Large deviation analysis of the CPD detection problem based on random tensor theory

The performance in terms of minimal Bayes' error probability for detection of a random tensor is a fundamental understudied difficult problem. In this work, we assume that we observe under the alternative hypothesis a noisy rank-ñ tensor admitting a Q-order Canonical Polyadic Decomposition (CPD) with large factors of size Nq × R, i.e., for 1 ≤ q ≤ Q, R,Nq → ∞ with R1/q/Nq converges to a finite constant. The detection of the random entries of the core tensor is hard to study since an analytic expression of the error probability is not easily tractable. To mitigate this technical difficulty, the Chernoff Upper Bound (CUB) and the error exponent on the error probability are derived and studied for the considered tensor-based detection problem. These two quantities are relied to a key quantity for the considered detection problem due to its strong link with the moment generating function of the log-likelihood test. However, the tightest CUB is reached for the value, denoted by s∗, which minimizes the error exponent. To solve this step, two methodologies are standard in the literature. The first one is based on the use of a costly numerical optimization algorithm. An alternative strategy is to consider the Bhattacharyya Upper Bound (BUB) for s∗ = 1/2. In this last scenario, the costly numerical optimization step is avoided but no guaranty exists on the optimality of the BUB. Based on powerful random matrix theory tools, a simple analytical expression of s∗ is provided with respect to the Signal to Noise Ratio (SNR) and for low rank CPD. Associated to a compact expression of the CUB, an easily tractable expression of the tightest CUB and the error exponent are provided and analyzed. A main conclusion of this work is that the BUB is the tightest bound at low SNRs. At contrary, this property is no longer true for higher SNRs.