Preprint Article Version 1 This version not peer reviewed

# Computational Information Geometry For Binary Classification of High-Dimensional Random Tensors

Version 1 : Received: 1 February 2018 / Approved: 1 February 2018 / Online: 1 February 2018 (16:32:04 CET)

How to cite: Pham, G..; Boyer, R..; Nielsen, F. Computational Information Geometry For Binary Classification of High-Dimensional Random Tensors. Preprints 2018, 2018020008 (doi: 10.20944/preprints201802.0008.v1). Pham, G..; Boyer, R..; Nielsen, F. Computational Information Geometry For Binary&nbsp;Classification of High-Dimensional&nbsp;Random Tensors. Preprints 2018, 2018020008 (doi: 10.20944/preprints201802.0008.v1).

## Abstract

The performance in terms of minimal Bayes’ error probability for detection of a
high-dimensional random tensor is a fundamental under-studied difficult problem. In this work, we
consider two Signal to Noise Ratio (SNR)-based detection problems of interest. Under the alternative
hypothesis, i.e., for a non-zero SNR, the observed signals are either a noisy rank-R tensor admitting a
Q-order Canonical Polyadic Decomposition (CPD) with large factors of size Nq  R, i.e, for 1  q  Q,
where R, Nq ! ¥ with R1/q/Nq converge towards a finite constant or a noisy tensor admitting
TucKer Decomposition (TKD) of multilinear (M1, . . . ,MQ)-rank with large factors of size Nq  Mq,
i.e, for 1  q  Q, where Nq,Mq ! ¥ with Mq/Nq converge towards a finite constant. The detection
of the random entries (coefficients) of the core tensor in the CPD/TKD is hard to study since the
exact derivation of the error probability is mathematically intractable. To circumvent this technical
difficulty, the Chernoff Upper Bound (CUB) for larger SNR and the Fisher information at low SNR
are derived and studied, based on information geometry theory. The tightest CUB is reached for
the value minimizing the error exponent, denoted by s?. In general, due to the asymmetry of the
s-divergence, the Bhattacharyya Upper Bound (BUB) (that is, the Chernoff Information calculated at
s? = 1/2) can not solve this problem effectively. As a consequence, we rely on a costly numerical
optimization strategy to find s?. However, thanks to powerful random matrix theory tools, a simple
analytical expression of s? is provided with respect to the Signal to Noise Ratio (SNR) in the two
schemes considered. A main conclusion of this work is that the BUB is the tightest bound at low
SNRs. This property is, however, no longer true for higher SNRs.

## Subject Areas

Optimal Bayesian detection, information geometry, minimal error probability, Chernoff/Bhattacharyya upper bound, large random tensor, Fisher information, large random sensing matrix