Tootfinder

A Low-Rank tensor framework for THB-Splines
Tom-Christian Riemer, Martin Stoll
https://arxiv.org/abs/2509.09434 https://arxiv.org/pdf/2509.09434

A Low-Rank tensor framework for THB-Splines
We introduce a low-rank framework for adaptive isogeometric analysis with truncated hierarchical B-splines (THB-splines) that targets the main bottleneck of local refinement: memory- and time-intensive matrix assembly once the global tensor-product structure is lost. The method interpolates geometry-induced weight and source terms in separable spline spaces and computes their tensor-train (TT) representations via the alternating minimal energy (AMEn) solver, enabling level-wise assembly of syst…

Minimality of Tree Tensor Network Ranks
Jana Jovcheva, Tim Seynnaeve, Nick Vannieuwenhoven
https://arxiv.org/abs/2509.09463 https://arxiv.org/pdf/2509.0946…

Minimality of Tree Tensor Network Ranks
For a given tree tensor network $G$, we call a tuple of bond dimensions minimal if there exists a tensor $T$ that can be represented by this network but not on the same tree topology with strictly smaller bond dimensions. We establish necessary and sufficient conditions on the bond dimensions of a tree tensor network to be minimal, generalizing a characterization of Carlini and Kleppe about existence of tensors with a given multilinear rank. We also show that in a minimal tree tensor network, t…

OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case
Ratna Khatri, Anthony Kolshorn, Colin Olson, Harbir Antil
https://arxiv.org/abs/2509.08169

OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case
This paper presents a novel, mathematically rigorous framework for autoencoder-type deep neural networks that combines optimal control theory and low-rank tensor methods to yield memory-efficient training and automated architecture discovery. The learning task is formulated as an optimization problem constrained by differential equations representing the encoder and decoder components of the network and the corresponding optimality conditions are derived via a Lagrangian approach. Efficient mem…

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…

The Nondecreasing Rank
Andrew McCormack
https://arxiv.org/abs/2509.00265 https://arxiv.org/pdf/2509.00265

The Nondecreasing Rank
In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity constraint. It is shown that for certain poset orderings finding an ND factorization of rank $r$ is equivalent to finding a nonnegative rank-r factorization of a transformed tensor. However, not every tensor that is monotonic has a finite ND rank. Theory is develope…

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…

TeRA: Vector-based Random Tensor Network for High-Rank Adaptation of Large Language Models
Yuxuan Gu, Wuyang Zhou, Giorgos Iacovides, Danilo Mandic
https://arxiv.org/abs/2509.03234

TeRA: Vector-based Random Tensor Network for High-Rank Adaptation of Large Language Models
Parameter-Efficient Fine-Tuning (PEFT) methods, such as Low-Rank Adaptation (LoRA), have significantly reduced the number of trainable parameters needed in fine-tuning large language models (LLMs). Subsequent developments of LoRA-style adapters have diverged into two main directions: (1) enhancing model expressivity with high-rank adapters, and (2) pushing for further parameter reduction, as exemplified by vector-based methods. However, these approaches present a trade-off, as achieving the exp…

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…

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…

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 …

Restarting the Numerical Flow Iteration through low rank tensor approximations
Rostislav-Paul Wilhelm, Katharina Kormann
https://arxiv.org/abs/2509.08474 https://

Restarting the Numerical Flow Iteration through low rank tensor approximations
The numerical flow iteration method has recently been proposed as a memory-slim solution method for the Vlasov-Poisson equation. It stores the temporal evolution of the electric field and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. If the number of time steps gets large, the computational cost of this reconstruction may get prohibitive. Given a representation of the intermediate sol…

Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions
Divyansh Agrawal, Venkateswaran P. Krishnan, Vladimir A. Sharafutdinov
https://arxiv.org/abs/2509.03046

Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions
The ray transform $I_m$ integrates a symmetric $m$ rank tensor field $f$ on $\mathbb{R}^n$ over lines. In the case of $n\ge3$, the range characterization of the operator $I_m$ on weighted Sobolev spaces $H^{s}_t({\mathbb R}^n;S^m{\mathbb R}^n)$ was obtained in [V. Krishnan and V. Sharafutdinov. Range characterization of ray transform on Sobolev spaces of symmetric tensor fields. Inverse Problems and Imaging, 18(6), 1272--1293, 2024]. Here we obtain a range characterization result in higher orde…

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…

Low-rank matrix and tensor approximations: advancing efficiency of machine-learning interatomic potentials
Igor Vorotnikov, Fedor Romashov, Nikita Rybin, Maxim Rakhuba, Ivan S. Novikov
https://arxiv.org/abs/2509.04440

Low-rank matrix and tensor approximations: advancing efficiency of machine-learning interatomic potentials
Machine-learning interatomic potentials (MLIPs) have become a mainstay in computationally-guided materials science, surpassing traditional force fields due to their flexible functional form and superior accuracy in reproducing physical properties of materials. This flexibility is achieved through mathematically-rigorous basis sets that describe interatomic interactions within a local atomic environment. The number of parameters in these basis sets influences both the size of the training datase…

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…

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 …

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 $12$.

An Inexact Tensor-Train Primal-Dual Interior-Point Method for Semidefinite Programs
Frederik Kelbel, Sergey Dolgov, Dante Kalise, Alessandra Russo
https://arxiv.org/abs/2509.11890

An Inexact Tensor-Train Primal-Dual Interior-Point Method for Semidefinite Programs
In this work, we introduce an interior-point method that employs tensor decompositions to efficiently represent and manipulate the variables and constraints of semidefinite programs, targeting problems where the solutions may not be low-rank but admit low-tensor-train rank approximations. Our method maintains approximate superlinear convergence despite inexact computations in the tensor format and leverages a primal-dual infeasible interior-point framework. In experiments on Maximum Cut, Maximu…

Sum-of-Gaussians tensor neural networks for high-dimensional Schr\"odinger equation
Qi Zhou, Teng Wu, Jianghao Liu, Qingyuan Sun, Hehu Xie, Zhenli Xu
https://arxiv.org/abs/2508.10454

Sum-of-Gaussians tensor neural networks for high-dimensional Schrödinger equation
We propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schrödinger equation. The SOG-TNN utilizes a low-rank tensor product representation of the solution to overcome the curse of dimensionality associated with high-dimensional integration. To handle the Coulomb interaction, we introduce an SOG decomposition to approximate the interaction kernel such that it is dimensionally separable, leading to a tensor re…

An Inexact Low-Rank Source Iteration for Steady-State Radiative Transfer Equation with Diffusion Synthetic Acceleration
Wei Guo, Zhichao Peng
https://arxiv.org/abs/2509.00805 ht…

An Inexact Low-Rank Source Iteration for Steady-State Radiative Transfer Equation with Diffusion Synthetic Acceleration
We propose an inexact low-rank source iteration with diffusion synthetic acceleration (SI-DSA) for solving the multidimensional steady-state radiative transfer equation (RTE) in the second-order formulation. The angular flux is represented in either a low-rank matrix or hierarchical Tucker tensor (HTT) format, enabling substantial reductions in computational resources. Each SI step is solved using a preconditioned low-rank conjugate gradient (CG) method with a diffusion preconditioner. To furth…

Worldline Formulations of Covariant Fracton Theories
Filippo Fecit, Davide Rovere
https://arxiv.org/abs/2508.14591 https://arxiv.org/pdf/2508.14591

Worldline Formulations of Covariant Fracton Theories
We develop worldline formulations of covariant fracton gauge theories. These are a one-parameter family of gauge theories of a rank-two symmetric tensor field, invariant under a scalar gauge transformation involving a double derivative. These theories, which can be interpreted as linearized gravity theories invariant under longitudinal diffeomorphisms, provide a covariant framework for studying Lorentz-breaking fracton quasiparticles, which are excitations with restricted mobility due to dipole…

Tensorized Multi-Task Learning for Personalized Modeling of Heterogeneous Individuals with High-Dimensional Data
Elif Konyar, Mostafa Reisi Gahrooei, Kamran Paynabar
https://arxiv.org/abs/2508.15676

Tensorized Multi-Task Learning for Personalized Modeling of Heterogeneous Individuals with High-Dimensional Data
Effective modeling of heterogeneous subpopulations presents a significant challenge due to variations in individual characteristics and behaviors. This paper proposes a novel approach to address this issue through multi-task learning (MTL) and low-rank tensor decomposition techniques. Our MTL approach aims to enhance personalized modeling by leveraging shared structures among similar tasks while accounting for distinct subpopulation-specific variations. We introduce a framework where low-rank d…

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…

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…

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-…

A geometry of cubic discriminants in 8 dimensions
Elitza Hristova, Ivan Minchev
https://arxiv.org/abs/2508.12014 https://arxiv.org/pdf/2508.12014

A geometry of cubic discriminants in 8 dimensions
This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to ${SO(4)}_{ir}\subset GL(8,\mathbb R)$, the image of an irreducible representation of $SO(4)$ on $\mathbb R^8$. We demonstrate that such a reduction can be described by an almost quaternion-Hermitian structure and a special rank-4 tensor field, which we call a cubic discriminant. This tensor field is pointwise linearly equivalent to the formula for the discriminant of a cubic polynomial. We show that the only…

Massive spin 3 and Metric-Affine Gravity
Roberto Percacci, Ergin Sezgin
https://arxiv.org/abs/2508.14211 https://arxiv.org/pdf/2508.14211

Massive spin 3 and Metric-Affine Gravity
Symmetric Metric-Affine Gravity is a theory of gravity with an independent non-metric connection, and zero torsion. It can be thought of as ordinary metric gravity coupled to a rank-three tensor Q, symmetric in one pair of indices. This field carries a spin-three degree of freedom. We find that, in contrast to the totally symmetric case, it is possible to arrange the parameters in the Lagrangian, so that, at the linearized level, the spin-three state, either alone or in combination with a spin-…

Goal-Oriented Low-Rank Tensor Decompositions for Numerical Simulation Data
Daniel M. Dunlavy, Eric T. Phipps, Hemanth Kolla, John N. Shadid, Edward Phillips
https://arxiv.org/abs/2508.11139

Goal-Oriented Low-Rank Tensor Decompositions for Numerical Simulation Data
We introduce a new low-dimensional model of high-dimensional numerical simulation data based on low-rank tensor decompositions. Our new model aims to minimize differences between the model data and simulation data as well as functions of the model data and functions of the simulation data. This novel approach to dimensionality reduction of simulation data provides a means of directly incorporating quantities of interests and invariants associated with conservation principles associated with the…

The low-rank tensor-train finite difference method for three-dimensional parabolic equations
Gianmarco Manzini, Tommaso Sorgente
https://arxiv.org/abs/2509.10142 https://…

The low-rank tensor-train finite difference method for three-dimensional parabolic equations
This paper presents a numerical framework for the low-rank approximation of the solution to three-dimensional parabolic problems. The key contribution of this work is the tensorization process based on a tensor-train reformulation of the second-order accurate finite difference method. We advance the solution in time by combining the finite difference method with an explicit and implicit Euler method and with the Crank-Nicolson method. We solve the linear system arising at each time step from th…

A Tensor Train-Based Isogeometric Solver for Large-Scale 3D Poisson Problems on Complex Geometries
Quoc Thai Tran, Duc P. Truong, Kim {\O}. Rasmussen, Boian Alexandrov
https://arxiv.org/abs/2509.13224 …

A Tensor Train-Based Isogeometric Solver for Large-Scale 3D Poisson Problems on Complex Geometries
We introduce a three-dimensional (3D) fully tensor train (TT)-assembled isogeometric analysis (IGA) framework, TT-IGA, for solving partial differential equations (PDEs) on complex geometries. Our method reformulates IGA discrete operators into TT format, enabling efficient compression and computation while retaining geometric flexibility and accuracy. Unlike previous low-rank approaches that typically rely on structured domains, our framework accommodates general 3D geometries through low-rank …