相关论文: The $F$-adjoined Gauss Map and Gaussian Likelihood…
We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, $\mathrm{MLD}_F(X)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that $\mathrm{MLD}_F(X)$ is equal to the…
Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by…
Gaussian graphical models are used throughout the natural sciences, social sciences, and economics to model the statistical relationships between variables of interest in the form of a graph. We here provide a pedagogic introduction to…
In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical…
We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum…
In algebraic statistics, the maximum likelihood degree of a statistical model refers to the number of solutions (counted with multiplicity) of the score equations over the complex field. In this paper, the maximum likelihood degree of the…
Let $\mathcal{X}$ be a projective algebraic curve and denote by $\mathcal{X}^{'}$ its strict dual curve. The map $\gamma:\mathcal{X} \longrightarrow \mathcal{X}^{'}$ is called (strict) Gauss map of $\mathcal{X}$. In this manuscript, we…
We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in…
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…
The hierarchical prior used in Latent Gaussian models (LGMs) induces a posterior geometry prone to frustrate inference algorithms. Marginalizing out the latent Gaussian variable using an integrated Laplace approximation removes the…
Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees…
We propose a family of multivariate Gaussian process models for correlated outputs, based on assuming that the likelihood function takes the generic form of the multivariate exponential family distribution (EFD). We denote this model as a…
With any \emph{surjective rational map} $f: \mathbb{P}^n \dashrightarrow \mathbb{P}^n$ of the projective space we associate a numerical invariant (\emph{ML degree}) and compute it in terms of a naturally defined vector bundle $E_f…
Graphical models with bi-directed edges (<->) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood…
We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is non-convex, yet it has been…
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of…
We study an optimization problem with the feasible set being a real algebraic variety $X$ and whose parametric objective function $f_u$ is gradient-solvable with respect to the parametric data $u$. This class of problems includes Euclidean…
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a…
We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its…
We study the probability distribution $F(u)$ of the maximum of smooth Gaussian fields defined on compact subsets of $\R^d$ having some geometric regularity. Our main result is a general formula for the density of $F$. Even though this is an…