Similarity Matching Networks: Hebbian Learning and Convergence Over Multiple Timescales

A recent breakthrough in biologically plausible normative frameworks for dimensionality reduction is based on the similarity matching cost function and the low-rank matrix approximation problem. Despite clear biological interpretation, successful application in several domains, and experimental validation, a formal complete convergence analysis remains elusive. Building on this framework, we consider and analyze a continuous-time neural network, the similarity matching network, for principal subspace projection. Derived from a min-max-min objective, this biologically plausible network consists of three coupled dynamics evolving at different timescales: neural dynamics, lateral synaptic dynamics, and feedforward synaptic dynamics at the fast, intermediate, and slow timescales, respectively. The feedforward and lateral synaptic dynamics consist of Hebbian and anti-Hebbian learning rules, respectively. By leveraging a multilevel optimization framework, we prove convergence of the dynamics in the offline setting. Specifically, at the first level (fast timescale), we show strong convexity of the cost function and global exponential convergence of the corresponding gradient-flow dynamics. At the second level (intermediate timescale), we prove strong concavity of the cost function and exponential convergence of the corresponding gradient-flow dynamics within the space of positive definite matrices. At the third and final level (slow timescale), we study a nonconvex and nonsmooth cost function, provide explicit expressions for its global minima, and prove almost sure convergence of the corresponding gradient-flow dynamics to the global minima. These results rely on two empirically motivated conjectures that are supported by thorough numerical experiments. Finally, we validate the effectiveness of our approach via a numerical example.