A Theoretical Framework for Discovering Groups and Unitary Representations via Tensor Factorization

• URL: http://arxiv.org/abs/2511.23152v1
• Date: Fri, 28 Nov 2025 12:58:13 GMT
• Title: A Theoretical Framework for Discovering Groups and Unitary Representations via Tensor Factorization
• Authors: Dongsung Huh, Halyun Jeong,
• Abstract summary: We provide a rigorous theoretical explanation for this inductive bias by decomposing its objective into a term regulating factor scales.<n>We prove two key results: (1) the global minimum is achieved by the unitary regular representation for groups, and (2) non-group operations incur a strictly higher objective value.
• Score: 9.572235167281685
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We analyze the HyperCube model, an \textit{operator-valued} tensor factorization architecture that discovers group structures and their unitary representations. We provide a rigorous theoretical explanation for this inductive bias by decomposing its objective into a term regulating factor scales ($\mathcal{B}$) and a term enforcing directional alignment ($\mathcal{R} \geq 0$). This decomposition isolates the \textit{collinear manifold} ($\mathcal{R}=0$), to which numerical optimization consistently converges for group isotopes. We prove that this manifold admits feasible solutions exclusively for group isotopes, and that within it, $\mathcal{B}$ exerts a variational pressure toward unitarity. To bridge the gap to the global landscape, we formulate a \textit{Collinearity Dominance Conjecture}, supported by empirical observations. Conditional on this dominance, we prove two key results: (1) the global minimum is achieved by the unitary regular representation for groups, and (2) non-group operations incur a strictly higher objective value, formally quantifying the model's inductive bias toward the associative structure of groups (up to isotopy).

Related papers

• Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory [45.88028371034407]
  We study a $mathbbZ$ lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell.<n>We show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability.<n>By performining a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization.
  arXiv  Detail & Related papers  (2026-03-02T22:59:25Z)
• Stabilizing Fixed-Point Iteration for Markov Chain Poisson Equations [49.702772230127465]
  We study finite-state Markov chains with $n$ states and transition matrix $P$.<n>We show that all non-decaying modes are captured by a real peripheral invariant subspace $mathcalK(P)$, and that the induced operator on the quotient space $mathbbRn/mathcalK(P) is strictly contractive, yielding a unique quotient solution.
  arXiv  Detail & Related papers  (2026-01-31T02:57:01Z)
• The metaplectic semigroup and its applications to time-frequency analysis and evolution operators [0.0]
  We develop a systematic analysis of the metaplectic semigroup $mathrmMp_+(d,mathbbC)$ associated with positive complex symplectic matrices.<n>We exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties.
  arXiv  Detail & Related papers  (2026-01-29T19:21:40Z)
• Random-Matrix-Induced Simplicity Bias in Over-parameterized Variational Quantum Circuits [72.0643009153473]
  We show that expressive variational ansatze enter a Haar-like universality class in which both observable expectation values and parameter gradients concentrate exponentially with system size.<n>As a consequence, the hypothesis class induced by such circuits collapses with high probability to a narrow family of near-constant functions.<n>We further show that this collapse is not unavoidable: tensor-structured VQCs, including tensor-network-based and tensor-hypernetwork parameterizations, lie outside the Haar-like universality class.
  arXiv  Detail & Related papers  (2026-01-05T08:04:33Z)
• Group-theoretical analysis of quantum complexity: the oscillator group case [0.0]
  We present the derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group.<n>The questions concerning the complexity of particular unitaries are solved by lifting the abstract structure to the operator level.<n>We compute, at least in principle, the complexity of any unitary operator belonging to the representation.
  arXiv  Detail & Related papers  (2025-12-19T13:17:18Z)
• Graph-based Clustering Revisited: A Relaxation of Kernel $k$-Means Perspective [73.18641268511318]
  We propose a graph-based clustering algorithm that only relaxes the orthonormal constraint to derive clustering results.<n>To ensure a doubly constraint into a gradient, we transform the non-negative constraint into a class probability parameter.
  arXiv  Detail & Related papers  (2025-09-23T09:14:39Z)
• Unified Structural Embedding of Orbifold Sigma Models [0.0]
  This study introduces a new unified structural framework for orbifold sigma models.<n>The formalism is shown to recover conventional sigma model results in the smooth limit where $G$ approaches the trivial group.<n>Examples demonstrate the applicability, including explicit calculations for the $mathbbC/mathbbZ$ orbifold.
  arXiv  Detail & Related papers  (2025-05-09T20:20:08Z)
• Representation theory of Gaussian unitary transformations for bosonic and fermionic systems [0.0]
  We analyze the behavior of the sign ambiguity that one needs to deal with when moving between the groups of the symplectic and special annihilation group. We show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space.
  arXiv  Detail & Related papers  (2024-09-18T01:22:38Z)
• Understanding Heterophily for Graph Neural Networks [42.640057865981156]
  We present theoretical understandings of the impacts of different heterophily patterns for Graph Neural Networks (GNNs) We show that the separability gains are determined by the normalized distance of the $l$-powered neighborhood distributions. Experiments on both synthetic and real-world data verify the effectiveness of our theory.
  arXiv  Detail & Related papers  (2024-01-17T11:01:28Z)
• Lie Group Decompositions for Equivariant Neural Networks [12.139222986297261]
  We show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the benchmark affine-invariant classification task.
  arXiv  Detail & Related papers  (2023-10-17T16:04:33Z)
• Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation [64.49871502193477]
  We propose a novel semi-supervised subspace clustering method, which is able to simultaneously augment the initial supervisory information and construct a discriminative affinity matrix. Comprehensive experimental results on six commonly-used benchmark datasets demonstrate the superiority of our method over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-05-21T01:47:17Z)
• Spatially relaxed inference on high-dimensional linear models [48.989769153211995]
  We study the properties of ensembled clustered inference algorithms which combine spatially constrained clustering, statistical inference, and ensembling to aggregate several clustered inference solutions. We show that ensembled clustered inference algorithms control the $delta$-FWER under standard assumptions for $delta$ equal to the largest cluster diameter.
  arXiv  Detail & Related papers  (2021-06-04T16:37:19Z)
• GroupifyVAE: from Group-based Definition to VAE-based Unsupervised Representation Disentanglement [91.9003001845855]
  VAE-based unsupervised disentanglement can not be achieved without introducing other inductive bias. We address VAE-based unsupervised disentanglement by leveraging the constraints derived from the Group Theory based definition as the non-probabilistic inductive bias. We train 1800 models covering the most prominent VAE-based models on five datasets to verify the effectiveness of our method.
  arXiv  Detail & Related papers  (2021-02-20T09:49:51Z)
• Manifold Topology, Observables and Gauge Group [0.0]
  The relation between manifold topology, observables and gauge group is clarified. The implications on the observability of the Permutation Group in Particle Statistics are discussed.
  arXiv  Detail & Related papers  (2021-02-17T18:05:13Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.