On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses (2503.10580v1)

Abstract: We prove an upper bound on the expected $\ell_p$ injective norm of sums of subgaussian random tensors. Our proof is simple and does not rely on any explicit geometric or chaining arguments. Instead, it follows from a simple application of the PAC-Bayesian lemma, a tool that has proven effective at controlling the suprema of certain ``smooth'' empirical processes in recent years. Our bound strictly improves a very recent result of Bandeira, Gopi, Jiang, Lucca, and Rothvoss. In the Euclidean case ($p=2$), our bound sharpens a result of Lata{\l}a that was central to proving his estimates on the moments of Gaussian chaoses. As a consequence, we obtain an elementary proof of this fundamental result.

Analysis of the Injective Norm of Sums of Random Tensors and Gaussian Chaoses

The paper under discussion rigorously explores the behavior of the expected $\ell_p$ injective norm of sums of subgaussian random tensors. This research is essential as it addresses gaps in our understanding of non-Euclidean tensor norms which frequently arise in probability theory, computer science, and various statistical models.

Core Contributions

One of the paper's pivotal achievements is establishing an upper bound for the expected $\ell_p$ injective norm of the sum of random subgaussian tensors. The derived bound improves upon recent results by Bandeira et al. by offering a more precise bound for the Euclidean case, refining Latała's earlier estimates on Gaussian chaoses' moments. A central innovation in the paper is the use of the PAC-Bayesian lemma, which circumvents traditional geometric arguments and chaining techniques, illustrating its effectiveness in managing the suprema of certain "smooth" empirical processes.

Detailed Analysis

Random tensors of form $T = \sum_{k=1}^n \xi_k T_k$ are considered, where $\xi_k$ are independent zero-mean subgaussian random variables, and $T_k$ are deterministic order $r$ tensors. The paper focuses on the $\ell_p$ injective norm of $T$, given by: $$\|T\|_p = \sup_{x_1 \in B_p^{d_1}, \dots, x_r \in B_p^{d_r}} T(x_1, \dots, x_r),$$ where $B_p^d$ is the $\ell_p$ unit ball in $\mathbb{R}^d$. This investigation is non-trivial, especially for higher-order tensors, due to the failure of conventional methods like matrix trace inequalities to extend beyond the matrix setting.

The paper introduces new variance parameters to support these findings. These parameters are pivotal in expressing the results in terms applicable to observed systems where Frobenius norms and other statistical measures play a significant role. The main theorem provides a bound dependent on these variance parameters and is particularly influential for symmetric tensors previously studied by Bandeira et al.

Theoretical and Practical Implications

The results theorized in this discussion have substantial implications. Practically, improved bounds on random tensor norms can impact machine learning, especially for models relying on high-dimensional data representations, offering more efficient algorithms and better analytical tools. Theoretically, these findings contribute to the foundational understanding of random tensors, extending the scope of tensor analysis in high-dimensional probability.

Further, the paper achieves an elementary and accessible proof of Latała’s estimates on Gaussian chaoses’ moments. This underscores the potential for new superficial analysis tools, like the PAC-Bayesian lemma, to provide clarity in other complex settings where classical methods might fail or be cumbersome.

Future Directions

The research opens various pathways for future exploration. Firstly, applying the PAC-Bayesian approach to other forms of empirical processes could yield similarly simplified and robust results. Furthermore, the gap between practical dimensional bounds and theoretical limits remains an area ripe for exploration, particularly in understanding how these findings translate to real-world application scenarios in modern AI systems where tensor computations are routine. Extending this analysis to explore interactions between high-order tensor structures and other random matrix forms might provide insights applicable across disciplines such as quantum computing and statistical mechanics.

In conclusion, this paper sets a significant milestone in the paper of random tensors and their norms, offering both practical tools and theoretical insights into high-dimensional probability and stochastic process analysis.