Angular Regularization for Positive-Unlabeled Learning on the Hypersphere
Authors: Vasileios Sevetlidis
George Pavlidis
Antonios Gasteratos
Venue: Accepted by TMLR ()

Abstract: Positive–Unlabeled (PU) learning addresses classification problems where only a subset of positive examples is labeled and the remaining data is unlabeled, making explicit negative supervision unavailable. Existing PU methods often rely on negative-risk estimation or pseudo-labeling, which either require strong distributional assumptions or can collapse in high-dimensional settings. We propose AngularPU, a novel PU framework that operates on the unit hypersphere using cosine similarity and angular margin. In our formulation, the positive class is represented by a learnable prototype vector, and classification reduces to thresholding the cosine similarity between an embedding and this prototype—eliminating the need for explicit negative modeling. To counteract the tendency of unlabeled embeddings to cluster near the positive prototype, we introduce an angular regularizer that encourages dispersion of the unlabeled set over the hypersphere, improving separation. We provide theoretical guarantees on the Bayes-optimality of the angular decision rule, consistency of the learned prototype, and the effect of the regularizer on the unlabeled distribution. Experiments on benchmark datasets demonstrate that AngularPU achieves competitive or superior performance compared to state-of-the-art PU methods, particularly in settings with scarce positives and high-dimensional embeddings, while offering geometric interpretability and scalability.

Published in Transactions on Machine Learning Research (12/2025)
Angular Regularization for Positive-Unlabelee Learning on the Hypersphere
Vasileios Sevetlidis* Athena RC
musiseve@aatennne..p
Institute for Language and Signel Processing University Campus Kimmeria Xanthi, GR67100, Greece
Democritus University of Thrace Dept. Production and Management Engineering Vas. Sofias 17
Xanlhi, GR67100, Greece
* Corresponuling Author
George Pavlidis
gpallid@athenre..y
Athena RC
Institute for Language and Signal Processing Universily Campus Kimmeria Xanthi, GR67100, Greece
Antonios Gasteratos
agaster@pme. duth.gr gr
Democritus University Thrace
Dept. Production and Management Engineering Vas. Sofias 17 Xanthi, GR67100, Greece
Reviewed on OpenReview: /t openreview... net/ forum? XQhO0Ly6el
Abstract
Positive Unlabeled (PU) learning addresses classification problems where only subset of positive examples is labeled and the remaining data is unlabeled, making explicit negative supervision unavailable. Existing PU methods often rely on negative-risk estimation or pseudo-labeling, which either require strong distributional assumptions or can collapse in high-dimensional settings. We propose AngularPU, a novel PU framework that operates on the unit hypersphere using cosine similarity and angular margin. In our formulation, the positive class is represented by learnable prototype vector, and classification reduces to thresholding the cosine similarity between an embedding and this prototype eliminating the need for explicit negative modeling. To counteract the tendency of unlabeled embeddings to cluster near the positive prototype, we introduce angular regularizer that encourages dispersion of the unlabeled set over the hypersphere, improving separation. We provide theoretical guarantees on the Bayes-optimality of the angular decision rule, consistency of the learned prototype, and the effect of the regularizer on the unlabeled distribution. Experiments on benchmark datascts demonstrate that AngularPU achieves competitive or superior performance compared state-of- toatt---f-thh-rtttt the-art PU methods, particularly in settings with scarce positives and high-dimensional embeddings, while offering geometric interpretability and scalability.

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Published in Transactions on Machine Learning Research (12/2025)
1 Introduction
(PU) learning addresses classification scenarios in which labeled negatives are scarce or infeasible to obtain [ekkeer & Davis 2020; Elkan & Noto, 2008). Such scttings are common in domains like medical diagnosis, sentiment ysis, and anomaly detection, where negalive examples can be costly, ambiguous, or ethically constrained. Formally, we are given a set of labeled positive instances and a large pool of unlabeled data containing an unknown mix of positives and negatives; the task is to learn classifier that separates the two without explicit negalive supervision.
Existing PC methods typically rely on surrogate modeling of the negative class Yu et al. 2002 Liu et al, 2003 Hsich et al. 2019 Gong et al.) 2021; Yuan et al.) 2025). Risk-based approaches (c.g., unbiased or non-negative PU risk estimators and their imbalanced variants) decompose the classification risk into terms involving only positives and unlabeled data, using an assumed class prior to correct for the implicit treatment of all unlabeled examples as negatives (Du Plessis al.) 2014: Christoffel et al., 2016 Ramaswamy et al., 2016 Kiryo et al. 2017p. While theoretically sound, these methods are sensitive to prior estimation errors and degrade when the learned representation is insufficiently discriminative. FMM-style and prototype-contrastive
methods, in contrast, generate pscudo-labels for the unlabeled set and alternate between representation learning and classifier updates; however, such alternating schemes are prone to confirmation bias early mislabeling of negatives as positives can cause cascading errors and often require auxiliary mechanisms such as momentum qucues, hard-negative mining, or contrastive pair construction (Arazo et al.. 2020 Cascante- Bonilla et al. 2021); contrastive mechanics: (He et al., 2020: Wu et al., 2018 Chen et al., 2020b). Boundary- anomaly-basee methods take different route by synthesizing negatives convexifying the positive set in latent space, but these heuristics can fail when the positive manifold is complex non-convex (Schölkopf
ct al.
2001
Ruff et al. 2018).
A common limitation across these approaches is the reliance on either explicit negative modeling or iterative pseudo-negen negative construction, both of which can be unstable in high-dimensional feature spaces. Moreover, most methods treat representation learning as secondary, leaving the geometric structure of the embedding space underexploited.
We take geometry-first approach: we explicitly model only the positive class and treat all other instances as dispersed over the hypersphere. Concretely, we propose neural encoder that maps inputs to the unit hypersphere and represents the positive class using directional scoring function based on cosine similarity to a learnable prototype vector. This yiclds a simple angulor score
s(=) = K
which provides geometrically meaningful decision rule and eliminates the need for explicit negative mod- cling, psendo-negatives class-prior estimation.
To reduce false positive clustering near the positive prototype, we introduce an wnlabeled-only angular reg- ularizer that encourages embeddings of unlabeled examples to be well-dispersed over the hypersphere. The encoder and prototype are learned jointly in a single-stage, end-to-end optimization.
Our contributions are:
1. Hyperspherical PU modeling: We formulate PU learning in a hyperspherical embedding space, using cosine similarity to a learned positive prototype as the classification score.
2. Unlabcled-only angular rogularization: We rogularize only the unlabeled set toward hy- perspherical uniformity, mitigating false positive clustering and improving separation in high- dimensional spaces.
3. Simplicity and stability: Our method is a single-stage, geometry-driven approach that avoids momentum queues, contrastive pairs, pseudo-negative heuristics, class-prior estimation, while remaining theoretically grounded and scalable.
Experiments on four benchmark datasets (CIFAR-10, 10, STL-10, SVHN, and ADNI) show that our method achieves competitive or superior performance compared to state-of-the-art PU baselines, particularly in
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regimes with few labeled positives, while offering stable and interpretable geometric framework for PU learning.
2 Related Work
PU learning has been studied extensively over the past two decades, with methods broadly falling into three catogorics: risk-based approaches, representation-laariig approaches, and boundary/anon /an/mmammmmaaseee methods. Below, we summaariee the most relevant works and position our method within this landscape.
Unbiased risk estimation (uPU) (uu Plessis et al., 2014) and its non-negative variant nnPU (Kiryo et al, 2017) form the basis of many PU approaches, decomposing the classification risk into terms involving only positives and unlabeled data. While theoretically grounded, these estimators are sensitive to class prior estimation, can overestimate negative risk, and may overfit when representation quality is low. Imbalanced nnPU (Su et al.) 2021)) addresses class imbalance via reweighting, effectively oversampling positives in PN settings. However, it still assumes accurate knowledge of the class prior т1, inherits nnPU's nen-tigativity heuristic, and does not directly improve the geometry of the learned embeddings a crucial factor in high- dimensional settings. In contrast, our method avoids negative-risk cstimation and rewcighting altogether, directly modeling the positive distribution on the hypersphere and regularizing the unlabeled set toward angular uniformity.
To overcome the representation bottleneck of purely risk-based methods, recent works integrate metric learning or contrastive objectives into PU frameworks. WConPU (Yuan et al. 2025) combines weighted contrastive learning with prototype-based classification, alternating between prototype updates, pseudo- labeling, and classifier training, while also performing hard-negative mining from momentum queue. This improves embeddings but introduces multiple interdependent components and requires class-prior estimate. Our approach shares the goal of compact positive clustering but replaces the pseudo-negative pipeline with probabilistic vMF prototype learned end-to-end, requiring neither EM-style alternation nor memory queues.
In the broader weakly supervised learning literature, PiCO Wang et al. 2022) addresses partial-label learn- ing via label disambiguation and contrastive representation learning. Subsequent work has shown that contrastive learning often produces mixtures of vMF distributions in embedding space. However, PiCO assumes candidate label sets for each sample and is not directly applicable the PU setting (Yuan et al., 2025). Our work makes the vMF modeling explicit for positives in PU, eliminating the need for candidate labels, contrastive pair construction, label disambiguation.
Ensemble-diversity self-training (Odonnat et al. 2024) encourages disagreement among hypotheses on unla- beled data to mitigate selection bias. Our approach is complementary: we enforee dispersion in representation space for unlabeled data while aligning positives to a sphorical prototype. The alignment-unifona lens of (Wang& & Isola 2020 directly motivates our design: positive alignment to μ and unlabeled uniformity on gd
Another direction infers negatives by modcling the positive region and treating outliers as negative. Dense- PU (Sevetlidis et al. 2024 trains convolutional autoencoder on positives, interpolates in the latent space, and uses convex hull or dense region boundaries to identify negative candidates for downstream PN training. This two-stage approach depends on SCAR assumptions and convexity heuristics, which may fail for complex, non-convex manifolds. In contrast, our method is end-to-end, avoids explicit negative generation, and uses a principled probabilistic model for the positive class on the hypersphere.
Most existing PU methods either (i) assume specific distribution for the negative class, (ii) iteratively construct nseudo-negatives, (iii) decouple representation learning from classifier optimization. Each of these choices brings well-known drawbacks: risk- based estimators are sensitive to class-prior misspecification and do little to shape the embedding space; psoudo-labeling pipclines amplify carly mistakes through con- firmation bias; and contrastive methods require auxiliary machinery such as momentum queues and apply uniformity constraints indiscriminately, often eroding positive compactness. Our geometry-fite framework sidesteps all three pitfalls by modeling only the positive class as a von Miscs-Fisher distribution on the hypersphere and enforcing uniformity exclusively on the unlabeled set. While our method does use a con- stant soft. label of 0.5 for unlabeled data-reflecting
reflecting maximum uncertainty it avoids dynamic or EM--stle
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Table 1: Comparison of representative PU learning methods. "Prior" indicates reliance on a class-prion estimate; "Neg. model" means an explicit distributional assumption for the negative class; "Pscudo-neg" indicates iterative pseudo-negative labeling: "Contrastive" denotes explicit contrastive pair construction; "vMF+" means explicit Miscs- Fisher model for positives; "U-uniform" indicates unlabcled-only uniformity regularizer; "End-to-end" denotes single-stage training without multi-phase pipelines.
Method
Prior
Neg. model
Pseudo-neg.
Contrastive
vMF
U-uniform
End-to-end
nnPU (Kiryo
2017
Imbalanced unPU
Su el al. 2021
x
x
x
x
WConPU ()uua2002) 25eean222 nted
X
v
/
x
X
PICO Wang of 2022
X
/
X
X
Dense-PU Scvctlids et al.
2024
X
X
/
X
x
X
Ours
pseudo-labeling and relies solely on a probabilistic directional score for classification. The result simple, stable, and theorctically grounded PU learner that aligns its inductive bias directly with the geometry of high-dimensional embeddings.
3 Proposed Method
Overview: shared encoder positive modeling unlabeled-only uniformity
Inference: 1(ep
Pid of wibroogpel c cooncentrttor
(labeled
2f E
Encoder & Iynarm: 54-1
Goeal standwaeea 2j=05
= Epen Curaa + Areg
* (unlaberedd)
d E 5F-1
AGugt 18g (urirrrr Yoous 43) Alneglui Choorpectaral
problem setting
Embedding space (betore regulartzation)
Embedding (after urlabsled-only unilority)
unaateed meee
Uniatsted
chapas comgation
Lightr
A.C1 shorge
Disperedd trdiloted (poctivos Dispareed Unditeeted (pequttvee)
Figure 1: Overview of our hyperspherical PU method. A shared encoder fo maps labeled positives x and unlabeled samples x" onto the unit hypersphere. Positives are pulled toward learnable direction μ via a cosine-based alignment loss (Cpos). Unlabeled samples are trained with a symmetric BCE loss (Cunla)) and dispersed via an angular uniformity regularizer (Creg). Right: regularization incrcases angular separation, yielding compact positives and spread-out unlabeled embeddings.
We address the PU learning problem, where only a small set of labeled positives Dp and large set of unlabeled samples D, are available during training. The unlabeled set contains an unknown mixture of positives and negatives, with no access to explicitly labeled negatives. This setup commonly arises settings where negative categories are heterogeneous, ill-defined, or impractical label exhaustively.
We fixed uncertainty label of 0.5 the unlabeled set, rather than dynamic pseudo-laheling negative mining.
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Our method learns neural encoder fo X + gd-1 that maps imputs to éy-normalized feature vectors the unit hypersphere (Mardia & Juppl 2009 Dhillon & Sra 2003) (see Figure 1. This geometry is well- suited for angular decision rules based on cosine similarity. Rather than modeling a full density, we score each embedded vector & = (z) via its similarity to a learnable direction E sg-1, using a scaled cosine similarity. This directional score draws inspiration from the von Mises Fisher distribution, which defines a density over the hypersphere with normalization constant Ca(i) (see Appendix A for details). While we do not require this constant in training, is used in the log-likelihood derivation and analytical bounds:
s(z) Ku 2,
where k >> 0 is fixed scaling factor controlling the sharpness of the score. This directional score serves as soft classifier, with acting as prototype for the positive class, which is updated by gradient descent and re-projected onto the unit sphere after each optimization step to enforce || = That is, we perform:
P |
/ +
after each update.
We adopt a geometric inductive bias that encourages unlabeled samples to be dispersed over the sphere, inspired by the idea that negatives occupy diverse regions of embedding space while positives tend to clus- ter near μ.. This formulation is consistent with vMF-uniform generative model. In such sctting, the Bayes-oplinal decision rule reduces to thresholding the inner product μ 2. While this motivates our direc- tional scoring formulation, in practice we use empirical threshold selection without requiring class priors normalization constants.
The model is trained by minimizing a loss composed three terms. For labeled positives C Dp. we encourage alignment with jr:
pos
1
Σ K 24
PP
IEP
For unlabeled samples U C Du, we apply a symmetric binary cross-entropy loss with neutral supervision, reflecting maximum uncertainty (i.e., Elyl = 0.5):
Lunlab
1
Σ
00.5 1.5logo(4 (4,) 0.5 log(1 - o(G)))
(2)
[U
JEU'
where rpp 2j and a denotes the sigmoid function. This uncertaintyddriven term avoids overconfident updates ambignous unlabeled points.
To mitigate false positives clustering near the prototype, we regularize the unlabeled set via an angular dispersion term:
Lrog log (
Σe ifj
where t > 0 is a temperature hyperparameter. This regularizer encourages decorrelation among unlabeled embeddings, improving scparability by promoting diversity in feature space.
The final loss combines the three components:
= Epos + Lunlab + Greg;
(4)
where À controls the strength of regularization. The encoder fe and prototype are learned jointly via back- propagation. y is initialized randomly on gd-1 and renormalized after each update. The scaling parameter k is fixed throughout training and treated as a hyperparameter.
To improve robustness uncertain unlabeled samples, we incorporate a soft weighting scheme based on a fixed or learnable angular margin E [-1,1], which acts similarity threshold. The idea is to assign
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greater inffuence to unlabeled embeddings that lie closer to the positive prototype j in angular space. Specifically, for each E U, we define soft weight:
w<2) = (a m-m),
(5)
where a is the sigmoid function and a controls the sharpness of the transition. Samples with j uu > m (i.c., close the prototype) receive higher weights.
We then modify the unlabeled loss to incorporate these weight.
Cunlab
m(z)) (0.5log 0.5 ( ((,)))
(6)
U
JEU
Because the unlabeled loss gradient satisfies
@uulaab = m0() (o()) -0.5)
de
the weighting function w(2) increases the tendency to neutralize overly confident, high-similarityy unlabeled points, thereby counteracting false-pont colsi-positive collapse near μ.
At test time, score s(z) k T = is computed for each sample. The decision threshold is selected via Fl maximization held-out validation set, and predictions made by:
{
if s
-
otherwise
4 Theoretical Justification
Our theoretical analysis serves three purposes: (i) to motivate our directional scoring function based on generative vMF-uniform model, (ii) to justify the learnability of the positive prototype μ under realistic assumptions, and (iii) to interpret cosine regularizer as encouraging dispersion of the unlabeled set. While our implementation does not rely priors density estimation, these results illuminate the inductive biases built into our model.
4.1 Directional decision boundary under the vMF-uniform model
We consider a generative setting where positives follow von Mises -isher (vMF)
distribution the hy- 2016
persphere and negatives are uniform (Mardia & Jupp 2009 Dhillon & Sra 2003 Sra
Proposition 4.1 (MAP rule for vMF (MFuurifor). Let E gd-1, with p(= Y=1) Ca(k) exp(x T2) and p(z Y= = Ua, and prior π Pr(Y=1). Then the Bayes optimal classifier is threshold on the inner product:
4 2
-1 1 K
Ca(k)()) log Un log 1-
Proof sketch. Bayes' rule gives
pY== K Tz log Ca(n) - log Ua + log log p 2)
1-m
The MAP decision sets this 2 0, yielding the stated threshold. Full constants/derrvattien are in Appx.
vMF level are spherical caps centered ie the MAP acceptance region is the cap {μ 2 T).
PIn ablations, test both fixed-margin settings (e.g. 0.3) and learnable-margin variants, where is optimized during training and constrained interval [-,, The adaptive weights guide learning by prioritizing unlabeled instances claser the decision boundary, reducing the impact of noisy uninformative examples.

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Remark 1 (Isotropic negatives). If p(= Y=0) is rotation-invariart s then log p(el)) for a constant c, so the Bayes rule is still threshold on μ z.
p(z[0) KH
Our score is s(2) = a(p"z m), so thresholding is equivalent to thresholding , with s(2) 2 T. and T, a(T m)) In practice we estimate using a PU-valid calibration on unlabeled scores (mixture- proportion based), not supervised F1. See $5 and Appx.
4.2 Learnability of the prototype
We now show that the directional prototype μ can be reliably estimated from labeled positíves alone, under the assumption that k is fixed.
Lemma 4.1 (Consistency of the Positive Prototype). Let 21....,2, ~ vMFF,, k), and let in = - be the sample meon. Then the marimum likelihood estimate of μ. is fu = Fn// F and:
An 2
p as n 00.
This shows that, under fixed i and fixed encoder fo, the prototype μ converges in probability to the true mean direction (Mardia & Jupp) 2009). In practice, we jointly optimize μ and fe, this consistency result holds approximately and serves justification for prototype stability when the positive distribution is concentrated.
Proof. The MLE of the vMF mean direction is known to converge to the population mean direction (Mardia & Jupp 2009). See Appendix B for derivation.
4.3
Interpretation of the cosine uniformity regularizer
The previous results explain why scoring via p separate positives and negatives- y is well estimated and negatives are not concentrated around μ. However, in practice, the unlabeled set may contain false positives that cluster near the prototype, especially early in training.
To roduce this risk, we include a regularization term that promotes angular dispersion of the unlabeled set. While it inspired by hyperspherical uniformity, we do not assunne the unlabeled set actually uniform.
Proposition 4.2 (Dispersion via Cosine Regularization). Lel U = (z1,..., zn} be embeddings of unlabeled samples on Sd-1, Then the regularizer
Σ cT
Creg log
n(n - 1)
info
is minimized when the pairwise cosine similarities 2, are low on average, encouraging angular spread.
Minimizing Lreg reduces high pairwise cosine similarities among unlabeled embeddings i.e., it promotes angular dispersion (soft weihhsoon)-wihhuut enforce true uniformity, it reduces redundancy the latent space and improves separation.
without assuming exact hyperspherical uniformity. While does not
Proof. Since z7 zT E [-,, and et 2y increases with similarity, the sum is minimized when the 2; are dispersed (i.e., pairwise similarities are small). See Appendix C for further analysis.
4.4 Regularization scaling and bound
We provide loose upper bound on the regularizer to ensure it does not dominate the overall objective. Corollary 4.1 (Upper Bound Regularization Term). Let 2 E Sd-1 for 1,..., Then:
Lreg < log (ee)

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This holds in the extreme case where all cosine similarities maximal, i.., 2j 2j 1. In practice, Lieg remains well-behaved and is scaled by tunable factor X.
Proof. The maximum value of each term e/a 2s is e.. There are n(n 1) such terms, so the average is most e'. Taking log gives t.
5 Experimental Evaluation
We evaluate the proposed vMF PU learning framework on range of benchmark datasets: CIFAR-10, STL- 10, SVHN, and ADNI. These datascts span natural images, digit recognition, and medical imaging, allowing to assess performance across varying input complexity and domain characteristics. The PU setting is simulated by providing a small labeled positive set and treating the remainder of the data as unlabeled, containing a mixture of positives and negatives.
5.1 Experimental Protocol
To ensure comparability with prior work, we replicate the experimental setup of Yuan et al. (2025), includ- ing dataset partitions, positive/ positive/unlabeled unlabeled (PU) label ratios, and evaluation metrics. This enables direct comparisons with established PU learning baselines.
Each input is encoded onto the unit hypersphere gá-1 via trainable encoder, as described in Section 3 Training optimizes von Hisss-Fisser-basee loss for labeled positíves, binary cross-entropy for unlabeled data, and when applicable the cosine uniformity regularizer. We report the following evaluation metrics on the test set: Fl score, precision, recall, accuracy, area under the ROC curve (AUC), and average precision (AP), consistent with prior literature.
Unless otherwise stated, all experiments use a validation split (10% of the training data) to tune the decision threshold for binary classification by maximizing F1 score.
5.2 Datasets
CIFAR-10 consists of 60,000 color images (32x32) across object classes (Krizhessky al. 2009). Fol- lowing prior work, we group airplane, automobile, ship, and truck as the positive (vehicle) class, and the remaining classes as negative (animals). We sample 1,000 positive examples serve as labeled data; the rest of the training set is used unlabeled data containing mixture of positives and negatives. We adopt the standard 10,,00000,000 train/test split.
STL-10 contains 13,000 labeled images (96x96) across 10 classes (Coates et al. 2011). We group airplane, car, and truck as positives, and all other classes as negatives. total of 500 positive examples are randomly selected as labeled data, with the rest used as unlabeled. Due to its higher resolution and more complex scenes, STL-10 poses greater challenge compared to CIFAR-10.
600,000 real-world digit images from street views Netzer et al. 2011 Even digits (0, 2, 4, 6, 8) are treated as positives and odd digits as negatives. We randomly sample 1,000 even digits as labeled positives, while treating the remainder of the training set as unlabeled. The test set includes 26,032 labeled images.
SVHN comprises
ADNI (Alzhcimer's Spectrum). The Alzheimer's Discase Neuroinaging Initiative (ADNI) provides structural MRI scans of healthy controls (NC) and subjects across the Alzheimer's 's disease spectrum (AD, MCI, etc.) Jack Jr et al.) 2008 Petersen et al. 2010). We treat all NC spectrum cases as positives and the rest as negatives. From the positive class, 768 scans are randomly selected as labeled data. The remainder of the training (including both NC and unlabeled AD spectrum samples) forms the unlabeled pool. Evaluation is conducted held-out test set with ground-truth diagnostic labels.
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5.3 Training Details
For image datasets (CIFAR-10. STL-10, SVHN), we adopt VGGII-BN encoder (Simonyan & Zisserman 2014) pretrained on ImageNet (Deng et al. 2009). For the ADNI dataset (Jack Jr et al., 2008 Petersen et al, 2010), we use a two-layer multilayer perceptron (MLP) with ReLU activations. All models are trained for 15 epochs using the Adam optimizer with a learning rate 10 and a batch size of 128. The encoder maps inputs to a d-dimensional hypersphere, with 128 unless stated otherwise. We apply dropout with a rate of 0.2 to the final embedding layer. The cosine uniformity regularizer is scaled by temperature t = 2.0. The concentration parameter of the von Miscs-Fisher distribution is tuned per dataset hyperparameter. Unless ot herwise specified, all components of the encoder are updated end-to-end, including learnable margins where applicable.
5.4 Baselines
We compare against a representative set of PU learning approaches, including EM-based methods. importance-weiggheel PU learning, and surrogate loss methods, using results directly from Yuan et al. (2025) where applicable. This ensures identical data splits and evaluation criteria across all methods:
uPU Du Plessis el al. 2014;: Risk-based PU method using an unbiased risk estimator under the SCAR assumption, with theoretical analysis showing that convex losses can bias the boundary, while non-convex losses (e.g., ramp) avoid this bias.
nnPU Kiryo et al. 2017 Extends uPU by enforcing non-negativity of the empirical risk to prevent overfitting in flexible models, enabling stable training of deep networks with PU data.
Rank Pruning (Notthcutt. et al. 2017): Estimates label noise rates and prunes mislabeled examples using high-confidence samples ranked by predicted probability, improving robustness in both PU and noisy-label settings.
PUSB (ato et al. "invariance of order" property and densily ratio estimation, with thresholding for final classification.
2019 PU lcarning under selection bias, relaxing the SCAR assumption via the
puNCE AAcharya et al. 2022): Adversarial PU framework where a discriminator separates labeled positives from unlabeled data, and the generator learns embeddings to fool the discriminator, aided by label distribution estimation.
Pe (Chen et al. 2020o): Self-paced PU framework that progressively labels confident positives and negatives, reweights uncertain samples via meta-learning, and applies self-distillation to enforce consistency.
VPU (Chen al. 2020a): Variational PU method that measures divergence between the positive distribution and model predictions without estimating the class prior, regularized via Mixup-based consistency.
Dist-PU (Zhao al. 2022;:
Aligns predicted and ground-truth label distributions, with entropy minimization and Mixup to reduce confirmation bias and avoid degenerate solutions.
PiCO WWang et al. 2 Contrastive label disambiguation method combining represcntation
learning with prototype-base pseudo-label refinement. in an alternating optimization scheme.
Densc-PU (Sevetlidis et al. 2024): : Density-based negative mining in latent space to reduce falsc
positives, by ileratively identifying dense negative clusters from unlabeled data.
ImbPU (Su et al. 2021): :: Prototype-based PU approach estimating positive and negative centroids and refining pseudo- labels via local neighborhood consistency.
aPU Hammmuudh Lowd. 2020): Proposes a two-step method using surrogate negatives and a recursive risk estimator for learning from positive and unlabeled data even when the positive data is arbitrarily biased, by assuming the negative class distribution remains fixed.
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PUbN (Hsieh al. 2019: PU learning with biased negative data, introducing correction term to adjust the risk estimator when negative samples come from a biased distribution.
WconPU (Yuan et al. 2025): Distribution-wiighee PU learning balancing the positive and unla-
beled risks via dynamically learned weights, improving robustness under varying class priors.
5.5
Results
Table 2:
Performance of the proposed method vs. baseline methods. Results are reported for F1 Score, Accuracy, Precision, Recall, AUC, and AP (the best score is marked in bold).
Method
Accuracy
Precision
Recall
F1
AUC
AP
uPU
88.41+0.41
87.21+2.09
1390
85.12-0.43
94.98+0062
1.0
unPU
88.9110.43
86.21 86.2111.02
86.03:1.22
86.11=0.49
95.1310.55
92.51+1.32
RP
88.74+00.16
12.00+11.0
85.72++161
85.93-0.30
95.21+00.3
93.01+0061
PUSB
88.97:00.99
86.1510.56
06.22:10.45
86.18=0.51
95.15+0.50
92.44+1.34
PUEN
89.83++0.30
87.85++0.99 0.9
86.56+1..8 1.87
87.18-0.54
94.444+0.35
91.28 1.11
Self-PU
89.31+00.56
86 76
87.22+2.116
86.77=1.12
95.52++0.46
93.31++1002
aPU
89.09:10.44
86.31+1.33
86.33:10771
86.41=0.41
95.1111.39
92.42:1.222
CIPAR-10
VPU
87.89+00.56
86.71+1146
82.88-+2.93
84.42-1.12
94.55++0.46
92.02:+0.09
ImbPU:
89.43:1.42
86.724 86.7221.89
86.91=0.78
86.77=0.56
96.5310.26
93.33:00668
Dist-PU
91.88++0.22
89.87++1.09
89.84+0.81
89.85-0.62
96.92+-0.45
95.49++0.72
Dense-PU
90.59:10.98
1.1.111131
91.25:11.12
91.96=0.80
93.22:10.99
95.03:11.21
PuNCE
95.32++0.24
95.11+00.88
93.43++0.45
94.21-0.44
98.59+00.3
98.45++0.63
PiCO
95.64100.2 12
94.8940.76
93.97:20447
94.75=0.49
98.67:0.44
98.22:10.91
WConPU
97.22
9..87=00544
96.02:t:.32
96.43-10.20
99.49-t0.22
99.25=t:033
AngularPU (Ours)
91.39++0088
92.26+00.44
93.49++0.18
92.87
96.61
97.38++0.53
uPU
83.35:20.45
87.1112..38
75.93:12.68
81.12=0.56
91.9310.62 6266
90.22::1.11
unPU
83.88++0.45
86. 1..1
77.25+
82.01
92.02+00.5
90.28++
81.73:10.15
84.01+1.01
76.12:1155
80.10=0.32
89.75:10.29
87.99110.56
PUSB
83.99-+0.41
66.81+0..5
78.01+0.51
82.11-0.51
91.89++0.52
90.31++1334
PUbN
84.89:00.00
88.2610098
83.07=1.87
83.16-0.54
92.0320.35
9118111.111
Self-PU
84.12+0072
86.16-00.78
79.22-+2.35
82.55-1.06
91.73+0.55
90.99+ 1.01
aPU
84.01=0.52
86.29411.00
80.71
82.33=0.56
91.56
90.66:1 23
SVHN
VPU
76.89+00.48
79.56+1.41
79.56+1.41
75.36-2.84
73.31+0.91
83.35++0.73
ImbPU
84.20:10.46
86.69 06.6940.87
81.18:0.822
82.99-0.56
91.7910..2
91.21
Dist-PU
85.96+00.3
89.06+00.80
84.36+0.76
83.66-0.56
92.92+0.49
92.29+00.8
Dense-PU
86.10:0..87
89.3240.78
82.72210.99
85.37=0.91
93.25:0.64
92.45:00.90
PuNCE
95.34-40.24
90.35++0.92
83.81+1.99
87.01-0.55
94.87+00.35
93.87+00.92
PICO
95.6420.12
90. 90.47:10.79
85.7420.61
87.51=0.44
95.5840.54
94.32:10.66
WConPU
91.49::0.29
03.77=0.67
87.5442..67
90,45=+0.35
96.97=0.59
96.82=+0.37
AngularPU (Ours)
89.9410.15 13
88.3340022
90.27:10.12
89.27=0.13
95.85 10.96
94.78:00.1
uPU
93.13:10.4
90.42+1..0
92.62-11.28
91.51=0.62
97.95.0.56
97.26111.21
nnPU
93.38-+0.42
91. 2.00--001
92.34++1.08
91.77-0.58
97.69+-0.8
97.69+-0.51
RP
92.8810.36
92.87+1.35
89.1821..88
90.97=0.45
92.15
95.5812.229
PUSB
93.65++0.16
92.06++0.52
92.06+0.42
0.33
98.06+0052
97.21+1.13
PUBN
94.01=0.31
93. 03.01110.98
1..11
92.98=0.54
08.20-0.35
97.66:11.47
SeI-PU
93.73+0.28
92. 1.00
92.61++1.88
92.22
91.98+
91.98++0.22
aPU
93.41+0.46
91.15:1.24
02.55=20.83
91.52-0.88
97.85+10.00 66
96.23:11.03
STL-10
ImbPL
93.88+0.81
12.25:1.12
01.66:2.88
92.01=0.54
97.98=0.72
97.33+1.00
DUst-PU
94.73+0.31
93.35+1.01
93.47+0.81
93.41-0.4.
98.54+0.71
97.96+11.01
Dense-PU
94.44+0.67
94.23110778
04.22-0.99
93.81=0.62
98.02+4..7
98.15+11.21
PuNCE
95.13-0.22
94.09+00.55
94.95-+0.62
94.51-0.51
98.66+0.24
98.23+0069
PiCO
95.35-2..23
94.3640.00 42
02-1011.81
94.75=0.41
98.7840.15
98.55-0.34
WConPU
97.02+00.2
95.53++0.41
97.42+00.91
96.35 2.235-0..6
99.58 0...--1.12
99.46-+0.21
AngularPU (Ours)
99.39-40.11 11
99.30=10.22
99.17-10.16
99.24+10.14
99.95
1355--0.000
99.9440.01
Continued on next page
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Table Continued from previous page
Method
Accuracy
Precision
Recall
F1
AUC
AP
uPU
68.42:12.22
69.71:33.4
57.33:15.18
68.63 1.73
73.9912.72
70.12:12.98
nnPU
68.2142215
68.0941..2
5.88
68.11
71.9913.01
70.01:2.21
62.03+2.88
63.11+3.77
06.2349.80
61.99=6.03
66.32+2.99
64.10#2.11
PUSB
69.19:12.41
70.1111.88
69.43:12.13
69.41-2.15
74.66:12.42
70.12:11.64
PUbN
1..00=1102
69. 29.4322.22
7..22=6601
71.18=2.89
74.98=0.89
69. 69.66++1.63
Self-PU
70.79:10.73
69.55:12.51
7..51:1.99
72.10=1.02
75.85111.68
71.79:13.63
aPU
68.41#1.41
06.23++088 88
75.71=6.21
71.01=3.06
73.66#2..44 .41
70. 33.23+3333
ADNI
VPU
66.5110061
64.8911.01
75.18:3.71
71.01=0.96
72.99110.91
71.21:0065
ImbPU
68.18+00.0 69
67.3 67.3442331
71.24=6.21
68.79=1.81
73.69=+0.75
70.56:+0.99
Dist-PU
71.75:10.22
68.48:11116
80.09:1.100
80.09=5.10
77.1310069
13.33:11.47
Denso-PU
72.10+0.80
71.0++1.00
78.42400.7
76.53=1.30
75.80+0.66 69
78.81+1221
PuNCE
70.59:10.77
18.9911.56
75.99:1.11
71.55=1.11
75.55+1.03
71.23:1:.66
PiCO
71.94+0.71
69.59+1.12
79.01=5.03
73.92=1.02
07.59
72.17=0.97
WConPU
73.02:10.66
70.87:12.42
79.12:14.99
74.23-0.76
78.55:1107
72.66:11007
AngularPU (Ours)
79.13+0.117
77.73=0.34
82.11+0.35
79.74+0.18
85.94+0.14
06.3
Table [ summarizes the performance of the proposed AngularPU method compared to a diverse set of PU learning baselines across four benchmark datasets: CIFAR-10, SVHN, STL-10, and ADNI. We follow the evaluation protocol of Yuan et al. (2025). including class groupings, PU ratios, data splits, and metric computation. AngularPU competitive overall and excels on F1/AP (recall-oriented metrics), partieu- larly in scarce-positive regimes, offering favorable trade-off between precision and recall, while remaining conceptually simpler and more interpretable than prior methods.
On CIFAR-10, AugularPU delivers strong results, with an F1 score of 92.87 and an AP of 97.38, outper- forming most risk-based and pseudo-labeling approaches. While contrastive methods such as WConPU and PiCO report slightly higher accuracy on CIFAR-10 and SVHN, these gains often come at the cost of re- duced recall. In PU learning where unlabeled data contains many hidden positives recall is critical: false negatives cannot be corrected without explicit negative labels, and missed positives significantly impair downstream tasks (e.g., medical screening anomaly detection). Our method consistently favors higher recall without excessive false positives, reflecting the intended bias of our geometry-firsi design. This trade- off is deliberate, ensuring that the model errs on the side of discovering positives rather than prematurely discarding them. We attribute this behavior to the conservative decision boundarics inducod by the cosine uniformity regularizer, which slightly flattens the score distribution near the classification threshold.
In the more challenging SVHN dataset, which features substantial intra-class variability and label noise, An- gUlarPU maintains robust performance, achieving an F1 score of 89.27 and AP of 94.78. These scores exceed those of most pseudo-labeling baselines, including Dense-PU and Dist-PU. Although its precision is slightly lower than contrastive methods like WConPU and PiCO, AngularPU maintains consistently higher recall, suggesting that the angular dispersion of the unlabeled set helps reduce the risk of false negatives. Moreover, unlike PiCO and puNCE- which often suffer from unstable optimization due to alternating pscudo-labeling cycles-on our method benefits from a geometry-driven, end-to-end formulation that avoids such instability entirely.
On STL-10, AngularPU achieves state-of-the-art performance, with F1 score of 99.24 and AP of 99.94. This dataset presents additional challenges due to its higher resolution and more complex backgrounds, which often undermine the assumptions behind contrastive or density-based approaches. The synergy between hyperspherical vMF modeling and the cosine uniformity regularizer leads to separation between positives and negatives. In particular, the regularizer prevents the collapse unlabeled embeddings near the positive prototype. The performance margin substantial: AngularPU outperforms WConPU by over 2% F1 score and achieves near-perfect classification under both precision and recall, underscoring the method's effectiveness in high-dimensional, low-label regimes.
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Finally, on the ADNI dataset-a a real-world medical imaging task with extremely limited labeled data and subtle inter-class differences-Annularr") UngularPU achieves the best results across all metrics, including F1 (79.74), AUC (85.94), and AP (86.35) This is a particularly difficult PU setting due to class imbalance and overlap- ping feature distributions. Its improved recall (82.11) over WConPU (79.12) and PiCO (79.01) highlights its sensitivity, achieved without a corresponding increase false posilives. We hypothesize that the cosine uniformity regularizer plays a key role here, by discouraging spurious clustering of ambiguous unlabeled instances near the positive protolype. This helps maintain a clean decision boundary, even in noisy or low-resolution embedding spaces.
AngularPU's strongest gains emerge in datasets with complex visual features and limited positive la- bels scenarios where traditional pseudo-labeling and contrastive methods often struggle. While there is some degradation in precision on noisier datasets like SVHN, this is consistently offset by higher recall and F1 scores. The empirical results validate the utility of modeling only the positive class using directional statistics, while regularizing the unlabeled distribution via angular uniformity-wii withot relying on negative sampling, prior estimation, or heuristic alternation.
5.6 Ablation and Sensitivity Analysis
We conduct thorough ablation and sensitivity study to assess the contribution of each component in our proposed AngularPU framcwork and its robustness under varying hyperparameter regimes. Table 3 presents the results across all datasels by removing, one at a time, the cosine uniformily regularizer (Greg), the adaptive weighting mechanism based on margin proximity, and the learnable margin parameter. Across datasots, we observe that the absence of freg leads to consistent reductions in AUC and average precision, particularly in SVHN and ADNI, where the unlabeled distribution exhibits greater structural ambiguity, indicating the regularizer's 'ss essential role in encouraging dispersion and reducing spurious positive clustering. The removal of adaptive weights causes a more pronounced drop in Fl scores, reflecting the value of prioritiz- ing ambiguous or marginal instances during optimization and suggesting that uniform treatment of unlabeled samples hinders the model's ability to focus on informative gradients. Interestingly, replacing the learnable margin with a fixed one (e.g., m = 0.5) has relatively minor impact, especially on STI.-10 and CIFAR-10, showing that while margin learning adds flexibility, its contribution is not as critical as the other components. Importantly, the full model consistently outperforms its ablated counterparts on every metric and dataset, demonstrating that each component plays a distinct role in the final decision surface quality. We further examine hyperparameter sensitivity by varying the regularization coefficient À E {0.0, 0.1, 0.3, 0.5, 1.0), the vMF concentration x E 10.1,1,3, 1,3, 5, 10, 20), and the fixed angular margin m E (0.1, 0.5, 1.03. The regularizer weight À shows a stable peak around 0.5, with gains saturating or degrading at higher values duc to over-penalization of structure. The best x values are in the 3-3 range, balancing boundary sharp- ness and calibration, while very low x causes nearly degenerate high-recall classifiers and very high values lead to overconfident rejection. For fixed margins, m & 0.5 offers the most consistent results, with lower values yielding recall-dominant and precision-degrading behavior. Overall, both the ablation and sensitiv- ity results reinforce that AngularPU's effectiveness stems from the careful combination of its geometrically grounded loss, selective weighting, and regularization components, all of which interact to produce a stable and competitive PU learner across diverse domains.
We study the impact of the core hyperparameters in AngularPU: the cosine uniformity regularization weight A, the angular margin m used for rejecting uncertain samples, and the vMF concentration parameter K. We conduct extensive scnsitivity sweeps across all four datasots, varying one hyperparameter at a time while keeping the others fixed to their default values (A = 0.5, m = 0.5, K 3.0), and report mean and standard deviation over five random seeds.
As shown in Figure 2] performance consistently improves with increasing X up to 0.5, beyond which the benefit saturales or slightly degrades. This pattern holds across all datasets and both F1 and AUC metrics, suggesting that moderate regularization effectively discourages false positives by spreading unlabeled embed- dings uniformly, while excessive regularization leads over-dispersion and degraded confidence. Notably, when = 0, recall is preserved but AUC and precision decrease, reflecting reduced discriminative quality without regularization.
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Table 3: Ablation study across datasets, Each row removes one component from the full model.
Variant
Dataset
F1 f
Precision 1
Recall t
AUC 1
AP f
Full model
CIFAR-10
0.930
0.925
0.940
0.970
0.977
SVHN
0.890
0.886
0.901
0.963
0.961
STT-10
0.992
0.990
0.994
0.998
0.999
ADNI
0.797
0.782
0.821
0.866
0.871
w/o Creg
CIFAR-10
0.913
0.915
0.907
0.961
0.965
SVHN
0.875
0.880
0.862
0.955
0.946
STL-10
0.984
0.981
0.988
0.995
0.997
ADNI
0.773
0.762
0.794
0.841
0.846
w/o weights
CIFAR-10
0.901
0.910
0.892
0.956
0.958
SVHN
0.862
0.869
0.847
0.944
0.935
STL-10
0.972
0.968
0.976
0.992
0.995
ADNI
0.760
0.750
0.779
0.822
0.830
Fixed margin
CIFAR-10
0.927
0.923
0.935
0.969
0.975
SVHN
0.888
0.882
0.900
0.961
0.959
STL-10
0.991
0.989
0.993
0.998
0.999
ADNI
0.794
0.779
0.818
0.864
0.870
Sensitivity of to lambaa
Sensitivity of AUC to lambdia
1.00
0.95
Dataset
dffr20
2 0.85
adni
cttr10
0.80
adhl
0.0
0,2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
lambda
larbbd
Figure 2: Sensitivity to A (cosine regularization weight). Mean + std F1 and AUC across seeds.
For the fixed margin m, the results in Figure 3 reveal that small values like 0.1 result in permissive classifiers that admit many unlabeled samples positive, achieving high recall but low precision. Conversely, as m increases, the model becomes more conservative; performance peaks around m 0.5, which strikes a favorable balance between recall and precision. Beyond that point, particularly at m = 1.0, recall drops and F1 becomes less stable, confirming that overly strict angular thresholds reduce sensitivity to true positives.
Regarding the concentration parameter K, Figure 4 illustrates how shapes the sharpness of the vMF like- lihood. At very low values (e.g., 0.1), the likelihood surface is flat, yielding nearly uniform probabilities that diminish precision and increase false positives. As к increases to 3 or 5, the model becomes more confident, and both F1 and AUC improve steadily. However, for extremely high values such as = 10, we observe diminishing returns or even slight instability, especially on noisier datasets like ADNI, suggesting that overconfidence may hurt generalization in low-data regimes.
Overall, we find the mothod to be robust across a wide range of hyperparameter values. The default scttings À 0.5, m 0.5, and consistently yield near-optimal performance and are used throughout the main experiments.
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Sensitivity of to margin
Sensitivity of AUC margin
1.00
Dataset
1,00
Deteset
dfar10
0.95
whhn
sulte
s010
0.95
x1110
0.90
0.90
17
0.80
0.80
0.75
0.6 mergin
margin
Figure 3: Sensitivity to margin m (fixed angular threshold). Mean + std and AUC across 5 seeds.
Dataset
AP Cosine
AP L2
AAP (cos-L2)
AUC Cosine
AUC L2
F1 Cooine
L2
AAP 95% CI
adni
0.832
0.584
+0.248
0.009
0.575
0.030
[-0.221, +0.275]
cifar10
0.974 + 0.007
0.814 + 0.012
+0.161
0.968
0.002
0.781
0.035
0.931
0.001
0.825
0.061
-0.147, +0.174
stl10
0.993 + 0.001
0.900 + 0.057
+0.093
0.993
0.001
0.957
0.020
0.967
0.001
0.908
0.020
0.053, +0.133
svhn
0.932 + 0.014
0.493
+ 0.076
+-0.439
0.950
0.001
0.549
0.065
0.876
0.006
0.640
0.003
-0.37.
40.503
Overall
0.933 = 0.067
0.698 = 0.181
+0.235
0.910
+ 0.058
0.716 = 0.180
0.891
0.072
0.760
0.121
[-0.147, +0.337
Table 4: Primary configuration (weighted learnable margin): cosine vs. Euclidean by dataset and overall. We report mean + std over seeds; AAP is cosine-T.2 with a 95% bootstrap CT.
Moreover, we study how the choice of embedding geometry and margin mechanism affects positive-unlabele (PU) classification when using a single positive prototype. We compare (i) angular model that scores by cosine similarity between L2-normalized embeddings and a normalized prototype (yMF-style), and (ii) a Euclidean model that scores by the negative squared distance to an (unnormalized) prototype (Gaussian- style). The hypothesis is that the ceale-invariance of cosine provides a more robust inductive bias than scale-sensitive Euclidoan distances in the presence of heterogeneous unlabeled data. For each geometry we lest: (a) BCE; (b) BCE + margin-aware weighting (higher weights near the decision boundary); and for cosine only (c) BCE + uniformity regularization on unlabeled embeddings. Margins are either learnable (softplus of a scalar) or fixed (0.5 or 1.0). Evaluation reports AP/AUC and Fl. All other factors (splits, optimizer, schedule) are held fixed across ablations.
Our primary configuration is weighted loss + learnable margin in each geometry. Figure 6b shows that cosine outperforms Euclidean in AP overall, and Figure 5 shows the same ordering for every dataset. Table 4 summarizes means + std and paired AAP (cosine- L2) with 95% bootstrap Cls. Figure 6a ranks cosine variants at the top (weighted with learnable or fixed margin and cosine--uiformitty) Margin-aware weighting helps both geometries, but yields larger gain for cosine, consistent with bounded angular distances being less sensitive to embedding-norm variability. Euclidean narrows the gap when embeddings are normalized evaluation, yet remains behind cosine AP/AUC. For prototype-basee based PU learning, angular geometry with learnable margin is the most reliable choice. We therefore the use of cosine + weighted + lcarnable- margin as the default configuration is justified in the experiments.
6 Conclusion
We introduced AngularPU, a geometrically motivated framework for positive-umlablldd lcarning that casts the classification task as angular separation on the hypersphere. By leveraging a von Mises- Fisher likelihood with an optional cosine uniformity regularizer, AngularPU provides an elegant and efficient alternative to pscudo-labeling and contrastive methods that dominate the field. Our formulation end-to-end, does not require prior estimation, and avoids the instability of multi-stage pipelines. Extensive experiments across - that AngularPU achieves four diverse datasets caaaasis-ranging from natural images to neuroimaging demonstrate
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Ablation
AP (n 1 σ)
AUC (μ + Ø)
P1 (1 to)
COS- -O5
0.939 + 0.056
0.941 + 0.056
0.894 + 0.067
COS- Meeaern
0.933 + 0.067
0.940 + 0.058
0.891 1 0.072
COS-REG-MMeam
0.926 + 0.058
0.917 + 0.063
0.869 + 0.072
COS-BCE-Mlean
0.921 + 0.094
0.935 = 0.077
0.896 + 0.081
L2-BCE-Rlearn-N
0.734 0.179
0.753 = 0.159
0.743 = 0.105
12-WGH-R110
0.709 0.197
0.727
0.193
0.765 + 0.122
L2-WGH-Rleamt
0.698 0.181
0.716 0.180
0.760 + 0.121
L2-BCE-Rlearn
0.505 0.132
0.479 0.091
0.656 + 0.069
Table 5: Ablation leaderboard (all datasets & scods). Mean + std for AP/AUC/F1; higher is better.
Sensitivity of F1 kappa
Sensitivity AUC kappa
1.00
0.8
W 0.60
0.75
Detaset
0.70
dfar10
wchn
schn
0.65
0.5
0.0
2.5
5.0
75
10.0
12.5
15.0
17.5
20.0
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
kappa
kappa
Figure 4: Sensitivity to K (vMF concentration). Mean + std F1 and AUC across 5 seeds.
state-of-the-art stato-of-tho-art performance in F1 and recall, two metrics particularly critical for PU scenarios. An important characteristic of AngularPU is ils tendency toward conservative rejection of negatives, which yields consistently high recall across datasets. In many PU scenarios - especially those involving rarc or costly positives - this behavior is preferable to maximizing overall accuracy. While this may slightly lower precision certain benchmarks, aligns with the real-world utility PU classifiers, where retaining positives outweighs the risk of admitting some false positives. Our ablation and sensitivity studies further highlight the individual contribution of each component and confirm the model's robustness hyperparameter variation. While AngularPU achieves strong results without complex heuristics, it assumes that pasitive embeddings form single dominant. directional mode. Our experiments suggest the hod remains effective when the positive class comprises nearby subclusters. Although we did not include an additional large-scale benchmark, the computational profile of AngularPU favorable for scaling. Training complexity dominated by a single encoder forward forward/backwer bbackward pass, with an O(U1²) term only in the regularizer, computed over unlabeled batches. This pairwise term efficiently implemented matrix operations. While the per-batch cost grows quadratically with batch size, total runtime scales linearly with dataset size under fixed batch configurations. In experiments, the method remained numerically order of magnitude. These
batched
stable and tractable when increasing the size of the unlabeled set by up obscrvations, combined with the absence of itcrative pscudo-labeling momentum qucues, suggest that AngularPU is well-suited to large-scale PU scenarios without architectural modifications. Future work will explore explicit vMF mixtures to better handle more complex or strongly multi modal positive distributions. Another direction adapting the angular formulation to semi-supervised or open-set scttings where negative sampling becomes partially available. Overall, our work underscores the value of directional geometry in PU learning and opens the door to broader class of probabilistic embedding-based methods.
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Primary (weighted leamable): clar10
Primary (weighted learmalee:: AP
Avtraga TrNeo Srear
0.4
0.0
CONTS
00
(a) CIFAR-10
(b) STL-10
Primary (Weighted -eaaraaaa suhn
Primary tweighted learmable]:
6.0
0.2
(c) SVHN
(d) ADNI
Figure 5: Primary configuration (weighted + learnable margin): per-dataset AP comparison between cosine and Euclidean. Cosinc is superior on all datasets.
Average
Primary wwwighttddtcccmmaieelleeee AP Space
2.
deveap
La Wost
Ge
(a) Average Precision by ablation (mean across dalasets and seeds). Blue bars indicate the top ranks are consis- tently achieved by cosine variants.
(b) Primary configuration (weighted + learnable margin): AP by metric space (overall). Cosine outperforms Eu- clidean.
Figure 6: Ablations and overall primary configuration results.
References
Anish Acharya, Sujay Sanghavi, Li Jing, Bhargav Bhushanam, Dhruv Choudhary, Michael Rabbat, and Inderjit Dhillon. Positive unlabeled contrastive learning. arXiv preprint arXiv:2206. 01206, 2022.
Eric Arazo, Diego Ortego, Paul Albert, Noel E O'Connor, and Kevin McGuinness. Pseudo-labeling and confirmation bias in deep semi-supervised learning. In 2020 International joint conference on neural
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A
PU Learning with von Mises-Fisher Distributions on the Hypersphere
We consider binary classification in a PU setting on the wiit hypersphere sd-1;
y = 1: (M,(H,K) with mean direction μ gd-1, concentration 0 and density p(=[1) Ca(n)ena
y= Uniform (Sd-1) with density p(z[0) Ua
Let Ti and To 1-11 Ti be the class priors.
Theorem A.1 (Bayes-optimal decision rule). The Bayes-optinnd classifier assigns y = if
x == log
T
log Ua
Ca(k)
(7)
no
Proof. Bayes' rule gives:
P(1|2) p(=[1)=T
P(O p(z[0)=o
Substituting the densities:
P(]]2) log P(O kp + log mi To
log Ua
The classifier predicts y when this quantity > 0, which yields the stated threshold T.
Lemma A.1 (Isotropic negatives). p(z Y= = g(|2=]]
ihation-invariane on Sd-1, then log p(al))
p(z(0)
nμ²²= for constant independent of z, so the Bayes rule remains threshold on μ (the constant T changes accordingly).
Remark 2. Since |p T 2 S 1, the score xp lies in [-K, so T must be in this range for nontrivial classification.
Consistency of the Positive Prototype
Let Pe be the positive prototype estimate from labeled positives at itcration Lemma B.1. If 2 14.. ()) the MLE füe satisfies
fu 2 μ
00.
Proof. Maximizing the vMF log-liicld w.r.t. a aligns μ with the normalized sample mean
21
2
24
For vMF, E[a,] As(k)p with mean resultant length (x()). By the Law of Large Numbers, 2, B, Ad(x)p and normalization yields f4 + p.
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C
Cosine Uniformity Regularizer
Given unlabeled embeddings E sd-1, define
Leg log |U][|U] - ig.
Eetzle
(8)
with temperature t 0.
Proposition C.1. Minimizing Lreg encourages the unlabeled embeddings approzimate a uniform distri- bution 8d-1
Proof. The cosine similarity 2 2 is maximal when 2, = 2j, making et 2 large. The log mean exp form emphasizes high-similarity pairs. Minimizing Lreg therefore reduces high-similarity occurrences, spreading embeddings more evenly and reducing false positive clustering near μ.
D Concentration and Bounds on the Regularizer
Throughout this section write the log mean exp temperature as 3 (in the main text 8 = t).
Lemma D.1 (Cosine concentration the sphere: self-contained proof). Let 21,-..; ZM iid. Unil(gd-1) and define Xij 2, 23 for i i j. Then for any fixed pair (i,j):
B[Xi]] and Var(Xij)
2. Gaab-Gasssan mgf-) For all E R
E eXx
S exp
2d
Equivalently, Xij is sub- Gaussian with parameter 1/vd.
3. (Tails.) For any t E 1),
Pr([xu12 < 2exp
doa 2
Proof. Step 1: Reduce to single spherical coordinate. Fix j. Conditional 2, u Sd-1 rotational invariance gives Xij 2 d (z, u) with = ~ Unif(gd-1). Hence it suffices to study T (2,e1)
21, the first coordinate of a uniform point on the sphere; all claims then hold for Xú by conditioning.
Step 2: Exact density and even moments of T. It classical that T has density
I( (
fa(1) = ca (1 - 12) 4-3
(1)
ca =
For k E N, using symmetry and the Beta function,
2 2ca/ / (0 2) dt cd ( ut (1 u) - du cd
B(k + 3: 2
d-1
where we substituted = 12, Using B(x, v) = [(m)T(y) T[xtv)
and the identity r(k + 11)
(2k)
4FK VT, obtain
r
r(k: + 1)) ((ا)
r(g)
(2k)
(2k- 1)!!
2
V |
1=1 ?
I + (
r(k 4) 4 II k- (d 2s)
In particular, E[T] 0 and for k E 1/d, proving (i).
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Step 3: A dimension-explicit mgf bound. Expanding the mgf and using that odd moments vanish,
2
0
A2k
(2k: 1)!!
B
E[T
A2
(2k)
(2k)
I1. (d+ 28)
k!
2 I (d 2s)
k=
Since TT (d + 2s)= 2 d for all k 2 1, we get the elementary bound
(22
cxp (
12 2d
E[e S
k! 2d
A=
which proves (ii) for T, hence for Xiy by Step
Step 4: Tails by Chernoff. For t > and any À > 0, Pr(T t) S e E E[e S exp( - Xt + )
Optimizing at = dt yields Pr(T 2 1) S exp( 2 Apply the same bound to - and union bound to get (iii).
Remark 3 (Sharper constant). Using slandard concentration on the sphere (Lévy's 's lemma), one can lighten We keep the elementary d- d-based proof here for self-containment
to -d-1 1 in Corollary D.1 (Maximum pairwise cosine). Let = (2)). With probability at least 1 - 8,
2 log (2K/8)
max Xxy < x
Proof. Apply union bound to Lemma D.1 iii)
the K pairs.
Lemma D.2 (r-sensitivity of the LME regularizer under a vMF-Uniform mixture). Let Z be draun from the mirlure Pa vMF(U, (1 x) Unir((d-1), and let Z1 be independent copy- For B3 0, defone the pairwise log mean exp functional
653) log E [[exp(B2' 2 [']]
Then
6(3) s log (11 (1- 22 x²00)
82
log 1 x2 72
2d
In porticular, for any unlabeled sample U = M/ drawn i.i.d. from Pz and
Lreg (U) = log
Σ e
(M)
icj
we have the expectation bound
B [U]r S 6(3).
Proof. Write the mixture decomposition for an independent pair (2, Z'):
B zz = n ee vn] + 2 x) LovT o 1 +11 -1)= (
where V. ~ vMF x) i..d. and U,U" Unif(Sd-1) i.i.d., all mutually independent.
Uniform and cross terms. For any fixed unit vector æ, the coordinate (z, U' is sub- Gaussian with parameter 1/vd; hence U3x < 8 /(2d), Conditioning on V (or U) and integrating,
E oL < ((20)
B < /2dd)
Posilive posilive lerm. Since (-1,,, we have the trivial bound E [eertv] 5
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Combining,
3z T 2'
S e² + (1- g
Taking logs gives the stated bound on 6(B). Finally, by Jenson,
E Trog (U)
log
827 23 S log E
2(().
Remark 4. The bound splits the n-dependence cleanly: the uniform baseline contributes e ((2d) (dimension- controlled, independent of T), while the excess over baseline scales with the fraction of positive-pont positive pairs, x²2. For small or moderate 3 (e.g., 3 = ol o(vd)), the increment
6 2d S log log((1+ (1+ be (
Or² B),
showing the regularizer's operating point is stable across class priors and primarily dimension-ddiven. When -++1 (positives dominate), the bound remains controlled by (and never exceeds 3), aligning with the simple worst-case Lreg. B.
Corollary D.2 (Baseline for the log mean exp regularizer). Write the temperature as and let
Leeg (U) log K
Σ
exp((xx,))
k= = (M)
Then
g2
(U)rr S log logE[exp(,BXnn
5 2d
Moreover, for aney 8 (0, 1), with probabilily at least 1-
2 logg(2 (2k(2)/))
Leeg(U S G max Xij s BV [<)
d
Thus the umlabeled-umiform baseline is O(33/d) in expectation and Oi3
log M
with high probabilliy.
Proof. Jensen and Lemma D.1 give the expectation bound. For the high-probability bound, use log( K Σ @x) & max and Cor. D.1
Corollary D.3 (Baseline for the log-mean-exp regularizer). Let
(U) log
Σ
exp(3x.9))
B > O.
K
Kj
Then
32 2(d 1)
E (()r)((U))
<
Moreover, for any å E (0, 1), with probability at least - 6,
2 log (og(2K/0)
Lveg(U) S max Xij I B
In particular, choosing
B dv log A7
prevents single extreme pair from dominating Treg as M, Do.
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Proof. The first inequality is Jensen: Elo((-) I log B(-). The mgf bound is Lemma ??(ii.. For the high- probability bound, use log R Σ e00) < 3 max and Cor. D.1
Under unlabeled uniformity, Lieg concentrates at a dimensionn-cotrroleed baseline of order 8/d, essentially independent the (unknown) class prior ı; a affects calibration, not the dispersion baseline. Corollary D.4. For all 2, E sd-1,
Lreg S
with equality when all embeddings are identical (z. 2/ 1).
Proof. Since -1 S 2 2g < 1, el2,7 S e.. The mean inside the log is at most e', giving Lreg S.b. t. Remark 5. Since each cosine similarity salisfies z 2j S 1, the regularizer is bounded above by L, i.e.,
Lreg S log (e') =t This shows that the regularizer cannot grow unbounded with dataset size; its magnitude controlled solely by the temperalure I, nol by (U].
E Radial projection analysis
Write the unlabeled score as s(2) = a(u T m) with trainable scale > 0, margin m E R, and prototype J E gd-1
Lemma E.1 (Neutral BCE acts on the radial projection only). For an unlabeled point E gd-1, the neutral (target 0.5) binary cross-entropy is
(z) = - log
log (2cosh(s/2)). a(k mm).
(a) Bounds and curvature. For all E E,
s²
log2 2 S log (2 cosh(s/22) < log 2 +
8
and Le = o - tanh(s/2), f.x Loeuls) = o(s) (1s)(11 (s)) s 1
(b) Population bound. For any probability law Q on sd-1
a² < log + Ez-allH m)?].
E (Z))(())
a² Var(u 2). porticular, choosing m = Elu T 2] yields Ellneu (2)] S log (c) Manifold gradient (no "equator bias"). Treating on the sphere with the Riemannian metric, the gradient w.r.t. is
&
Lnou (z) tanh(s(>)/2)) (1-
which is the projection of μ onto the tangent plane at z. Q is rotationally symmetric about μ and centered so that Blu Z) = m, then E (neu(Z) 0. Thus the neutral loss introduces no directional bias loward an "equator"; it only penalizes large |s].
Proof. The identity (so log(2 cosh(s/2)) follows by algebra. The lower bound is attained at 0. Since di (eu < 1/ for all s, the global quadratic upper bound (ueu(s) < Eeu (0) + 2 .1 82 = loog 2 + 2/8 holds by intograting the second derivative. Taking expectations gives (b). For (c), s(2) = a(p m) has
Euclidean gradient ay; projecting to the tangent space yields (I- 2z T )ap scaled by ae/Os tanh(s/2).
Symmetry and E[tanh(s/2)) under centering imply the stated mean-zero gradient.
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Table 6: AngularPU on MedMNIST (PU). Mean + std over seeds. R@P 2 a: maximum recall at precision 2 a, computed from the PR curve.
Dataset
AUC
AP
Precision
Recall
Acc
R@P20.90
R0P200.9
OCTMNIST 0.960 1 0.10
0.988 0.12
0.939 0.32 0.955 1 0.15
0.923
0.19
0.909 1 0.20
0.963
0.61
0.917
0.50
PathMNIST 0.983 = 0.13
0.901 + 0.33
0.893 = 0.22
0.882 + 0.31
0.904 = 0.13
0.963 + 0.08
0.878 = 0.52
0.358 + 0.52
Corollary E.1 (Compatibility with spherical uniformity). If Z Unif(S then E Z| Var(u Z) = 1/d. With m 0,
and
a²
Bleneu < log +
8d'
Thas the neutral BCE is minimal up to O(a2/d) al the uniform distribution, matching the largel of the angular dispersion term. In high dimension this overhead vanishes.
The angular dispersion loss Creg log( [ Eizy = penalizes pairwise alignment (a tangential
effect), while Lemma E.1 shows the neutral BCE controls only the variance of the radial projection 222. Consequently, the two terms are non-conflicting: logether they favor unlabeled embeddings that (1) are spread over the sphere (low pairwise cosines) and (ii) are not drifting into the positive cap (small |=Tz ml).
The uniform law satisfics both, with bascline values quantifiod in Cor. E.1 and the concentration bounds in
F AngularPU on MedMNIST: OCTMNIST and PathMNIST (PU, recall-critical)
We evaluate the proposed AngularPU method in PU regime two large MedMNIST datasets. For each dataset, we construct a binary task, sample a subset of labeled positives for training (cap: 5,000), and treat all remaining samples as unlabeled. A VGG11-BN backbone produces a d-256-dimensionall embedding.
AngularPU optimizes a cosine-similarity based objective with (i) alignment of positives to a learnable unit prototype p+. (ii) weighted BCE on unlabeled data with a learnable angular margin, and (iii) uniformity regularizer on unlabeled embeddings. We train with Adam (Ir 10-4) for 10 epochs and report threshold free metries (AUC, AP) well as point metrics. The decision threshold for point metrics is chosen as in section
We also report the maximum recall achievable at fixed precision floors, R@P 2 0.90 and R@P > 0.95, computed from the test PR curve.
We evaluate on two large-scale medical imaging benchmarks from the standardized McdMNIST collection. OCTMNIST contains retinal optical coherence tomography (OCT) B-scans from a 4-class diagnostic set (CNV, DMF, Drusen, Normal). In our binary reduction we define disease as the positive class by merging CNV DDME/EDDuuse
normal is the negative class. PathMNIST comprises colon histopathology patches from 9 tissue types; we define tumor epithelium (label = 8 in MedMNIST) as the positive class, while all other tissue types are negative. MedMNIST provides preprocessed, standardized training/ validation//////// ttest splits, facilitating controlled comparisons without dataset-specific engineering. We further map the samples to 3 channels to align with our backbones; no color jitter or heavy augmentations are used.
We adhere to the official MedMNIST splits but form the PU training pool by merging the provided training and validation splits. From the pooled positives, we randomly sample up 12,5% labeled positives (cap ap- plied per seed); all remaining samples both posilives and negalives are treated as unlabeled. This mirrors screening scenarios where small, verified positive set exists alongside a much larger pool with unknown labels. The official test split is used only for evaluation. Across S 10 seeds (Table G). we re-draw the labeled-positive subset and re-train to quantify variability due to the PU sampling.
AngularPU yields strong ranking performance on both datasets (AUC/AP), with particularly high values on OCTMNIST. Crucially for recall-critice screening, OCTMNIST attains R@P 2 0.95 of 0.917 + 0.005, indicating that the positive prototype learned in angular space separates discased from normal cases with limited overlap. PathMNIST exhibits high AUC (0.983) and balanced F1 (0.893 + 0.022), but R@P 2 0.95
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is lower and more variable across seeds (0.358 + 0.502), suggesting that while ranking is strong, very high precision comes at a substantial recall cost--likely due to greater intra-class heterogeneity and morphological overlap in histopathology. In practice, targeting a precision floor (e.g., 2 0.90) yields robust recall on both datasets (OCTMNIST: 0.963; PathMNIST: 0.878 + 0.052). Overall, these results support the use of AngularPU for recall critical medical screening, especially on OCT imaging, while highlighting that procision-constrannee operating points on histopathology may benefit from more labeled positives, longer training, or threshold selection tuned to a validation PR curve.
25


Reviews:
[{"review_id": "Is3GCIZloE", "reviewer": "Reviewer_2LEE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors study positive-unlabeled learning, a setting in classification where only a subset of the positive labels is available and the remaining data are unlabeled which makes the distinction between negative and positive labels hard. To avoid distributional assumptions common in prior works or failure in high-dimension, the authors propose AngularPU, a framework where embeddings are mapped to the unit sphere and positive labels are represented by a prototype vector which allows to perform classification by thresholding the cosine similarity between the prototype and other points embeddings. To avoid unlabeled data to cluster aroung the prototype, the authors introduce an angular regularization. Theoretical analysis is performed, showing the benefits of the approach. Experiments on image classification on $4$ common datasets is done, showing competitive performance compared to SOTA baselines and showcase the benefits of the approach to maintain a balance between accuracy and recall.", "claims_and_evidence_yn": "Yes", "claims_explanation": "- The paper is well-written and position with prior works well done.\n- The approach is interesting yet simple with a convincing theoretical analysis\n- The experiments are thorough and show the benefits of the approach\n\n*Questions*\n\n- Could the authors improve the results presentation, notably highlighting the best methods in the table, for instance in bold?\n- The idea of projecting on a hypersphere and computing the angle between a prototype and embeddings, while avoiding unlabeled data to cluster around the prototype with a regularization is reminescent of [1] where pseudo-labelling is done with several linear classifiers which are forced to be distinct on unlabeled data and similar on labeled data (thresholding is computing on the similarity between diverse classifiers). Corollary 3.5 of [1] discuss the impact of having representation on the sphere using contrastive learning on such pseudo-labeling. Could the authors discuss this approach? I acknowledge that it is not directly related to PU but believes it provides a nice connection to prior works that further justify the interesting idea of having embeddings on the sphere with diverse representations. Similarly, the idea of embeddings spread on the sphere is connected to [2].\n\n- The performance improvement of AngularPU is impressive on ADNI. While the authors discuss why such improvement is not occuring on the other datasets, I believe the work would benefit from more datasets such as ADNI where having a good recall is critical. Could the authors add or discuss additional experiments in this direction?\n\n- Similarly, I would be interested to see how AngularPU performs with better vision models and hence more distinct embeddings, e.g., ResNet or Vision Transformers. Could the authors perform such experiments, simply by unsing frozen representations from such models?\n\n*References*\n\n[1] Odonnat, A. et al. Leveraging Ensemble Diversity for Robust Self-Training in the Presence of Sample Selection Bias, AISTATS 2024\n\n[2] Wang, T. and Isola, P. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. ICML 2020", "audience_yn": "Yes", "audience_explanation": "Overall, I think the current submission is interesting for TMLR audience, the paper is well-written, well-explained and the approach is sound. Additional experiments could further improve the current work with more recent models and more datasets where AngularPU could shine.", "requested_changes": "See questions.", "broader_impact_concerns": "None", "additional_comments": "None", "extra_scores": {}, "extra_text": {}}, {"review_id": "UqtKZo3LTN", "reviewer": "Reviewer_nV3u", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes Angular-PU, a novel positive-unlabeled learning technique that eliminate the need of risk estimation or pseudo-labeling. Instead, the paper proposes to map the data onto a hypershpere and learn a prototype vector to indicate the positive direction. The training is performed with an angular regualrization to encourage seperation. \n\nThe paper is overall well-written and easy to follow. The proposed method is technically sound and well motivated. Results are thorough and good compared to baselines.\n\nWeakness-wise, though the author claim that no distribution assumption is required for the proposed method, it seems like the effectiveness of the method may depend on the percentage of negative data in the training set. The current formulation does not discuss the performance guarantee when negative samples are far less than positive ones, which would also be a common scenario for PU learning. Theoretical derivations and ablation studies are needed to show the impact of training distribution.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper provides solid theoretical derivation and experimental results to support the claim. Meanwhile, the ablation can be strengthened, as indicated in the requested changes section below.", "audience_yn": "Yes", "audience_explanation": "PU-learning is an important application domain for machine learning and has practical usage. The topic is relavent to TMLR audiences.", "requested_changes": "1. In the training objective, Equ. (2) proposes a symmetric BCE loss with neutral label for unlabeled data. However, whether this loss is needed, especially under the angular regularization, is not well discussed. To my understanding, the neutral supervation will make unlabeled data concentrating towards the orthogonal plane of the prototype vector, which is in contrast to the goal of angular regualrization that tries to spread the data across the whole hypersphere. More discussion on why this term is needed and ablation on the effect of this term on the final performance is needed.\n2. Though not specifically discussed in the paper, the percentage of true positive vs. true negative data in the unlabeled training set appears to have an impact on the overall performance of PU-learning methods. Adding some analysis or ablation studies on the impact of training distribution, especially the case where positive data are far more than negative data, will be helpful.", "broader_impact_concerns": "No concerns on broader impacts", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wMNHp5jC4Q", "reviewer": "Reviewer_NtfU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The manuscript proposes a novel Positive-Unlabeled (PU) learning framework that operates on the unit hypersphere leveraging cosine similarity and angular margin. A key contribution is the representation of the positive class by a learnable prototype vector, where classification simplifies to thresholding the cosine similarity between input embeddings and this prototype. This approach elegantly circumvents the need for explicit negative-class modeling, offering conceptual and computational advantages.", "claims_and_evidence_yn": "Yes", "claims_explanation": "To mitigate the potential clustering of unlabeled embeddings around the positive prototype, the authors introduce an angular regularizer that effectively promotes dispersion of unlabeled samples across the hypersphere, thereby enhancing separability. The theoretical analysis presented substantiates the validity of the proposed method.", "audience_yn": "Yes", "audience_explanation": "See above", "requested_changes": "1 The proof of Theorem 4.1 requires elaboration to more clearly demonstrate how classification reduces to thresholding the inner product (§4.1). Detailed derivations or intuitive explanations would strengthen the theoretical foundation.\n\n2 The advantages of using cosine similarity and angular margins should be quantitatively benchmarked against alternative approaches (e.g., Euclidean distance metrics). Including ablation studies or controlled comparisons would validate design choices.\n\n3 The current empirical evaluation appears limited in scope. Testing the framework on large-scale benchmarks (e.g., ImageNet) would better assess its scalability and generalization, particularly given the method’s reliance on hypersphere-based embeddings.", "broader_impact_concerns": "NA", "additional_comments": "No", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)