[Paper Review] Further Results on Cauchy Tensors and Hankel Tensors

Haibin Chen, Guoyin Li+1 more|arXiv (Cornell University)|Jan 27, 2015

Tensor decomposition and applications27 references4 citations

TL;DR

This paper introduces generalized Cauchy tensors and establishes that positive semi-definite Cauchy tensors are equivalent to SOS (sum-of-squares) tensors and completely positive tensors, meaning they can be decomposed into nonnegative rank-1 tensors. It further proves that even-order Cauchy-Hankel tensors are positive definite if and only if their generating parameters satisfy specific positivity conditions, and links positive semi-definiteness of Hankel tensors to their associated plane tensors and SOS properties under rank constraints.

ABSTRACT

In this article, we present various new results on Cauchy tensors and Hankel tensors. { We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries.} As a consequence, we show that Cauchy tensors are positive semi-definite if and only if they are SOS (Sum-of-squares) tensors.} Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed { as} the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. { We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor $\mathcal{A}$ is less than the dimension of the underlying space, then positive semi-definiteness of $\mathcal{A}$ is equivalent to the fact that $\mathcal{A}$ is a complete Hankel tensor, and so, is further equivalent to the SOS property of $\mathcal{A}$. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously.} Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.

Motivation & Objective

• To extend the theory of Cauchy tensors by introducing generalized Cauchy tensors with nonzero entries.
• To characterize positive semi-definiteness and complete positivity of generalized Cauchy tensors.
• To establish the equivalence between positive semi-definiteness, SOS property, and complete positivity for even-order Cauchy tensors.
• To investigate the relationship between Hankel tensor positive semi-definiteness and its associated plane tensor, especially under Vandermonde rank constraints.
• To define and analyze Cauchy-Hankel tensors, establishing necessary and sufficient conditions for their positive definiteness.

Proposed method

• Introduce generalized Cauchy tensors as an extension of standard Cauchy tensors using generating vectors with specific functional forms.
• Use algebraic tensor decomposition and positivity analysis to prove that positive semi-definite Cauchy tensors are SOS tensors.
• Establish that even-order Cauchy tensors are completely positive by showing they admit nonnegative rank-1 decompositions.
• Define and analyze Hankel tensors via their associated plane tensors and Vandermonde structure, linking positive semi-definiteness to plane tensor properties.
• Introduce Cauchy-Hankel tensors as a hybrid class combining Cauchy and Hankel structures, and derive parameter-based conditions for positive definiteness.
• Apply monotonicity analysis of homogeneous polynomials to characterize positive definiteness via strict monotonicity in the positive orthant.

Experimental results

Research questions

• RQ1 Under what conditions is a generalized Cauchy tensor with nonzero entries positive semi-definite?
• RQ2 Is every positive semi-definite Cauchy tensor equivalent to an SOS tensor or a completely positive tensor?
• RQ3 What is the relationship between the positive semi-definiteness of an even-order Hankel tensor and the positive semi-definiteness of its associated plane tensor?
• RQ4 When is the positive semi-definiteness of a Hankel tensor equivalent to it being a complete Hankel tensor or an SOS tensor?
• RQ5 What are the necessary and sufficient conditions for an even-order Cauchy-Hankel tensor to be positive definite?

Key findings

• All positive semi-definite Cauchy tensors are SOS tensors, meaning they can be expressed as sums of squares of homogeneous polynomials.
• An even-order Cauchy tensor is positive semi-definite if and only if it is completely positive, i.e., decomposable into nonnegative rank-1 tensors.
• All H-eigenvalues of nonnegative Cauchy tensors are nonnegative, confirming spectral nonnegativity.
• An even-order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite.
• If the Vandermonde rank of a Hankel tensor is less than the dimension of the underlying space, then its positive semi-definiteness is equivalent to it being a complete Hankel tensor and an SOS tensor.
• An even-order Cauchy-Hankel tensor is positive definite if and only if the parameters satisfy $ g + mh > 0 $ and $ g + nmh > 0 $.