Related papers: Ridges, Neural Networks, and the Radon Transform
We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of…
Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal…
Characterizing the function spaces defined by neural networks helps understanding the corresponding learning models and their inductive bias. While in some limits neural networks correspond to function spaces that are Hilbert spaces, these…
The main purpose of the paper is to present some recent results on metric characterizations of superreflexivity and the Radon-Nikod\'ym property.
Overparameterized neural networks often contain many removable neurons, yet what makes a neuron redundant remains poorly understood. Existing pruning criteria commonly rely on local quantities such as weight magnitude, activation strength,…
Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon…
The classical kernel ridge regression problem aims to find the best fit for the output $Y$ as a function of the input data $X\in \mathbb{R}^d$, with a fixed choice of regularization term imposed by a given choice of a reproducing kernel…
Transformer-based language models have achieved state-of-the-art performance in natural language generation (NLG), yet their internal mechanisms for synthesizing task-relevant information remain insufficiently understood. While prior…
We study the inversion of the conical Radon which integrates a function in three-dimensional space from integrals over circular cones. The conical Radon recently got significant attention due to its relevance in various imaging applications…
Neural radiance fields (NeRFs) produce state-of-the-art view synthesis results. However, they are slow to render, requiring hundreds of network evaluations per pixel to approximate a volume rendering integral. Baking NeRFs into explicit…
We study the least-square regression problem with a two-layer fully-connected neural network, with ReLU activation function, trained by gradient flow. Our first result is a generalization result, that requires no assumptions on the…
In this paper, we focus on the interpretability of deep neural network. Our work is motivated by the renormalization group (RG) in statistical mechanics. RG plays the role of a bridge connecting microscopical properties and macroscopic…
We introduce a methodology for designing and training deep neural networks (DNN) that we call "Deep Regression Ensembles" (DRE). It bridges the gap between DNN and two-layer neural networks trained with random feature regression. Each layer…
The proposed method, Neural Radiance Projection (NeRP), addresses the three most fundamental shortages of training such a convolutional neural network on X-ray image segmentation: dealing with missing/limited human-annotated datasets;…
It is difficult to describe in mathematical terms what a neural network trained on data represents. On the other hand, there is a growing mathematical understanding of what neural networks are in principle capable of representing.…
We investigate the properties of random feature ridge regression (RFRR) given by a two-layer neural network with random Gaussian initialization. We study the non-asymptotic behaviors of the RFRR with nearly orthogonal deterministic…
Recent methods in geometric deep learning have introduced various neural networks to operate over data that lie on Riemannian manifolds. Such networks are often necessary to learn well over graphs with a hierarchical structure or to learn…
We introduce an original method of multidimensional ridge penalization in functional local linear regressions. The nonparametric regression of functional data is extended from its multivariate counterpart, and is known to be sensitive to…
This article studies divergence of multivector fields on Banach manifolds with a Radon measure. The proposed definition is consistent with the classical divergence from finite-dimensional differential geometry. Certain natural properties of…
In this manuscript, we show that any neural network with any activation function can be represented as a decision tree. The representation is equivalence and not an approximation, thus keeping the accuracy of the neural network exactly as…