Equivariant machine learning is an approach for designing deep learning models that respect the symmetries of the problem, with the aim of reducing model complexity and improving generalization. In this paper, we focus on an extension of shift equivariance, which is the basis of convolution networks on images, to general graphs. Unlike images, graphs do not have a natural notion of domain translation. Therefore, we consider the graph functional shifts as the symmetry group: the unitary operators that commute with the graph shift operator. Notably, such symmetries operate in the signal space rather than directly in the spatial space.We remark that each linear filter layer of a standard spectral graph neural network (GNN) commutes with graph functional shifts, but the activation function breaks this symmetry. Instead, we propose nonlinear spectral filters (NLSFs) that are fully equivariant to graph functional shifts and show that they have universal approximation properties. The proposed NLSFs are based on a new form of spectral domain that is transferable between graphs. We demonstrate the superior performance of NLSFs over existing spectral GNNs in node and graph classification benchmarks.

This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_\omega(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_\omega(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

Real-world geometry and 3D vision tasks are replete with challenging symmetries that defy tractable analytical expression. In this paper, we introduce Neural Isometries, an autoencoder framework which learns to map the observation space to a general-purpose latent space wherein encodings are related by isometries whenever their corresponding observations are geometrically related in world space. Specifically, we regularize the latent space such that maps between encodings preserve a learned inner product and commute with a learned functional operator, in the same manner as rigid-body transformations commute with the Laplacian. This approach forms an effective backbone for self-supervised representation learning, and we demonstrate that a simple off-the-shelf equivariant network operating in the pre-trained latent space can achieve results on par with meticulously-engineered, handcrafted networks designed to handle complex, nonlinear symmetries. Furthermore, isometric maps capture information about the respective transformations in world space, and we show that this allows us to regress camera poses directly from the coefficients of the maps between encodings of adjacent views of a scene.

One of the most analyzed problems in Graph Neural Networks (GNNs) is over-smoothing, that is usually described as the exponential convergence of node embeddings to a common vector through the GNN layers. One of the more frequently used metrics to analyze both theoretically and empirically over-smoothing is the Dirichlet energy, that is induced by the graph Laplacian, with different possibilities as analyzed in the next section. A formal axiomatic definition of over-smoothing, based on the definition of a "total over-smoothing" state where all node embeddings are identical, has been proposed in [1]. A key axiom of the proposal is that a smoothness measure should be zero if and only if this state is reached. In this paper, we point out that the widely-used Dirichlet Energy induced by the normalized graph Laplacian does not satisfy this axiom. Recently, some other issues in adopting Dirichlet energies in order to measure over-smoothing were pointed out in [2], where it is explained that Dirichlet energy induced by the normalized Laplacian tends to zero when node embeddings tend to its dominant eigenvector v s.t.

The key limitation of the verification performance lies in the ability of error detection. With this intuition we designed several variants of pessimistic verification, which are simple workflows that could significantly improve the verification of open-ended math questions. In pessimistic verification we construct multiple parallel verifications for the same proof, and the proof is deemed incorrect if any one of them reports an error. This simple technique significantly improves the performance across many math verification benchmarks without incurring substantial computational resources. Its token efficiency even surpassed extended long-CoT in test-time scaling. Our case studies further indicate that the majority of false negatives in stronger models are actually caused by annotation errors in the original dataset, so our method's performance is in fact underestimated. Self-verification for mathematical problems can effectively improve the reliability and performance of language model outputs, and it also plays a critical role in enabling long-horizon mathematical tasks. We believe that research on pessimistic verification will help enhance the mathematical capabilities of language models across a wide range of tasks.

Unlike images, graphs do not have a natural notion of domain translation.

We propose CatEquiv, a category-equivariant neural network for Human Activity Recognition (HAR) from inertial sensors that systematically encodes temporal, amplitude, and structural symmetries. We introduce a symmetry category that jointly represents cyclic time shifts, positive gain scalings, and the sensor-hierarchy poset, capturing the categorical symmetry structure of the data. CatEquiv achieves equivariance with respect to the categorical symmetry product. On UCI-HAR under out-of-distribution perturbations, CatEquiv attains markedly higher robustness compared with circularly padded CNNs and plain CNNs. These results demonstrate that enforcing categorical symmetries yields strong invariance and generalization without additional model capacity.

Human activity recognition is challenging because sensor signals shift with context, motion, and environment; effective models must therefore remain stable as the world around them changes. We introduce a categorical symmetry-aware learning framework that captures how signals vary over time, scale, and sensor hierarchy. We build these factors into the structure of feature representations, yielding models that automatically preserve the relationships between sensors and remain stable under realistic distortions such as time shifts, amplitude drift, and device orientation changes. On the UCI Human Activity Recognition benchmark, this categorical symmetry-driven design improves out-of-distribution accuracy by approx.