group representation design

This work proposes a novel method based on the generalized Fourier transform to discover continuous symmetries in scientific and machine learning problems, where such symmetries are often unknown and challenging to identify. By analyzing the spectral structure of functions under irreducible representations of Lie groups, the approach leverages symmetry-induced spectral sparsity to detect continuous one-parameter subgroups, circumventing conventional reliance on generator optimization or data augmentation. The method integrates representation theory, multidimensional Fourier analysis, and sparsity detection on maximal tori, offering strong theoretical grounding and enhanced interpretability. Experiments on the double pendulum system and top-quark tagging successfully recover one-dimensional continuous symmetries, demonstrating the efficacy and robustness of the proposed spectral framework.

This work addresses the challenge of missing symmetry priors in physical system modeling and machine learning by proposing a novel paradigm for directly discovering latent symmetry structures from high-dimensional data (e.g., 2D/3D point clouds). Methodologically, it formulates symmetry discovery as a flow matching problem on Lie groups and introduces LieFlow—a unified framework capable of handling both continuous and discrete symmetry groups, including non-compact and disconnected groups such as complex-domain reflections. To overcome convergence failure of conventional flow matching at critical time points, LieFlow introduces a novel complex-domain interpolation strategy. Experiments demonstrate that LieFlow accurately identifies rotational and reflective symmetries, significantly improving distribution matching accuracy and robustness in symmetry identification—even under few-shot settings. The framework provides a learnable, generalizable tool for group-structure discovery, enabling symmetry-driven generative modeling and physics-informed learning.

Existing 2D continuous representations struggle to simultaneously preserve continuity and satisfy arbitrary plane group symmetries, particularly because non-reflection operations often disrupt continuity. This work proposes the first general-purpose symmetrization framework that rigorously enforces full plane group symmetry—including non-reflection operations—while maintaining continuity in 2D continuous representations. By integrating group-theoretic modeling with approximation theory for continuous functions, the method transforms any 2D continuous representation into one that strictly adheres to prescribed symmetries without compromising smoothness. The approach is validated across four diverse applications: pattern design, kirigami art, stylized topology, and material design, demonstrating high-fidelity, controllable generation of symmetric patterns. This study thus achieves, for the first time, full compatibility between general plane group symmetries and continuous 2D representations.

This work develops a novel algorithmic information theory within the framework of symmetric groups to characterize string complexity induced by symmetries. By introducing symmetry groups generated by computable bijections, the authors define a “symmetric prior” and, under the fix-retractable condition, prove it constitutes a universal lower-semicomputable semimeasure, thereby establishing a geometric coding theorem. The central innovation lies in the first unified integration of algorithmic information theory with group theory, proposing a new paradigm for complexity measures grounded in symmetry. Furthermore, the study reveals a structural correspondence between subgroups and sets of binary strings via a Galois connection. This theoretical foundation advances computational algorithmic statistics (CAS) and opens new avenues for analyzing structured data through algebraic and informational lenses.

This work overcomes the fundamental limitation of the Cohn–Umans group-theoretic framework—which has hitherto applied only to finite groups—by extending it to infinite groups, particularly Lie groups, thereby circumventing intrinsic barriers posed by finite groups of Lie type in matrix multiplication algorithm design. Methodologically, we generalize the triple product property and integrate Lie group representation theory with structural analysis to establish a complete theoretical framework that directly derives matrix multiplication algorithms from irreducible representations of Lie groups. Our main contributions are threefold: (1) We prove that Lie groups achieve asymptotic exponent parameters surpassing those attainable by any finite group; (2) we obtain a new upper bound on the matrix multiplication exponent ω, improving upon all prior finite-group constructions and providing a viable pathway toward ω < 2.37286; (3) we demonstrate the substantive algorithmic relevance of infinite groups in algebraic complexity theory, yielding fast matrix multiplication algorithms with concrete computational significance.

This work addresses the limitation of standard spectral embedding methods, which neglect invariance under symmetry groups such as rotations and thus fail to accurately recover the intrinsic geometry of symmetric data. The authors propose incorporating compact Lie group symmetries directly into the affinity kernel to construct a group-invariant graph Laplacian. Under the assumption of an underlying Riemannian manifold, they establish—for the first time—the convergence of graph Laplacians derived from three classes of invariant kernels to second-order differential operators on the corresponding quotient space. Notably, this convergence occurs at an accelerated rate due to the reduced effective dimensionality induced by the group action. Experiments demonstrate that, on data with SO(2) or SO(3) symmetries, the proposed method successfully recovers the intrinsic geometry, whereas conventional spectral embeddings remain inconsistent even in the infinite-sample limit.

This work addresses the problem of constructing minimal sufficient invariant representations for likelihood families invariant under Lie group actions. By introducing spherical harmonic analysis on compact homogeneous spaces, the authors establish a connection between the spherical Fourier coefficients of finite-bandwidth harmonic exponential families and minimal sufficient statistics. Leveraging Clebsch–Gordan decomposition, they isolate the trivial representation component to derive an algebraic expression for the partition function. The study theoretically proves that empirical harmonic coefficients constitute minimal sufficient statistics, thereby providing—for the first time—an explicit algebraic form of the partition function for this class of models. This result establishes a computable harmonic representation framework for invariant statistical inference.

Homogeneous spaces—realized as quotients of Lie groups—pose a challenge for existing flow-matching methods due to the absence of explicit metrics and geodesic structures. This work proposes an intrinsic framework that lifts the target distribution to the underlying Lie group and performs Euclidean flow matching on the corresponding Lie algebra, thereby entirely circumventing the need for a predefined metric or geodesics. The approach requires only a Lie group action and a local section, eliminating any reliance on Riemannian geometric computations and substantially simplifying the generative modeling pipeline. Experiments demonstrate that this method yields more efficient, concise, and scalable generative models on homogeneous spaces while preserving intrinsic geometric fidelity.

This work addresses the problem of determining the continuous symmetries—specifically, the symmetry Lie algebra—of a parametrized algebraic variety directly from its parametric representation, without explicitly computing its vanishing ideal. We propose the first method that derives the symmetry Lie algebra directly from the parametrization, introducing a polynomial-time Monte Carlo algorithm that circumvents the high computational complexity of traditional approaches involving vanishing ideals. By integrating techniques from algebraic geometry, Lie theory, and randomized algorithms, we construct an efficient computational framework. The method is successfully applied to parametrized varieties arising in staged tree models and colored Gaussian graphical models, confirming the binomial nature of their ideals under coordinate transformations and elucidating the symmetry structures of several classes of secant varieties.

In three-dimensional graphic statics, editing complex polyhedral graphs often compromises their symmetry, thereby undermining engineering applicability. This work introduces crystallographic point group theory into the field for the first time and establishes length consistency among equivalent edge sets as a necessary and sufficient condition for preserving symmetry. By integrating symmetry detection algorithms from spglib and pymatgen, the authors develop an efficient fingerprinting method to automatically classify equivalent edges and enforce corresponding constraints. Implemented in the PolyFrame 2 plugin, this approach significantly reduces the dimensionality of the solution space while effectively maintaining the symmetry of polyhedral graphs, thereby enhancing both design feasibility and computational efficiency.