SPMLL: Single-Positive Multi-Label Learning

• SPMLL is a weakly supervised multi-label learning paradigm where each instance has one confirmed positive label while the full set of relevant labels remains latent.
• Key methodologies include loss design adaptations, latent-label estimation, pseudo-label completion, and class-prior correction to counter sparse supervision challenges.
• Empirical studies show that calibrated approaches and tailored pseudo-labeling enable SPMLL methods to approach or match fully supervised performance on benchmark datasets.

Searching arXiv for recent and foundational papers on single-positive multi-label learning. Single-Positive Multi-Label Learning (SPMLL), also called Single-Positive Multi-Label Learning (SPML) in much of the literature, denotes a weakly supervised multi-label regime in which each training instance has exactly one observed positive label while the latent target remains a full multi-label assignment. In the canonical formulation, the learner observes the input, one confirmed positive class, no confirmed negatives, and no annotation for the remaining classes. The setting is therefore neither ordinary multiclass classification nor ordinary fully supervised multi-label learning: supervision is instance-wise, positive-only, and maximally sparse. Work from 2021 onward established SPMLL as a distinct problem within weakly supervised multi-label learning and then expanded it along several axes, including loss design, latent-label estimation, pseudo-label completion, class-prior correction, calibration, and task-specific reformulations (Cole et al., 2021, Arroyo et al., 2023).

1. Formal problem setting and relation to nearby paradigms

In standard multi-label classification, a training example is written as

$(\mathbf{x}, \mathbf{y}),$

where $\mathbf{x}\in\mathbb{R}^p$ is an image and $\mathbf{y}\in\{0,1\}^L$ is the binary label vector over $L$ classes. In SPMLL, the latent object of interest is still this full multi-label target, but the observed supervision is reduced to one positive label per instance. One common notation is a dataset $\{(x_n,z_n)\}_{n=1}^N$ with

$z_n \in \{1,\varnothing\}^L,$

such that exactly one category satisfies $z_{ni}=1$ and every other entry is unobserved rather than negative. Another notation uses $\mathbf{z}\in\{0,1,\varnothing\}^L$, with exactly one observed positive and no observed negatives. Across these formulations, the essential constraint is unchanged: one confirmed positive, all other labels unknown (Cole et al., 2021, Arroyo, 2023).

This supervision pattern differs from several neighboring settings. It differs from conventional multi-label learning because the learner does not observe the full $\mathbf{y}$. It differs from positive-unlabeled learning because SPMLL is multi-label and instance-wise: each image may have several true positives, but only one is revealed. It differs from partial-label learning because the observed label is known to be correct; the ambiguity lies not in which observed candidate is true, but in which additional labels are missing. It also differs from generic missing-label multi-label learning because SPMLL is the extreme case in which only one positive is retained per sample and no reliable negatives are given (Arroyo et al., 2023, Hagos et al., 12 Sep 2025).

A recurring formulation emphasizes a supervision mismatch: single-positive supervision during training, but full multi-label prediction during evaluation. This mismatch is central to later methodological developments. It motivates objectives that avoid collapsing all unobserved labels to zero, estimators that try to reconstruct missing positives, and calibration procedures that restrict test-time outputs to labels supported by sufficiently reliable evidence (Hagos et al., 12 Sep 2025).

2. Benchmark construction and the label-bias problem

The standard synthetic SPMLL benchmark is constructed from a fully labeled multi-label dataset by selecting one true positive label per training image and discarding the rest. The 2021 study that systematized the setting used the assumption that one true positive is selected uniformly at random and then fixed for training. Under that protocol, the observed single positive is not re-sampled during optimization, while validation and test remain fully labeled (Cole et al., 2021).

A 2023 empirical critique argued that this benchmark assumption is systematically unrealistic. It formalized the usual protocol as

$P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$

but noted that realistic annotators do not usually choose among present objects uniformly. Larger objects, centrally located objects, semantically salient categories, or categories that are more likely to be named may be disproportionately observed. To study this effect, the paper introduced three additional annotation protocols on COCO: size bias,

$\mathbf{x}\in\mathbb{R}^p$0

location bias,

$\mathbf{x}\in\mathbb{R}^p$1

and semantic bias,

$\mathbf{x}\in\mathbb{R}^p$2

For each bias model, the study generated three independent realizations with fixed random seeds; training labels were fixed once generated, and validation and test remained fully labeled (Arroyo et al., 2023).

The empirical consequence was that conclusions drawn under uniform selection did not necessarily persist under biased selection. On COCO, for example, $\mathbf{x}\in\mathbb{R}^p$3 achieved $\mathbf{x}\in\mathbb{R}^p$4 MAP under $\mathbf{x}\in\mathbb{R}^p$5, but $\mathbf{x}\in\mathbb{R}^p$6 under $\mathbf{x}\in\mathbb{R}^p$7, $\mathbf{x}\in\mathbb{R}^p$8 under $\mathbf{x}\in\mathbb{R}^p$9, and $\mathbf{y}\in\{0,1\}^L$0 under $\mathbf{y}\in\{0,1\}^L$1. Likewise, $\mathbf{y}\in\{0,1\}^L$2 achieved $\mathbf{y}\in\{0,1\}^L$3 under $\mathbf{y}\in\{0,1\}^L$4, but $\mathbf{y}\in\{0,1\}^L$5 under $\mathbf{y}\in\{0,1\}^L$6, $\mathbf{y}\in\{0,1\}^L$7 under $\mathbf{y}\in\{0,1\}^L$8, and $\mathbf{y}\in\{0,1\}^L$9 under $L$0 (Arroyo et al., 2023).

This suggests that SPMLL is not only a missing-label problem but also an annotation-process problem. Reported performance depends not merely on the learning objective, but on the stochastic mechanism by which a single positive becomes visible.

3. Loss design, latent-label estimation, and risk correction

A central early observation is that the obvious baselines behave pathologically in the single-positive regime. If unobserved labels are ignored, the loss admits a trivial all-positive solution because there are no negative penalties. If all unobserved labels are treated as negative, the learner receives $L$1 assumed negatives for every observed positive, thereby injecting false negatives whenever the hidden label set contains additional positives. The 2021 study therefore organized SPMLL objectives around several adaptations: Assume Negative (AN), Weak Assume Negative (WAN), Assume Negative with Label Smoothing (AN-LS), Expected Positive Regularization (EPR), and Regularized Online Label Estimation (ROLE). ROLE combines classifier predictions with a learned latent soft-label table $L$2 and an expected-positive-count prior

$L$3

using a batch regularizer

$L$4

This line of work showed that SPMLL can approach fully supervised performance on some datasets, especially in linear settings and with strong pretrained features (Cole et al., 2021).

A related but conceptually distinct response is to treat unobserved labels as unknown rather than negative. The 2022 paper on “Acknowledging the Unknown” proposed the Entropy-Maximization (EM) loss,

$L$5

where

$L$6

EM maximizes entropy on unknown labels, producing a gradient regime intended to avoid the aggressive suppression of hidden positives. The same paper added Asymmetric Pseudo-Labeling (APL), which generates only negative pseudo-labels in a self-paced manner and leaves positive pseudo-labeling unused because it is too error-prone in this regime (Zhou et al., 2022).

A more explicitly theoretical route was taken by SMILE, which derived an SPMLL risk estimator under the annotation model

$L$7

and rewrote the fully supervised risk as an expectation over $L$8. SMILE then replaced latent binary labels with soft relevance scores $L$9 and estimated them through a variational label-enhancement model in which each $\{(x_n,z_n)\}_{n=1}^N$0 is approximated by a Beta posterior. This yields a label-enhancement view of SPMLL: the missing label structure is recovered as latent soft relevance rather than by direct hard completion (Xu et al., 2022).

A further correction addressed label prevalence. Crisp argued that many SPMLL methods implicitly assume identical class priors, which is unrealistic when class frequencies differ substantially. It therefore estimated per-class priors $\{(x_n,z_n)\}_{n=1}^N$1 from single-positive data and used them in the risk

$\{(x_n,z_n)\}_{n=1}^N$2

This replaces equal-prior heuristics with class-specific prevalence calibration (Liu et al., 2023).

A 2024 study generalized the expected-risk viewpoint further by assigning each missing label a soft pseudo-label weight

$\{(x_n,z_n)\}_{n=1}^N$3

and introducing a generalized robust loss in which missing labels contribute a mixture of positive and negative risk terms. In that formulation, $\{(x_n,z_n)\}_{n=1}^N$4 is instantiated as a logistic function, and the method is explicitly motivated by balancing false positives, false negatives, and class imbalance (Chen et al., 2024).

Representative method	Main mechanism	Paper
ROLE	Expected-positive regularization plus online label estimation	(Cole et al., 2021)
EM + APL	Entropy maximization on unknown labels plus asymmetric pseudo-labeling	(Zhou et al., 2022)
SMILE	Unbiased risk estimator with variational Beta label enhancement	(Xu et al., 2022)
Crisp	Class-prior estimation and prior-guided risk	(Liu et al., 2023)
Generalized robust loss	Soft pseudo-label expected-risk framework	(Chen et al., 2024)

Taken together, these methods define a core SPMLL taxonomy: hard negative assumption, weak negative weighting, entropy-based acknowledgment of unknown labels, count-prior regularization, soft-label enhancement, and class-prior correction.

4. Pseudo-label completion, semantic guidance, and foundation-model priors

A major SPMLL branch abandons direct risk correction and instead tries to complete the missing label matrix. The simplest version is Pseudo Multi-Labels: train a teacher on single-positive data, obtain per-class predictions $\{(x_n,z_n)\}_{n=1}^N$5, threshold them independently at $\{(x_n,z_n)\}_{n=1}^N$6, and train a student on the resulting hard binary pseudo-labels

$\{(x_n,z_n)\}_{n=1}^N$7

The paper emphasizes that the pseudo labels are hard binary labels, not soft targets, and reports that the student beats Assume Negative but does not beat Entropy Maximization (Arroyo, 2023).

A more elaborate completion pipeline is Scob, or Semantic Contrastive Bootstrapping. Scob uses class activation maps as semantic guidance, a recurrent semantic masked transformer to extract class-aware object-level features, contrastive learning over those features, and an EM-style loop in which current masks and pseudo-labels refine the next training round. In the single-positive setting with ResNet-50, it reports mAP of $\{(x_n,z_n)\}_{n=1}^N$8 on VOC 2007, $\{(x_n,z_n)\}_{n=1}^N$9 on VOC 2012, $z_n \in \{1,\varnothing\}^L,$0 on CUB, and $z_n \in \{1,\varnothing\}^L,$1 on MS COCO, outperforming generic contrastive methods such as BYOL, MoCo, SimCLR, DetCo, SoCo, and UP-DETR in that setting (Chen et al., 2023).

Foundation-model priors have also been incorporated directly. Vision-Language Pseudo-Labeling (VLPL) uses a fixed CLIP-style image-text branch to score image-label relevance with the prompt template “A photo of $z_n \in \{1,\varnothing\}^L,$2.” The method can generate pseudo-positive and pseudo-negative labels, but its main empirical conclusion is that pseudo-negative labels consistently hurt performance, so the final version keeps only pseudo-positive labels and treats the rest as unknown under EM. With a ResNet-50 backbone, VLPL reports $z_n \in \{1,\varnothing\}^L,$3 mAP on VOC, $z_n \in \{1,\varnothing\}^L,$4 on COCO, $z_n \in \{1,\varnothing\}^L,$5 on NUS-WIDE, and $z_n \in \{1,\varnothing\}^L,$6 on CUB (Xing et al., 2023).

Other architectural variants exploit label-conditioned structure rather than only label completion. A patch-based PU architecture assumes that a small enough image patch contains only one label, builds label-specific image representations with patch attention, and estimates soft negative labels through similarity among those representations. Its weak-negative loss reaches $z_n \in \{1,\varnothing\}^L,$7 mAP on COCO and $z_n \in \{1,\varnothing\}^L,$8 on VOC when trained from scratch (Jouanneau et al., 2022). SigRL, a semantic-guided representation framework built around a graph-based multi-label correlation module and semantic visual feature reconstruction, is not SPMLL-specific but is evaluated under single-positive supervision; on COCO, SigRL + EM reports $z_n \in \{1,\varnothing\}^L,$9 mAP, and on VOC the best mAP is $z_{ni}=1$0 with IUN while the best Top-3 and Top-5 F1 are $z_{ni}=1$1 and $z_{ni}=1$2 with EM (Zhang et al., 4 Apr 2025).

This suggests that SPMLL has increasingly been treated not only as a loss-design problem but also as a representation-learning problem. Missing labels can be approached through teacher-student completion, CAM-guided object discovery, vision-language transfer, or semantically structured local-global alignment.

5. Calibration, task reformulation, and later extensions

A notable later development is a shift from latent-label recovery toward calibrated decision making. LAMC does not try to reconstruct all missing labels during training. Instead, it trains a base SPMLL predictor, holds out a fully labeled calibration set $z_{ni}=1$3, computes class-conditional score sets

$z_{ni}=1$4

and derives label-specific thresholds $z_{ni}=1$5 from empirical quantiles. Test-time prediction is then the set-valued output

$z_{ni}=1$6

In the reported experiments, the base predictor is a two-layer MLP trained with WAN for 25 epochs, Adam, batch size 16, and $z_{ni}=1$7, with a 70% train, 10% calibration, 10% validation, 10% test split and 10 instances per label for the calibration set (Hagos et al., 12 Sep 2025).

SPMLL has also been exported into tasks that were previously treated as single-label. In situation recognition, verb classification was reframed as SPMLL on the grounds that many images support multiple valid verb labels, while datasets provide only one positive verb. The proposed GE-VerbMLP combines a frozen CLIP image encoder, a verb MLP, a GCN over label nodes, cosine-similarity classification, and adversarial training; the paper reports more than $z_{ni}=1$8 MAP improvement while remaining competitive on top-1 and top-5 accuracy metrics (Lin et al., 29 Aug 2025).

Remote sensing work adapted SPMLL to scene classification, where full multi-label annotation is costly. AdaGC uses an AN warm-up stage, an adaptive trigger based on noisy validation mAP, dual-EMA pseudo-labels, Mixup, and a gradient-calibration regularizer intended to counteract harmful AN gradients on likely hidden positives. The paper reports state-of-the-art performance on two benchmark remote-sensing datasets under two distinct label noise types (Liu et al., 9 Oct 2025).

A structurally different extension is hyperbolic classification. The hyperbolic structured framework represents each label as a hyperbolic ball rather than a point or vector, with ball inclusion, overlap, and separation encoding hierarchy, co-occurrence, and independence. Its adaptive score is

$z_{ni}=1$9

and the full loss combines classification, a double-well regularizer, and a uniformity term. On four SPMLL benchmarks it is competitive but below SCPNet overall, while the paper emphasizes superior interpretability and reports a Pearson correlation $\mathbf{z}\in\{0,1,\varnothing\}^L$0 with $\mathbf{z}\in\{0,1,\varnothing\}^L$1 between co-occurrence probability and learned label-center distance on COCO (Lin et al., 17 Oct 2025).

SPMLL-adjacent work has even extended the single-positive regime into conditional generation. The S2M sampling framework does not train a discriminative SPMLL classifier; instead, it uses density-ratio estimation and MCMC to sample plausible multi-label joint classes from single-positive annotations in cGANs. This is best read as adjacent rather than central SPMLL methodology, but it underscores that single-positive supervision has been treated as a general weak-label primitive rather than as a problem confined to discriminative recognition (Cho et al., 2022).

6. Empirical behavior, limitations, and unresolved questions

A stable empirical pattern is that SPMLL can be surprisingly effective, but only under certain data and modeling conditions. In the 2021 linear setting, fully supervised BCE achieved $\mathbf{z}\in\{0,1,\varnothing\}^L$2 MAP on VOC12 and ROLE achieved $\mathbf{z}\in\{0,1,\varnothing\}^L$3; on COCO the corresponding numbers were $\mathbf{z}\in\{0,1,\varnothing\}^L$4 and $\mathbf{z}\in\{0,1,\varnothing\}^L$5; on NUS $\mathbf{z}\in\{0,1,\varnothing\}^L$6 and $\mathbf{z}\in\{0,1,\varnothing\}^L$7; on CUB $\mathbf{z}\in\{0,1,\varnothing\}^L$8 and $\mathbf{z}\in\{0,1,\varnothing\}^L$9. The same study therefore showed both the promise of the regime and the dependence of performance on label cardinality: high-cardinality CUB remained substantially harder (Cole et al., 2021).

Entropy-based methods sharpened this picture. With full supervision, BCE reached $\mathbf{y}$0 on VOC, $\mathbf{y}$1 on COCO, $\mathbf{y}$2 on NUS, and $\mathbf{y}$3 on CUB; under single-positive supervision, EM + APL reached $\mathbf{y}$4, $\mathbf{y}$5, $\mathbf{y}$6, and $\mathbf{y}$7, respectively. The paper also reported that negative pseudo-labels mined by APL had precision $\mathbf{y}$8 on VOC, $\mathbf{y}$9 on COCO, $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$0 on NUS, and $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$1 on CUB, whereas attempted positive pseudo-labeling had much lower precision: $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$2, $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$3, $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$4, and $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$5 (Zhou et al., 2022).

Several methodological cautions recur across the literature. Treating all unknown labels as negatives is simple but biased. Ignoring them can degenerate to trivial solutions. Positive pseudo-labeling can help, but some papers explicitly report that pseudo-negatives hurt when class imbalance is already severe, as in VLPL’s ablation where “Pos. only” reached $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$6 on VOC while “Pos.+Neg.” ranged from $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$7 to $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$8 as more pseudo-negatives were added (Xing et al., 2023). Other methods require additional side information or assumptions: ROLE and EPR use an estimate of the expected number of positives $P_\mathrm{uniform}(i)=\mathbbm{1}_{[y_i=1]}\times\frac{1}{|\{i:y_i=1\}|},$9; Crisp requires reliable class-prior estimation; LAMC requires a fully labeled calibration split; VLPL and SigRL rely on transferable vision-language priors; and the label-bias critique shows that the uniform positive-selection assumption built into standard benchmarks is itself a source of distortion (Arroyo et al., 2023).

Several limitations are stated directly in the papers. ROLE stores an $\mathbf{x}\in\mathbb{R}^p$00 soft-label matrix and therefore faces a memory limitation at larger scales (Cole et al., 2021). Pseudo Multi-Labels does not discuss per-class calibration or rare labels and notes that hard thresholding can propagate teacher errors (Arroyo, 2023). VLPL notes that domain shift, prompt mismatch, rare or fine-grained labels, and long-tail effects can degrade CLIP-derived pseudo-label quality (Xing et al., 2023). LAMC does not present a formal conformal guarantee theorem in the text and should therefore be read as conformal-inspired empirical calibration rather than a full theorem-driven CP construction (Hagos et al., 12 Sep 2025).

The field’s open questions follow directly from these fault lines. This suggests that realistic annotation modeling, especially non-uniform positive selection, remains unresolved. A plausible implication is that future SPMLL work will need to integrate annotator behavior, calibration, prevalence shift, and semantic structure more tightly than current uniform-retention benchmarks permit. Another plausible implication is that SPMLL will continue to split into at least two subproblems: latent-label recovery during training and reliability control at inference time. The existing literature already contains both trajectories, but no single framework yet eliminates the tension among false negatives, false positives, label bias, calibration, and scalability.