The Hessian of f(Z) can be viewed as an KN KN matrix by vectorizing the matrix Z. For deeper linear networks, it can be shown that flat saddle points exist at the origin, but there are no spurious local minima [34,37]. While most of these results based on the bottom-up approach explain optimization and generalization of certain types of deep neural networks, they provided limited insights into the practice of deep learning. In fact, our proof techniques are inspired by recent results on low-rank matrix recovery [77,80]. Some of the metrics are similar to those presented in [1]. Figure 7 depicts the learning curves in terms of both the training and test accuracy for all three optimization algorithms (i.e., SGD, Adam, and LBFGS).

Object, attribute or stylistic changes can be learned from visually static datasets. On the other hand, high-quality data for action and reasoning-centric edits is scarce and has to come from entirely different sources that cover e.g.

Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the adjoint method, many downstream tasks such as active learning, exploration in reinforcement learning, robust control, or filtering require accurate estimates of predictive uncertainties.