# A scaled Bregman theorem with applications

arXiv: https://arxiv.org/abs/1607.00360 · HTML: https://mlsanity.com/paper/1607.00360/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2016.

## Built on by (most cited first, 7 of 7)

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- [A Unified Framework for Multi-distribution Density Ratio Estimation](https://mlsanity.com/paper/2112.03440/) (2021)
- [Coherence Mechanisms for Provable Self-Improvement](https://mlsanity.com/paper/2511.08440/) (2025)
- [Calibration without Ground Truth](https://mlsanity.com/paper/2601.19862/) (2026)
- [Density-Ratio Losses for Post-Hoc Learning to Defer](https://mlsanity.com/paper/2605.19557/) (2026)
- [Seasoning Generative Models for a Generalization Aftertaste](https://mlsanity.com/paper/2603.18817/) (2026)

## Abstract (arXiv)

Bregman divergences play a central role in the design and analysis of a range of machine learning algorithms. This paper explores the use of Bregman divergences to establish reductions between such algorithms and their analyses. We present a new scaled isodistortion theorem involving Bregman divergences (scaled Bregman theorem for short) which shows that certain "Bregman distortions'" (employing a potentially non-convex generator) may be exactly re-written as a scaled Bregman divergence computed over transformed data. Admissible distortions include geodesic distances on curved manifolds and projections or gauge-normalisation, while admissible data include scalars, vectors and matrices. Our theorem allows one to leverage to the wealth and convenience of Bregman divergences when analysing algorithms relying on the aforementioned Bregman distortions. We illustrate this with three novel applications of our theorem: a reduction from multi-class density ratio to class-probability estimation, a new adaptive projection free yet norm-enforcing dual norm mirror descent algorithm, and a reduction from clustering on flat manifolds to clustering on curved manifolds. Experiments on each of these domains validate the analyses and suggest that the scaled Bregman theorem might be a worthy addition to the popular handful of Bregman divergence properties that have been pervasive in machine learning.