Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.

通用线性浓度模型的最大似然阈值

Maximum likelihood thresholds of generic linear concentration models

本文研究了通用线性浓度模型的最大似然阈值，发现其与直观的维度计数一致，且通过几何方法解释了模型可能失效的情况。

Varieties with maximum likelihood degree one

We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood estimator corresponding to such a variety is Kapranov's Horn uniformization. This extends Kapranov's characterization of A-discriminantal hypersurfaces to varieties of arbitrary codimension.

The maximum likelihood degree of sparse polynomial systems

We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the model. We prove the maximum likelihood degree of a sparse polynomial system is determined by its Newton polytopes and equals the mixed volume of a related Lagrange system of equations.

Toric invariant theory for maximum likelihood estimation in log-linear models

We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.

Bounding the maximum likelihood degree

Maximum likelihood estimation is a fundamental computational problem in statistics. In this note, we give a bound for the maximum likelihood degree of algebraic statistical models for discrete data. As usual, such models are identified with special very affine varieties. Using earlier work of Franecki and Kapranov, we prove that the maximum likelihood degree is always less or equal to the signed intersection-cohomology Euler characteristic. We construct counterexamples to a bound in terms of the usual Euler characteristic conjectured by Huh and Sturmfels.

Maximum Likelihood Estimation from a Tropical and a Bernstein--Sato Perspective

In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein--Sato theory. We investigate the critical points of very affine varieties and study their asymptotic behavior. We relate these asymptotics to particular rays in the tropical variety as well as to Bernstein--Sato ideals and give a connection to Maximum Likelihood Estimation in Statistics.

The maximum likelihood degree of Fermat hypersurfaces

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 projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed t…

基于群作用的多元与矩阵变量对称拉普拉斯分布的极大似然估计

Maximum Likelihood Estimation in the Multivariate and Matrix Variate Symmetric Laplace Distributions through Group Actions

本文针对不属于指数分布族的多元与矩阵变量对称拉普拉斯分布，提出基于群作用的极大似然估计方法，将其转化为群上范数最小化问题，建立了数据群作用稳定性与似然函数性质的对应关系。

Maximum likelihood estimation for tensor normal models via castling transforms

In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answer for both real and complex models. One consequence of our results is that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE. Our techniques are based on invariant theory and castling transforms.

直线格拉斯曼流形上的最大似然估计

Maximum Likelihood Estimation on the Grassmannian of Lines

研究正格拉斯曼流形，通过代数统计视角给出直线格拉斯曼流形最大似然度封闭公式，推测概率单纯形有唯一局部最大值并提供计算证据。

关于Drton, Sturmfels和Sullivant关于环的高斯图模型极大似然度猜想的证明

Proof of a Conjecture of Drton, Sturmfels and Sullivant on the maximum likelihood degree of the Gaussian graphical model of a cycle

本文通过计算环的高斯图模型极大似然度的精确值，证实了Drton, Sturmfels和Sullivant提出的猜想。