Phase transition in the spiked random tensor with Rademacher prior

Abstract

We consider the problem of detecting a deformation from a symmetric Gaussian random ptensor (p ≥ 3) with a rank-one spike sampled from the Rademacher prior. Recently in Lesieur et al. [30], it was proved that there exists a critical threshold β p so that when the signal-to-noise ratio exceeds β p , one can distinguish the spiked and unspiked tensors and weakly recover the prior via the minimal mean-square-error method. On the other side, Perry, Wein, and Bandeira [44] proved that there exists a β p < β p such that any statistical hypothesis test can not distinguish these two tensors, in the sense that their total variation distance asymptotically vanishes, when the signa-to-noise ratio is less than β p . In this work, we show that β p is indeed the critical threshold that strictly separates the distinguishability and indistinguishability between the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model with Ising spin, arising initially from the field of spin glasses. In particular, we identify the signal-to-noise criticality β p as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure p-spin mean-field spin glass model.

An Optimal Statistical and Computational Framework for Generalized Tensor Estimation

“…This work is related to a broad range of literature on tensor analysis. For example, tensor decomposition/SVD/PCA focuses on the extraction of low-rank structures from noisy tensor observations (Zhang and Golub, 2001;Richard and Montanari, 2014;Anandkumar et al, 2014;Hopkins et al, 2015;Montanari et al, 2015;Lesieur et al, 2017;Johndrow et al, 2017;Zhang and Xia, 2018;Chen, 2019). Correspondingly, a number of methods have been proposed and analyzed under either deterministic or random Gaussian noise, such as the maximum likelihood estimation (Richard and Montanari, 2014), (truncated) power iterations (Anandkumar et al, 2014;Sun et al, 2017), higher-order SVD (De Lathauwer et al, 2000a;Zhang and Xia, 2018), higher-order orthogonal iteration (HOOI) (De Lathauwer et al, 2000b;Zhang and Xia, 2018), sequential-HOSVD (Vannieuwenhoven et al, 2012), STAT-SVD (Zhang and Han, 2019).…”

Free energy in multi-species mixed $p$-spin spherical models

“…By generalizing this mechanism, Panchenko obtained variational formulas for the free energy of Potts spin glass models [61] and mixed p-spin models with vector spins [60]. The synchronization technique has since been pivotal in a variety of related models [38,27,25,43,52,47]. Using the formula produced by Panchenko in [59], the authors together with Sloman [19] studied symmetry breaking for multi-species SK models (see also [37] from the physics literature).…”

Sparse random tensors: Concentration, regularization and applications

“…Tensors have been popular data formats in machine learning and network analysis. The statistical models on tensors and the related algorithms have been widely studied in the last ten years, including tensor decomposition [1,29], tensor completion [34,48], tensor sketching [49], tensor PCA [53,17,7], and community detection on hypergraphs [36,30,50,20]. This raises the urgent demand for random tensor theory, especially the concentration inequalities in a non-asymptotic point of view.…”

“…Instead, the exact asymptotic spectral norm was given in [3] using techniques from spin glasses. This is also the starting point for a line of further research: tensor PCA and spiked tensor models under Gaussian noise, see for example [53,44,17,7]. The tools from spin glasses rely heavily on the assumption of Gaussian distribution and cannot be easily adapted to non-Gaussian cases.…”