Complex-Valued Unitary Representations as Classification Heads for Improved Uncertainty Quantification in Deep Neural Networks
arXiv:2602.15283v1 Announce Type: new 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).
arXiv:2602.15283v1 Announce Type: new 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.
Executive Summary
This article proposes a novel classification head architecture for deep neural networks, inspired by quantum mechanics. The method, called Complex-Valued Unitary Representations (CVUR), projects backbone features into a complex-valued Hilbert space and evolves them under a learned unitary transformation. Experimental results on CIFAR-10 demonstrate a significant improvement in calibration, with an Expected Calibration Error (ECE) of 0.0146. The method also performs well on human-uncertainty benchmarks, indicating its potential for capturing the structure of human perceptual ambiguity. However, the article notes negative results on out-of-distribution detection and sentiment analysis, highlighting the method's limitations. The authors provide theoretical analysis connecting norm-preserving unitary dynamics to calibration and discuss practical implications for safety-critical applications.
Key Points
- ▸ Proposes a novel classification head architecture, CVUR, inspired by quantum mechanics.
- ▸ Demonstrates significant improvement in calibration on CIFAR-10 with ECE of 0.0146.
- ▸ Performs well on human-uncertainty benchmarks, capturing the structure of human perceptual ambiguity.
Merits
Improved Calibration
CVUR demonstrates a significant reduction in Expected Calibration Error (ECE), indicating improved calibration.
Quantum-Inspired Approach
The method's use of complex-valued Hilbert space and unitary transformation draws on quantum mechanics, offering a novel perspective on classification.
Human Uncertainty Benchmark Performance
CVUR performs well on human-uncertainty benchmarks, indicating its potential for capturing human perceptual ambiguity.
Demerits
Negative Results on Out-of-Distribution Detection
CVUR performs poorly on out-of-distribution detection, highlighting its limitations in certain applications.
Negative Results on Sentiment Analysis
CVUR also performs poorly on sentiment analysis, indicating its potential limitations in certain areas.
Expert Commentary
The article presents a novel and intriguing approach to classification in deep neural networks, drawing on quantum mechanics. While the results are promising, the limitations of the method, particularly in out-of-distribution detection and sentiment analysis, should be carefully considered. Furthermore, the article's theoretical analysis provides valuable insights into the connection between norm-preserving unitary dynamics and calibration. As the field of quantum-inspired machine learning continues to grow, this work offers a significant contribution, warranting further investigation and exploration.
Recommendations
- ✓ Future research should focus on expanding the scope of CVUR to other applications, such as natural language processing and reinforcement learning.
- ✓ The authors should investigate the use of other quantum-inspired techniques, such as quantum circuit learning and quantum annealing, to further improve the performance of CVUR.
