The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.
The large deviation principle (LDP) is studied for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the"bad"cut norm, and sufficient conditions for uniqueness of the optimizer and existence of constant optimizers are given.
Reihaneh Malekian, Sohom Bhattacharya, Nabarun Deb et al.· 0 citations
The concept of tensor eigenpairs has attracted increasing research attention in the past decades. Recent works have focused on a special class termed regular simplex tensors, which are constructed from an equiangular tight frame of n vectors in (n-1) dimensional space for n>= 3 and order m>= 3. Existing works focus on...
We study a diffusive nonlocal interaction equation generated by the squared volumes of random $n$-simplices. The nonlinear drift depends on the solution only through its mean and covariance matrix, which yields a closed finite-dimensional covariance system and an explicit representation of the measure-valued solution a...
The construction of a reliable renormalization group (RG) flow for discrete gravity models that preserves their underlying symmetry group, typically $U(N)$ or $O(N)$, remains an open problem. For random matrix models, which are the focus of this paper, this symmetry is intrinsically tied to the interactions encoding th...
We develop and implement a positive tensor-network parameterization for computing Ricci-flat K\"ahler metrics on Calabi-Yau manifolds. It replaces the large Hermitian coefficient matrix of a high-degree algebraic metric by a matrix-product factorization. For the immersed source spaces used here, the resulting metric is...
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed...
Xu-Yang Hou, Hao Guo· 0 citations
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