiv:1710.08308v1

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