Applied Physical Sciences Colloquium, Nina Miolane, University of California, Santa Barbara

Title: The Geometry of Neural Manifolds

Abstract: 

In machine learning, the manifold hypothesis states that many real-world high-dimensional data sets actually lie along low-dimensional latent manifolds inside the high-dimensional space. In neurosciences, the neural manifold hypothesis postulates that the activity of a neural population forms a low-dimensional manifold whose structure reflects that of the encoded task variables. In this work, we combine topological deep generative models and extrinsic Riemannian geometry to quantify the structure of (neural) manifolds. This approach (i) computes an explicit parameterization of the manifolds and (ii) estimates their local extrinsic curvature--hence quantifying their shape within the high-dimensional neural state space. Importantly, our methodology is invariant with respect to transformations that do not bear meaningful neuroscience information, such as permutation of the order in which neurons are recorded. We show empirically that we correctly estimate the geometry of synthetic manifolds generated from smooth deformations of circles, spheres, and tori, using realistic noise levels. We additionally validate our methodology on simulated and real neural recordings, and show that we recover geometric structure known to exist in hippocampal place cells. We expect this approach to open new avenues of inquiry into geometric neural correlates of perception and behavior, and quantify manifolds' geometry in natural and artificial neural networks.