Tootfinder

Scalable inference of large-scale random kronecker graphs via tensor decomposition and Einstein summation
Sanaa Khobizy
https://arxiv.org/abs/2506.22292 ht…

Scalable inference of large-scale random kronecker graphs via tensor decomposition and Einstein summation
In this paper, we extend the analysis of random Kronecker graphs to multi-dimensional networks represented as tensors, enabling a more detailed and nuanced understanding of complex network structures. We decompose the adjacency tensor of such networks into two components: a low-rank signal tensor that captures the essential network structure and a zero-mean noise tensor that accounts for random variations. Building on recent advancements in tensor decomposition and random tensor theory, we intr…

Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning
Marius Willner, Marco Trenti, Dirk Lebiedz
https://arxiv.org/abs/2507.21726 https://

Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning
Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation. In this work, we present a formal analysis of the differential geometry underlying TTNs. Building on this foundation, we develop efficient first- and second-order optimization algorithms that exploit the intrinsic quotient structure of TTNs. Additionally, we devise a backpropagation algorithm for training TTNs in a kernel learning setting. We validate our methods through numerical experiments…

Simple $\mathcal{W}\ltimes\widehat{H_4}$-modules from tensor products
Dashu Xu
https://arxiv.org/abs/2507.21927 https://arxiv.org/pdf/2507.21927

Simple $\mathcal{W}\ltimes\widehat{H_4}$-modules from tensor products
This paper investigates simple modules of the semi-direct product algebra $\mathcal{W}\ltimes\widehat{H_4}$, where $\mathcal{W}$ is the Witt algebra and $\widehat{H_4}$ is the loop Diamond algebra. We first use simple modules over the Weyl algebra to construct a family of simple $\mathcal{W}\ltimes\widehat{H_4}$-modules. Then, we classify simple $\mathcal{W}\ltimes\widehat{H_4}$-modules that are free $U(\mathbb{C}L_0\oplus\mathbb{C} a_0)$-modules of rank 1. Finally, we give a necessary and suff…

Global anomalies in $6D$ gauged supergravities
Katrin Becker, Ergin Sezgin, David Tennyson, with a mathematical appendix by Yuji Tachikawa
https://arxiv.org/abs/2507.22127 https…

Global anomalies in $6D$ gauged supergravities
There exists a rare class of R-symmetry gauged $N=(1,0)$ supergravities in six dimensions with gauge group $G\times U(1)_R$, where $G$ is semisimple with rank greater than one, and the number of tensor multiplets $n_T=1$, which are free from all local anomalies. We find new members of this family, in which $G$ contains up to four factors. We study the global anomalies of these models in a framework in which the Dirac quantization of the anomaly coefficients and the well-definedness of the Green…

Change Point Localization and Inference in Dynamic Multilayer Networks
Fan Wang, Kyle Ritscher, Yik Lun Kei, Xin Ma, Oscar Hernan Madrid Padilla
https://arxiv.org/abs/2506.21878

Change Point Localization and Inference in Dynamic Multilayer Networks
We study offline change point localization and inference in dynamic multilayer random dot product graphs (D-MRDPGs), where at each time point, a multilayer network is observed with shared node latent positions and time-varying, layer-specific connectivity patterns. We propose a novel two-stage algorithm that combines seeded binary segmentation with low-rank tensor estimation, and establish its consistency in estimating both the number and locations of change points. Furthermore, we derive the l…

On the construction of a counterexample to Strassen's rank additivity conjecture
Viktoriia Borovik, Cosimo Flavi, Pawe{\l} Pielasa, Anatoli Shatsila, Jeyoung Song
https://arxiv.org/abs/2507.17890

On the construction of a counterexample to Strassen's rank additivity conjecture
The rank additivity conjecture, first formulated by Volker Strassen in 1973, states that the rank of the direct sum of two independent tensors is equal to the sum of their individual ranks. In the last decades, this conjecture has been a central topic in tensor rank theory and its implications for computational complexity. In 2019, Yaroslav Shitov disproved this conjecture in its general form by showing the existence of a counter-example using a dimension counting argument. In this paper, we pr…

Bounds for Geometric rank in Terms of Subrank
Qiyuan Chen, Ke Ye
https://arxiv.org/abs/2506.16132 https://arxiv.org/pdf/2506.16132

Bounds for Geometric rank in Terms of Subrank
For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is bounded by a function in its subrank in some field extension of bounded degree. Secondly, we show that, over any algebraically closed field, the geometric rank of a tensor is bounded by a function in its subrank. Lastly, we prove that, for any order three tens…

Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection
Afra Kilic, Kim Batselier
https://arxiv.org/abs/2507.11136 …

Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection
Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabil…

Accelerating Large-Scale Regularized High-Order Tensor Recovery
Wenjin Qin, Hailin Wang, Jingyao Hou, Jianjun Wang
https://arxiv.org/abs/2506.09594 https:/…

Accelerating Large-Scale Regularized High-Order Tensor Recovery
Currently, existing tensor recovery methods fail to recognize the impact of tensor scale variations on their structural characteristics. Furthermore, existing studies face prohibitive computational costs when dealing with large-scale high-order tensor data. To alleviate these issue, assisted by the Krylov subspace iteration, block Lanczos bidiagonalization process, and random projection strategies, this article first devises two fast and accurate randomized algorithms for low-rank tensor approx…

Block Tensor Decomposition: A dual grid scheme with formal O(N3) for THC decomposition of molecular systems
Yueyang Zhang, Xuewei Xiong, Wei Wu, Peifeng Su
https://arxiv.org/abs/2506.19392

Block Tensor Decomposition: A dual grid scheme with formal O(N3) for THC decomposition of molecular systems
Accurate and fast treatment of electron-electron interactions remains a central challenge in electronic structure theory because post-Hartree-Fock methods often suffered from the computational cost for 4-index electron repulsion integrals (ERIs). Low-rank approaches such as tensor hyper-contraction (THC) and interpolative separable density fitting (ISDF) have been proposed for Hartree-Fock exchange and correlation's calculations. Their application to molecular systems remains inefficient due to…

Fast and Provable Hankel Tensor Completion for Multi-measurement Spectral Compressed Sensing
Jinsheng Li, Xu Zhang, Shuang Wu, Wei Cui
https://arxiv.org/abs/2507.04847

Fast and Provable Hankel Tensor Completion for Multi-measurement Spectral Compressed Sensing
In this paper, we introduce a novel low-rank Hankel tensor completion approach to address the problem of multi-measurement spectral compressed sensing. By lifting the multiple signals to a Hankel tensor, we reformulate this problem into a low-rank Hankel tensor completion task, exploiting the spectral sparsity via the low multilinear rankness of the tensor. Furthermore, we design a scaled gradient descent algorithm for Hankel tensor completion (ScalHT), which integrates the low-rank Tucker deco…

A parametric tensor ROM for the shallow water dam break problem
Md Rezwan Bin Mizan, Maxim Olshanskii, Ilya Timofeyev
https://arxiv.org/abs/2506.20007 http…

A parametric tensor ROM for the shallow water dam break problem
We develop a variant of a tensor reduced-order model (tROM) for the parameterized shallow-water dam-break problem. This hyperbolic system presents multiple challenges for model reduction, including a slow decay of the Kolmogorov $N$-width of the solution manifold, shock formation, and the loss of smooth solution dependence on parameters. These issues limit the performance of traditional Proper Orthogonal Decomposition based ROMs. Our tROM approach, based on a low-rank tensor decomposition, buil…

Beyond one-loop: higher-order effects on Gross-Neveu-Yukawa tensorial criticality
SangEun Han, Igor F. Herbut
https://arxiv.org/abs/2506.20710 https://

Beyond one-loop: higher-order effects on Gross-Neveu-Yukawa tensorial criticality
We study the Gross-Neveu-Yukawa field theory for the SO($N$) symmetric traceless rank-two tensor order parameter coupled to Majorana fermions using the $ε$-expansion around upper critical dimensions of $3+1$ to two loops. Previously we established in the one-loop calculation that the theory does not exhibit a critical fixed point for $N \geq 4$, but that nevertheless the stable fixed point inevitably emerges at a large number of fermion flavors $N_f$. For $N_f < N_{f,c1} \approx N/2$, no criti…

The spectrum of global representations for families of bounded rank and VI-modules
Miguel Barrero, Tobias Barthel, Luca Pol, Neil Strickland, Jordan Williamson
https://arxiv.org/abs/2506.21525

The spectrum of global representations for families of bounded rank and VI-modules
A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family $\mathscr{U}$. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this paper we begin a systematic study of the derived category $\mathsf{D}(\mathscr{U};k)$ of global representations over fields $k$ of characteristic zero, from the point-of-view of tensor-triangular geomet…

Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor
Daiki Kawabe
https://arxiv.org/abs/2507.13126 https://

Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor
This note provides a detailed proof of Conner--Gesmundo--Landsberg--Ventura's result that the border rank of the Kronecker square of the little Coppersmith--Winograd tensor is $(q+2)^{2}$.We also indicate how the same ideas seem to extend to the case of the Kronecker cube, pointing toward the conjectural value $(q+2)^{m}$ for $m\ge 4$, although a full proof is left for future work.

$\mathbf{O}(D,D)$-Symmetric Box Operator and $\alpha^{\prime}$-Corrections with Riemann Curvature
Kawon Lee, Jeong-Hyuck Park
https://arxiv.org/abs/2506.21143

$\mathbf{O}(D,D)$-Symmetric Box Operator and $α^{\prime}$-Corrections with Riemann Curvature
Within the framework of Double Field Theory, we construct an $\mathbf{O}(D,D)$-symmetric d'Alembertian, or box operator, that is applicable to tensors of arbitrary rank. Parameterized by the Riemannian metric and the $B$-field, the operator naturally incorporates the Riemann curvature tensor and the $H$-flux. When applied to the massless string sector, it produces a consistent stringy wave equation under an $\mathbf{O}(D,D)$-symmetric harmonic gauge condition. Furthermore, the one-loop integral…

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…

Resonant dynamics of dipole-conserving Bose-Hubbard model with time-dependent tensor electric fields
Jiali Zhang, Shaoliang Zhang
https://arxiv.org/abs/2506.13159

Resonant dynamics of dipole-conserving Bose-Hubbard model with time-dependent tensor electric fields
Recently, tensor gauge fields and their coupling to fracton phases of matter have attracted more and more research interest, and a series of novel quantum phenomena arising from the coupling has been predicted. In this article, we propose a theoretical scheme to construct a time-dependent rank-2 tensor electric field by introducing a periodically driving quadratic potential in a dipole-conserving Bose-Hubbard model, and investigate the dynamics of dipole and fracton excitations when the drive f…

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…

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…

Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds
Markus Bachmayr, Huqing Yang
https://arxiv.org/abs/2506.19584 https…

Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial expansion in the remaining parametric variables, it addresses in particular classes of elliptic problems where a direct polynomial expansion is inefficient, such as those arising from random diffusion coefficients with short correlation length. A convergent adaptive …

Property $(UW_E)$ and Consistent invertibility
Simi Thomas, T Prasad, Shery Fernandez
https://arxiv.org/abs/2506.18596 https://arxiv.…

Property $(UW_E)$ and Consistent invertibility
In this paper, we study the stability of Property $(UW_E)$ under commuting finite rank perturbation in terms of spectrum induced by consistent in invertibility for operators and a variant of upper semi-Weyl spectrum. Also we discuss the stability of Property $(UW_{E})$ under direct sum $T\oplus S$ and tensor product $T\otimes S$ for Hilbert space operators $T$ and $S$.

Interaction between skew-representability, tensor products, extension properties, and rank inequalities
Krist\'of B\'erczi, Bogl\'arka Geh\'er, Andr\'as Imolay, L\'aszl\'o Lov\'asz, Carles Padr\'o, Tam\'as Schwarcz
https://arxiv.org/abs/2507.10709 …

Interaction between skew-representability, tensor products, extension properties, and rank inequalities
Skew-representable matroids form a fundamental class in matroid theory, bridging combinatorics and linear algebra. They play an important role in areas such as coding theory, optimization, and combinatorial geometry, where linear structure is crucial for both theoretical insights and algorithmic applications. Since deciding skew-representability is computationally intractable, much effort has been focused on identifying necessary or sufficient conditions for a matroid to be skew-representable. …

Functional Tensor Regression
Tongyu Li, Fang Yao, Anru R. Zhang
https://arxiv.org/abs/2506.09358 https://arxiv.org/pdf/2506.09358

Functional Tensor Regression
Tensor regression has attracted significant attention in statistical research. This study tackles the challenge of handling covariates with smooth varying structures. We introduce a novel framework, termed functional tensor regression, which incorporates both the tensor and functional aspects of the covariate. To address the high dimensionality and functional continuity of the regression coefficient, we employ a low Tucker rank decomposition along with smooth regularization for the functional m…

Statistical Limits for Finite-Rank Tensor Estimation
Riccardo Rossetti, Galen Reeves
https://arxiv.org/abs/2506.06749 https://arxiv.o…

Statistical Limits for Finite-Rank Tensor Estimation
This paper provides a unified framework for analyzing tensor estimation problems that allow for nonlinear observations, heteroskedastic noise, and covariate information. We study a general class of high-dimensional models where each observation depends on the interactions among a finite number of unknown parameters. Our main results provide asymptotically exact formulas for the mutual information (equivalently, the free energy) as well as the minimum mean-squared error in the Bayes-optimal sett…

A Note on Lower Bounds in Szemer\'edi's Theorem with Random Differences
Jason Zheng
https://arxiv.org/abs/2508.01187 https://arxiv.org/pdf/2508.011…

A Note on Lower Bounds in Szemerédi's Theorem with Random Differences
In this note, we consider Szemerédi's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a generalization of an argument of Altman with Moshkovitz--Zhu's bounds for the partition rank of a tensor in terms of its analytic rank, we (slightly) improve the best known lower bounds (due to Briët) on the size $|S|$ required for Szemerédi's theorem…

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…

Hollow Lattice Tensor Gauge Theories with Bosonic Matter
Jos\'e M. Cruz, Masafumi Udagawa, Pedro Bicudo, Pedro Ribeiro, Paul A. McClarty
https://arxiv.org/abs/2508.02326 htt…

Hollow Lattice Tensor Gauge Theories with Bosonic Matter
Higher rank gauge theories are generalizations of electromagnetism where, in addition to overall charge conservation, there is also conservation of higher rank multipoles such as the total dipole moment. In this work we study a four dimensional lattice tensor gauge theory coupled to bosonic matter which has second rank tensor electric and magnetic fields and charge conservation on individual planes. Starting from the Hamiltonian, we derive the lattice action for the gauge fields coupled to $q=1…

Quantics Tensor Train for solving Gross-Pitaevskii equation
Aleix Bou-Comas, Marcin P{\l}odzie\'n, Luca Tagliacozzo, Juan Jos\'e Garc\'ia-Ripoll
https://arxiv.org/abs/2507.03134

Quantics Tensor Train for solving Gross-Pitaevskii equation
We present a quantum-inspired solver for the one-dimensional Gross-Pitaevskii equation in the Quantics Tensor-Train (QTT) representation. By evolving the system entirely within a low-rank tensor manifold, the method sidesteps the memory and runtime barriers that limit conventional finite-difference and spectral schemes. Two complementary algorithms are developed: an imaginary-time projector that drives the condensate toward its variational ground state and a rank-adapted fourth-order Runge-Kutt…

Fast adaptive tubal rank-revealing algorithm for t-product based tensor approximation
Qiaohua Liu, Jiehui Gu
https://arxiv.org/abs/2508.08557 https://arxiv…

Fast adaptive tubal rank-revealing algorithm for t-product based tensor approximation
Color images and video sequences can be modeled as three-way tensors, which admit low tubal-rank approximations via convex surrogate minimization. This optimization problem is efficiently addressed by tensor singular value thresholding (t-SVT). To mitigate the computational burden of tensor singular value decomposition (t-SVD) in each iteration, this paper introduces an adaptive randomized algorithm for tubal rank revelation in data tensors \(\mathcal{A}\). Our method selectively captures the p…

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…

This https://arxiv.org/abs/2411.06188 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_grqc_…

The thermodynamic profile of AdS black holes in Lorentz-violating Bumblebee and Kalb-Ramond gravity
Lorentz invariance violation (LIV) is a topic of significant interest in quantum gravity and in extensions of the Standard Model of particle physics. Recently, new classes of black hole solutions have been proposed, involving vector fields and rank-two antisymmetric tensor fields that acquire nontrivial vacuum expectation values, resulting in the Bumblebee and Kalb-Ramond (KR) gravity models, respectively. These models exhibit novel geometric structures and differ in notable ways from standard …

Some criteria concerning the rational vanishing of Whitehead groups
Wolfgang Lueck, Bob Oliver
https://arxiv.org/abs/2506.22906 https://

Some criteria concerning the rational vanishing of Whitehead groups
We give several examples of finite groups $G$ for which the rank of the tensor product $\mathbb{Z} \otimes_{\mathbb{Z}\mathrm{Aut}(G)}$ Wh$(G)$ is or is not zero. This is motivated by an earlier theorem of the first author, which implies as a special case that when this group has nonzero rank, the Whitehead group of any other group (finite or infinite) that contains $G$ as a normal subgroup is rationally nontrivial.

This https://arxiv.org/abs/2501.15401 has been replaced.
link: https://scholar.google.com/scholar?q=a

Splitting Property of quasitriangular Hopf algebras
We investigate the splitting property of quasitriangular Hopf algebras through the lens of twisted tensor products. Specifically, we demonstrate that an infinite-dimensional quasitriangular Hopf algebra possesses the splitting property if it admits a factorizable quotient Hopf algebra. We also establish that the splitting property holds if there exists a full rank quotient Hopf algebra where the left and right coinvariants coincide. As a consequence, we obtain a new obstruction for quasitriangu…

Alternating steepest descent methods for tensor completion with applications to spectromicroscopy
Oliver Townsend, Sergey Dolgov, Silvia Gazzola, Misha Kilmer
https://arxiv.org/abs/2506.10661

Alternating steepest descent methods for tensor completion with applications to spectromicroscopy
In this paper we develop two new Tensor Alternating Steepest Descent algorithms for tensor completion in the low-rank $\star_{M}$-product format, whereby we aim to reconstruct an entire low-rank tensor from a small number of measurements thereof. Both algorithms are rooted in the Alternating Steepest Descent (ASD) method for matrix completion, first proposed in [J. Tanner and K. Wei, Appl. Comput. Harmon. Anal., 40 (2016), pp. 417-429]. In deriving the new methods we target the X-ray spectromic…

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-…

Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory
Milan Zlatanovi\'c, Vladimir Rovenski
https://arxiv.org/abs/2508.08021 https://

Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory
Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where $F$ can be defined as an almost complex or almost contact metric (a.c.m.) structure. In the paper, we first study more general models, where $F$ has constant rank and is …

Hierarchical Tucker Low-Rank Matrices: Construction and Matrix-Vector Multiplication
Yingzhou Li, Jingyu Liu
https://arxiv.org/abs/2508.05958 https://arxiv…

Hierarchical Tucker Low-Rank Matrices: Construction and Matrix-Vector Multiplication
In this paper, a hierarchical Tucker low-rank (HTLR) matrix is proposed to approximate non-oscillatory kernel functions in linear complexity. The HTLR matrix is based on the hierarchical matrix, with the low-rank blocks replaced by Tucker low-rank blocks. Using high-dimensional interpolation as well as tensor contractions, algorithms for the construction and matrix-vector multiplication of HTLR matrices are proposed admitting linear and quasi-linear complexities respectively. Numerical experime…

Theory of Electronic Nematic Criticality Constrained by Elastic Compatibility
W. Joe Meese, Rafael M. Fernandes
https://arxiv.org/abs/2507.23754 https://ar…

Theory of Electronic Nematic Criticality Constrained by Elastic Compatibility
The defining property of electronic nematicity -- the spontaneous breaking of rotational symmetry -- implies an unavoidable coupling between the nematic order parameter and elastic strain fields, known as nemato-elasticity. While both quantities are rank-2 tensors, the strain tensor is constrained through the Saint Venant compatibility relations. These three coupled second-order partial differential equations arise from the lattice displacement vector's role as a potential field, and they refle…

Norming Sets for Tensor and Polynomial Sketching
Yifan Zhang, Joe Kileel
https://arxiv.org/abs/2506.05174 https://arxiv.org/pdf/2506.…

Norming Sets for Tensor and Polynomial Sketching
This paper develops the sketching (i.e., randomized dimension reduction) theory for real algebraic varieties and images of polynomial maps, including, e.g., the set of low rank tensors and tensor networks. Through the lens of norming sets, we provide a framework for controlling the sketching dimension for \textit{any} sketch operator used to embed said sets, including sub-Gaussian, fast Johnson-Lindenstrauss, and tensor structured sketch operators. Leveraging norming set theory, we propose a ne…

D1-brane correction to a line operator index
Yosuke Imamura, Akihiro Sei
https://arxiv.org/abs/2506.05788 https://arxiv.org/pdf/2506.…

D1-brane correction to a line operator index
Wilson line operators in the rank $k$ totally symmetric tensor representation of ${\cal N}=4$ $U(N)$ sypersymmetric Yang-Mills theories are expected to be realized as D3-branes expanded in $AdS_5$. Although there is a mismatch between the corresponding line operator indices even in the large $N$ and large $k$ limit, it is possible to calculate the finite $k$ correction on the AdS side as the contribution from D1-branes. We analyze D1-brane fluctuation modes and calculate the leading finite $k$ …

Weak metric structures on generalized Riemannian manifolds
Vladimir Rovenski, Milan Zlatanovi\'c
https://arxiv.org/abs/2506.23019 https://

Weak metric structures on generalized Riemannian manifolds
Linear connections with torsion are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate (0,2)-tensor and $F$ is the skew-symmetric part. Some space-time models in theoretical physics are based on $(M,G=g+F)$, where $F$ is defined using a complex structure. In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the first author and R. Wolak), wh…

Efficient randomized algorithms for the fixed Tucker-rank problem of Tucker decomposition with adaptive shifts
Maolin Che, Yimin Wei, Chong Wu, Hong Yan
https://arxiv.org/abs/2506.04840

Efficient randomized algorithms for the fixed Tucker-rank problem of Tucker decomposition with adaptive shifts
Randomized numerical linear algebra is proved to bridge theoretical advancements to offer scalable solutions for approximating tensor decomposition. This paper introduces fast randomized algorithms for solving the fixed Tucker-rank problem of Tucker decomposition, through the integration of adaptive shifted power iterations. The proposed algorithms enhance randomized variants of truncated high-order singular value decomposition (T-HOSVD) and sequentially T-HOSVD (ST-HOSVD) by incorporating dyna…

Transverse diffeomorphism invariant spin-2 theories via gauge invariants
D. Dalmazi, Luiz G. M. Ramos
https://arxiv.org/abs/2506.24047 https://

Transverse diffeomorphism invariant spin-2 theories via gauge invariants
We analyze the particle spectrum of a second-order (in derivatives) theory based on a rank-2 tensor field with both symmetric and antisymmetric components. By demanding the existence of a propagating massless spin-2 particle and invariance under transverse diffeomorphisms, we derive a new class of stable models with two massless scalars and a single massless spin-2 particle. A natural nonlinear completion is proposed in terms of a dynamical metric field. The identification of the spectrum is ca…