Yes, there are notable connections between the rank conditions characterizing critical points in deep linear networks and the underlying data distribution or task complexity. The rank of the product of weight matrices, denoted as ( r = rk(WH \\cdots W1) ), plays a crucial role in determining the nature of critical points, including whether they are global minimizers, strict saddle points, or non-strict saddle points.
The rank conditions are inherently linked to the data distribution through the empirical risk minimization framework. For instance, the rank of the critical point is influenced by the rank of the data matrices ( \\Sigma_{XX} ) and ( \\Sigma_{XY} ). If the data is well-conditioned and has full rank, it is more likely that the optimization landscape will exhibit a richer structure with a variety of critical points. Conversely, if the data is poorly conditioned or has low rank, the optimization landscape may be flatter, leading to a higher likelihood of encountering non-strict saddle points.
Additionally, task complexity, which can be defined by the number of parameters relative to the amount of training data, also affects the rank conditions. In scenarios where the model capacity exceeds the complexity of the task (e.g., overparameterization), the optimization landscape may contain many non-strict saddle points, complicating the training dynamics. This relationship suggests that understanding the rank conditions in the context of the data distribution and task complexity can provide valuable insights into the optimization behavior of deep linear networks and inform strategies for improving convergence and generalization in practice.