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Relationship between the tensorial moment measure and tensor eigenvalues

Determine the relationship between the probability measure associated to a given tensor via its trace-invariant moments and the eigenvalues of that tensor, clarifying whether and how this measure corresponds to any natural notion of tensor spectrum.

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Background

The paper introduces a probability measure built from tensor trace invariants and develops tensorial free convolution tools. However, the authors state that the connection between this measure and tensor eigenvalues is not yet understood.

Clarifying this link would connect the combinatorial/invariant-based measure μ_T with spectral notions for tensors (for which multiple eigenvalue concepts exist), and would be vital for spectral interpretation and applications.

References

These developments will be the subject of further works, jointly with a better understanding of the link between this measure and the eigenvalues of a given tensor, which is still unclear.

— Tensorial free convolution, semicircular, free Poisson and R-transform in high order (2412.02572 - Bonnin, 3 Dec 2024) in Section 1 (Introduction), end of Subsection “Main results”