Related papers: Manifold Function Encoder: Identifying Different F…
We first investigate on the asymptotics of the Kolmogorov metric entropy and nonlinear n-widths of approximation spaces on some function classes on manifolds and quasi-metric measure spaces. Secondly, we develop constructive algorithms to…
Statistically correcting measured cross sections for detector effects is an important step across many applications. In particle physics, this inverse problem is known as unfolding. In cases with complex instruments, the distortions they…
We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…
We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear and rotated Q1 polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are…
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…
A recurring task in image processing, approximation theory, and the numerical solution of partial differential equations is to reconstruct a piecewise-smooth real-valued function f(x) in multiple dimensions from its truncated Fourier…
In this paper, we present a method of embedding physics data manifolds with metric structure into lower dimensional spaces with simpler metrics, such as Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful step in…
Sparse autoencoders (SAEs) have been used widely to decompose and interpret neural network activations, especially those of transformer language models. One key issue with SAEs is their inability to directly model multidimensional features.…
Idempotent Boolean functions form a highly structured subclass of Boolean functions that is closely related to rotation symmetry under a normal-basis representation and to invariance under a fixed linear map in a polynomial basis. These…
How to represent a face pattern? While it is presented in a continuous way in our visual system, computers often store and process the face image in a discrete manner with 2D arrays of pixels. In this study, we attempt to learn a continuous…
We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
Parametric feature grid encodings have gained significant attention as an encoding approach for neural fields since they allow for much smaller MLPs, which significantly decreases the inference time of the models. In this work, we propose…
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove the proposed multiscale method achieves…
A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential…
The adoption of Transformer-based architectures in the medical domain is growing rapidly. In medical imaging, the analysis of complex shapes - such as organs, tissues, or other anatomical structures - combined with the often anisotropic…
A unified framework for the characterization of continuous electromagnetic (EM) manifolds for arbitrary multipleinput multiple-output (MIMO) system geometries is presented. The EM manifold refers to the set of all physically realizable…
Many scientific areas are faced with the challenge of extracting information from large, complex, and highly structured data sets. A great deal of modern statistical work focuses on developing tools for handling such data. This paper…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
A central goal in understanding human vision is to uncover the visual features that drive neuronal activity. A growing body of work has used artificial neural networks as encoding models to predict cortical responses to natural images,…