In this work, we explore the trade-offs of explicit structural priors, particularly group-equivariance. We address this through theoretical analysis and a comprehensive empirical study focusing on point clouds. To enable controlled and fair comparisons, we introduce Rapidash, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit symmetry breaking via geometric reference frames consistently improves performance, while breaking equivariance through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection.

In this section, we provide a brief introduction to the concepts from Group Theory which we need in our derivations. A group is a pair (G,)containing a set Gand a binary operation: G G! G,(h,g) 7! h g which satisfies the group axioms: Associativity: 8a,b,c 2 Ga (b c)=( a b) c Identity: 9e 2 G: 8g 2 Gg e = e g = g Inverse: 8g 2 G 9g 1 2 G: g g 1 = g 1 g = e The operation is the group law of G. The inverse elements g 1 of an element g, and the identity element e are unique. In addition, if the group law is also commutative, the group G is an abelian group. To simplify the notation, we commonly write ab instead of a b. It is also common to denote the group (G,) just with the name of its underlying set G. The order of a group G is the cardinality of its set and is indicated by |G|. A group G is finite when |G|2 N, i.e., when it has a finite number of elements. A compact group is a group that is also a compact topological space with continuous group operation. Given a group G, its action on a set X is a map . A simple example of group action is the group law itself: G G! Gwhich defines an action of G on its own elements (X = G). Another important action is the one defined on signals overs the group G. Given a signal x: G! R, the action of an element g 2 G maps x 7! g.x, [g.x](h):= x(g 1h).

Wepresentgroupequivariant capsule networks,aframeworktointroduce guaranteed equivariance and invariance properties to the capsule network idea.

Frequently,transformations occurring in data can be better represented by a subset of a group than by agroup asawhole, e.g., rotations in[ 90,90 ]. Insuch cases, amodel that respects equivariancepartially is better suited to represent the data.

Neural decoding, a critical component of Brain-Computer Interface (BCI), has recently attracted increasing research interest. Previous research has focused on leveraging signal processing and deep learning methods to enhance neural decoding performance. However, the in-depth exploration of model architectures remains underexplored, despite its proven effectiveness in other tasks such as energy forecasting and image classification. In this study, we propose NeuroSketch, an effective framework for neural decoding via systematic architecture optimization. Starting with the basic architecture study, we find that CNN-2D outperforms other architectures in neural decoding tasks and explore its effectiveness from temporal and spatial perspectives. Building on this, we optimize the architecture from macro- to micro-level, achieving improvements in performance at each step. The exploration process and model validations take over 5,000 experiments spanning three distinct modalities (visual, auditory, and speech), three types of brain signals (EEG, SEEG, and ECoG), and eight diverse decoding tasks. Experimental results indicate that NeuroSketch achieves state-of-the-art (SOTA) performance across all evaluated datasets, positioning it as a powerful tool for neural decoding. Our code and scripts are available at https://github.com/Galaxy-Dawn/NeuroSketch.

We present group equivariant capsule networks, a framework to introduce guaranteed equivariance and invariance properties to the capsule network idea.