Infinite Tensor Framework

• Infinite Tensor Framework is a unified structure combining mathematical, quantum, and machine learning techniques to analyze and compute infinite-dimensional systems.
• It provides rigorous tensor constructs that enable exact analysis, careful entanglement evaluation, and efficient tensor network contractions in physics and data science.
• The framework offers scalable algorithms such as iMPS, iTEBD, and Tensor Programs, extending classical methods for accurate infinite-dimensional problem solving.

The Infinite Tensor Framework encompasses a collection of mathematical, algorithmic, and physical methodologies unifying the treatment of tensor algebra, tensor network states, and infinite-dimensional limit theories across quantum physics, operator algebras, and high-dimensional machine learning. By generalizing classical finite tensor representations to infinite systems and infinite-width limits, this framework enables exact analysis and efficient computation in settings such as quantum many-body systems, continuous data, quantum information, and neural networks at scale. Rigorous infinite tensor constructs underlie phenomena from entanglement in quantum spin chains to non-Gaussian limiting laws in neural architectures, to inductive-limited operator algebras governing emergent symmetries and superselection structure.

1. Algebraic and Analytic Foundations

Infinite tensor products of vector spaces and Hilbert spaces are defined as universal objects encoding infinite multilinearity, subject to an equivalence on sequences that agrees on all but finitely many locations. For a family of vector spaces $\{X_i\}_{i\in I}$ over $\mathbb{C}$, the infinite tensor product $\bigotimes_{i\in I} X_i$ carries the property that

$$\operatorname{Hom}\left(\bigotimes_{i\in I} X_i, Y\right) \cong \{\varphi : \prod_{i\in I} X_i \to Y\,|\,\varphi \text{ multilinear}\}$$

for all $Y$ (Ng, 2011). For Hilbert spaces, the construction is completed under a norm induced from infinite products of local inner products, possibly breaking into orthogonal "superselection sectors" when infinite divergence occurs (Svozil, 10 Sep 2024).

Infinite tensor algebras admit additional structure when the components have *-algebra or Hilbert-module structure. The algebraic direct sum decomposes over an index set of “infinite signatures” $\Omega_{I;X}$ defined by equivalence modulo finitely many differing entries, yielding

$$\bigotimes_{i\in I} X_i = \bigoplus_{\omega\in\Omega_{I;X}} V_\omega$$

where $V_\omega$ collects tensors equivalent to a fixed representative (Ng, 2011). Special subalgebras, such as the unitary-tensor algebra $\bigotimes_{i\in I}^{\mathrm{ut}} A_i$, reflect group-like symmetries and crossed product structures.

2. Operator Algebras, Superselection, and Sectorization

Infinite tensor products in the Hilbert-space setting naturally realize representations of operator algebras (e.g., the von Neumann algebra $\bigotimes_{n=1}^\infty \mathcal{B}(\mathcal{H}_n)$), with the sector structure corresponding to inequivalent physical phases or "pointer bases" in quantum measurement theory (Svozil, 10 Sep 2024). Given two infinite product vectors $\Psi=\bigotimes_i|\psi_i\rangle$ and $\Phi=\bigotimes_i|\phi_i\rangle$, their inner product may vanish unless the sum of $1-\langle\psi_i|\phi_i\rangle$ converges, partitioning the Hilbert space into mutually orthogonal sectors $H_\alpha$.

Within the operator-algebraic formalism, such infinite tensor products generate factors of type I, II, or III, depending on how trace and unit structures persist under the inductive limit. This manifests in physical contexts such as the emergence of hyperfinite type-II$_1$ factors in infinite tensor network representations of holographic codes and type-III factors in models lacking complementary recovery (Chemissany et al., 31 Mar 2025). Additionally, crossed-product constructions encode symmetries and cocycle actions, as in

$$\bigotimes_{i\in I}^{\mathrm{ut}}A_i \cong V_{[1]} \rtimes_{(\alpha, m)} \Omega_{I;A}^{\mathrm{ut}}$$

(Ng, 2011).

3. Infinite Tensor Networks in Quantum Many-Body Systems

In quantum physics, infinite tensor networks provide scalable, translationally invariant representations of states and operators in the thermodynamic limit. Infinite Matrix Product States (iMPS) and their associated algorithms, such as infinite Time-Evolving Block Decimation (iTEBD), yield explicit constructions for ground states and dynamics (Roy et al., 2018). The transfer matrix formalism allows computation of reduced density matrices of all small blocks by contracting a finite window of tensors between fixed-point boundary vectors. Quantities such as the generalized geometric measure (GGM) for multisite entanglement become evaluable as: $$\mathcal{G}_\infty = 1 - \max\{\max_k\nu_k^{(1)}, \max_k\nu_k^{(2)}\}$$ where $\nu_k^{(m)}$ are eigenvalues of reduced states over $m$ consecutive sites.

In open quantum system theory, infinite MPO (Matrix Product Operator) evolution schemes efficiently contract environment-induced process tensors, supporting access to long-time asymptotics, steady states, and dynamical phase transitions (Link et al., 2023). These methods compress memory and dynamical influence into finite bond-dimension objects via SVD truncations, exploiting translational invariance in time or spatial direction.

4. Infinite Tensor Programs and Deep Learning Limits

The Tensor Programs formalism systematizes the analysis of neural architectures in the infinite-width regime, providing exact limiting laws for layerwise functionals and gradients (Yang, 2020). In this context, the "infinite tensor framework" refers to the recursive Gaussian (or conditional mixture) laws obtained for the statistics of activations and outputs. The language leverages random matrix-vector and nonlinear operations, yielding convergence of empirical distributions to deterministic Gaussian processes or, for self-attention layers, to hierarchical mixture distributions (Sakai et al., 1 Jun 2025): $$\text{Law}(y) = \int N(0, \Sigma(\mathfrak{p})) \, d\text{Law}(\mathfrak{p})$$ with $\mathfrak{p}$ denoting a jointly Gaussian family of attention scores.

Extensions accommodate general optimizers (e.g., Adam, RMSProp) and complex architectures using "outer-nonlinear" instructions and Dirac bra-ket notation, enabling a programmatic framework for deriving neural tangent kernel (NTK) and feature-learning limits (Yang et al., 2023). The asymptotic regime is controlled via abcd-parametrizations, with a dichotomy between pure kernel dynamics ($r>0$) and feature learning ($r=0$).

5. Infinite-Dimensional Tensor Decomposition and Quasitubal Theory

Infinite tensor methods underpin scalable probabilistic and algebraic models for multiway and functional data. The Infinite Tucker (InfTucker) decomposition enables nonparametric Bayesian factorization in infinite-dimensional feature space via kernelized mappings and tensor-variate Gaussian processes (Xu et al., 2011). The approach leverages variational inference over Kronecker-structured covariances, supporting efficient Bayesian regression, classification, and tensor completion.

The quasitubal tensor algebra generalizes the t-product structure to matrices whose entries are infinite-dimensional Hilbert-space elements (Mor et al., 22 Apr 2025). Key innovations involve replacing the non-unital ring of tubes by a commutative unital C$^*$-algebra (the bounded operators), permitting SVD and best low-rank approximation theory to extend to the infinite-dimensional field. Truncations in the transformed domain yield concrete finite-dimensional approximations with explicit convergence rates.

6. Numerical Methods and Computational Algorithms

Numerical frameworks tailored for infinite tensor problems include the Tensor Infinite Arnoldi Method (TIAR) for nonlinear eigenvalue problems (Mele et al., 2016), which encodes polynomial and exponential basis functions in a low-rank tensor structure. Restart and compression schemes exploit the rapid factorial decay of coefficients, ensuring efficient memory usage and rigorously bounded residuals upon projection. For infinite 2D tensor networks, fixed-point corner methods and variational uniform MPS approaches accelerate infinite-PEPS contraction, crucially outperforming classic power-method variants near criticality (Fishman et al., 2017).

7. Physical and Mathematical Implications

The Infinite Tensor Framework is foundational in modeling irreversibility, emergent symmetries, and superselection rules in quantum theory (Svozil, 10 Sep 2024). Nested measurement schemes and inductive-limited tensor constructions exhibit transitions to orthogonal sectors and the breakdown of unitary equivalence, paralleling fundamental irreversibility and entropy generation. In holographic models and error-correcting codes, infinite tensor network algebra precisely determines the emergent factor structure (type II, III, or hyperfinite) and reflects underlying entanglement spectra (Chemissany et al., 31 Mar 2025). In functional data analysis, signal processing, and operator learning, infinite tensor methods expand the analytic toolbox for inherently infinite-dimensional objects.

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Principal References:

• "Tensor-network approach to compute genuine multisite entanglement in infinite quantum spin chains" (Roy et al., 2018)
• "From Unitarity to Irreversibility: The Role of Infinite Tensor Products and Nested Wigner's Friends" (Svozil, 10 Sep 2024)
• "Open Quantum System Dynamics from Infinite Tensor Network Contraction" (Link et al., 2023)
• "Infinite Tucker Decomposition: Nonparametric Bayesian Models for Multiway Data Analysis" (Xu et al., 2011)
• "On genuine infinite algebraic tensor products" (Ng, 2011)
• "Infinite-Width Limit of a Single Attention Layer: Analysis via Tensor Programs" (Sakai et al., 1 Jun 2025)
• "Tensor Programs II: Neural Tangent Kernel for Any Architecture" (Yang, 2020)
• "Faster Methods for Contracting Infinite 2D Tensor Networks" (Fishman et al., 2017)
• "Quasitubal Tensor Algebra Over Separable Hilbert Spaces" (Mor et al., 22 Apr 2025)
• "On Infinite Tensor Networks, Complementary Recovery and Type II Factors" (Chemissany et al., 31 Mar 2025)
• "Restarting for the Tensor Infinite Arnoldi method" (Mele et al., 2016)
• "Tensor Programs IVb: Adaptive Optimization in the Infinite-Width Limit" (Yang et al., 2023)