We derive a general system of ODEs and associated explicit solutions in a special case for geodesics between full rank matrices in the deep linear network geometry. In the process, we characterize all horizontal straight lines in the invariant balanced manifold that remain geodesics under Riemannian submersion.

On manifolds with metrics, minimizing geodesics, or shortest paths, are minimum-length curves connecting points. Various real-world tasks can be reduced to the generation of geodesics on manifolds. Examples include time-optimal path planning on sloping ground [Matsumoto, 1989], robot motion planning under various constraints [LaValle, 2006, Ratliff et al., 2015], physical systems [Pfeifer, 2019], the Wasserstein distance [Agueh, 2012], and image morphing [Michelis and Becker, 2021, Effland et al., 2021]. Typically, metrics are only known infinitesimally (a form of a Riemannian or Finsler metric), and their distance functions are not known beforehand. Computation of geodesics by solving optimization problems or differential equations is generally computationally costly and requires an explicit form of the metric, or at least, values of its differentials.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density. The resulting locally adaptive normal distribution (LAND) is a generalization of the normal distribution to the "manifold" setting, where data is assumed to lie near a potentially low-dimensional manifold embedded in R^D. The LAND is parametric, depending only on a mean and a covariance, and is the maximum entropy distribution under the given metric. The underlying metric is, however, non-parametric. We develop a maximum likelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, and provide the corresponding EM algorithm. We demonstrate the efficiency of the LAND to fit non-trivial probability distributions over both synthetic data, and EEG measurements of human sleep.