2026· Uniform Distribution Theory· Vol 21, pp. 1-15· 0 citations
TL;DR
This work wants to approximate the representative matrices of the Watts–Strogatz networks using tensor methods and compare the accuracy and the computational cost involved in operating with the original matrices and the matrices written in the approximate tensor form.
Abstract
Small-world networks are characterized by the existence, on
average, of shortest paths between any arbitrary pair of nodes with only a few
edges. In order to preserve the local clustering while permitting the existence of
the small-world phenomenon, Watts and Strogatz introduced their celebrated
network model (Nature, 1998) exemplifying what they observed in different
types of networks, such as the neural network of the C. elegans, the power grid
network, or the collaboration network in cinema.
As the number of data increases, the networks and matrices that model it also
do so, which makes it increasingly expensive to manipulate them. Recent work
has shown the efficiency of tensor-based structures when performing, for
example, matrix products, reducing the number of operations performed.
In the present work, we want to approximate the representative matrices of the
Watts–Strogatz networks using tensor methods and compare the accuracy and
the computational cost involved in operating with the original matrices and the
matrices written in the approximate tensor form.
This work presents the tree tensor network Nyström (TTNN), an algorithm that extends recent research on streamable tensor approximation to the more general tree tensor network format, enabling a unified treatment of various existing methods.
Alberto Bucci, Gianfranco Verzella· Numerical Linear Algebra wit...· 3 citations
Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to $2^{70}$ configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.
Xiao Wang, Tomohiro Hashizume, Pia Siegl et al.· 2 citations
A new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies is introduced, and it is found that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.
This work shows how to efficiently perform arbitrary contractions of the derivative tensor of arbitrary order for multivariate functions by properly constructing the input tangents to univariate high-order AD, which can be used to randomize any differential operator efficiently.
†. ZekunShi, †. ZheyuanHu, Min Lin et al.· 0 citations
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 globally positive for every parameter value and, at fixed local and bond dimensions, its number of parameters grows only linearly with the algebraic degree. We test the construction on three generalized complete-intersection Calabi-Yau (gCICY) threefolds, constructing chart by chart the generalized sections, holomorphic volume forms and sampling measures that define and train the metric there. From a common low-degree metric, the tensor network outperforms a parameter-matched neural potential using the same section data, reducing both bulk errors and the one-percent tail conditional mean in every paired run. It also reaches a substantially lower error than direct optimization of an unrestricted Hermitian metric of the same degree from the same start, with both methods optimized to validation convergence under their respective schedules. On a second geometry, a higher-degree network with fewer parameters than a lower-degree unrestricted Hermitian baseline substantially reduces the same-sample errors. We further observe saturation within the tested calculations: at fixed bond dimension, increasing the degree eventually plateaus; increasing the bond dimension at fixed optimization effort gives no resolved gain; and the outcome depends strongly on initialization and optimization path.