Explore a lecture on the implicit regularization in deep learning through the lens of rank minimization. Delve into the theoretical analysis of matrix and tensor factorizations, equivalent to certain linear and non-linear neural networks. Discover how gradient-based optimization tends to fit training data with predictors of low complexity, leading to generalization. Examine dynamical characterizations establishing implicit regularization towards low matrix and tensor ranks, challenging prior beliefs about norm minimization. Consider the implications of these findings for both theory and practice in modern deep learning, highlighting the potential of ranks to explain and improve generalization. Learn about matrix completion, tensor completion, and their connections to linear and non-linear neural networks. Investigate practical applications, including rank minimization in neural network layers and potential explanations for generalization on natural data.