A moment-based approach to the injective norm of random tensors

Abstract

In this paper, we present a technically simple method to establish upper bounds on the expected injective norm of real and complex random tensors. Our approach is somewhat analogous to the moment method in random matrix theory, and is based on a deterministic upper bound on the injective norm of a tensor which might be of independent interest. Compared to previous approaches to these problems (spin-glass methods, epsilon-net techniques, Sudakov-Fernique arguments, and PAC-Bayesian proofs), our method has the benefit of being nonasymptotic, relatively elementary, and applicable to non-Gaussian models. We illustrate our approach on various models of random tensors, recovering some previously known (and conjecturally tight) bounds with simpler arguments, and presenting new bounds, some of which are provably tight. From the perspective of statistical physics, our results yield rigorous estimates on the ground-state energy of real and complex, possibly non-Gaussian, spin glass models. From the perspective of quantum information, they establish bounds on the geometric entanglement of random bosonic states and of random states with bounded multipartite Schmidt rank, both in the thermodynamic limits as well as the regimes of large local dimensions.

Figures (5)

• Figure 1: The above plot superimposes: in red and solid lines, the result of computing (a lower bound to) the injective norm by gradient ascent for Steinhaus random tensors for various values of the dimension (the average result over $64$ realizations for each value of the local dimension $d$); and, in blue and dashed lines, the corresponding optimal non-asymptotic upper bound (see Appendix \ref{['app:non-asymptotic-upper-bound']} and Proposition \ref{['prop:finite']}).
• Figure 2: The above plot displays, for each value of $d$: in dashed lines and light blue, the values of the finite non asymptotic upper bounds; in solid lines and light red, the numerics over $40$ realizations and $35$ restarts. It also shows the limiting expectation of \ref{['eq:asymptotic_bound_bounded_rank']} as the purple line.
• Figure 3: Numerics for the average injective norm of random bounded-rank tensors at rank $R=25$. Two methods are compared: alternating least squares (red) and (noisy projected) gradient ascent method (blue). Bars show the standard deviations. The two methods obtain nearly identical values, so the red data is almost hidden behind the blue data. For each value of the dimension the average of the results is computed over $40$ realizations. Each algorithm uses $35$ restarts to improve the quality of the estimates.
• Figure 4: Plot of the relative difference $\Delta_{\text{KR-LM}}:=\frac{\lvert\sqrt{p}E_0(p)-\psi_p(\alpha_0(p)) \rvert}{\sqrt{p}E_0(p)}$ of the bounds obtained by Kac--Rice method versus our large moments approach against $p$, in the case $d_1=\ldots=d_p=d\to\infty$.
• Figure 5: In this figure, we informally summarize three upper bounds for the quantity $\mathbb{E}[\lvert \!\vert T \vert \! \rvert_{\mathrm{inj},\mathbb{R}}]$ in the case that $T$ comes from \ref{['model:ak']}, when $p$ is fixed and $d_1 = \cdots = d_p = d$ (in the $d \to +\infty$ table, the bound is for $\limsup_{d \to \infty} \mathbb{E}[\lvert \!\vert T \vert \! \rvert_{\mathrm{inj},\mathbb{R}}]$). The "Kac--Rice bound" is a numerical evaluation of the quantity referred to as $\alpha(p)$ in dartois2024injective; the "moment bound" is a numerical evaluation of the right-hand side of \ref{['eqn:sudakov-fernique-our-bound']} in the $d \to \infty$ table, and is a numerical evaluation of the right-hand side of \ref{['eqn:real-case-finite-bound']} (divided by $\sqrt{d}$) in the $d = 100$ tables; the "Sudakov--Fernique bound" is the right-hand side of \ref{['eqn:sudakov-fernique']}.

Theorems & Definitions (61)

• Definition 1.1
• Example 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of real inequality \ref{['eq:real_inj_norm_bounded']}.
• Remark 2.3
• Remark 2.4