Uncovering Hidden Symmetries in Data

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Author: Denis Avetisyan

A new approach called LieFlow learns underlying symmetries directly from data, offering a powerful way to analyze complex systems without pre-defined assumptions.

Flow matching exposes hidden symmetries within data by transforming complex distributions into simpler, more manageable forms, effectively revealing underlying order from apparent chaos.
Flow matching exposes hidden symmetries within data by transforming complex distributions into simpler, more manageable forms, effectively revealing underlying order from apparent chaos.

LieFlow utilizes flow matching on Lie groups to provide a unified framework for discovering both continuous and discrete symmetries from data.

Understanding the underlying symmetries of data is crucial across diverse fields, yet discovering these symmetries often requires strong prior assumptions. This paper, ‘Discovering Lie Groups with Flow Matching’, introduces LieFlow, a novel approach that learns symmetries directly from data by formulating symmetry discovery as a flow matching problem on Lie groups. This allows for a unified framework capable of identifying both continuous and discrete symmetries without predefining the group structure. By addressing challenges like ‘last-minute convergence’ with a new interpolation scheme, can LieFlow unlock more robust and interpretable generative models for complex datasets?

Decoding Hidden Order: Symmetries in Data

Data, in many forms, frequently exhibits inherent symmetries – predictable patterns of equivalence that remain consistent even under transformations like rotations, reflections, or permutations. Recognizing and capitalizing on these symmetries offers substantial benefits in data analysis and machine learning. When a model is designed to acknowledge these underlying structures, it can achieve comparable performance with significantly less data, reducing computational costs and preventing overfitting. This principle stems from the idea that the model isn’t learning entirely new information for each variation, but rather generalizing from a smaller, symmetrical representation of the data. Consequently, models built upon symmetry-aware architectures demonstrate improved efficiency and a greater capacity to generalize to unseen data, making them particularly valuable in fields dealing with complex, high-dimensional datasets such as image recognition, signal processing, and particle physics.

Conventional machine learning approaches frequently operate as “black boxes,” treating data points as isolated instances without recognizing inherent relational structures. This oversight leads to inefficiencies, as models must learn redundant patterns instead of capitalizing on existing symmetries within the data. Consequently, these methods require substantially larger datasets to achieve comparable performance to techniques that explicitly incorporate symmetry awareness. The need for extensive data not only increases computational cost and storage demands, but also limits applicability in scenarios where data acquisition is expensive or restricted. By neglecting these underlying symmetries, traditional algorithms often produce models that generalize poorly to unseen data and exhibit diminished predictive power, hindering their practical utility and broader impact.

Generated data samples closely replicate the C4 or D4 symmetries present in the original datasets.
Generated data samples closely replicate the C4 or D4 symmetries present in the original datasets.

The Language of Symmetry: Lie Groups Defined

Lie groups are mathematical structures, specifically smooth manifolds equipped with a group operation, that formally describe continuous symmetries. These groups allow for the representation of transformations which leave certain properties of an object or system invariant; for example, rotations in 2D or 3D space are described by the Lie groups $SO(2)$ and $SO(3)$ respectively. The utility of Lie groups lies in their ability to provide a consistent framework for analyzing and manipulating data exhibiting such symmetries; this is achieved through the group operation, which defines how symmetries combine, and the smooth manifold structure, which allows for the differentiation of transformations and the calculation of infinitesimal changes. This formalism is broadly applicable in physics, engineering, and computer vision, where understanding and exploiting symmetry is often crucial for simplifying models and improving efficiency.

The TangentSpace at the identity element of a Lie group provides a vector space approximation of the group’s local behavior. This allows for the application of linear algebra techniques to study the group’s infinitesimal transformations. The ExponentialMap, denoted as $exp: \mathfrak{g} \rightarrow G$, is a crucial function that maps elements from the Lie algebra $\mathfrak{g}$ (the tangent space at the identity) to the Lie group $G$. Specifically, it provides a way to “exponentiate” an infinitesimal transformation, represented by an element in the Lie algebra, to obtain a finite, continuous transformation within the Lie group. This map is essential for moving between the linear representation in the Lie algebra and the non-linear manifold of the Lie group, enabling calculations of group elements and their properties from algebraic data.

Several Lie groups serve as foundational examples and appear frequently in applications. $SO(2)$ represents the group of 2D rotations, while $GL(2)$ denotes the general linear group of 2×2 invertible matrices. The cyclic group $C_4$ and the dihedral group $D_4$ represent discrete symmetries common in crystallography and geometry. Further examples arise from the symmetries of polyhedra; the TetrahedralGroup corresponds to the rotational symmetries of a regular tetrahedron, and the OctahedralGroup describes the rotational symmetries of an octahedron or cube. These groups provide concrete instances for understanding the abstract properties of Lie groups and their applications in fields like physics and computer graphics.

Visualization of 100 samples demonstrates the transformation of SO(2) group elements into C4 symmetry over time.
Visualization of 100 samples demonstrates the transformation of SO(2) group elements into C4 symmetry over time.

LieFlow: Unveiling Symmetry Through Dynamic Systems

LieFlow introduces a new methodology for symmetry discovery by directly applying flow matching techniques to Lie groups. Traditional approaches often require pre-defined symmetry constraints; LieFlow, however, learns these symmetries directly from the data. This is achieved by training a transformation that maps samples from a prior distribution to the target data distribution, all while operating within the constraints of the Lie group’s inherent structure. By performing flow matching directly on the Lie group manifold, the method avoids the need for explicit symmetry parameterization and allows for the identification of complex, data-driven symmetries.

LieFlow employs flow matching to learn a continuous transformation that maps samples from a predefined $PriorDistribution$ to the observed $DataDistribution$. This learning process is constrained to operate directly on the Lie group associated with the data’s symmetries. Specifically, the flow is constructed such that it preserves the group structure, ensuring that transformations within the $PriorDistribution$ are consistently mapped to corresponding transformations in the $DataDistribution$. This direct operation on the Lie group avoids the need for reparameterization or approximations of the symmetry transformations, enabling accurate and efficient learning of the underlying data manifold while respecting inherent symmetries.

Evaluations of LieFlow on multi-object datasets demonstrate a Wasserstein-1 distance of 0.100, indicating a high degree of similarity between the generated and real data distributions. This performance represents a substantial improvement over the LieGAN method, achieving reductions of 2.10 and 1.58 in Wasserstein-1 distance when utilizing 1 and 6 Lie generators, respectively. These quantitative results confirm LieFlow’s enhanced capacity for accurately modeling and generating complex, multi-object scenes while preserving underlying symmetries, as measured by the $W_1$ distance metric.

Tracing the Flow: Dynamics and the Emergence of Order

The VelocityField, central to the flow matching process, functions as a detailed map of the transformations learned by the model and, crucially, exposes the inherent structure within the data itself. By visualizing this field, researchers gain insight into how the model navigates the data landscape, identifying key trajectories and dependencies. A consistently aligned VelocityField suggests the model has effectively learned a smooth and meaningful representation, while disruptions or irregularities can pinpoint areas where the data is complex or the learning process is struggling. This allows for targeted refinement of the model and a deeper understanding of the underlying data distribution, ultimately enhancing the quality and interpretability of the generated outputs. The field essentially translates the abstract process of data transformation into a visually accessible format, revealing patterns and relationships that might otherwise remain hidden.

A curious dynamic emerges during flow matching, termed ‘LastMinuteModeConvergence,’ wherein the velocity field-representing the learned transformation-remains consistently near zero for the majority of the training process. This suggests the model doesn’t immediately commit to a specific transformation, but rather delays the development of a strong velocity field until the later stages of learning. The observation implies a unique strategy where the model initially explores the data distribution with minimal directional change, only solidifying a defined transformation pathway as training progresses. This delayed commitment may contribute to the stability and robustness of the learned flow, allowing for fine-tuning and preventing premature convergence on suboptimal solutions, and potentially enhancing generalization to unseen data.

The methodology demonstrates considerable versatility, attaining a Wasserstein-1 distance of 0.282 when applied to structured-preserving transformation tasks-a significant benchmark in assessing the fidelity of generated data. This outcome suggests the approach effectively captures and maintains the inherent structure within complex datasets during transformations. Ongoing research focuses on refinements, notably through techniques like ConditionalFlowMatching, which aim to further enhance both the stability and efficiency of the learning process. These advancements seek to address existing challenges and optimize performance, paving the way for more robust and accurate data transformations across a wider range of applications, ultimately minimizing discrepancies between the learned and true data distributions, as measured by metrics like the Wasserstein distance.

The entropy of the posterior remains largely uniform initially, but the generated samples rapidly converge to the nearest target mode based on the initial state.
The entropy of the posterior remains largely uniform initially, but the generated samples rapidly converge to the nearest target mode based on the initial state.

The work detailed in this paper embodies a spirit of rigorous exploration, mirroring the approach of Carl Friedrich Gauss, who once stated, “I prefer a hard problem to an easy one.” LieFlow doesn’t merely apply existing symmetry detection methods; it challenges the fundamental assumptions of how symmetries are discovered. By operating directly on Lie groups, the framework bypasses the need for predefined symmetry constraints, effectively reverse-engineering the underlying principles governing the data. This echoes Gauss’s preference for confronting complexity; LieFlow isn’t content with simple solutions but delves into the intricate structure of symmetry itself, revealing hidden patterns through a method grounded in the core concept of manifold learning and generative modeling.

What’s Next?

The assertion that LieFlow unlocks symmetry discovery from data feels less like a destination and more like a carefully constructed demolition. The system doesn’t merely reveal symmetries; it actively probes for the cracks in the data’s apparent order, exploiting the inherent tension between representation and invariance. Future work, then, isn’t about polishing the technique, but about deliberately stressing it. Can LieFlow be made to fail in predictable ways, thus revealing the limits of its underlying assumptions about data structure? A bug, after all, is the system confessing its design sins.

The current framework elegantly handles both continuous and discrete symmetries, but this unification begs the question: are these truly fundamental distinctions, or merely points on a spectrum of invariance? The next iteration shouldn’t aim for broader coverage of symmetry types, but for a deeper understanding of the transition between them. What mechanisms govern the emergence of discrete symmetries from continuous ones, and can LieFlow be adapted to model this process dynamically?

Ultimately, the true test lies in application to genuinely complex systems – those where symmetries are not cleanly defined, but rather approximate and emergent. To apply this approach to systems where the ‘ground truth’ is unknown will necessitate a new methodology for interpreting LieFlow’s output. The goal isn’t to find the ‘correct’ symmetry, but to identify the most useful one – the one that best simplifies the system’s behavior, even if it doesn’t perfectly reflect its underlying physics.

Original article: https://arxiv.org/pdf/2512.20043.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-24 19:00