Invariant Theory and Scaling Algorithms for Maximum Likelihood Estimation

Nonlinear algebra and applications

“…Invariant theory has shown useful in computing maximum likelihood estimates and degrees [19,67,75]. Another approach for tackling 3.2 is to compute the point in a model that minimizes some distance of the sample point from the model [43,74].…”

“…The prevalence of outliers calls for robust estimation techniques, among which heuristics based on RANSAC (RANdom SAmpling and Consensus [92]) have been ubiquitous in computer vision. The key to RANSAC is using the minimal number of measurements-for the equations in (19), this means m = 5-to generate several "hypotheses" for E, which are then checked against the remaining data. An ingenious scheme for solving the m = 5 case of (19) was proposed and implemented by Nistér [183].…”

“…The key to RANSAC is using the minimal number of measurements-for the equations in (19), this means m = 5-to generate several "hypotheses" for E, which are then checked against the remaining data. An ingenious scheme for solving the m = 5 case of (19) was proposed and implemented by Nistér [183]. As in previous sections, this solution exploits genericity.…”

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

“…In our companion work [1], we establish a connection between finding the MLE and norm minimization along an orbit under a group action. We focus there on the setting of Gaussian group models, centered multivariate Gaussian models whose concentration matrices are of the form g T g, where g lies in a group.…”

“…We compare iterative proportional scaling (IPS), a classical method to find the MLE for log-linear models, with approaches to norm minimization in Section 5. We conclude the paper with a comparison with the multivariate Gaussian setting of [1] in Section 6 and outline a possible generalization for future research.…”