Follow the Geometry, Not the Model: Cold Start Semi-Supervised Learning

Itai David; Daphna Weinshall

cs.LGSep 13, 2026

Follow the Geometry, Not the Model: Cold Start Semi-Supervised Learning

Authors: Itai David, Daphna Weinshall

Organizations: School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Abstract

Modern semi-supervised learning (SSL) couples pseudo-label generation and classifier training, using the classifier's own confidence to select the pseudo-labels that are then used to update the model. In the cold-start regime, where at most a few labels per class are available, this coupling is ill-posed, since the classifier cannot supervise itself before it has learned. To address this problem, we propose VAST (Veracity-Aware Semi-Supervised Training), which decouples these two stages. Probabilistic beliefs over the unlabeled set are first inferred directly from the geometry of a frozen self-supervised embedding and only then distilled into an inductive classifier. The construction rests on the Veracity Matrix, a kernel-based structure that aggregates label evidence across the data manifold and admits an interpretation as a Dirichlet posterior under a per-observation powered-likelihood model. Additionally, we introduce Veracity Propagation, a self-terminating belief-spreading step that extends coverage beyond the kernel neighborhood of the labeled set. Under a controlled protocol in which all methods receive identical frozen embeddings and labeled sets, VAST outperforms the strongest graph-based SSL baselines at every operating point across three datasets, with statistically significant gains in 7 of 9 comparisons, while producing a deployable inductive classifier rather than requiring transductive graph inference. Compared with end-to-end confidence-gated SSL, we further find that these methods underperform in this setting and, in our experiments, do not consistently exceed labeled-only performance.

Figures & tables

Figure 2: Receiver-threshold (

\tau_{r}

) sensitivity on CIFAR-100 (BYOL): test accuracy (mean

\pm

SEM, 5 seeds) at 100/200/400 labels. Dashed line marks the default

\tau_{r}=0.75

	CIFAR-100			TinyImageNet			ImageNet100
Method	1/cls	2/cls	4/cls	1/cls	2/cls	4/cls	1/cls	2/cls	4/cls
MLP (labeled only)	$29.4_{\pm 0.5}$	$38.8_{\pm 0.5}$	$47.6_{\pm 0.5}$	$12.76_{\pm 0.46}$	$17.89_{\pm 0.89}$	$23.30_{\pm 0.96}$	$19.7_{\pm 1.1}$	$30.8_{\pm 1.1}$	$38.4_{\pm 0.5}$
1-NN	$31.0_{\pm 0.6}$	$37.4_{\pm 0.4}$	$43.4_{\pm 0.6}$	$14.59_{\pm 0.75}$	$18.22_{\pm 0.70}$	$21.58_{\pm 0.69}$	$26.7_{\pm 2.2}$	$33.2_{\pm 1.5}$	$37.3_{\pm 0.5}$
Laplace	$29.4_{\pm 0.9}$	$40.2_{\pm 0.6}$	$47.7_{\pm 0.3}$	$11.18_{\pm 1.65}$	$17.44_{\pm 1.13}$	$22.82_{\pm 0.94}$	$18.0_{\pm 1.7}$	$31.2_{\pm 0.3}$	$39.0_{\pm 0.6}$
Poisson	$\mathbf{38.6_{\pm 1.0}}$	$45.1_{\pm 0.7}$	$50.1_{\pm 0.4}$	$16.60_{\pm 0.68}$	$21.06_{\pm 0.68}$	$24.36_{\pm 0.51}$	$31.7_{\pm 0.9}$	$38.5_{\pm 0.8}$	$42.0_{\pm 0.6}$
C&S	$33.6_{\pm 0.5}$	$44.4_{\pm 0.5}$	$51.2_{\pm 0.3}$	$14.51_{\pm 0.91}$	$20.81_{\pm 0.57}$	$\mathbf{26.89_{\pm 1.03}}$	$28.3_{\pm 0.8}$	$37.8_{\pm 0.5}$	$44.8_{\pm 0.5}$

Table 1: Cross-dataset summary: test accuracy (%) by method and label budget. CIFAR-100/ImageNet100 use BYOL embeddings; TinyImageNet uses SimCLR embeddings. † taken directly from the official USB benchmark ( Wang et al. 2022 ) .

Method	100	200	400
FlexMatch (WRN-28-2)	16.2	33.4	–
FlexMatch (WRN-28-8)	9.4	40.8	$49.85_{\pm 1.5}^{\dagger}$
FreeMatch (WRN-28-8)	$9.8_{\pm 0.3}$	$43.4_{\pm 0.9}$	$50.8_{\pm 1.0}^{\dagger}$
CoMatch	30.7	38.9	$39.02_{\pm 0.77}^{\dagger}$
SimMatch	17.2	46.9	$51.18_{\pm 1.07}^{\dagger}$
ExMatch ∗	$32.49_{\pm 1.3}$	$43.71_{\pm 0.7}$	$51.13_{\pm 1.7}$

Table 2: Confidence-based pseudo-labeling vs. VAST , CIFAR-100 cold-start regime. ∗ ExMatch results are taken from the original paper. † taken directly from the official USB benchmark ( Wang et al. 2022 ) . ‡

\tau

selected per budget to maximize Conf-ST accuracy.

Figure 3: Full-pipeline Wall-clock cost, VAST vs. FreeMatch (log scale, hours). VAST includes one-time BYOL pretraining; FreeMatch pays full training cost per run.

Configuration	1/cls	2/cls	4/cls
VAST (distill + prop.)	$\mathbf{39.2_{\pm 0.9}}$	$\mathbf{47.8_{\pm 0.3}}$	$\mathbf{54.4_{\pm 0.4}}$
Distill only (matched kernel)	$30.5_{\pm 0.7}$	$38.9_{\pm 0.6}$	$47.3_{\pm 0.7}$
Distill only (local RBF, best)	$34.9_{\pm 1.1}$	$42.0_{\pm 0.9}$	$48.8_{\pm 0.3}$

Table 3: Propagation ablation on CIFAR-100: test accuracy (%) by label budget (labels per class).

Configuration	1/cls	2/cls	4/cls
VAST (distill + prop.)	$\mathbf{39.2_{\pm 0.9}}$	$\mathbf{47.8_{\pm 0.3}}$	$\mathbf{54.4_{\pm 0.4}}$
VAST w/o degree normal.	$37.8_{\pm 0.7}$	$\mathbf{46.8_{\pm 0.2}}$	$53.7_{\pm 0.4}$

Table 4: Degree-normalization ablation on CIFAR-100: test accuracy by label budget.

Method	SimCLR	Barlow	BYOL	DINOv2	VAE2
1 label per class
1-NN	$24.3_{\pm 0.5}$	$27.64_{\pm 0.8}$	$31.0_{\pm 0.6}$	$58.4_{\pm 0.6}$	–
Poisson	$26.83_{\pm 0.5}$	$36.37_{\pm 0.8}$	$\mathbf{38.6_{\pm 1.0}}$	$\mathbf{69.8_{\pm 0.9}}$	$4.5_{\pm 0.2}$
C&S	$23.70_{\pm 0.46}$	$32.76_{\pm 0.66}$	$33.6_{\pm 0.5}$	$67.6_{\pm 0.8}$	–
VAST	$\mathbf{28.9_{\pm 0.4}}$	$\mathbf{37.3_{\pm 0.5}}$	$\mathbf{39.2_{\pm 0.9}}$	$\mathbf{70.1_{\pm 0.7}}$	–
4 labels per class

Table 5: Solver against embedding on CIFAR-100; mean

\pm

SEM over 5 seeds ( VAST C&S and 1-NN not run on VAE2). VAST is highest in every column; bold marks a significant margin over the strongest baseline (

p<0.05

, paired

t

-test).

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Architecture	Parameters
ResNet-18 backbone ( VAST /BYOL)	${\sim}11.2$ M
BYOL projector (2-layer MLP, 512 $\to$ 256 $\to$ 256)	${\sim}0.2$ M
BYOL predictor (same architecture)	${\sim}0.2$ M
Full BYOL network (backbone + projector + predictor)	${\sim}11.6$ M
Labeled-only MLP classifier head	${\sim}0.16$ – $0.18$ M
WRN-28-2 ( FlexMatch )	${\sim}1.47$ M

Appendix

Table 6: Parameter counts by architecture.

Figure 4: Receiver-threshold sensitivity on TinyImageNet (SimCLR). Test accuracy (mean

\pm

SD over 5 seeds) as a function of

\tau_{r}

at 200, 400, and 800 labels. The dashed line marks the default

\tau_{r}=0.75

Figure 5: Dirichlet prior (

\alpha_{0}

) sensitivity on CIFAR-100 (BYOL), in a narrow band around the a priori value

\alpha_{0}=1/M=10^{-2}

(dashed line). Mean

\pm

SEM over 5 seeds.

Figure 6: Dirichlet prior (

\alpha_{0}

) sensitivity on TinyImageNet (SimCLR), in a narrow band around the a priori value

\alpha_{0}=1/M=5\times 10^{-3}

(dashed line). Mean

\pm

SD over 5 seeds.

Figure 7: Distillation weight (

\lambda

) sensitivity on CIFAR-100 (BYOL), log 2 scale; dashed line marks the default

\lambda=64

. Mean

\pm

SEM over 5 seeds.

	CIFAR-100	ImageNet100	TinyImageNet
Encoder	BYOL	BYOL	SimCLR
$N$	50,000	130,000	100,000
PR	138.1	60.6	75.5
$n$ @90%	252	164	214
$n$ @95%	339	256	306
$n$ @99%	460	425	441

Appendix

Table 7: PCA intrinsic dimensionality. CIFAR-100/ImageNet100 on BYOL embeddings; TinyImageNet on SimCLR.

N

is the labeled+unlabeled pool size; PR is the participation ratio; the last three rows report the number of principal components needed to explain the given fraction of variance.

Dataset	Encoder	$k^{*}$	mean	max	p99	mean	max	p99
			Hard degree			Soft degree
CIFAR-100	BYOL	13	20.3	518	65	12.2	280.5	37.1
ImageNet100	BYOL	5	8.0	78	23	4.8	44.3	13.7
TinyImageNet	SimCLR	3	5.1	50	17	3.0	28.7	9.9

Appendix

Table 8: Sparse-graph degree statistics at each dataset’s connectivity-selected

k

: hard degree (neighbor count) and soft degree (sum of RBF neighbor weights), over the full labeled+unlabeled pool. CIFAR-100/ImageNet100 on BYOL embeddings; TinyImageNet on SimCLR. (Cited as Table 7 in the main paper’s Section 3.3.)

Figure 8: Hard degree (top row) and soft degree (bottom row) distributions on the sparse $k$ -NN graph at each dataset’s connectivity-selected $k$ . Dashed lines mark the mean. The additional TinyImageNet–BYOL panel uses the same $k{=}3$ selected on the SimCLR embedding (Table 8 ), so all panels share a common $k$ per dataset.

$k$	CIFAR-100	ImageNet100	TinyImageNet
1	0.616	0.611	0.343
3	0.581	0.572	0.312
5	0.563	0.551	0.295
10	0.536	0.522	0.272
15	0.520	0.505	0.258

Appendix

Table 9:

k

-NN edge purity (fraction of same-class neighbor edges) at matched

k

. CIFAR-100/ImageNet100 on BYOL embeddings; TinyImageNet on SimCLR.

Figure 9:

k

-NN edge purity as a function of neighborhood size

k

. Dashed vertical lines mark each dataset’s connectivity-selected

k

Figure 10: TinyImageNet

k

-NN edge purity under BYOL vs. SimCLR, as a function of neighborhood size

k

. Dashed line marks the connectivity-selected

k=3

Figure 11: TinyImageNet 1-NN test accuracy (mean

\pm

SEM over 5 seeds) under BYOL vs. SimCLR, at 1/2/4 labels per class.

Figure 12: Per-iteration util_ratio (solid, left axis) and unsupervised loss (dashed, right axis, gray) for FreeMatch and CoMatch , CIFAR-100, 100 labels. CoMatch ’s ungated contrastive graph loss is not shown.

Gate	100	200	400
$\tau=0.60$	$22.3_{\pm 8.0}$	$39.4_{\pm 0.5}$	$47.4_{\pm 0.7}$
$\tau=0.75$	$10.9_{\pm 6.4}$	$38.4_{\pm 0.6}$	$\mathbf{48.2_{\pm 0.7}}$
$\tau=0.90$	$11.7_{\pm 0.9}$	$38.1_{\pm 0.4}$	$46.7_{\pm 0.8}$
$\tau=0.95$	$\mathbf{30.4_{\pm 0.3}}$	$\mathbf{39.9_{\pm 1.0}}$	$47.2_{\pm 0.6}$
adaptive	$27.2_{\pm 0.2}$	$37.1_{\pm 0.6}$	$46.6_{\pm 0.7}$
MLP (labeled only)	$29.4_{\pm 0.5}$	$38.8_{\pm 0.5}$	$47.6_{\pm 0.5}$

Appendix

Table 10: Conf-ST threshold sweep on CIFAR-100 (BYOL), mean

\pm

SEM over 5 seeds. Bold marks the best gate per budget, the values reported in Table 2 of the main paper.

Stage	Time	Space	Frequency
Graph construction	$O(N^{2}d)$ with brute-force pairwise distances; reduced to sub-quadratic in practice by any standard exact or approximate nearest-neighbor index (e.g. FAISS, HNSW) without changes to the rest of the algorithm	$O(Nk)$ , sparse	once
Veracity Matrix seeding	$O\!\left(\sum_{j\in\mathcal{L}}\deg(x_{j})\right)$ , averaging to $O(N_{L}k)$ under the sparse $k$ -NN graph	$O(NM)$ for the matrix itself the dominant memory structure for typical $N,M$ (e.g. $5{\times}10^{6}$ floats on CIFAR-100, vs. ${\sim}6.5{\times}10^{5}$ pre-symmetrization directed edges, ${\sim}5{\times}10^{5}$ after symmetrization)	once
Propagation	$O\!\left(M\sum_{t}\|\mathcal{R}_{t}\|k\right)$ only active receivers are updated each round, and $\|\mathcal{R}_{t}\|$ monotonically shrinks as points cross $\tau_{r}$ and freeze; $O(TNkM)$ is a loose upper bound	reuses seeding	once
Classifier training	$O(E\cdot\|\mathcal{T}_{\text{train}}\|\cdot F)$ for $E$ epochs at per-example cost $F$ identical in form to any baseline of the same architecture	model-dependent	every epoch
Inference (new test point)	$O(F)$ one forward pass through $f_{\theta}$	model-dependent	every query

Appendix

Table 11: Asymptotic time and space complexity of VAST , decomposed by stage. The connectivity criterion selected

k\in\{3,5,13\}

across the three evaluated datasets.

	CIFAR-100			ImageNet100
Method	1/cls	2/cls	4/cls	1/cls	2/cls	4/cls
Poisson ( $k{=}50$ )	$117.7_{\pm 5.9}$	$158.7_{\pm 1.2}$	$203.7_{\pm 3.7}$	$595.7_{\pm 0.3}$	$472.3_{\pm 117.1}$ †	$628.3_{\pm 2.9}$
Laplace (LP)	$270.0_{\pm 2.1}$	$246.0_{\pm 4.0}$	$192.7_{\pm 4.9}$	$287.3_{\pm 3.9}$	$323.3_{\pm 34.7}$	$298.3_{\pm 38.4}$
C&S	$158.3_{\pm 1.8}$	$140.3_{\pm 7.3}$	$131.7_{\pm 0.9}$	$223.0_{\pm 2.7}$	$240.7_{\pm 15.6}$	$213.0_{\pm 33.5}$
LP-DSL	$78.7_{\pm 1.2}$	$77.0_{\pm 1.0}$	$79.3_{\pm 0.9}$	$223.7_{\pm 0.9}$	$260.3_{\pm 8.1}$	$235.0_{\pm 16.1}$
VAST (distill + prop.)	$\mathbf{18.6_{\pm 0.8}}$	$\mathbf{22.3_{\pm 0.1}}$	$\mathbf{22.5_{\pm 0.8}}$	$\mathbf{142.3_{\pm 7.3}}$	$\mathbf{142.8_{\pm 6.3}}$	$\mathbf{139.8_{\pm 2.3}}$

Appendix

Table 12: Track 1 timing: end-to-end time in seconds (training plus test-set prediction) on identical BYOL embeddings, averaged over 3 seeds. Label budgets are per class.

	CIFAR-100			ImageNet100
Configuration	1/cls	2/cls	4/cls	1/cls	2/cls	4/cls
VAST (distill + prop.)	$\mathbf{39.2_{\pm 0.9}}$	$\mathbf{47.8_{\pm 0.3}}$	$\mathbf{54.4_{\pm 0.4}}$	$\mathbf{34.9_{\pm 1.1}}$	$\mathbf{43.1_{\pm 0.9}}$	$\mathbf{49.3_{\pm 0.2}}$
Distill only (matched kernel)	$30.5_{\pm 0.7}$	$38.9_{\pm 0.6}$	$47.3_{\pm 0.7}$	$25.0_{\pm 0.9}$	$34.5_{\pm 0.7}$	$40.7_{\pm 0.6}$
Distill only (local RBF, best-effort)	$34.9_{\pm 1.1}$	$42.0_{\pm 0.9}$	$48.8_{\pm 0.3}$	$27.1_{\pm 0.9}$	$36.0_{\pm 1.2}$	$42.9_{\pm 0.2}$

Appendix

Table 13: Propagation ablation on CIFAR-100 and ImageNet100: test accuracy (%) by configuration and label budget (labels per class).

CIFAR-100
Weighting	1/cls	2/cls	4/cls
Per-sample $c_{i}$ (reported)	$39.2_{\pm 0.9}$	$47.8_{\pm 0.3}$	$54.4_{\pm 0.4}$
Uniform ( $w_{i}=1$ )	$39.06_{\pm 0.76}$	$47.54_{\pm 0.16}$	$54.23_{\pm 0.26}$
TinyImageNet
Weighting	200 lbl	400 lbl	800 lbl
Per-sample $c_{i}$ (reported)	$17.71_{\pm 1.26}$	$22.83_{\pm 1.40}$	$27.55_{\pm 1.61}$

Appendix

Table 14: Distillation weighting ablation, mean

\pm

dispersion over 5 seeds (SEM for CIFAR-100; sample SD for TinyImageNet, Supp. C.1 ). CIFAR-100 uses BYOL; TinyImageNet uses SimCLR.

CIFAR-100 (BYOL)
Peakedness function	1/cls	2/cls	4/cls
Max, $\tilde{V}_{k,(1)}$ (reported)	$39.2_{\pm 0.9}$	$47.8_{\pm 0.3}$	$54.4_{\pm 0.4}$
Margin, $\tilde{V}_{k,(1)}-\tilde{V}_{k,(2)}$	$39.41_{\pm 0.93}$	$47.49_{\pm 0.21}$	$54.29_{\pm 0.44}$
Entropy, $1-H(\tilde{\mathbf{V}}_{k})/\log M$	$37.52_{\pm 0.78}$	$46.19_{\pm 0.27}$	$52.94_{\pm 0.52}$
Poisson (reference)	$38.6_{\pm 1.0}$	$45.1_{\pm 0.7}$	$50.1_{\pm 0.4}$

Appendix

Table 15: Peakedness function ablation on CIFAR-100, mean

\pm

SEM over 5 seeds, current pipeline. Max and Margin are within one SEM of each other at every budget and are therefore left unbolded; Entropy trails both at every budget. Poisson Learning is repeated from Table 1 of the main paper as a reference point.

Embed.	Method	100	200	400
BYOL	VAST	$\mathbf{39.2_{\pm 0.9}}$	$\mathbf{47.8_{\pm 0.3}}$	$\mathbf{54.4_{\pm 0.4}}$
Barlow Tw.	VAST	$\mathbf{37.3_{\pm 0.5}}$	$\mathbf{44.7_{\pm 0.4}}$	$\mathbf{51.8_{\pm 0.4}}$
	Poisson	$36.37_{\pm 0.8}$	$43.2_{\pm 0.5}$	$48.49_{\pm 0.1}$
	PoissonMBO	$34.8_{\pm 0.8}$	$43.4_{\pm 0.6}$	$49.6_{\pm 0.4}$
	Laplace	$24.2_{\pm 1.1}$	$37.4_{\pm 0.5}$	$45.9_{\pm 0.2}$
SimCLR	VAST	$\mathbf{28.9_{\pm 0.4}}$	$\mathbf{34.9_{\pm 0.3}}$	$\mathbf{41.3_{\pm 0.4}}$

Appendix

Table 16: Test accuracy (%) on CIFAR-100 with BYOL, Barlow Twins, and SimCLR embeddings, random label selection.

Embedding	100	200	400
BYOL	$38.6_{\pm 1.0}$	$45.1_{\pm 0.7}$	$50.1_{\pm 0.4}$
VAE2	$4.5_{\pm 0.2}$	$5.4_{\pm 0.3}$	$7.1_{\pm 0.2}$

Appendix

Table 17: Poisson Learning test accuracy (%) on BYOL vs. VAE2 embeddings, CIFAR-100. Same solver, dataset, and labeled sets; only the embedding differs. VAE2 is the Poisson Learning library’s own recommended default, so this is not an adversarially chosen embedding.

Method	100 lbl	200 lbl	400 lbl
Laplace	$57.12_{\pm 0.77}$	$70.16_{\pm 0.55}$	$77.77_{\pm 0.37}$
LP-DSL	$46.52_{\pm 6.90}$ †	$71.20_{\pm 1.37}$	$77.80_{\pm 0.73}$
C&S	$67.60_{\pm 0.75}$	$\mathbf{77.74_{\pm 0.13}}$	$\mathbf{82.07_{\pm 0.16}}$
Poisson	$\mathbf{69.81_{\pm 0.89}}$	$\mathbf{77.41_{\pm 0.13}}$	$79.73_{\pm 0.34}$
VAST (distill + prop.)	$\mathbf{70.07_{\pm 0.68}}$	$\mathbf{78.28_{\pm 0.79}}$	$\mathbf{82.83_{\pm 0.54}}$

Appendix

Table 18: Track 1 comparison on DINOv2 (ViT-B/14) features, CIFAR-100; mean

\pm

SEM over 5 seeds with identical labeled sets per seed. Among the baselines, bold marks the strongest baseline in each column together with any baseline not significantly distinguishable from VAST (

p\geq 0.05

, paired

t

-test); Laplace and LP-DSL are never bolded, since no significance test was run for those two comparisons. Significance values are discussed in the text.

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Graph-Based Semi-Supervised Learning Pseudo-Labeling Semi-Supervised Learning

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Semi-supervised learning faces significant challenges in realistic scenarios where labeled data is scarce and unlabeled data follows unknown, arbitrary distributions. We formalize this critical yet under-explored paradigm as Universal Semi-supervised Learning (UniSSL). Existing methods typically leverage unlabeled data via pseudo-labeling. However, they often rely on the idealized assumption of a uniform unlabeled data distribution or require sufficient labeled data to estimate it. In the UniSSL setting, such dependencies lead to numerous erroneous pseudo-labels, thereby triggering representation confusion. Fortunately, we observe that inter-sample relations captured by representations are more reliable than pseudo-labels. Leveraging this insight, we shift our focus to representation-level structural inference to bypass distribution estimation. Accordingly, we propose Simplex Anchored Graph-state Equipartition (SAGE), which captures high-order inter-sample dependencies to establish structural consensus for guiding representation learning. Meanwhile, to mitigate representation confusion, we employ vectors that satisfy a simplex equiangular tight frame to serve as a coordinate frame for guiding inter-class representation separation. Finally, we introduce a weighting strategy based on distribution-agnostic metrics to prioritize reliable pseudo-labels and an auxiliary branch to isolate potentially erroneous pseudo-labels. Evaluations on five standard benchmarks show that SAGE consistently outperforms state-of-the-art methods, with an average accuracy gain of

\textbf{8.52%}

Abstract

Semi-Supervised Learning

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