Geometric Regularization for Long-Tailed Semi-Supervised Learning via Gaussian Feature Bridges

Hongyang He; Xinyuan Song; Yan Zhong; Daizong Liu; Yanbin Li; Yang-fan He; Wenqiao Zhang

cs.LGAug 21, 2026

Geometric Regularization for Long-Tailed Semi-Supervised Learning via Gaussian Feature Bridges

Authors: Hongyang He, Xinyuan Song, Yan Zhong, Daizong Liu, Yanbin Li, Yang-fan He, Wenqiao Zhang

Organizations: University of Warwick, Coventry, United Kingdom · Wuhan University, Wuhan, China · Emory University, Atlanta, GA, USA · Peking University, Beijing, China ·

Abstract

Real-world semi-supervised learning (SSL) often encounters significant challenges with long-tailed label distributions and noisy pseudo-labels, which hinder generalization and amplify confirmation bias. In this work, we introduce a novel framework, Gaussian Bridge Consistency (GBC), to address these challenges by constructing semantic interpolation paths between unlabeled samples and high-quality class anchors. Our method maintains a dynamic Prototype Atlas that stores a diverse and evolving set of labeled and pseudo-labeled exemplars per class. For each unlabeled instance, GBC forms a class-conditional Gaussian Feature Bridge in the latent space, enabling the student model to traverse a smooth trajectory from uncertain predictions to reliable class prototypes. A bridge consistency loss is applied along this path to enforce alignment with a geometrically interpolated target distribution. Furthermore, we propose BridgeMix, a confidence-aware feature mixing strategy that interpolates both sample and anchor pairs to amplify cross-sample generalization. Extensive experiments on CIFAR10-LT and ImageNet-LT (USB benchmarks) validate the robustness and effectiveness of GBC under realistic long-tailed SSL settings, consistently improving long tail-class performance without sacrificing scalability.

Figures & tables

Figure 1: Overview of our proposed framework.

Figure 2: Overview of BridgeMix. Confidence-guided weights $\lambda_{i}$ facilitate smooth semantic transitions between paired unlabeled samples and their anchors.

Table 3

	consistent		uniform		reversed		middle		head-tail
$\gamma_{\ell}$	150	100	150	100	150	100	150	100	150	100
$\gamma_{u}$	150	100	1	1	$1/150$	$1/100$	150	100	150	100
FixMatch [ 49 ]	62.91 $\pm$ 0.87	65.48 $\pm$ 0.81	71.03 $\pm$ 0.69	72.49 $\pm$ 0.61	59.94 $\pm$ 1.02	62.53 $\pm$ 0.95	64.29 $\pm$ 0.85	65.95 $\pm$ 0.74	58.33 $\pm$ 1.01	60.52 $\pm$ 0.93
w/ CReST+ [ 57 ]	67.98 $\pm$ 0.73	70.45 $\pm$ 0.63	82.01 $\pm$ 0.51	84.96 $\pm$ 0.43	65.97 $\pm$ 0.76	69.03 $\pm$ 0.71	69.00 $\pm$ 0.65	72.47 $\pm$ 0.54	62.98 $\pm$ 0.82	65.95 $\pm$ 0.76
w/ DASO [ 43 ]	73.01 $\pm$ 0.49	74.48 $\pm$ 0.51	87.97 $\pm$ 0.31	89.52 $\pm$ 0.29	73.99 $\pm$ 0.58	75.51 $\pm$ 0.52	74.99 $\pm$ 0.48	78.53 $\pm$ 0.41	69.97 $\pm$ 0.66	72.48 $\pm$ 0.62
w/ ACR $\dagger$ [ 58 ]	76.99 $\pm$ 0.41	81.61 $\pm$ 0.33	91.32 $\pm$ 0.22	92.09 $\pm$ 0.19	81.82 $\pm$ 0.27	85.03 $\pm$ 0.28	77.90 $\pm$ 0.49	73.62 $\pm$ 0.46	78.96 $\pm$ 0.37	79.82 $\pm$ 0.39

Table 3: Top-1 accuracy (%) on CIFAR10-LT (

N_{1}=500,M_{1}=4000

) with different labeled/unlabeled imbalance ratios

\gamma_{\ell},\gamma_{u}

under five unlabeled class distributions (SimPro protocol [ 10 ] ).

\dagger

denotes reproduced results without anchor distributions for fair comparison [ 58 ] . All results are reported in alignment with established SimPro performance trends; best in each column is bold . Pink highlights our GBC, and Peach indicates the baseline (SimPro). Manifold MixUp is a strong interpolation-based baselines

	ImageNet-127 ( $\gamma_{t}{\approx}286$ )		ImageNet-127 ( $\gamma_{t}{=}1$ )		ImageNet-1k ( $\gamma_{t}{=}1$ )
Method	$32{\times}32$	$64{\times}64$	$32{\times}32$	$64{\times}64$	$32{\times}32$	$64{\times}64$
FixMatch [ 49 ]	29.7 $\pm$ 0.37	42.3 $\pm$ 0.41	38.7 $\pm$ 0.46	46.7 $\pm$ 0.44	–	–
w/ DARP (NeurIPS’22) [ 33 ]	30.5 $\pm$ 0.28	42.5 $\pm$ 0.40	–	–	–	–
w/ CReST+ (CVPR’21) [ 57 ]	32.5 $\pm$ 0.33	44.7 $\pm$ 0.39	–	–	–	–
w/ CoSSL (CVPR’22) [ 11 ]	43.7 $\pm$ 0.35	53.9 $\pm$ 0.38	–	–	–	–
w/ SAW (ICML’22) [ 35 ]	58.3 $\pm$ 0.31	65.5 $\pm$ 0.29	54.9 $\pm$ 0.27	62.7 $\pm$ 0.25	18.9 $\pm$ 0.33	24.4 $\pm$ 0.36

Table 4: Top-1 accuracy (%) on ImageNet-127 (

\gamma_{\ell}{\approx}286

N_{1}{\approx}28

M_{1}{\approx}250

k) and ImageNet-1k (

\gamma_{\ell}{=}256

N_{1}{=}256

M_{1}{=}1024

) under different test imbalance

\gamma_{t}

and resolutions.

\dagger

indicates results reproduced for ACR without anchor distributions. All baselines follow the SimPro protocol [ 10 ] . Pink indicates GBC while Peach indicates the ReaLTSSL baseline (SimPro).

Figure 3: Many/Medium/Few breakdown and t-SNE visualization. Left: Per-split Top-1 accuracy on ImageNet-127 ( $\gamma_{t}{\approx}286$ , $64{\times}64$ ). Classes are grouped by labeled training samples under the long-tailed distribution ( $\gamma_{\ell}$ ): Many $>$ 100, Medium 20–100, Few $<$ 20. The split is fixed across all methods and accuracy is averaged per class within each group. The same class split is fixed across all compared methods, and group accuracy is computed as the unweighted average over classes within each group. Right: t-SNE visualization on CIFAR10-LT (uniform setting).

Figure 4: Mean KL divergence on CIFAR10-LT ( $N_{1}=500$ , $M_{1}=4000$ , SimPro protocol). $\mathrm{KL}(p_{\text{teacher}}\|p_{\text{student}})$ is plot on weak views, averaged over 3 seeds and smoothed by EMA(0.9). Curves correspond to three unlabeled-distribution regimes: consistent ( $\gamma_{\ell}{=}\gamma_{u}{=}150$ ), uniform ( $\gamma_{u}{=}1$ ), reversed ( $\gamma_{u}{=}1/\gamma_{\ell}$ ). GBC exhibits faster and smoother KL divergence decay across all regimes, especially under reversed , indicating stable bridge-consistency alignment under severe pseudo-label noise.

Figure 5: Effect of the confidence threshold $\tau_{c}$ on performance.

Table 9

Table 10

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Backbone	Params (M)	CIFAR10-LT ( $32^{2}$ )		CIFAR100-LT ( $32^{2}$ )
Backbone	Params (M)	Baseline	+GBC	Baseline	+GBC
ResNet-50	25.6	80.7 $\pm$ 0.3	83.5 $\pm$ 0.2 (+2.8)	52.2 $\pm$ 0.2	53.0 $\pm$ 0.2 (+0.8)
ViT-B/16 [ 9 ]	186.4	82.1 $\pm$ 0.3	84.0 $\pm$ 0.2 (+1.9)	53.0 $\pm$ 0.3	54.0 $\pm$ 0.2 (+1.0)
Vision Mamba-B [ 64 ]	48.5	81.0 $\pm$ 0.3	83.0 $\pm$ 0.2 (+2.0)	51.8 $\pm$ 0.3	52.9 $\pm$ 0.2 (+1.1)
		ImageNet-127 ( $64^{2}$ )		ImageNet-1k ( $64^{2}$ )
		Baseline	+GBC	Baseline	+GBC

Appendix

Table 9: Additional adaptation experiments on diverse backbones. Top-1 accuracy (%) with/without GBC on three representative benchmarks. Backbones: ResNet-50, ViT-B/16 [ 9 ] , Vision Mamba-B [ 64 ] . All runs follow the same SimPro/ReaLTSSL protocol and hyperparameters as the main paper.

\Delta

reports the absolute gain from adding GBC.

	CIFAR10-LT ( $32^{2}$ )			CIFAR100-LT ( $32^{2}$ )			ImageNet-127 ( $64^{2}$ )
Method	1-shot	5-shot	10-shot	1-shot	5-shot	10-shot	1-shot	5-shot	10-shot
FixMatch [ 49 ]	80.2 $\pm$ 0.8	89.0 $\pm$ 0.5	91.1 $\pm$ 0.4	42.9 $\pm$ 0.6	55.6 $\pm$ 0.4	61.2 $\pm$ 0.5	34.1 $\pm$ 0.5	49.5 $\pm$ 0.4	54.8 $\pm$ 0.3
SAW [ 35 ]	81.4 $\pm$ 0.5	89.6 $\pm$ 0.4	91.6 $\pm$ 0.3	43.7 $\pm$ 0.5	56.1 $\pm$ 0.3	61.7 $\pm$ 0.4	35.0 $\pm$ 0.4	50.2 $\pm$ 0.4	55.3 $\pm$ 0.4
DePL [ 55 ]	81.9 $\pm$ 0.5	90.1 $\pm$ 0.6	91.8 $\pm$ 0.3	44.2 $\pm$ 0.4	56.9 $\pm$ 0.4	62.0 $\pm$ 0.5	35.4 $\pm$ 0.4	50.8 $\pm$ 0.3	55.8 $\pm$ 0.5
BaCon [ 12 ]	82.5 $\pm$ 0.4	90.3 $\pm$ 0.5	92.0 $\pm$ 0.4	44.9 $\pm$ 0.3	57.4 $\pm$ 0.5	62.3 $\pm$ 0.3	35.8 $\pm$ 0.5	51.1 $\pm$ 0.4	56.2 $\pm$ 0.3
CPE [ 38 ]	82.7 $\pm$ 0.5	90.4 $\pm$ 0.3	92.3 $\pm$ 0.4	45.2 $\pm$ 0.5	57.7 $\pm$ 0.3	62.6 $\pm$ 0.4	36.1 $\pm$ 0.4	51.4 $\pm$ 0.4	56.7 $\pm$ 0.3

Appendix

Table 10: Few-shot semi-supervised performance (k-shot per class). We evaluate GBC and recent SSL methods under the ReaLTSSL few-shot protocol, where only

k\!\in\!\{1,5,10\}

labeled samples per class are provided. Results report Top-1 accuracy (%) averaged over three random seeds.

Noise Strength $\nu$	Top-1 Acc. (%)	KL Decay $\downarrow$ (epoch 300)	Anchor Recall (%)
0.00 (no noise)	91.82 $\pm$ 0.11	0.164	78.3
0.05	92.11 $\pm$ 0.10	0.151	81.0
0.10 (default)	92.30 $\pm$ 0.17	0.143	83.7
0.20	91.97 $\pm$ 0.13	0.158	80.1

Appendix

Table 11: Ablation on Gaussian noise strength

\nu

in the Gaussian Feature Bridge. We evaluate GBC on CIFAR10-LT under the consistent setting (

\gamma_{\ell}{=}100

\gamma_{u}{=}100

\nu

controls the magnitude of stochastic perturbation

\sigma(t)=\nu t(1-t)

injected along the bridge. We report Top-1 accuracy, KL divergence decay (lower is better), and anchor recall (ratio of correct PA retrieval). Moderate noise (

\nu{=}0.10

) gives the best trade-off.

PA Capacity $C$	Top-1 Acc. (%)	Anchor–Proto Cosine $\uparrow$	Refresh Rate (%) $\downarrow$
0 (EMA only)	80.81 $\pm$ 0.18	0.71	42.3
16	85.05 $\pm$ 0.15	0.76	38.5
32	85.72 $\pm$ 0.14	0.79	33.1
64 (default)	92.30 $\pm$ 0.17	0.82	29.7
128	92.18 $\pm$ 0.19	0.80	46.2

Appendix

Table 12: Ablation on Prototype Atlas (PA) capacity on CIFAR10-LT under the consistent setting.

C

denotes the maximum number of anchors per class;

C{=}0

corresponds to using only the EMA prototype without storing exemplar anchors. Moderate capacity (

C{=}64

) achieves the best trade-off between representation diversity and stability.

Method	Avg. Epoch Time (s)	Total Time (300 ep)	Overhead vs. FixMatch
FixMatch	42.1	3.51 h	0%
SimPro	43.0	3.58 h	+2.1%
GBC (ours)	42.8	3.56 h	+1.7%

Appendix

Table 13: Wall-clock comparison on CIFAR10-LT (consistent setting) using a single A100 GPU. We report the average epoch time over the last 50 epochs and the total training time for 300 epochs. GBC introduces minimal computational overhead (

<2\%

) compared with FixMatch and remains comparable to SimPro.

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Long-tailed distributions are prevalent in real-world semi-supervised learning (SSL), where pseudo-labels tend to favor majority classes, leading to degraded generalization. While many long-tailed semi-supervised learning (LTSSL) methods have been proposed, the mechanisms by which they implicitly debias logits remain poorly understood. In this work, we revisit LTSSL through the lens of learning dynamics and provide a theoretical characterization of logits debiasing. Specifically, we derive a step-wise decomposition of the logits updates, showing that predictions are dominated by class-imbalance bias that reliably reflects label priors. To expose this effect, we use the logits of a task-irrelevant baseline image as an indicator of accumulated bias and prove that they converge to the class prior. This provides a unified view where LTSSL remedies such as logit adjustment, reweighting, and resampling correspond to reshaping gradient dynamics. Based on this insight, we propose DyTrim, a principle-based dynamic pruning framework that reallocates gradient budget through class-aware pruning on labeled data and confidence-based soft pruning on unlabeled data. We provide theoretical guarantees that DyTrim reduces class bias and improves generalization. Extensive experiments on standard LTSSL benchmarks show consistent gains across architectures and methods. Code available at: https://jiajun0425.github.io/DyTrim

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