Topic: metric spaces

The hippocampus is the episodic memory and spatial navigation center of the brain. The cognitive map metaphor identifies episodic memory with ``navigation'' through a space of arbitrary learned relationships. However, what structures are admissible as ``cognitive maps'' remains unclear, particularly whether these maps are constrained to obey the same rules as physical space. We hypothesize that the hippocampus learns metric spaces as its cognitive maps. We test this hypothesis by training mice to learn and solve relational queries in a finite metric space which cannot be embedded in 2D Euclide

This talk establishes a general reduction from nonlinear spectral gaps of metric spaces to data-dependent space partitions in the form of locality-sensitive hashing. The reduction yields an approach to high-dimensional approximate near-neighbor search and a data structure for any d-dimensional norm whose query algorithm makes a sublinear number of probes. Its approximation factor is O(log d), improving on the square-root-of-d factor available from the generic approach based on John’s ellipsoid. The presentation connects spectral properties, geometric partitions, and efficient approximate searc