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

2026-02-16

AI Summary

This work addresses the poor calibration of deep neural networks, where predicted confidence often misaligns with actual accuracy. The authors propose a quantum-inspired complex-valued classification head that maps backbone features into a Hilbert space and evolves them via Cayley-parameterized unitary transformations—introducing, for the first time, complex-valued unitary representations into classifier design. This approach effectively models uncertainty and aligns with the fuzzy structure inherent in human perception. Theoretical analysis reveals a geometric connection between unitary dynamics and calibration performance. Empirical results demonstrate significant improvements: on CIFAR-10, the expected calibration error (ECE) is reduced to 0.0146, a 2.4× improvement over standard softmax; on CIFAR-10H, the method achieves a KL divergence of 0.336, substantially outperforming existing approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Representation Learning for Vision
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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.

Problem

• uncertainty quantification
• model calibration
• deep neural networks
• confidence reliability
• expected calibration error

Innovation

• complex-valued representations
• unitary transformation
• uncertainty calibration
• Cayley map
• quantum-inspired neural networks