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Implicit Gaussian process representation of vector fields over arbitrary latent manifolds
Sep 28, 2023
Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.
Interpretable statistical representations of neural population dynamics and geometry
Apr 06, 2023
The dynamics of neuron populations during diverse tasks often evolve on low-dimensional manifolds. However, it remains challenging to discern the contributions of geometry and dynamics for encoding relevant behavioural variables. Here, we introduce an unsupervised geometric deep learning framework for representing non-linear dynamical systems based on statistical distributions of local phase portrait features. Our method provides robust geometry-aware or geometry-agnostic representations for the unbiased comparison of dynamics based on measured trajectories. We demonstrate that our statistical representation can generalise across neural network instances to discriminate computational mechanisms, obtain interpretable embeddings of neural dynamics in a primate reaching task with geometric correspondence to hand kinematics, and develop a decoding algorithm with state-of-the-art accuracy. Our results highlight the importance of using the intrinsic manifold structure over temporal information to develop better decoding algorithms and assimilate data across experiments.