--- title: Positive-Only Binary Classification url: https://www.emergentmind.com/topics/binary-classification-from-positive-only-samples type: topic --- # Positive-Only Binary Classification Binary classification from positive-only samples denotes learning regimes in which training information is restricted to positive examples, while negative information is absent, latent, unlabeled, confidence-weighted, or only indirectly observed. In one formulation, the learner must “learn a binary classifier from only positive data, without any negative data or unlabeled data” by exploiting positive-confidence scores [1710.07138]. In another, “binary classification from positive-only samples is a variant of PAC learning in which the learner receives i.i.d. samples from the positive region of an unknown target concept, but is evaluated under the original distribution” [2606.28309]. A broader applied literature places the problem in positive-unlabeled, exposure-biased, or positive-labeling-source settings, but the common core is the same asymmetry: positives are observed directly, negatives are not. ## 1. Observation models and problem formulations The most restrictive observation model is positive-confidence (Pconf) classification. One observes only positive examples \(x\sim p(x\mid y=+1)\), each equipped with a confidence score \(r(x)=P(y=+1\mid x)\), and learns a real-valued decision function whose sign predicts \(y\in\{+1,-1\}\) [1710.07138]. This differs from ordinary supervised learning because neither negative examples nor unlabeled examples are available at training time. A second family is positive-unlabeled (PU) learning. Here the input consists of a labeled positive set \(P\) and an unlabeled set \(U\), where the unlabeled distribution is a mixture of positive and negative classes, for example \(P_u=\alpha P_p+(1-\alpha)P_n\) or \(p(x)=\pi p_p(x)+(1-\pi)p_n(x)\) depending on notation [2208.12477] [2004.09820]. The task remains binary classification, but the missingness pattern shifts from “no negatives and no unlabeled data” to “no verified negatives.” Several extensions refine how positives become observed. In positive-unlabeled-exposure (PUE) classification, the observed indicator is \(Z=Y\cdot S\), where \(S\) is an exposure or selection variable, so observed positives are jointly filtered by exposure and labeling [2303.04797]. In weak supervision with positive labeling sources only, one has labeling functions \(\lambda_j:\mathcal X\to\{0,1\}\) that either emit a positive vote or abstain; they never output a negative label [2208.01704]. In the positive-only PAC model, the learner receives \(n\) i.i.d. positives from the conditional distribution \(D_+\) and is evaluated under the original distribution \(D\), with error \(D(h\Delta c^*)\) [2606.28309]. A common misconception is that these settings collapse to one-class classification. The Pconf formulation explicitly distinguishes them: one-class classification is aimed at “describing” the positive class by clustering-related methods, but it “does not have the ability to tune hyper-parameters and their aim is not on ‘discriminating’ positive and negative classes” [1710.07138]. This suggests that positive-only binary classification is best understood as a discriminative problem under asymmetric observation, rather than as novelty detection alone. ## 2. Risk rewriting and estimators from positive information The central technical device in positive-only learning is a risk identity that replaces unavailable negative expectations by quantities computable from positive observations. In Pconf classification, the ordinary risk \[ R(g)=\mathbb E_{p(x,y)}[\ell(y\,g(x))] \] can be rewritten as \[ R(g)=\pi_+\,\mathbb E_{x\sim p(x\mid y=+1)}\!\left[\ell(g(x))+\frac{1-r(x)}{r(x)}\,\ell(-g(x))\right], \] which yields the empirical estimator \[ \widehat R(f)=\frac{1}{n}\sum_{i=1}^n\left[\ell(f(x_i))+\frac{1-c_i}{c_i}\,\ell(-f(x_i))\right]. \] This estimator is unbiased when the reported confidences are exact; by contrast, the naive alternative \[ \frac{1}{n}\sum_i \big[c_i\ell(f(x_i))+(1-c_i)\ell(-f(x_i))\big] \] is “biased and inconsistent” [1710.07138]. PU learning uses a related decomposition. One standard unbiased PU estimator writes \[ R(g)=\pi\,E_p[\ell(g,+1)]-\pi\,E_p[\ell(g,-1)]+E_u[\ell(g,-1)], \] and its empirical forms under exponential, logistic, or cross-entropy surrogates underlie methods such as uPU, nnPU, AdaPU, and iFPU [2205.09485] [2605.14467]. The nnPU correction replaces the raw negative-risk estimate by a non-negative version, typically through a term of the form \(\max\{0,\cdot\}\), to control overfitting in flexible models [2605.14467]. A distinct route avoids explicit prior-dependent negative-risk reconstruction. Variational PU learning introduces \[ \mathcal L_{\mathrm{var}}(\Phi)=\log\big(\mathbb E_{x\sim f}[\Phi(x)]\big)-\mathbb E_{x\sim f_P}[\log \Phi(x)], \] with empirical version \[ \widehat{\mathcal L}_{\mathrm{var}}(\Phi)=\log\Big(\tfrac1N\sum_{j=1}^N\Phi(x_j)\Big)-\tfrac1M\sum_{i=1}^M\log\Phi(x_i). \] Minimizing \(\mathcal L_{\mathrm{var}}\) is equivalent to minimizing a KL gap between the positive density and the model-induced density \(f_\Phi(x)\propto \Phi(x)f(x)\), and “no \(\pi_P\) or negative density \(f_N\) appears” in the objective [1906.00642]. These estimators establish the main methodological divide in the literature. Some methods reconstruct the supervised risk exactly or unbiasedly under structural assumptions; others replace that objective by a surrogate criterion that is computable from positive-only observations and still targets the Bayesian classifier. ## 3. Confidence distortion, selection bias, and identifiability Positive-only formulations are highly sensitive to how the positive signal is produced. In Pconf classification, the reported confidences may be systematically distorted by annotation bias. “Binary Classification from Positive Data with Skewed Confidence” models this by the power-law transformation \[ \hat r=r^\alpha,\qquad \alpha>0, \] or equivalently \(r=\hat r^{1/\alpha}\), and in the paper’s notation uses adjusted confidences \(r_i^{(k)}=(\hat r_i)^k\) [2001.10642]. The corresponding adjusted empirical risk is \[ \widehat R_k(g)=\frac{1}{n}\sum_{i=1}^n\left[\ell(g(x_i))+\frac{1-r_i^{(k)}}{r_i^{(k)}}\,\ell(-g(x_i))\right]. \] Since negative examples are unavailable for validation, the hyperparameter is selected by matching the empirical false-negative rate on positives, \[ \widehat\phi(k)=\frac{1}{n}\sum_{i=1}^n \ell_{01}(g_k(x_i)), \] to known prior knowledge \(\phi=P_+\{g(x)<0\}\) through \[ k^*=\arg\min_{k>0}\big(\widehat\phi(k)-\phi\big)^2 \] [2001.10642]. Class-prior estimation is another identifiability bottleneck. The mixture-proportion view of PU learning writes the unlabeled density as \(f(x)=\alpha f_1(x)+(1-\alpha)f_0(x)\), with \(f_1\) observed and \(\alpha\) unknown. “Nonparametric semi-supervised learning of class proportions” shows that estimation is generally ill-defined unless one adopts the “max-canonical form” \[ \alpha=\alpha^*=\inf R(f,f_1), \] equivalently \(\alpha^*=\inf_{x:f_1(x)>0} f(x)/f_1(x)\) in the absolutely continuous case, and proposes the AlphaMax procedure based on an elbow in a constrained likelihood curve \(\ell(c_j)\) [1601.01944]. This suggests that identifiability in positive-only learning is often a modeling choice rather than a direct consequence of the data. Distribution shift further complicates the problem. “Learning from Positive and Unlabeled Data with Arbitrary Positive Shift” allows \(P_{\mathrm{tr}}^+(X)\) and \(P_{\mathrm{te}}^+(X)\) to be arbitrarily different while assuming only \[ P_{\mathrm{tr}}^-(X)=P_{\mathrm{te}}^-(X), \] and develops statistically consistent estimators based on weighted unlabeled-unlabeled learning and a recursive risk estimator with absolute-value correction [2002.10261]. In PUE classification, the issue is not only class-prior ambiguity but also selection bias in the observed positives. Under strong ignorability, \[ Y\perp\!\!\!\perp S\mid X, \] one has \[ P(Y=1\mid X=x)=\frac{P(Z=1\mid x)}{\theta(1\mid x)}, \] with \(Z=Y\cdot S\), and ADPUE constructs an iterative empirical risk that “automatically” debiases without explicit propensity estimation [2303.04797]. ## 4. Methodological families beyond classical ERM A large methodological literature extends positive-only binary classification beyond direct risk minimization. Boosting-based PU learning is represented by AdaPU, which minimizes a PU analogue of the empirical exponential loss \[ L_{\exp}(H)=\frac{\pi}{n_p}\sum_{x\in P}e^{-H(x)}+\frac{1}{n_u}\sum_{x\in U}e^{H(x)}-\frac{\pi}{n_p}\sum_{x\in P}e^{H(x)}. \] Its derived weighted PN dataset includes pseudo-negative examples with negative weight, and the weak learner selection requires both \(e_t(h)\in[0,0.5)\) and \(e_t^{nn}(h)\ge 0\) to prevent dangerous exploitation of negative weights [2205.09485]. Representation-learning and generative approaches replace handcrafted risk decompositions by learned structure. Observer-GAN introduces a generator \(G\), discriminator \(D\), and observer \(O\); \(D\) enforces that \(G(z)\) lie within the support of the unlabeled data, while \(O\) prevents \(G(z)\) from falling into the positive distribution and is returned as the final classifier [2208.12477]. Positive Unlabeled Contrastive Learning defines a PU-specific InfoNCE-style loss \(L_{\mathrm{puNCE}}=(1/2b)(L_P+L_U)\) in which unlabeled anchors are treated as a \(\pi\)-weighted mixture of positive and negative contributions, and then trains a linear head with a cost-sensitive PU risk such as nnPU [2206.01206]. Tensor-network PU learning constructs positive and negative projectors \(P^p\) and \(P^n\) in a feature-space tensor embedding and predicts \(+1\) when \(\|\hat y^p(x)\|_2>\|\hat y^n(x)\|_2\); the same model is also generative and can sample new positive and negative instances [2211.14085]. Other methods use ranking, graph structure, or repeated pseudo-labeling. “Beyond Myopia” trains on balanced positive-unlabeled minibatches, records full score trajectories \(p_1,\dots,p_T\), interprets them as a temporal point process, and computes a robust trend score \[ \hat S=\frac{2}{T(T-1)}\sum_{1\le i0\), one has \[ R(f_n)-R(f^*)=O_p(\mathfrak R_n(\mathcal F)+1/\sqrt n), \] and for linear models in an RKHS, \[ R(f_n)-R(f^*)=O_p(1/\sqrt n), \] so minimizing the Pconf empirical risk is statistically consistent [1710.07138]. The skewed-confidence extension preserves unbiasedness when the correction exponent matches the true skew parameter, and its tuning rule uses only positives plus a scalar false-negative prior [2001.10642]. The most general structural theory is currently given for the positive-only PAC model. “Surprises in Proper Positive-Only Learning” proves the characterization \[ \text{proper positive-only learnability} \iff \text{finite VC dimension and uniform exterior-separability}, \] where, for every finite \(S\subseteq X\), the closure \[ \mathrm{Clo}_{\mathcal C}(S)=\bigcap_{c\in\mathcal C_S} c,\qquad \mathcal C_S=\{c\in\mathcal C:S\subseteq c\}, \] captures the points forced to be positive by consistency with \(S\) [2606.28309]. The paper also introduces exact exterior separation, uniform exterior separability (UES), and distributional exterior separability (DES), with \[ \text{exact ES}\Rightarrow \text{UES}\Rightarrow \text{DES}\Rightarrow \text{finite ES}, \] and shows that when \(\mathrm{VCdim}(\mathcal C)<\infty\), \(\mathrm{DES}\Leftrightarrow \mathrm{UES}\) [2606.28309]. This characterization yields several separations absent from ordinary PAC learning. Proper and improper learning are separated; randomized and deterministic proper learning are separated; there are classes for which no deterministic proper ERM rule succeeds; and finite VC dimension does not suffice even for non-uniform learning [2606.28309]. A common oversimplification is therefore incorrect: finite VC dimension alone is not the full story for positive-only learning. Complementary asymptotic theory also appears in broader PU settings. The trend score in holistic PUL concentrates around \(\alpha E[\Delta p]\) and satisfies a high-probability bound of order \[ O\!\left(\alpha\sigma\sqrt{\frac{\log(1/\varepsilon)}{T(T-1)}}\right), \] so \(\hat S/\alpha\to E[\Delta p]\) in probability [2310.04078]. Variational PU learning derives its objective from a KL divergence and states that the minimizer of \(\mathcal L_{\mathrm{var}}\) on \([0,1]\)-valued \(\Phi\) is, up to a positive scale, the true posterior \(\Phi^*\) [1906.00642]. These results indicate that positive-only theory now ranges from finite-sample estimator analysis to abstract learnability characterizations. ## 6. Empirical behavior, applications, and recurrent issues Empirical work consistently shows that positive-only methods can approach or match supervised baselines when their assumptions are approximately satisfied. On synthetic 2D Gaussians, Pconf classification “achieves accuracy within 1–2% of fully supervised,” and on Fashion-MNIST and CIFAR-10 it “beat the naive weighted-confidence baseline on 70–80% of tasks,” with several cases matching a fully supervised network [1710.07138]. When confidence is skewed, original Pconf can deteriorate sharply, but adjusted Pconf “recovers performance nearly to that of fully supervised learning” on synthetic problems and “restores >90 % accuracy on Fashion-MNIST” while substantially outperforming original Pconf on CIFAR-10 [2001.10642]. Several papers emphasize robustness under severe imbalance or bias. Holistic PUL reports improvements of “up to \(11.3\%\) in key metrics,” including Credit-Card-Fraud Recall from \(\sim 87.7\%\) to \(99.0\%\), \(F_1\) from \(\sim 91.9\%\) to \(99.1\%\), and AUC from \(\sim 97.5\%\) to \(99.7\%\) [2310.04078]. ProbTagging can “increase the AUC by up to 10%” on industrial and artificial PU data sets and degrades more slowly as the fraction of observed positives decreases [2004.09820]. Focused PU learning from imbalanced data reports that XGB+iFPU attains \(19.95\%\) R-precision on financial misstatement detection, compared with \(19.1\%\) for Calibrated-iFPU and \(18.2\%\) for PUHRF [2605.14467]. Real-world applications span multiple modalities. Adjusted Pconf was applied to drivers’ drowsiness prediction using “7 heart-rate-variability features” and expert-rated “sleepiness” scores for positive samples only; raw Pconf predicted “all-Alert” and yielded zero F-measure, whereas adjusted Pconf obtained F-measures of \(48\)–\(68\%\) [2001.10642]. Observer-GAN reports observer accuracies of \(89.6\pm0.7\%\) on CIFAR-10 and \(91.0\pm1.1\%\) on AFHQ [2208.12477]. Tensor-network PU learning reports average one-vs-one MNIST test accuracy \(0.99\pm0.01\) for \(N_p=100\), and an average \(F_1\) of \(0.90\) over \(45\) categorical or mixed-data tasks [2211.14085]. The 2-HNC method achieves the top accuracy on \(7/10\) real PU benchmarks and the top balanced accuracy on \(8/10\) [2505.08212]. Recurrent limitations are equally consistent across the literature. Pconf requires reliable estimates of \(p(y=+1\mid x)\); when \(c\to 0\), the weight \((1-c)/c\) can inflate and destabilize training [1710.07138]. Many PU methods assume a known or estimable class prior \(\pi\), and several papers focus explicitly on prior estimation or robustness to misspecification [1601.01944] [2605.14467]. Shift assumptions such as SCAR, representative positives, or exposure ignorability are often essential and often violated, motivating later work on arbitrary positive shift, SAR, and exposure-biased observation [2002.10261] [2303.04797] [2605.14467]. The overall empirical record therefore supports a narrow conclusion: binary classification from positive-only samples is feasible and often competitive, but only through estimators and models that explicitly encode how positive evidence is generated, distorted, or selected. Source: https://www.emergentmind.com/topics/binary-classification-from-positive-only-samples