Related papers: Manifold Density Estimation via Generalized Dequan…
The generalized density is a product of a density function and a weight function. For example, the average local brightness of an astronomical image is the probability of finding a galaxy times the mean brightness of the galaxy. We propose…
Object counting and localization are key steps for quantitative analysis in large-scale microscopy applications. This procedure becomes challenging when target objects are overlapping, are densely clustered, and/or present fuzzy boundaries.…
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…
Denoising is intuitively related to projection. Indeed, under the manifold hypothesis, adding random noise is approximately equivalent to orthogonal perturbation. Hence, learning to denoise is approximately learning to project. In this…
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data…
We introduce boundary quotients and present a framework for learning densities on manifolds that arise as boundary quotients of simpler domains. We show that this framework can be used to construct normalizing flows on quotient manifolds…
Kernel density estimation is a popular method for estimating unseen probability distributions. However, the convergence of these classical estimators to the true density slows down in high dimensions. Moreover, they do not define meaningful…
Crowd counting problem aims to count the number of objects within an image or a frame in the videos and is usually solved by estimating the density map generated from the object location annotations. The values in the density map, by…
Starting from the geometrical concept of a 4-dimensional de-Sitter configuration of spheres in Euclidean 3-space and modelling voids in the Universe as spheres, we show that a uniform distribution over this configuration space implies a…
To model manifold data using normalizing flows, we employ isometric autoencoders to design embeddings with explicit inverses that do not distort the probability distribution. Using isometries separates manifold learning and density…
Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…
Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic…
This article presents a novel method to sampling on manifolds based on the Dirichlet distribution. The proposed strategy allows to completely respect the underlying manifold around which data is observed, and to do massive samplings with…
A new formalism is presented for analytically obtaining the probability density function, \( P_{n}(s) \), for the distance between two random points in an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \( P_{n}(s) \) to…
Diffusion models often generate novel samples even when the learned score is only \emph{coarse} -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the…
In this paper we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the…
Learning disentangled representations is regarded as a fundamental task for improving the generalization, robustness, and interpretability of generative models. However, measuring disentanglement has been challenging and inconsistent, often…
We propose a method for learning and sampling from probability distributions supported on the simplex. Our approach maps the open simplex to Euclidean space via smooth bijections, leveraging the Aitchison geometry to define the mappings,…
Many approaches in the field of machine learning and data analysis rely on the assumption that the observed data lies on lower-dimensional manifolds. This assumption has been verified empirically for many real data sets. To make use of this…
Quantitative dynamical models of galaxies require deprojecting the observed surface brightness to determine the luminosity density of the galaxy. Existing deprojection methods for axisymmetric galaxies assume that a unique deprojection…