# Learning Log-Determinant Divergences for Positive Definite Matrices

arXiv: https://arxiv.org/abs/2104.06461 · HTML: https://mlsanity.com/paper/2104.06461/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 74 references that are in the map.
- First posted on arXiv 13 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Two-Stream Convolutional Networks for Action Recognition in Videos](https://mlsanity.com/paper/1406.2199/) (2014)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Compact Bilinear Pooling](https://mlsanity.com/paper/1511.06062/) (2015)
- [Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods](https://mlsanity.com/paper/1605.06182/) (2016)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [Bregman Divergences for Infinite Dimensional Covariance Matrices](https://mlsanity.com/paper/1403.4334/) (2014)
- [Second-order Democratic Aggregation](https://mlsanity.com/paper/1808.07503/) (2018)
- [Kernel Sparse Subspace Clustering on Symmetric Positive Definite Manifolds](https://mlsanity.com/paper/1601.00414/) (2016)
- [Statistically Motivated Second Order Pooling](https://mlsanity.com/paper/1801.07492/) (2018)
- [Higher-order Pooling of CNN Features via Kernel Linearization for Action Recognition](https://mlsanity.com/paper/1701.05432/) (2017)
- [Second-order Temporal Pooling for Action Recognition](https://mlsanity.com/paper/1704.06925/) (2017)
- [Learning the Information Divergence](https://mlsanity.com/paper/1406.1385/) (2014)

## Built on by (most cited first, 2 of 2)

- [Understanding and Mitigating Hyperbolic Dimensional Collapse in Graph Contrastive Learning](https://mlsanity.com/paper/2310.18209/) (2023)
- [The alpha-beta divergence for real and complex data](https://mlsanity.com/paper/2508.03272/) (2025)

## Abstract (arXiv)

Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the αβ-log-det divergence, which is a meta-divergence parametrized by scalars α and β, subsuming a wide family of popular information divergences on SPD matrices for distinct and discrete values of these parameters. Our key idea is to cast these parameters in a continuum and learn them from data. We systematically extend this idea to learn vector-valued parameters, thereby increasing the expressiveness of the underlying non-linear measure. We conjoin the divergence learning problem with several standard tasks in machine learning, including supervised discriminative dictionary learning and unsupervised SPD matrix clustering. We present Riemannian gradient descent schemes for optimizing our formulations efficiently, and show the usefulness of our method on eight standard computer vision tasks.