Tootfinder

The border rank of the $4 \times 4$ determinant tensor is twelve
Jong In Han, Jeong-Hoon Ju, Yeongrak Kim
https://arxiv.org/abs/2510.11051 https://arxiv.or…

The border rank of the $4 \times 4$ determinant tensor is twelve
We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.

Stress Response of Jammed Solids: Prestress and Screening
Surajit Chakraborty, Jishnu N. Nampoothiri, Subhro Bhattacharjee, Bulbul Chakraborty, Kabir Ramola
https://arxiv.org/abs/2509.14336

Stress Response of Jammed Solids: Prestress and Screening
Unlike classical elasticity, where stresses arise from deformations relative to a stress-free reference configuration, rigidity in amorphous systems is maintained by disordered force networks that generate internal prestress. Previously, we introduced a ''stress-only'' formulation, where mechanical equilibrium resembles Gauss's law in a rank-2 tensor electrostatics with vector charges, and demonstrated that the mechanical response of jammed solids is described by the dielectric response of this…

Tensor Completion via Monotone Inclusion: Generalized Low-Rank Priors Meet Deep Denoisers
Peng Chen, Deliang Wei, Jiale Yao, Fang Li
https://arxiv.org/abs/2510.12425 https://

Tensor Completion via Monotone Inclusion: Generalized Low-Rank Priors Meet Deep Denoisers
Missing entries in multi dimensional data pose significant challenges for downstream analysis across diverse real world applications. These data are naturally modeled as tensors, and recent completion methods integrating global low rank priors with plug and play denoisers have demonstrated strong empirical performance. However, these approaches often rely on empirical convergence alone or unrealistic assumptions, such as deep denoisers acting as proximal operators of implicit regularizers, whic…

A continued fraction approximation for the effective elasticity tensor of two-dimensional polycrystals as a function of the crystal elasticity tensor
Graeme W. Milton
https://arxiv.org/abs/2510.10297

A continued fraction approximation for the effective elasticity tensor of two-dimensional polycrystals as a function of the crystal elasticity tensor
For two-dimensional polycrystals the effective elasticity tensor $C_*$ as a function $C_*(C_0)$ of the elasticity tensor $C_0$ of the constituent crystal is considered. It is shown that this function can be approximated by one with a continued fraction expansion resembling that associated with a class of microstructure known as sequential laminates. These are hierarchical microstructures defined inductively. Rank 0 sequential laminates are simply rotations of the pure crystal. Rank $j$ sequenti…

LR-WaveHoltz: A Low-Rank Helmholtz Solver
Andreas Granath, Daniel Appel\"o, Siyang Wang
https://arxiv.org/abs/2510.09352 https://arxiv.org/pdf/2510.09…

LR-WaveHoltz: A Low-Rank Helmholtz Solver
We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave equation is discretized by high-order multiblock summation-by-parts finite differences. In two dimensions we use the singular value decomposition and in three dimensions we use tensor trains to compress the numerical solution. To control rank growth we use step-trun…

Scalable Quantum Monte Carlo Method for Polariton Chemistry via Mixed Block Sparsity and Tensor Hypercontraction Method
Yu Zhang
https://arxiv.org/abs/2510.11634 https://…

Scalable Quantum Monte Carlo Method for Polariton Chemistry via Mixed Block Sparsity and Tensor Hypercontraction Method
We present a reduced-scaling auxiliary-field quantum Monte Carlo (AFQMC) framework designed for large molecular systems and ensembles, with or without coupling to optical cavities. Our approach leverages the natural block sparsity of Cholesky decomposition (CD) of electron repulsion integrals in molecular ensembles and employs tensor hypercontraction (THC) to efficiently compress low-rank Cholesky blocks. By representing the Cholesky vectors in a mixed format, keeping high-rank blocks in block-…

Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold
Ke Xu, Yuefeng Han
https://arxiv.org/abs/2510.00569 https://arxiv.org…

Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold
Recovering a low-CP-rank tensor from noisy linear measurements is a central challenge in high-dimensional data analysis, with applications spanning tensor PCA, tensor regression, and beyond. We exploit the intrinsic geometry of rank-one tensors by casting the recovery task as an optimization problem over the Segre manifold, the smooth Riemannian manifold of rank-one tensors. This geometric viewpoint yields two powerful algorithms: Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (R…

Compressed Bayesian Tensor Regression
Roberto Casarin, Radu Craiu, Qing Wang
https://arxiv.org/abs/2510.01861 https://arxiv.org/pdf/2510.01861

Compressed Bayesian Tensor Regression
To address the common problem of high dimensionality in tensor regressions, we introduce a generalized tensor random projection method that embeds high-dimensional tensor-valued covariates into low-dimensional subspaces with minimal loss of information about the responses. The method is flexible, allowing for tensor-wise, mode-wise, or combined random projections as special cases. A Bayesian inference framework is provided featuring the use of a hierarchical prior distribution and a low-rank re…

Tensor rank of the determinant and periodic triangulations of $\mathbb{R}^n$
Sergey Avvakumov, Roman Karasev
https://arxiv.org/abs/2509.22333 https://arxiv…

Tensor rank of the determinant and periodic triangulations of $\mathbb{R}^n$
We prove that in any $\mathbb{Z}^n$-periodic triangulation of $\mathbb{R}^n$ the number of $\mathbb{Z}^n$-orbits of $n$-dimensional simplices is at least the tensor rank of the $n$th determinant tensor. The latter is known to be at least $\frac{n^{n-1}}{(n-1)!}$, which is approximately $\frac{e^n}{\sqrt{2πn}}$ for large $n$. The triangulation is not assumed to be geometric, meaning that its simplices can be ``curved''. We also provide lower bounds for general spaces. A simplicial cell comple…

Apparent Lorentz violation from disformally coupled ultralight dark matter
Guillem Dom\`enech, Apostolos Tsabodimos, Nathaniel Sherrill, Alexander Ganz, Fiona Kirk
https://arxiv.org/abs/2510.07490

Apparent Lorentz violation from disformally coupled ultralight dark matter
We study the impact of general disformal metric transformations on fermions, which shift the gravitational metric by an additional rank-2 tensor. This tensor can in principle be constructed from scalar-field gradients, vector fields, or field-strength contractions. We show this transformation results in the conventional Dirac action being modified by additional kinetic and axial-current couplings that are quadratic in the shifted field. When the field sourcing the metric shift takes on a non-tr…

Horospherical mean curvature functions and D'Atri spaces
Gerhard Knieper, JeongHyeong Park, Norbert Peyerimhoff
https://arxiv.org/abs/2510.04572 https://

Horospherical mean curvature functions and D'Atri spaces
We consider simply connected Riemannian manifolds without conjugate points for which the horospherical mean curvature function is continuous, reversible and invariant under the geodesic flow. We show under mild additional curvature tensor conditions that rank one manifolds in this family are automatically asymptotically harmonic. In particular, compact rank one manifolds of this kind must be locally symmetric spaces of negative curvature. Moreover, we show under the same conditions that rank on…

Tensor Product Decompositions, Limits in Excellent Filtrations, Affine Weyl Group Orbits, and Tableaux Counting
Laura Estivalez, Adriano Moura
https://arxiv.org/abs/2510.04697 h…

Tensor Product Decompositions, Limits in Excellent Filtrations, Affine Weyl Group Orbits, and Tableaux Counting
We express the outer multiplicities in the tensor products of two fundamental simple modules for an affine Kac-Moody algebra of type $A$ in terms of counting certain sets of multipartitions by exploring the stabilizing limits of certain excellent filtrations. This extends for all ranks a previously obtained result by Jakelić and the second author for rank $1$. The same outer multiplicities were previously computed by Misra and Wilson in terms of counting certain sets of tableaux. By comparing …

Bias-variance Tradeoff in Tensor Estimation
Shivam Kumar, Haotian Xu, Carlos Misael Madrid Padilla, Yuehaw Khoo, Oscar Hernan Madrid Padilla, Daren Wang
https://arxiv.org/abs/2509.17382

Bias-variance Tradeoff in Tensor Estimation
We study denoising of a third-order tensor when the ground-truth tensor is not necessarily Tucker low-rank. Specifically, we observe $$ Y=X^\ast+Z\in \mathbb{R}^{p_{1} \times p_{2} \times p_{3}}, $$ where $X^\ast$ is the ground-truth tensor, and $Z$ is the noise tensor. We propose a simple variant of the higher-order tensor SVD estimator $\widetilde{X}$. We show that uniformly over all user-specified Tucker ranks $(r_{1},r_{2},r_{3})$, $$ \| \widetilde{X} - X^* \|_{ \mathrm{F}}^2 = O …

Symmetric Persistent Tensors and their Hessian
Masoud Gharahi, Giorgio Ottaviani
https://arxiv.org/abs/2510.07404 https://arxiv.org/pdf/2510.07404

Symmetric Persistent Tensors and their Hessian
Persistent tensors, introduced in [Quantum 8 (2024), 1238], and inspired by quantum information theory, form a recursively defined class of tensors that remain stable under the substitution method and thereby yield nontrivial lower bounds on tensor rank. In this work, we investigate the symmetric case-namely, symmetric persistent tensors, or equivalently, persistent polynomials. We establish that a symmetric tensor in $\mathrm{Sym}^n \mathbb{C}^d$ is persistent if the determinant of its Hessian…