# On the relation between Loss Functions and T-Norms

arXiv: https://arxiv.org/abs/1907.07904 · HTML: https://mlsanity.com/paper/1907.07904/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jul 2019.

## Built on by (most cited first, 5 of 5)

- [Learning with Logical Constraints but without Shortcut Satisfaction](https://mlsanity.com/paper/2403.00329/) (2024)
- [Evaluating Relaxations of Logic for Neural Networks: A Comprehensive Study](https://mlsanity.com/paper/2107.13646/) (2021)
- [Understanding the Logic of Direct Preference Alignment through Logic](https://mlsanity.com/paper/2412.17696/) (2024)
- [Reduced Implication-bias Logic Loss for Neuro-Symbolic Learning](https://mlsanity.com/paper/2208.06838/) (2022)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)

## Abstract (arXiv)

Deep learning has been shown to achieve impressive results in several domains like computer vision and natural language processing. A key element of this success has been the development of new loss functions, like the popular cross-entropy loss, which has been shown to provide faster convergence and to reduce the vanishing gradient problem in very deep structures. While the cross-entropy loss is usually justified from a probabilistic perspective, this paper shows an alternative and more direct interpretation of this loss in terms of t-norms and their associated generator functions, and derives a general relation between loss functions and t-norms. In particular, the presented work shows intriguing results leading to the development of a novel class of loss functions. These losses can be exploited in any supervised learning task and which could lead to faster convergence rates that the commonly employed cross-entropy loss.