Single Positive Multi-Label Learning (SPML)

• SPML is a weakly supervised paradigm where each instance is annotated with one positive label despite multiple true positives, addressing label sparsity in multi-label settings.
• It employs varied risk estimators and loss functions such as entropy-maximization and generalized robust loss to mitigate label bias and improve estimation accuracy.
• SPML has broad applications in image recognition, remote sensing, and surgical vision, showing competitive performance on benchmarks like VOC and MS-COCO.

Single Positive Multi-Label Learning (SPML) is a weakly supervised learning paradigm in which each training instance is annotated with only a single relevant label, despite the potential presence of multiple true positives per instance. This extreme form of label sparsity presents unique optimization, statistical, and algorithmic challenges, fundamentally distinguishing SPML from fully supervised multi-label learning. SPML is motivated by the prohibitive costs and practical impediments associated with fully exhaustive multi-label annotation, especially for images and other modalities where the number of relevant categories per instance is often unknown and highly variable.

1. Formal Problem Statement and Theoretical Foundations

In SPML, let $\mathcal{X}$ denote the instance space and $\mathcal{Y} = \{0,1\}^L$ the multi-label space over $L$ categories. The canonical fully supervised setting assumes access to i.i.d. pairs $(x_n, y_n)$ with $y_n \in \{0,1\}^L$. The learner minimizes a surrogate loss, typically binary cross-entropy (BCE), across all $L$ labels. In contrast, in SPML, for each example $x_n$ only a single positive label is observed: $z_n \in \{0,1,\varnothing\}^L$ with exactly one $z_{ni}=1$ and $z_{nj} = \varnothing$ for $j \ne i$; no confirmed negatives are provided (Cole et al., 2021).

The objective is to train a classifier $h: \mathcal{X} \to [0,1]^L$ such that the $i$-th output, $h_i(x)$, estimates $P(y_i=1 \mid x)$, approaching the performance of a model trained with fully observed $y \in \{0,1\}^L$, despite the single-positive training constraint (Arroyo, 2023).

Several theoretically grounded risk estimators have been derived for this setting:

• The SMILE estimator recovers an unbiased risk for the full multi-label objective by leveraging the observation that each positive in the SPML regime can be viewed as a random draw from the true positive set $Y \subseteq \mathcal{Y}$, and marginalizing out the unknowns via variational inference over soft label posteriors (Xu et al., 2022).
• Class-priors estimation, as in CRISP, offers a provably unbiased risk estimator and minimizer by leveraging per-class marginal probabilities rather than uniform assumptions, which is especially beneficial under heavy label imbalance (Liu et al., 2023).
• Conformal set-valued predictors, as in LAMC, wrap any pre-trained base predictor to yield calibrated confidence sets for predicted positives under finite-sample guarantees, bridging the gap between single-label supervision and multi-label evaluation (Hagos et al., 12 Sep 2025).

2. Loss Functions, Regularization, and Label Inference

A broad literature has developed SPML-specific loss functions to address the inherent ambiguity and extreme positive-negative imbalance:

Loss Family	Key Principle	Reference
Assume-Negative (AN)	Treat all unknowns as negative	(Cole et al., 2021, Arroyo, 2023)
Entropy-Maximization (EM)	Unknowns pushed to max-entropy	(Zhou et al., 2022, Arroyo, 2023)
Weak Negatives (WAN)	Down-weight unknown negatives	(Cole et al., 2021)
Role (ROLE)	Online, symmetrized label estimation	(Cole et al., 2021)
Label Smoothing (LS)	Add constant smoothing to targets	(Cole et al., 2021)
Pseudo-labeling	Harvest pseudo-positives/-negatives from a teacher/classifier	(Arroyo, 2023, Xing et al., 2023, Jouanneau et al., 2022)
High-rankness reg.	Avoid collapse by maximizing label space rank	(Li et al., 2023)
Hill Loss	Down-weight hard negatives, focus on hardest pairs	(Li et al., 2023, Walimbe et al., 7 Jul 2025)
Generalized Robust (GR)	Soft pseudo-labels, per-instance reweighting, robust surrogates	(Chen et al., 2024, Tran et al., 28 Aug 2025)

Notable strategies include EM loss, which imposes an entropy penalty $H(f_j(x))$ on unknowns to discourage premature commitment, and APL, which bootstraps negatives in a self-paced, asymmetric manner based on model confidence (Zhou et al., 2022). High-rankness (e.g., via a log-det penalty) regularizes the output matrix to prevent low-dimensional collapse and to enhance retrieval of missing positives (Li et al., 2023). GR Loss (Chen et al., 2024) and GPR Loss (Tran et al., 28 Aug 2025) unify and generalize prior approaches by parameterizing soft pseudo-labeling and per-class weighting within an expected-risk framework, allowing for greater robustness and sensitivity to class imbalance.

Label inference—especially pseudo-labeling—has been advanced by techniques such as using pre-trained vision-LLMs (e.g., CLIP) to assign high-confidence positives and negatives to unannotated labels (Xing et al., 2023, Tran et al., 28 Aug 2025), as well as teacher-student paradigms in which a teacher network trained on SPML labels provides synthetic labels for a conventional multi-label student (Arroyo, 2023).

3. Algorithmic Architectures and Hybrid Approaches

In addition to loss design, architectural modifications have been leveraged to enhance the expressivity and label-disambiguation capability of models under SPML:

• Patch-based attention models that explicitly model image regions with a per-label codebook and attention-pooling, allowing estimation of negatives by self-similarity between label-specific embeddings, and enabling positive-unlabeled learning formulations (Jouanneau et al., 2022).
• Graph-based modules for modeling and propagating label correlations: the GMC in SigRL employs graph attention to encode label inter-dependencies into both textual and visual features (Zhang et al., 4 Apr 2025). The GE-VerbMLP integrates a lightweight GCN atop a frozen CLIP visual backbone, enhanced with adversarial training to define sharp boundaries in the label space, as demonstrated for ambiguous visual verbs in situation recognition (Lin et al., 29 Aug 2025).
• Hyperbolic embedding methods in SPML cast each label as a geometric ball in the Poincaré model, enabling the learned space to capture inclusion (hierarchy), overlap (co-occurrence), and separation (independence) between classes with explicit geometric interpretation (Lin et al., 17 Oct 2025).
• Expectation-Maximization bootstrapping and contrastive instance retrieval, as in Scob, iteratively refine object localization via CAMs and semantic-masked transformers, feeding better semantic masks into successive learning rounds (Chen et al., 2023).

Recent generalization includes the extensibility of SPML to multi-task settings with mixed annotation completeness, as in surgical video understanding with label-graph GCNs over prompt-based CLIP embeddings (Walimbe et al., 7 Jul 2025), or to remote sensing with adaptive gradient calibration and Mixup-based pseudo-labeling (Liu et al., 9 Oct 2025).

4. Empirical Insights: Label Bias, Pseudo-Labeling, and Performance Bounds

A fundamental concern in SPML is label-selection bias: practical annotations do not yield single positives uniformly at random. Studies have established the sensitivity of leading SPML approaches (AN, LS, ROLE, EM) to realistic biases (size, location, semantic prominence) in which annotators prefer salient, large, or central objects. While entropy-maximization losses and EM-style algorithms remain most performant overall, methods such as ROLE and AN exhibit greater robustness under non-uniform label selection (Arroyo et al., 2023).

Empirically, SPML-trained multi-label classifiers can approach full-supervision mAP performance while using $\sim L$ times fewer annotation labels in the uniform SPML regime, with differences shrinking to $\lesssim 1$ mAP point on canonical datasets (VOC, COCO) for advanced methods such as ROLE, GR Loss, or pseudo-label students (Cole et al., 2021, Arroyo, 2023, Zhang et al., 4 Apr 2025, Li et al., 2023, Chen et al., 2024, Tran et al., 28 Aug 2025). Pseudo-labeling, especially high-confidence positives from teacher models or vision-LLMs, consistently outperforms the naive Assume-Negative baseline and can account for a substantial portion ($\sim$40%) of the gap to full-label supervision. Careful threshold tuning is critical to control the label-noise/coverage trade-off (Arroyo, 2023, Xing et al., 2023, Tran et al., 28 Aug 2025).

The explicit modeling of class-priors instead of assuming uniform marginals further reduces bias and yields an unbiased statistical estimation of multi-label risk as shown in CRISP (Liu et al., 2023). Pseudo-labels are best restricted to high-confidence positives, as aggressive negative pseudo-labeling can degrade performance (Xing et al., 2023).

5. Applications, Benchmarks, and Domain Extensions

SPML has found broad application in domains where annotation is expensive, ambiguous, or partial:

• Standard vision benchmarks: PASCAL VOC, MS-COCO, NUS-WIDE, CUB-200-2011, and others, with experimental protocols systematically masking out all but one positive per image during training (Cole et al., 2021, Arroyo, 2023, Mankovich et al., 2023, Chen et al., 2024, Tran et al., 28 Aug 2025).
• Remote sensing imagery, where land cover annotation is cost-prohibitive. AdaGC leverages Mixup, dual-EMA teacher-student networks, and adaptive gradient calibration for robust recovery of missing positives under SPML noise (Liu et al., 9 Oct 2025).
• Surgical computer vision, where combining multi-task labels (phases, CVS, actions) under SPML enables scaling to multi-task settings with mixed and noisy annotation (Walimbe et al., 7 Jul 2025).
• Generative modeling: S2M enables multi-label conditional sampling using only a marginal (single-positive) annotated dataset by post-hoc MCMC-based reweighting of unconditional or class-conditional GANs (Cho et al., 2022).
• Natural language and zero-shot recognition: SigRL extends SPML to zero-shot MLR by infusing semantic guidance from text via label graphs and visual feature reconstruction (Zhang et al., 4 Apr 2025).

Recent work has begun to provide more realistic multi-label benchmarks for tasks like verb recognition in situation recognition (imSitu), revealing the intrinsic multi-label nature of such problems and enabling SPML evaluation at scale (Lin et al., 29 Aug 2025).

6. Limitations, Open Problems, and Future Directions

Several limitations of current SPML approaches persist:

• Even the best algorithms under SPML do not fully close the gap to full-label performance in highly ambiguous, fine-grained, or severely imbalanced domains (Li et al., 2023, Lin et al., 29 Aug 2025).
• The performance of pseudo-labeling and contrastive/retrieval-based methods depends heavily on the initial model and the representativeness of the prior or teacher network, especially under shift from the pre-trained domain (Xing et al., 2023, Tran et al., 28 Aug 2025).
• Label selection bias remains a critical challenge, warranting methods that either explicitly counteract bias or model annotator decision processes (Arroyo et al., 2023).
• Hyperparameter tuning, such as pseudo-label thresholds, regularizer weights, and balancing coefficients, often requires held-out data and is dataset-specific (Chen et al., 2024, Xing et al., 2023, Tran et al., 28 Aug 2025).
• For SPML in extremely large label spaces ($L \gg 10^2$), the computational cost of high-rank penalties, graph modules, or calibration may become prohibitive, motivating scalable approximations (Li et al., 2023, Lin et al., 17 Oct 2025).
• Joint learning of dynamic label correlations, instance-dependent priors, or curriculum pseudo-labeling strategies are promising and remain active research directions (Zhang et al., 4 Apr 2025, Liu et al., 2023, Lin et al., 29 Aug 2025).

Broader methodological challenges include extending SPML to continuous multi-labels, instance-dependent or structured outputs, and broader modalities (video, text), as well as investigating connections with positive-unlabeled (PU) learning, semi-supervised learning, and open-world recognition.

Key technical references (by arXiv id):

• (Cole et al., 2021) Cole et al., "Multi-Label Learning from Single Positive Labels"
• (Zhou et al., 2022) Zhou et al., "Acknowledging the Unknown for Multi-label Learning with Single Positive Labels"
• (Xu et al., 2022) Xie et al., "One Positive Label is Sufficient: Single-Positive Multi-Label Learning with Label Enhancement"
• (Cho et al., 2022) Shinoda et al., "Mining Multi-Label Samples from Single Positive Labels"
• (Jouanneau et al., 2022) Fini et al., "A patch-based architecture for multi-label classification from single label annotations"
• (Li et al., 2023) Zhang et al., "Pushing One Pair of Labels Apart Each Time in Multi-Label Learning: From Single Positive to Full Labels"
• (Arroyo et al., 2023) Cole et al., "Understanding Label Bias in Single Positive Multi-Label Learning"
• (Arroyo, 2023) Wu et al., "Pseudo Labels for Single Positive Multi-Label Learning"
• (Chen et al., 2023) Chen et al., "Semantic Contrastive Bootstrapping for Single-positive Multi-label Recognition"
• (Liu et al., 2023) Liu et al., "Can Class-Priors Help Single-Positive Multi-Label Learning?"
• (Xing et al., 2023) Chen et al., "Vision-Language Pseudo-Labels for Single-Positive Multi-Label Learning"
• (Chen et al., 2024) Zhang et al., "Boosting Single Positive Multi-label Classification with Generalized Robust Loss"
• (Zhang et al., 4 Apr 2025) Chen et al., "Semantic-guided Representation Learning for Multi-Label Recognition"
• (Walimbe et al., 7 Jul 2025) Suraj et al., "Adaptation of Multi-modal Representation Models for Multi-task Surgical Computer Vision"
• (Tran et al., 28 Aug 2025) Zhang et al., "More Reliable Pseudo-labels, Better Performance: A Generalized Approach to Single Positive Multi-label Learning"
• (Lin et al., 29 Aug 2025) Xiong et al., "The Demon is in Ambiguity: Revisiting Situation Recognition with Single Positive Multi-Label Learning"
• (Hagos et al., 12 Sep 2025) Fei et al., "Least-Ambiguous Multi-Label Classifier"
• (Liu et al., 9 Oct 2025) Meng et al., "Adaptive Gradient Calibration for Single-Positive Multi-Label Learning in Remote Sensing Image Scene Classification"
• (Lin et al., 17 Oct 2025) Zhang et al., "Hyperbolic Structured Classification for Robust Single Positive Multi-label Learning"