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

Akbar Anbar Jafari; Cagri Ozcinar; Gholamreza Anbarjafari

cs.LGFeb 17, 2026

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

Authors: Akbar Anbar Jafari, Cagri Ozcinar, Gholamreza Anbarjafari

Organizations: University of Tartu Tartu, Estonia · 3S Holding OÜ Tartu 51011, Estonia

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.

Figures & tables

Panel of Figure 1 — Figure 1: Architecture of the proposed complex-valued unitary classification head. Real-valued backbone features are projected into a complex Hilbert space, normalised to unit norm, evolved under a learned Cayley unitary, and read out via either magnitude–softmax (proposed best variant) or Born rule measurement.

Variant	Complex	Unitary	Born	Readout
Full WaveHead	$\checkmark$	$\checkmark$	$\checkmark$	Born rule
NoBorn (Mag)	$\checkmark$	$\checkmark$	$\times$	Magnitude + softmax
NoUnitary	$\checkmark$	$\times$	$\checkmark$	Born rule
ComplexLinear	$\checkmark$	$\times$	$\times$	Magnitude + softmax
Softmax	$\times$	$\times$	$\times$	Linear + softmax
Softmax (2x)	$\times$	$\times$	$\times$	Wider linear + softmax

Table 1: Ablation head variants. Each row indicates which architectural components are active (

\checkmark

) or absent (

\times

Method	Experiments	Type
Softmax Head	2, 3, 5	Baseline head
WaveFunction Head	2, 3, 5	Proposed (Born)
NoBorn (Magnitude) Head	5	Proposed (best)
MC-Dropout Head	2, 3	Bayesian approx.
Evidential Head	2, 3	Dirichlet-based
Temperature Scaling	2	Post-hoc

Table 2: Overview of compared methods across experiments. Exps. 2, 3, 5 use the hybrid backbone–head design; Exps. 1, 4 train end-to-end.

Figure 2: Experiment 1: Predicted samples overlaid on ground-truth data. Wave MLP captures both branches through superposition (Qual=0.214), outperforming Real MLP (0.043) and matching Deep Ensemble diversity, though MDN (0.431) achieves the sharpest concentration near true modes.

Figure 3: Experiment 1: Mode coverage (left) and quality (right). All models achieve full coverage; Wave MLP’s quality is $5\times$ that of Real MLP, demonstrating the superposition advantage for multi-modality, though MDN’s parametric mixture dominates.

Figure 4: Experiment 1: Predicted density profiles at four $x$ values. Wave MLP (left, red) produces broad multi-peaked distributions via Born rule. Real MLP and Deep Ensemble produce sharp unimodal peaks at each mode. MDN captures both modes with tight Gaussian components.

Panel of Figure 5 — Figure 5: Experiment 1: (a) Training loss convergence. Wave MLP plateaus at higher loss, indicating optimisation difficulty in complex-valued density estimation. (b) Phase structure of the evolved wave function at four $x$ values, showing non-trivial phase diversity across Hilbert space dimensions.

Head	Acc. $\uparrow$	ECE $\downarrow$	NLL $\downarrow$	Brier $\downarrow$	KL-Div $\downarrow$
Softmax	0.913	0.036	0.287	0.132	0.404
WaveFunction	0.909	0.082	0.349	0.148	0.336
MC-Dropout	0.913	0.048	0.325	0.137	0.486
Evidential	0.096	0.004	2.303	0.900	2.148
TempScaling	0.913	0.052	0.355	0.139	0.568

Table 3: Experiment 2: CIFAR-10H results with shared backbone (91.3% backbone accuracy). Bold indicates best; underline indicates second best.

\downarrow

= lower is better.

Figure 6: Experiment 2: Reliability diagrams for each head. The WaveFunction head (ECE=0.082) shows overconfidence at high confidence levels, while Softmax (ECE=0.036) adheres more closely to the diagonal. Despite its higher ECE, the WaveFunction head achieves the lowest KL-divergence to human soft labels (see text).

Figure 7: Experiment 2: Metric comparison across heads. The WaveFunction head’s best-in-class KL-divergence coexists with its worst ECE among functioning heads, revealing that calibration and human-alignment measure distinct properties.

Head	SVHN	CIFAR-100	Gaussian	Uniform
WaveFunction	0.853	0.841	0.837	0.863
Softmax	0.881	0.854	0.849	0.831
MC-Dropout	0.888	0.850	0.846	0.827
Energy	0.809	0.862	0.767	0.804

Table 4: Experiment 3: OOD detection AUROC (

\uparrow

) with shared backbone. Bold = best, underline = second best. The Evidential head is excluded due to its degenerate training.

Figure 8: Experiment 3: ROC curves for OOD detection. All heads receive features from the same backbone. The WaveFunction head is competitive but does not outperform simpler baselines, indicating that the Born rule does not confer an inherent OOD advantage.

Model	Acc. $\uparrow$	Neg. Acc. $\uparrow$	ECE $\downarrow$	Neg. ECE $\downarrow$
WaveTransformer	0.745	0.654	0.208	0.285
Transformer	0.762	0.709	0.176	0.224
LSTM	0.762	0.722	0.201	0.254
TextCNN	0.757	0.715	0.127	0.166

Table 5: Experiment 4: Sentiment analysis results. The WaveTransformer underperforms all baselines on all metrics.

\uparrow

= higher is better;

\downarrow

= lower is better.

Figure 9: Experiment 4: Detailed NLP results. The WaveTransformer (red) underperforms across all four metrics, with the largest deficit on negation accuracy—the metric most relevant to the interference hypothesis.

Variant	Acc. $\uparrow$	ECE $\downarrow$	NLL $\downarrow$	Brier $\downarrow$	Params
Full WaveHead	0.909	0.082	0.349	0.148	17.2K
NoBorn (Mag)	0.911	0.015	0.280	0.131	17.2K
NoUnitary	0.909	0.082	0.350	0.148	17.3K
ComplexLinear	0.913	0.053	0.359	0.140	17.3K
Softmax	0.913	0.036	0.287	0.132	0.7K
Softmax (2x)	0.913	0.051	0.346	0.139	9.6K

Table 6: Experiment 5: Ablation results with shared backbone. Bold = best, underline = second best. The NoBorn head achieves the best calibration (ECE

0.0146

), while the Full WaveHead achieves the worst (

0.0819

Figure 10: Experiment 5: Accuracy versus ECE scatter plot. The NoBorn (Magnitude) head occupies the ideal lower region: high accuracy with substantially lower ECE than all alternatives. Error bars show $\pm 1$ standard deviation across three seeds.

Figure 11: Experiment 5: Horizontal bar chart of ablation results. Accuracy is comparable across variants ( $\sim$ 0.91), but ECE and NLL reveal large differences. NoBorn achieves the lowest ECE and NLL.

Figure 12: Experiment 5: Parameter counts and per-epoch training time. Complex heads have $\sim$ 17K parameters and $\sim$ 0.6s/epoch, approximately $3\times$ the compute of a standard softmax head.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

	Seed 42		Seed 123		Seed 456
Variant	Acc.	ECE	Acc.	ECE	Acc.	ECE
Full WaveHead	0.907	0.080	0.909	0.082	0.910	0.084
NoBorn (Mag)	0.910	0.013	0.911	0.015	0.913	0.016
NoUnitary	0.907	0.081	0.909	0.082	0.910	0.084
ComplexLinear	0.912	0.052	0.913	0.053	0.914	0.054
Softmax	0.911	0.034	0.913	0.036	0.914	0.036

Appendix

Table 7: Extended ablation results: per-seed accuracy and ECE for Experiment 5.

	$d=16$	$d=32$	$d=64$	$d=128$
$T_{w}=0$ (no warmup)	0.031	0.024	0.019	0.020
$T_{w}=4$	0.027	0.020	0.016	0.017
$T_{w}=8$ (default)	0.025	0.018	0.015	0.016
$T_{w}=15$	0.026	0.019	0.015	0.016

Appendix

Table 8: Sensitivity analysis: NoBorn ECE varies with Hilbert dimension and warmup epochs.

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Abstract

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