Electroencephalography (EEG) provides access to neuronal dynamics noninvasively with millisecond resolution, rendering it a viable method in neuroscience and healthcare. However, its utility is limited as current EEG technology does not generalize well across domains (i.e., sessions and subjects) without expensive supervised re-calibration. Contemporary methods cast this transfer learning (TL) problem as a multi-source/-target unsupervised domain adaptation (UDA) problem and address it with deep learning or shallow, Riemannian geometry aware alignment methods. Both directions have, so far, failed to consistently close the performance gap to state-of-the-art domain-specific methods based on tangent space mapping (TSM) on the symmetric, positive definite (SPD) manifold. Here, we propose a machine learning framework that enables, for the first time, learning domain-invariant TSM models in an end-to-end fashion. To achieve this, we propose a new building block for geometric deep learning, which we denote SPD domain-specific momentum batch normalization (SPDDSMBN). ASPDDSMBN layer can transform domain-specific SPD inputs into domain-invariant SPD outputs, and can be readily applied to multi-source/-target and online UDA scenarios. In extensive experiments with 6 diverse EEG brain-computer interface (BCI) datasets, we obtain state-of-the-art performance in inter-session and -subject TL with a simple, intrinsically interpretable network architecture, which we denote TSMNet.

We push each class to its supernode, then apply majority vote to determine the final class.

Machine learning models--including prominent zero-shot models--are often trained on datasets whose labels are only a small proportion of a larger label space. Such spaces are commonly equipped with a metric that relates the labels via distances between them.

We also consider in detail the specific case where the summary is the Fréchet mean of data residing on a manifold.

Fine-grained visual classification (FGVC) tasks, such as insect and bird identification, demand sensitivity to subtle visual cues while remaining robust to spatial transformations. A key challenge is handling geometric biases and noise, such as different orientations and scales of objects. Existing remedies rely on heavy data augmentation, which demands powerful models, or on equivariant architectures, which constrain expressivity and add cost. Canonicalization offers an alternative by shielding such biases from the downstream model. In practice, such functions are often obtained using canonicalization priors, which assume aligned training data. Unfortunately, real-world datasets never fulfill this assumption, causing the obtained canonicalizer to be brittle. We propose a bootstrapping algorithm that iteratively re-aligns training samples by progressively reducing variance and recovering the alignment assumption. We establish convergence guarantees under mild conditions for arbitrary compact groups, and show on four FGVC benchmarks that our method consistently outperforms equivariant, and canonicalization baselines while performing on par with augmentation.

Real world data often exhibits unknown, instance-specific symmetries that rarely exactly match a transformation group $G$ fixed a priori. Class-pose decompositions aim to create disentangled representations by factoring inputs into invariant features and a pose $g\in G$ defined relative to a training-dependent, arbitrary canonical representation. We introduce RECON, a class-pose agnostic $\textit{canonical orientation normalization}$ that corrects arbitrary canonicals via a simple right-multiplication, yielding $\textit{natural}$, data-aligned canonicalizations. This enables (i) unsupervised discovery of instance-specific symmetry distributions, (ii) detection of out-of-distribution poses, and (iii) test-time canonicalization, granting group invariance to pre-trained models without retraining and irrespective of model architecture, improving downstream performance. We demonstrate results on 2D image benchmarks and --for the first time-- extend symmetry discovery to 3D groups.

This paper introduces the concept of a mean for trajectories and multi-object trajectories (defined as sets or multi-sets of trajectories) along with algorithms for computing them. Specifically, we use the Fréchet mean, and metrics based on the optimal sub-pattern assignment (OSPA) construct, to extend the notion of average from vectors to trajectories and multi-object trajectories. Further, we develop efficient algorithms to compute these means using greedy search and Gibbs sampling. Using distributed multi-object tracking as an application, we demonstrate that the Fréchet mean approach to multi-object trajectory consensus significantly outperforms state-of-the-art distributed multi-object tracking methods.