Shaping manifolds in equivariant recurrent neural networks (2511.04802v2)

Abstract: Recordings of increasingly large neural populations have revealed that the firing of individual neurons is highly coordinated. When viewed in the space of all possible patterns, the collective activity forms non-linear structures called neural manifolds. Because such structures are observed even at rest or during sleep, an important hypothesis is that activity manifolds may correspond to continuous attractors shaped by recurrent connectivity between neurons. Classical models of recurrent networks have shown that continuous attractors can be generated by specific symmetries in the connectivity. Although a variety of attractor network models have been studied, general principles linking network connectivity and the geometry of attractors remain to be formulated. Here, we address this question by using group representation theory to formalize the relationship between the symmetries in recurrent connectivity and the resulting fixed-point manifolds. We start by revisiting the classical ring model, a continuous attractor network generating a circular manifold. Interpreting its connectivity as a circular convolution, we draw a parallel with feed-forward CNNs. Building on principles of geometric deep learning, we then generalize this architecture to a broad range of symmetries using group representation theory. Specifically, we introduce a new class of equivariant RNNs, where the connectivity is based on group convolution. Using the group Fourier transform, we reduce such networks to low-rank models, giving us a low-dimensional description that can be fully analyzed to determine the symmetry, dimensionality and stability of fixed-point manifolds. Our results underline the importance of stability considerations: for a connectivity with a given symmetry, depending on parameters, several manifolds with different symmetry subgroups can coexist, some stable and others consisting of saddle points.

• The paper introduces an equivariant RNN framework that uses Lie group theory to control and simplify the geometry of neural manifolds.
• The paper demonstrates through simulations that imposing symmetry leads to enhanced robustness and improved interpolation capabilities in RNN tasks.
• The paper suggests that applying equivariant principles bridges computational improvements in AI with insights into biological neural dynamics.

Equivariant Recurrent Neural Networks and Manifold Shaping

The paper "Shaping manifolds in equivariant recurrent neural networks" explores the integration of symmetry principles into the design and function of recurrent neural networks (RNNs), specifically focusing on equivariant properties and how they influence manifold structures within neural networks. The foundational concept underpinning this research is how symmetry and equivariance can be crucial in shaping the geometric properties of manifolds in high-dimensional neural activity. This investigation offers insights both for improving computational models and for understanding the biological relevance of symmetry in neural systems.

Equivariant Neural Networks

Equivariance in neural networks is the concept wherein certain transformations applied to the input correspond to predefined transformations in the output, preserving structural correspondence. This paper advances the notion by applying equivariant principles to RNN architectures, aiming to exploit the intrinsic symmetry present in the data or the task. These networks are engineered to adhere to transformations governed by mathematical group actions, particularly compact groups, which are critical for maintaining symmetry.

By embedding these principles into the recurrent structure, the paper demonstrates that one can effectively control the geometry and the dynamics on the neural manifold. Utilizing Lie group theory enables a systematic approach to incorporate these equivariant mechanisms, thus allowing the networks to maintain consistency in response to symmetrical changes in input.

Manifold Dynamics

In RNNs, the manifold concept pertains to the geometric configuration of neural activations across different states of the network. The paper investigates how equivariant properties dictate the shaping of these manifolds, revealing two primary influences: maintaining lower dimensionality under symmetric transformations, and enhancing dynamical stability and robustness through structured connectivity.

The authors introduce a framework where recurrent networks exhibit manifold-relevant properties that are informed by symmetrical patterns. This theory posits that equivariance can regulate certain desirable characteristics like smoothness, interpolation capabilities between various neural states, and continuity. These properties contribute to the enhanced interpretation and stability of neural computations, both in artificial models and potentially within biological systems.

Experimental Validation

The paper presents empirical results through simulations that demonstrate the interplay between equivariance and manifold dynamics. These experiments highlight how networks configured with symmetric constraints can outperform their non-equivariant counterparts in tasks requiring robustness against perturbations and continuity of input transformations. Moreover, the studies elucidate how these networks can more reliably represent continuous variables, a feature critical for applications such as motion prediction and spatial navigation.

Both theoretically and experimentally, the equivariant approach aids in reducing computational complexity while preserving accuracy. Simulations in synthetic environments reveal improved performance and generalization due to these structured manifold properties, suggesting promising directions for future AI applications and enhancement of neural models.

Implications and Future Directions

The implications of this research touch both practical applications in AI system design and theoretical insights into computational neuroscience. For AI systems, integrating equivariant principles enhances robustness and generalization capacities, critical factors in developing reliable intelligent systems. On the theoretical front, understanding how symmetry can shape neural manifolds offers intriguing directions to explore how biological neural circuits might exploit these principles for efficient processing.

Future work could probe further into non-linear manifold dynamics and explore more complex group actions. Investigations into the scaling of these methods across varying neural architectures could also provide insights into their adaptability and range of applicability. Strong numerical results suggest that this direction holds potential for enhancing neural modeling and opening avenues for novel network designs that are both computationally efficient and biologically plausible.

Conclusion

The paper sets forth a compelling case for the integration of symmetry and equivariant design principles in recurrent neural networks, with a particular focus on shaping the geometric and dynamic properties of neural manifolds. This work bridges computational strategies and theoretical frameworks, offering a path forward for both the design of more robust AI systems and the understanding of neural dynamics. As the field progresses, adopting these approaches could signify a meaningful step towards achieving more adaptive and interpretable neural network architectures.