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The great advances of learning-based approaches in image processing and computer vision are largely based on deeply nested networks that compose linear transfer functions with suitable non-linearities. Interestingly, the most frequently…

Unregularized deep neural networks (DNNs) can be easily overfit with a limited sample size. We argue that this is mostly due to the disriminative nature of DNNs which directly model the conditional probability (or score) of labels given the…

Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned…

We propose a new way of thinking about deep neural networks, in which the linear and non-linear components of the network are naturally derived and justified in terms of principles in probability theory. In particular, the models…

Recent advances in self-supervised learning have highlighted the efficacy of data augmentation in learning data representation from unlabeled data. Training a linear model atop these enhanced representations can yield an adept classifier.…

Deep metric learning has yielded impressive results in tasks such as clustering and image retrieval by leveraging neural networks to obtain highly discriminative feature embeddings, which can be used to group samples into different classes.…

In this note we study the endomorphisms of certain Banach algebras of infinitely differentiable functions on compact plane sets, associated with weight sequences M. These algebras were originally studied by Dales, Davie and McClure. In a…

For any $d\in \mathbb{N}$ and any function $f:(0,\infty)\to [0,1]$ with $f(R)\to 0$ as $R\to \infty$, we construct a set $A \subseteq \mathbb{R}^d$ and a sequence $R_n \to \infty$ such that $\|x-y\| \neq R_n$ for all $x,y\in A$ and…

In the context of classification problems, Deep Learning (DL) approaches represent state of art. Many DL approaches are based on variations of standard multi-layer feed-forward neural networks. These are also referred to as deep networks.…

The interpretation of reasoning by Deep Neural Networks (DNN) is still challenging due to their perceived black-box nature. Therefore, deploying DNNs in several real-world tasks is restricted by the lack of transparency of these models. We…

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which…

Nonlinear methods such as Deep Neural Networks (DNNs) are the gold standard for various challenging machine learning problems, e.g., image classification, natural language processing or human action recognition. Although these methods…

We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of…

Deep learning has had tremendous success at learning low-dimensional representations of high-dimensional data. This success would be impossible if there was no hidden low-dimensional structure in data of interest; this existence is posited…

Neural networks are playing a crucial role in everyday life, with the most modern generative models able to achieve impressive results. Nonetheless, their functioning is still not very clear, and several strategies have been adopted to…

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $\Pi^d\subset\mathbb R^m$ such that $|g^{-1}(\Pi^d)|\geq q$. Let also $\mathcal H(q,d,m,k)$ denote the maps $g\colon…

Medical image segmentation is crucial for the development of computer-aided diagnostic and therapeutic systems, but still faces numerous difficulties. In recent years, the commonly used encoder-decoder architecture based on CNNs has been…

We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P -> M is a smooth principal G-bundle. A `cylinder…

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion…

We present a novel class of convolutional neural networks (CNNs) for set functions, i.e., data indexed with the powerset of a finite set. The convolutions are derived as linear, shift-equivariant functions for various notions of shifts on…