# Complex-Valued Unitary Representations as Classification Heads for Improved Uncertainty Quantification in Deep Neural Networks

arXiv: https://arxiv.org/abs/2602.15283 · HTML: https://mlsanity.com/paper/2602.15283/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 69 references that are in the map.
- First posted on arXiv 17 Feb 2026.

## Builds on (most cited first, 30 of 39)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Convolutional Neural Networks for Sentence Classification](https://mlsanity.com/paper/1408.5882/) (2014)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Measuring Calibration in Deep Learning](https://mlsanity.com/paper/1904.01685/) (2019)
- [Revisiting the Calibration of Modern Neural Networks](https://mlsanity.com/paper/2106.07998/) (2021)
- [Calibrating Deep Neural Networks using Focal Loss](https://mlsanity.com/paper/2002.09437/) (2020)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [On Mixup Training: Improved Calibration and Predictive Uncertainty for Deep Neural Networks](https://mlsanity.com/paper/1905.11001/) (2019)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Human uncertainty makes classification more robust](https://mlsanity.com/paper/1908.07086/) (2019)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)

## Abstract (arXiv)

Modern deep neural networks achieve high predictive accuracy but remain poorly calibrated: their confidence scores do not reliably reflect the true probability of correctness. We propose a quantum-inspired classification head architecture that projects backbone features into a complex-valued Hilbert space and evolves them under a learned unitary transformation parameterised via the Cayley map. Through a controlled hybrid experimental design - training a single shared backbone and comparing lightweight interchangeable heads - we isolate the effect of complex-valued unitary representations on calibration. Our ablation study on CIFAR-10 reveals that the unitary magnitude head (complex features evolved under a Cayley unitary, read out via magnitude and softmax) achieves an Expected Calibration Error (ECE) of 0.0146, representing a 2.4x improvement over a standard softmax head (0.0355) and a 3.5x improvement over temperature scaling (0.0510). Surprisingly, replacing the softmax readout with a Born rule measurement layer - the quantum-mechanically motivated approach - degrades calibration to an ECE of 0.0819. On the CIFAR-10H human-uncertainty benchmark, the wave function head achieves the lowest KL-divergence (0.336) to human soft labels among all compared methods, indicating that complex-valued representations better capture the structure of human perceptual ambiguity. We provide theoretical analysis connecting norm-preserving unitary dynamics to calibration through feature-space geometry, report negative results on out-of-distribution detection and sentiment analysis to delineate the method's scope, and discuss practical implications for safety-critical applications. Code is publicly available.