Multiclass Classification without Labels via Posterior Simplex Geometry

2026-07-27

AI Summary

This work addresses multiclass classification under the challenging setting of multi-source mixed data without instance-level labels and with unknown mixing proportions. It extends the Classification without Labels (CWoLa) principle to the multiclass regime for the first time, revealing that the posterior outputs of an optimal classifier naturally form a (K−1)-dimensional simplex whose vertices correspond to the latent classes. Leveraging this geometric structure, the authors propose a novel class discovery method that requires no prior knowledge, combining standard classifier training, posterior simplex fitting, and a bottleneck network architecture, using only the mixture source identity as the supervisory signal. Experiments on MNIST, CIFAR-10, and Galaxy10 DECaLS datasets demonstrate successful recovery of both latent classes and their mixing proportions, substantially narrowing the performance gap between weakly supervised and fully supervised learning.

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.

Problem

• multiclass classification
• classification without labels
• unlabeled mixtures
• latent class discovery
• weak supervision

Innovation

• Classification without Labels
• multiclass learning
• posterior simplex geometry
• mixture models
• weak supervision