# Foundations of gauge and perspective duality

arXiv: https://arxiv.org/abs/1702.08649 · HTML: https://mlsanity.com/paper/1702.08649/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2017.

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

- [The Geometry and Calculus of Losses](https://mlsanity.com/paper/2209.00238/) (2022)
- [Gauges and Accelerated Optimization over Smooth and/or Strongly Convex Sets](https://mlsanity.com/paper/2303.05037/) (2023)

## Abstract (arXiv)

We revisit the foundations of gauge duality and demonstrate that it can be explained using a modern approach to duality based on a perturbation framework. We therefore put gauge duality and Fenchel-Rockafellar duality on equal footing, including explaining gauge dual variables as sensitivity measures, and showing how to recover primal solutions from those of the gauge dual. This vantage point allows a direct proof that optimal solutions of the Fenchel-Rockafellar dual of the gauge dual are precisely the primal solutions rescaled by the optimal value. We extend the gauge duality framework to the setting in which the functional components are general nonnegative convex functions, including problems with piecewise linear quadratic functions and constraints that arise from generalized linear models used in regression.