This work provides a unique normal form and associated polynomial-time reduction strategy for tensor trains over arbitrary fields, and crucially provides the first tensor train form with the uniqueness property.
Abstract
Tensor trains (or Matrix-Product States) are a data structure used in many fields of computer science and physics. They were recently shown to generalise binary decision diagrams when used over the 2-element Galois field, prompting the question of their reducibility in such a context, when the standard approach, over real or complex number, is not amenable to finite fields. We provide here a unique normal form and associated polynomial-time reduction strategy for tensor trains over arbitrary fields. We also show how to directly extract a normal form out of a full tensor, how to get the leading index and value of a normal form, and an upper bound on the size of a fully-reduced tensor train relative to a naive storage of the full tensor. On the one hand, this work strengthens the use of tensor trains as a relevant formal tool. On the other hand, from the perspective of tensor networks, it extends the formalism to more general settings than the well-studied real and complex fields, and crucially provides the first tensor train form with the uniqueness property.
TFMs are introduced, realization-level Mathematical Structures in which a learned Operator maps a product of admissible component-section families to a prescribed family of time-dependent tangent sections on a Generative State Manifold.
In this paper, we study the large deviation principle (LDP) 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. Using the LDP, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. As applications, we focus on several concrete examples, which include monochromatic subgraph counts in sparse random graphs, Erd\H{o}s-R\'enyi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.
Reihaneh Malekian, Sohom Bhattacharya, Nabarun Deb et al.· 0 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
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.
Tensor network methods provide powerful analytical and numerical tools for characterizing quantum phases of matter. While the mathematical structure of matrix product states (MPS) is well understood through the MPS fundamental theorem, an analogous understanding for mixed-state tensor networks remains largely absent: if two purification tensors generate the same density matrix, how are they related? In this work, we initiate the study of a fundamental theorem for matrix product locally purifiable density operators (LPDOs) and focus on sequentially generated LPDOs (sLPDOs), a broad subclass admitting an interpretation in terms of successive applications of quantum channels on an initial state. We prove that, under suitable invertibility or cyclic conditions, two sLPDO representations generate the same density matrix for arbitrary system sizes if and only if they are related by a matrix product isometry acting on the purification bonds. Beyond the sLPDO setting, we provide a counterexample that suggests an obstruction to a general fundamental theorem for LPDOs with periodic boundary conditions. Finally, we discuss implications for mixed-state symmetry-protected topological phases, including the possibility of nontrivial phases protected only by weak symmetry conditions.
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.
Chun-Yin Siu· 0 citations
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