Related papers: Algebraic Identifiability of Gaussian Mixtures
We consider the problem of identifying a mixture of Gaussian distributions with same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special…
The points of a moment variety are the vectors of all moments up to some order of a family of probability distributions. We study this variety for mixtures of Gaussians. Following up on Pearson's classical work from 1894, we apply current…
We show that the parameters of a $k$-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first $3k-1$ moments, and rationally identifiable from the first $3k+2$ moments. Our proofs are based on…
We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that the parameters of a generic mixture of $ m\leq\mathcal{O}(n^4) $ Gaussians on $ \mathbb R^n $ can…
This paper establishes the theoretical foundations for the asymptotic separability of Gaussian Mixture Models (GMMs) in high dimensions by extending the classical Feldman-H\'ajek theorem. We first prove that a countable mixture of Gaussian…
Motivated by previous work on moment varieties for Gaussian distributions and their mixtures, we study moment varieties for two other statistically important two-parameter distributions: the inverse Gaussian and gamma distributions. In…
In this paper, we prove the Geometric Arveson-Douglas Conjecture for a special case which allow some singularity on $\partial{\mathbb{B}_n}$. More precisely, we show that if a variety can be decomposed into two varieties, each having nice…
The paper introduces a methodology for visualizing on a dimension reduced subspace the classification structure and the geometric characteristics induced by an estimated Gaussian mixture model for discriminant analysis. In particular, we…
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…
Mixtures of Gaussian (or normal) distributions arise in a variety of application areas. Many heuristics have been proposed for the task of finding the component Gaussians given samples from the mixture, such as the EM algorithm, a…
The method of moments is a classical statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding…
This work is a modern revisitation of a classical paper by Alessandro Terracini, going back to 1915, which suggests an elementary but powerful method for studing Grassmann defective varieties. In particular, the case of Veronese surfaces is…
Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has a running time, and data requirement polynomial in the dimension and…
We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and…
Expanding a lower-dimensional problem to a higher-dimensional space and then projecting back is often beneficial. This article rigorously investigates this perspective in the context of finite mixture models, namely how to improve inference…
We consider the problem of identifying the parameters of an unknown mixture of two arbitrary $d$-dimensional gaussians from a sequence of independent random samples. Our main results are upper and lower bounds giving a computationally…
In this paper, we establish the exact Fourier dimensions of all standard sub-critical Gaussian multiplicative chaos on the unit interval, thereby confirming the Garban-Vargas conjecture. The proof relies on a significant improvement of the…
The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to…
In this paper we show that very large mixtures of Gaussians are efficiently learnable in high dimension. More precisely, we prove that a mixture with known identical covariance matrices whose number of components is a polynomial of any…
We present an alternative proof of the Alexander-Hirschowitz Theorem in dimension 3 using degenerations of toric varieties.