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We derive concentration inequalities for the supremum norm of the difference between a kernel density estimator (KDE) and its point-wise expectation that hold uniformly over the selection of the bandwidth and under weaker conditions on the…
Probability mass curves the data space with horizons. Let f be a multivariate probability density function with continuous second order partial derivatives. Consider the problem of estimating the true value of f(z) > 0 at a single point z,…
Theoretical inverse problems are often studied in an ideal infinite-dimensional setting. The well-posedness theory provides a unique reconstruction of the parameter function, when an infinite amount of data is given. Through the lens of…
Fixed-order perturbative calculations for differential cross sections can suffer from non-physical artifacts: they can be non-positive, non-normalizable, and non-finite, none of which occur in experimental measurements. We propose a…
Let $G$ be a $3$-connected ordered graph with $n$ vertices and $m$ edges. Let $\mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer range $\{1, 2,3, \ldots, 2^b\}$ for $b\ge m^2$. Let $\ell$ be the vector of $m$…
The necessity of renormalization arises from the infinite integrals which are caused by the discrepancy between the orders of differential and integral operators in the four dimensional QFTs. Therefore in view of the fact that finiteness…
Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbours to each point of a dataset to perform their tasks. These proximity relations define a so-called…
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic…
The inductive dimension dim(G) of a finite undirected graph G=(V,E) is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the…
Single-view 3D object reconstruction is a challenging fundamental problem in computer vision, largely due to the morphological diversity of objects in the natural world. In particular, high curvature regions are not always captured…
The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding…
Diffusion-based generative models employ stochastic differential equations (SDEs) and their equivalent probability flow ordinary differential equations (ODEs) to establish a smooth transformation between complex high-dimensional data…
Diffusion reconstruction plays a critical role in various applications such as image editing, restoration, and style transfer. In theory, the reconstruction should be simple - it just inverts and regenerates images by numerically solving…
Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…
Many data analysis problems can be cast as distance geometry problems in \emph{space forms} -- Euclidean, spherical, or hyperbolic spaces. Often, absolute distance measurements are often unreliable or simply unavailable and only proxies to…
Computational image reconstruction algorithms generally produce a single image without any measure of uncertainty or confidence. Regularized Maximum Likelihood (RML) and feed-forward deep learning approaches for inverse problems typically…
Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance,…
We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces…
Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a…
A Poisson coalgebra analogue of a (non-standard) quantum deformation of sl(2) is shown to generate an integrable geodesic dynamics on certain 2D spaces of non-constant curvature. Such a curvature depends on the quantum deformation parameter…