# Enhancing Diversity in Bayesian Deep Learning via Hyperspherical Energy Minimization of CKA

arXiv: https://arxiv.org/abs/2411.00259 · HTML: https://mlsanity.com/paper/2411.00259/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 47 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Builds on (most cited first, 30 of 31)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Deep Exploration via Bootstrapped DQN](https://mlsanity.com/paper/1602.04621/) (2016)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning](https://mlsanity.com/paper/2002.06715/) (2020)
- [HyperStyle: StyleGAN Inversion with HyperNetworks for Real Image Editing](https://mlsanity.com/paper/2111.15666/) (2021)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Improved Online Conformal Prediction via Strongly Adaptive Online Learning](https://mlsanity.com/paper/2302.07869/) (2023)
- [HyperTransformer: Model Generation for Supervised and Semi-Supervised Few-Shot Learning](https://mlsanity.com/paper/2201.04182/) (2022)
- [HyperGAN: A Generative Model for Diverse, Performant Neural Networks](https://mlsanity.com/paper/1901.11058/) (2019)
- [A Unified Particle-Optimization Framework for Scalable Bayesian Sampling](https://mlsanity.com/paper/1805.11659/) (2018)
- [Hypernetworks in Meta-Reinforcement Learning](https://mlsanity.com/paper/2210.11348/) (2022)
- [Learning with Hyperspherical Uniformity](https://mlsanity.com/paper/2103.01649/) (2021)
- [Recomposing the Reinforcement Learning Building Blocks with Hypernetworks](https://mlsanity.com/paper/2106.06842/) (2021)
- [Active Learning Using Uncertainty Information](https://mlsanity.com/paper/1702.08540/) (2017)
- [Riemannian Stein Variational Gradient Descent for Bayesian Inference](https://mlsanity.com/paper/1711.11216/) (2017)
- [Recurrent Hypernetworks are Surprisingly Strong in Meta-RL](https://mlsanity.com/paper/2309.14970/) (2023)
- [Blurs Behave Like Ensembles: Spatial Smoothings to Improve Accuracy, Uncertainty, and Robustness](https://mlsanity.com/paper/2105.12639/) (2021)
- [COPlanner: Plan to Roll Out Conservatively but to Explore Optimistically for Model-Based RL](https://mlsanity.com/paper/2310.07220/) (2023)
- [Variational Inference for Infinitely Deep Neural Networks](https://mlsanity.com/paper/2209.10091/) (2022)
- [Hypernetwork-Based Adaptive Image Restoration](https://mlsanity.com/paper/2206.05970/) (2022)

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

- [Energy-Regularized Sequential Model Editing on Hyperspheres](https://mlsanity.com/paper/2510.01172/) (2025)
- [SphUnc: Hyperspherical Uncertainty Decomposition and Causal Identification via Information Geometry](https://mlsanity.com/paper/2603.01168/) (2026)

## Abstract (arXiv)

Particle-based Bayesian deep learning often requires a similarity metric to compare two networks. However, naive similarity metrics lack permutation invariance and are inappropriate for comparing networks. Centered Kernel Alignment (CKA) on feature kernels has been proposed to compare deep networks but has not been used as an optimization objective in Bayesian deep learning. In this paper, we explore the use of CKA in Bayesian deep learning to generate diverse ensembles and hypernetworks that output a network posterior. Noting that CKA projects kernels onto a unit hypersphere and that directly optimizing the CKA objective leads to diminishing gradients when two networks are very similar. We propose adopting the approach of hyperspherical energy (HE) on top of CKA kernels to address this drawback and improve training stability. Additionally, by leveraging CKA-based feature kernels, we derive feature repulsive terms applied to synthetically generated outlier examples. Experiments on both diverse ensembles and hypernetworks show that our approach significantly outperforms baselines in terms of uncertainty quantification in both synthetic and realistic outlier detection tasks.