Tensor Cumulants for Statistical Inference on Invariant Tensor Distributions

Core Concepts

The paper introduces a new set of objects called "tensor cumulants" that provide an explicit, near-orthogonal basis for invariant polynomials of tensors. These tensor cumulants generalize aspects of free probability theory for random matrices and enable new results for tensor PCA and distinguishing Wigner from Wishart tensors.

Abstract

The paper introduces a new set of mathematical objects called "tensor cumulants" that are useful for analyzing low-degree polynomial algorithms for statistical inference problems over invariant tensor distributions. The key insights are:

Tensor networks and graph moments provide a natural class of invariant polynomials that can be used as the building blocks for low-degree algorithms.

The tensor cumulants give an explicit, near-orthogonal basis for this space of invariant polynomials. They generalize the free cumulants from free probability theory on matrices to the tensor setting.

Using the tensor cumulants, the paper provides new results for two concrete problems:
a. Tensor PCA: The paper unifies and strengthens previous results on the computational hardness of tensor PCA below a critical signal-to-noise ratio, and gives a new tight analysis of reconstruction with low-degree polynomials.
b. Distinguishing Wigner from Wishart tensors: The paper establishes a sharp computational threshold for this problem, providing evidence of a new statistical-computational gap in the central limit theorem for random tensors.

More broadly, the tensor cumulants are shown to be valuable mathematical objects that generalize key properties of free probability theory from matrices to tensors.

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Quotes

"We unify and extend this work by considering tensor networks, orthogonally invariant polynomials where multiple copies of Y are "contracted" to produce scalars, vectors, matrices, or other tensors."
"We define a new set of objects, tensor cumulants, which provide an explicit, near-orthogonal basis for invariant polynomials of a given degree."
"Finally, we believe these cumulants are valuable mathematical objects in their own right: they generalize the free cumulants of free probability theory from matrices to tensors, and share many of their properties, including additivity under additive free convolution."

Key Insights Distilled From

Tensor cumulants for statistical inference on invariant distributions

by Dmitriy Kuni... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18735.pdf

Deeper Inquiries

What other statistical inference problems over invariant tensor distributions could benefit from the tensor cumulant framework

The tensor cumulant framework can be beneficial for various statistical inference problems over invariant tensor distributions. One such problem is tensor decomposition, where the goal is to decompose a tensor into a sum of simpler tensors. By using tensor cumulants, we can develop algorithms that can efficiently decompose tensors into their constituent parts. Additionally, problems related to tensor completion, tensor factorization, and tensor regression could also benefit from the tensor cumulant framework. These problems often involve high-dimensional data and complex relationships that can be better understood and analyzed using tensor cumulants.

How do the tensor cumulants relate to the eigenvalue structure of tensors, and can this connection be further exploited for algorithm design and analysis

Tensor cumulants have a close relationship with the eigenvalue structure of tensors. Just as eigenvalues provide important information about matrices, tensor cumulants offer insights into the structure and properties of tensors. By analyzing the tensor cumulants, we can extract valuable information about the underlying tensor, such as its rank, signal-to-noise ratio, and latent factors. This connection can be further exploited for algorithm design and analysis by incorporating tensor cumulants into optimization algorithms, spectral methods, and machine learning models. Leveraging the eigenvalue-like properties of tensor cumulants can lead to more efficient and accurate tensor analysis techniques.

Are there applications of the tensor cumulants beyond the statistical inference setting, for example in areas like quantum computing or combinatorics

Beyond statistical inference, tensor cumulants have applications in various fields such as quantum computing and combinatorics. In quantum computing, tensor cumulants can be used to analyze and manipulate multi-qubit states, tensor networks, and quantum entanglement. By understanding the cumulant structure of quantum tensors, researchers can develop quantum algorithms, error-correction techniques, and quantum information processing protocols. In combinatorics, tensor cumulants can be applied to study graph theory, network analysis, and discrete structures. The use of tensor cumulants in combinatorics can provide new insights into graph properties, connectivity patterns, and combinatorial optimization problems. Overall, the versatility of tensor cumulants makes them valuable tools in a wide range of applications beyond statistical inference.