Self-supervised learning (SSL) has empirically shown its data representation learnability in many downstream tasks. There are only a few theoretical works on learnability, and of those focus final representation, treating the nonlinear neural network as ``black box". However, accurate results networks crucial for describing distribution features learned by SSL models. Our paper is first to anal...

Jack D. Cowan Department of Math University of Chicago Chicago, IL 60637 [email protected] We couple the tasks of source separation and density estimation by extracting the local geometrical structure of distributions obtained from mixtures of statistically independent sources. Our modifications of the self-organizing map (SOM) algorithm results in purely digital learning rules which perform...

Some oscillation criteria are established for the second order nonlinear neutral differential equations of the form ((x(t) + ax(t− τ1)− bx(t+ τ2)) ) ′′ = q(t)x(t− σ1) + p(t)x (t+ σ2), and (x(t) − ax(t− τ1) + bx(t+ τ2) )′′ = q(t)x(t− σ1) + p(t)x (t+ σ2) where α and β are the ratios of odd positive integers with β ≥ 1. Examples are provided to illustrate the main results.