SIGEST

SIAM Review, Volume 66, Issue 4, Page 719-719, November 2024.
The SIGEST article in this issue, “A Bridge between Invariant Theory and Maximum Likelihood Estimation,” by Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal, uncovers the deep connections between geometric invariant theory and statistical methods, specifically maximum likelihood estimation (MLE) by connecting it to norm minimization over group orbits. The authors develop a dictionary relating stability notions in geometric invariant theory to the existence and uniqueness of MLEs, which applies to both Gaussian and log-linear models. In comparison to the original 2021 version of the paper that appeared in the SIAM Journal on Applied Algebra and Geometry, for the SIGEST version, the authors added new content on log-linear models, simplified technical proofs, removed detailed appendices, and incorporated new examples and figures for accessibility. In particular, the focus was primarily on Gaussian models, whereas this updated SIGEST version expands the coverage by incorporating results from the authors' companion paper on log-linear models. Furthermore, a new figure (Fig. 1) visually illustrates the two core concepts of invariant theory and MLE. Significant changes include the removal of technical details and appendices to streamline the content and make it more accessible to a broader audience. The introduction of examples, particularly for the Kempf--Ness Theorem, further aids understanding. This paper makes several key contributions of broad mathematical interest. MLE is a key statistical technique that is widely used. Having a new handle on its well-posedness analysis deepens the understanding of the mechanisms behind this technique as well as potentially paves the way to extending existing theory for MLE models. Also, on the computational side, algorithms from the optimization over orbits can be used for MLE, and vice versa, which could possibly lead to new and more efficient algorithms in both fields. Overall, the work beautifully highlights how techniques from one field can be applied to the other, with applications to generalization bounds, group actions, and optimization landscapes. In the last section of their SIGEST paper the authors discuss possible future research directions that capitalize on the dictionary they have uncovered.