On conformal divergences and their population minimizers (1311.5125v2)

Abstract: Total Bregman divergences are a recent tweak of ordinary Bregman divergences originally motivated by applications that required invariance by rotations. They have displayed superior results compared to ordinary Bregman divergences on several clustering, computer vision, medical imaging and machine learning tasks. These preliminary results raise two important problems : First, report a complete characterization of the left and right population minimizers for this class of total Bregman divergences. Second, characterize a principled superset of total and ordinary Bregman divergences with good clustering properties, from which one could tailor the choice of a divergence to a particular application. In this paper, we provide and study one such superset with interesting geometric features, that we call conformal divergences, and focus on their left and right population minimizers. Our results are obtained in a recently coined $(u, v)$-geometric structure that is a generalization of the dually flat affine connections in information geometry. We characterize both analytically and geometrically the population minimizers. We prove that conformal divergences (resp. total Bregman divergences) are essentially exhaustive for their left (resp. right) population minimizers. We further report new results and extend previous results on the robustness to outliers of the left and right population minimizers, and discuss the role of the $(u, v)$-geometric structure in clustering. Additional results are also given.

• The paper defines conformal divergences, a generalization of Bregman divergences, and characterizes their left and right population minimizers.
• It characterizes left minimizers as weighted u-means robust against outliers and describes right minimizers via orthogonal projections in total Bregman divergences.
• The findings provide a theoretical framework for improving clustering algorithms with enhanced adaptability and robustness, particularly in geometric data applications.

Overview of Conformal Divergences in Population Minimization

The paper "On Conformal Divergences and their Population Minimizers" by Richard Nock, Frank Nielsen, and Shun-ichi Amari explores the theoretical foundations and applications of conformal divergences, a broad class of distance metrics extending beyond ordinary and total Bregman divergences. It seeks to address the need for effective population minimizers in clustering and related fields, which can adapt to varied applications through better geometric and statistical properties.

Key Contributions

1. Definition of Conformal Divergences:
   • Conformal divergences are characterized by a continuous, strictly convex function $\varphi$ and an auxiliary function $g$ which introduces a transformation via the gradient of $\varphi$. The divergences are defined over a $(u,v)$-geometric structure, generalizing the dually flat affine connections found in information geometry.
2. Population Minimizers:
   • Left population minimizers for conformal divergences are characterized as weighted $u$-means, providing a robust mechanism against outliers through the $(u,v)$-geometric structure.
   • Right population minimizers are more intricate, obtained through the orthogonal projection in total Bregman divergences. The paper proves exhaustiveness for total Bregman divergences in yielding right population minimizers parallel to classical minimizers like sample means.
3. Robustness and Structure Relations:
   • Robustness to outliers is analytically extended to the left population minimizers, revealing that certain assumptions on the scaling functions and geometric coordinates can mitigate the influence of outliers.
   • A novel discussion on the tolerance and equivalence relations within the geometric structures, deriving compact subgroups for clustering tasks.

Practical and Theoretical Implications

The renewed focus on conformal divergences provides compelling avenues for improving clustering algorithms, especially in applications like medical imaging and computer vision, where geometric invariance under transformations is crucial. Conformal divergences encapsulate the flexibility required to tailor a divergence to specific datasets, enhancing practical adaptability without compromising computational efficiency.

Speculations for Future Directions

Further research could investigate the integration of conformal divergences into large-scale machine learning systems, exploring their applicability in dynamic real-world datasets where geometric and statistical properties evolve. Additionally, scaling these frameworks to handle multidimensional data in complex environments and expanding their use in unsupervised learning are promising directions. The unification of conformal divergences with deep neural networks may yield powerful tools, underpinning robust and adaptable learning models.

In conclusion, this paper sets a crucial academic cornerstone for advancing clustering techniques by enhancing the adaptability and robustness through comprehensive theoretical frameworks and illustrating potential practical implementations across varied domains.