Related papers: Learning on Manifolds: Universal Approximations Pr…

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do…

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…

We suggest a universal map capable to recover a behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare…

Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…

Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…

We propose a novel second-order ODE as the continuous-time limit of a Riemannian accelerated gradient-based method on a manifold with curvature bounded from below. This ODE can be seen as a generalization of the ODE derived for Euclidean…

Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of…

Smooth autonomous dynamical systems modeled by ordinary differential equations (ODEs) cannot robustly and globally stabilize a point in compact, boundaryless manifolds. This obstruction, which is topological in nature, implies that…

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both…

Deep Neural Networks (DNNs) are widely used for their ability to effectively approximate large classes of functions. This flexibility, however, makes the strict enforcement of constraints on DNNs an open problem. Here we present a framework…

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

Understanding the inner workings of neural networks, including transformers, remains one of the most challenging puzzles in machine learning. This study introduces a novel approach by applying the principles of gauge symmetries, a key…

Modifications to a neural network's input and output layers are often required to accommodate the specificities of most practical learning tasks. However, the impact of such changes on architecture's approximation capabilities is largely…

Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple…

It is shown that applying manifold learning techniques to Poincar\'e sections of high-dimensional, chaotic dynamical systems can uncover their low-dimensional topological organization. Manifold learning provides a low-dimensional embedding…

We investigate the expressive power of deep residual neural networks idealized as continuous dynamical systems through control theory. Specifically, we consider two properties that arise from supervised learning, namely universal…

We explore the use of a topological manifold, represented as a collection of charts, as the target space of neural network based representation learning tasks. This is achieved by a simple adjustment to the output of an encoder's network…