On the Error Exponent of a Random Tensor with Orthonormal Factor Matrices

In signal processing, the detection error probability of a random quantity is a fundamental and often difficult problem. In this work, we assume that we observe under the alternative hypothesis a noisy tensor admitting a Tucker decomposition parametrized by a set of orthonormal factor matrices and a random core tensor of interest with fixed multilinear ranks. The detection of the random entries of the core tensor is hard to study since an analytic expression of the error probability is not tractable. To cope with this difficulty, the Chernoff Upper Bound (CUB) on the error probability is studied for this tensor-based detection problem. The tightest CUB is obtained for the minimal error exponent value, denoted by s(star), that requires a costly numerical optimization algorithm. An alternative strategy to upper bound the error probability is to consider the Bhattacharyya Upper Bound (BUB) by prescribing s(star) = 1/2. In this case, the costly numerical optimization step is avoided but no guarantee exists on the tightness optimality of the BUB. In this work, a simple analytical expression of s(star) is provided with respect to the Signal to Noise Ratio (SNR). Associated to a compact expression of the CUB, an easily tractable expression of the tightest CUB is provided and studied. A main conclusion of this work is that the BUB is the tightest bound at low SNRs but this property is no longer true at higher SNRs.