Classification without Labels (CWoLa) shows that, in the binary case, a classifier trained to distinguish two impure mixtures with different class proportions can recover an optimal class discriminator without knowing the mixture proportions, and proposes prior-free procedures that train a standard classifier to distinguish mixture identities and then extract latent class structure using either post-hoc simplex fitting or a bottleneck architecture.
Abstract
In many classification problems, reliable instance-level labels are unavailable. However, it is often possible to construct weakly enriched unlabeled samples: datasets selected by different cuts, sources, populations, or experimental conditions that change latent class proportions without revealing them. Classification without Labels (CWoLa) shows that, in the binary case ($K=2$), a classifier trained to distinguish two impure mixtures with different class proportions can recover an optimal class discriminator without knowing the mixture proportions. We extend this principle to multiclass learning from several unlabeled mixtures ($K>2$), where the learner observes only mixture identity and neither latent class labels nor class-prior matrices. We prove that, for a multiclass mixture model, the Bayes-optimal mixture classifier $g^\star$ maps data points into a $(K-1)$-simplex embedded in mixture-posterior space. The $K$ vertices of this simplex are induced by the latent classes through the unknown mixing matrix. Leveraging this geometry, we propose prior-free procedures that train a standard classifier to distinguish mixture identities and then extract latent class structure using either post-hoc simplex fitting or a bottleneck architecture. Experiments on MNIST, CIFAR-10, and Galaxy10 DECaLS show that mixture identity alone can recover latent classes and their fractions in the mixture. By narrowing the gap between weakly supervised and fully supervised performance, we provide a mathematically grounded, scalable tool for multiclass discovery in label-scarce domains.
This work studies a different regime in which the probability of label missingness depends on posterior classification uncertainty, so that the observed missing-label indicators can themselves carry information about the Bayes decision boundary.
Huanchao Zhou, Jinran Wu, Fariborz Setoudehtazang et al.· 0 citations
We study how a limited labeling budget should be allocated to minimize multiclass zero-one classification risk. We consider parametric classification problems in which features are observed for all sampling units while class labels can be acquired selectively. By combining the Fisher information supplied by an acquired label with the local geometry of multiclass excess risk, we derive an acquisition criterion that minimizes the leading asymptotic coefficient of expected multiclass excess risk. The resulting rule values a label according to how strongly its information is aligned with parameter directions that perturb the active Bayes decision boundary, rather than according to posterior uncertainty or global parameter information alone. We characterize the oracle acquisition design, establish its threshold structure, and derive face-specific and cost-sensitive extensions. An analytic example shows that posterior uncertainty and classification value can produce different, and even reversed, acquisition rankings. We further develop a two-stage adaptive procedure that attains the oracle leading-risk criterion under regularity conditions and provide explicit results for Gaussian discriminant analysis. Three-class QDA experiments illustrate the resulting acquisition geometry, while an application to the six-class Statlog Landsat Satellite data shows that classification-risk acquisition can differ materially from both uncertainty-based acquisition and the complete-classification-information comparator. The adaptive classification-risk design attains lower mean error than this Fisher comparator across the labeling budgets considered, although it does not uniformly outperform entropy or margin sampling and differences among the targeted strategies become small as the labeling budget increases.
F. Setoudehtanzangi, Geoffrey J. McLachlan· 0 citations
Gaussian-mixture calculations and a medical diagnosis example illustrate how uncertainty-dependent labeling mechanisms can improve estimation and classification under a fixed labeling budget.
You‐Gan Wang, Jinran Wu, Geoffrey J. McLachlan· 0 citations
Learning with class-conditional label noise often relies on a transition model from latent clean classes to observed annotations. Forward correction embeds this transition in the likelihood, yet finite-sample networks may still memorize corrupted labels. The corrected likelihood also induces a reverse posterior over the clean classes that could explain each annotation. When its leading class differs from the annotation, the model and transition matrix provide evidence against that annotation, but the leading alternatives can remain nearly tied. We introduce Selective Posterior Margin Regularization (SPMR), which preserves the Forward objective and converts this disagreement into a graded update on the clean classifier. SPMR selects the leading reverse-posterior class, scales a detached pairwise margin by the separation between the two leading posterior classes, and assigns correspondingly little influence to diffuse conflicts. The gap factorizes into transition- adjusted pairwise separation and the posterior mass carried by the leading pair. The active margin follows the locally minimum-norm logit direction that enlarges the selected pairwise margin. Across five known-transition benchmarks, SPMR improves full-length Forward by 2.5-7.0 percentage points and remains 0.7-2.5 percentage points above Forward with Mixup and early stopping. Matched interventions support distinct gains from the posterior-space coefficient, transition-adjusted target, and pairwise action. The same design transfers to estimated transitions, human annotations, architectural changes, and stronger Forward recipes. The formulation uses latent-class evidence already available inside Forward correction without promoting every posterior conflict to a corrected label.
Ze-Xing Zhang, Ji-Chao Li, Tian-Yang Lei et al.· 0 citations
We consider semi-supervised classification from a partially classified sample arising from a two-component Weibull mixture. The feature is observed for all data, whereas some class labels are missing. The probability of a missing label is modelled as a function of classification uncertainty, giving a feature-dependent missing-at-random (MAR) mechanism that shares parameters with the Weibull-mixture classifier. The missing-label indicators can therefore provide information about the classifier in addition to the observed features and available class labels. Under a common Weibull shape, a Bayes'rule has at most one positive decision boundary, which is unique when the rule is nonconstant; under unequal shapes, it can have two. We characterise these decision regions, derive the Fisher information for the classifier after adjustment for nuisance parameters in the missingness model, and obtain a decision-boundary expansion of the expected error rate of the plug-in sample rule relative to the Bayes error. The expansion yields classification-specific asymptotic relative efficiency formulas for the one- and two-boundary cases and shows that a positive-definite increase in Fisher information is sufficient, but not necessary, for a smaller first-order expected error rate. Numerical studies and a semi-synthetic analysis based on hard-drive failure data illustrate potential reductions in expected error rate and improvements in decision-boundary estimation from modelling feature-dependent label missingness.
Jinran Wu, You‐Gan Wang, Geoffrey J. McLachlan· 0 citations
A novel PML method, namely Wasserstein Partial Multi-Label Learning with dual Label Correlation Perspectives (Wpml3cp), solved by the gradient descent with an augmented Lagrange multiplier technique, and empirical results demonstrate that Wpml3cp and Wpml3cp-D can outperform the PML baselines in various noisy levels.
Ximing Li, Yuanchao Dai, Bing Wang et al.· ACM Transactions on Knowledg...· 0 citations
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