Karoline Wiesner

Sönke Beier, Paula Pirker-Díaz, Friedrich Pagenkopf et al.

Diffusion Map is a spectral dimensionality reduction technique which is able to uncover nonlinear submanifolds in high-dimensional data. And, it is increasingly applied across a wide range of scientific disciplines, such as biology, engineering, and social sciences. But data preprocessing, parameter settings and component selection have a significant influence on the resulting manifold, something which has not been comprehensively discussed in the literature so far. We provide a practice oriented review of the Diffusion Map technique, illustrate pitfalls and showcase a recently introduced technique for identifying the most relevant components. Our results show that the first components are not necessarily the most relevant ones.

Ibrahim Talha Ersoy, Andrés Fernando Cardozo Licha, Karoline Wiesner

Training Deep Neural Networks relies on the model converging on a high-dimensional, non-convex loss landscape toward a good minimum. Yet, much of the phenomenology of training remains ill understood. We focus on three seemingly disparate observations: the occurrence of phase transitions reminiscent of statistical physics, the ubiquity of saddle points, and phenomenon of mode connectivity relevant for model merging. We unify these within a single explanatory framework, the geometry of the loss and error landscapes. We analytically show that phase transitions in DNN learning are governed by saddle points in the loss landscape. Building on this insight, we introduce a simple, fast, and easy to implement algorithm that uses the L2 regularizer as a tool to probe the geometry of error landscapes. We apply it to confirm mode connectivity in DNNs trained on the MNIST dataset by efficiently finding paths that connect global minima. We then show numerically that saddle points induce transitions between models that encode distinct digit classes. Our work establishes the geometric origin of key training phenomena in DNNs and reveals a hierarchy of accuracy basins analogous to phases in statistical physics.

Ibrahim Talha Ersoy, Karoline Wiesner

Increasing the L2 regularization of Deep Neural Networks (DNNs) causes a first-order phase transition into the under-parametrized phase -- the so-called onset-of learning. We explain this transition via the scalar (Ricci) curvature of the error landscape. We predict new transition points as the data complexity is increased and, in accordance with the theory of phase transitions, the existence of hysteresis effects. We confirm both predictions numerically. Our results provide a natural explanation of the recently discovered phenomenon of '\emph{grokking}' as DNN models getting stuck in a local minimum of the error surface, corresponding to a lower accuracy phase. Our work paves the way for new probing methods of the intrinsic structure of DNNs in and beyond the L2 context.