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We propose a conjugate and calibrated Gaussian process (GP) model for multi-class classification by exploiting the geometry of the probability simplex. Our approach uses Aitchison geometry to map simplex-valued class probabilities to an…
We present a framework for learning probability distributions on topologically non-trivial manifolds, utilizing normalizing flows. Current methods focus on manifolds that are homeomorphic to Euclidean space, enforce strong structural priors…
Diffusion models learn to reverse the progressive noising of a data distribution to create a generative model. However, the desired continuous nature of the noising process can be at odds with discrete data. To deal with this tension…
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…
The sequential nature of autoregressive next-token prediction imposes a fundamental speed limit on large language models. While continuous flow models offer a path to parallel generation, they traditionally demand expensive iterative…
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…
Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible…
A new methodology is proposed for generating realizations of a random vector with values in a finite-dimensional Euclidean space that are statistically consistent with a data set of observations of this vector. The probability distribution…
We propose models and algorithms for learning about random directions in two-dimensional simplex data, and apply our methods to the study of income level proportions and their changes over time in a geostatistical area. There are several…
Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of…
We propose models and algorithms for learning about random directions in simplex-valued data. The models are applied to the study of income level proportions and their changes over time in a geostatistical area. There are several notable…
We propose Dirichlet Simplex Nest, a class of probabilistic models suitable for a variety of data types, and develop fast and provably accurate inference algorithms by accounting for the model's convex geometry and low dimensional…
In this paper, we provide an explicit probability distribution for classification purposes. It is derived from the Bayesian nonparametric mixture of Dirichlet process model, but with suitable modifications which remove unsuitable aspects of…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…
In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for…
The classification loss functions used in deep neural network classifiers can be grouped into two categories based on maximizing the margin in either Euclidean or angular spaces. Euclidean distances between sample vectors are used during…
With the widespread success of deep neural networks in science and technology, it is becoming increasingly important to quantify the uncertainty of the predictions produced by deep learning. In this paper, we introduce a new method that…
Diffusion models typically operate in the standard framework of generative modelling by producing continuously-valued datapoints. To this end, they rely on a progressive Gaussian smoothing of the original data distribution, which admits an…
Discrete diffusion or flow models could enable faster and more controllable sequence generation than autoregressive models. We show that na\"ive linear flow matching on the simplex is insufficient toward this goal since it suffers from…