Neural Information Processing Systems http://nips.cc/

The ratio of two probability densities, called a density-ratio, is a vital quantity in machine learning.

Partial-label learning(PLL) is a multi-class classification problem, where each training example isassociated withasetofcandidate labels.

Given only positive examples and unlabeled examples (from both positive and negative classes), we might hope nevertheless to estimate an accurate positiveversus-negative classifier. Formally, this task is broken down into two subtasks: (i) Mixture Proportion Estimation(MPE)--determining the fraction of positive examples in the unlabeled data; and (ii)PU-learning--given such an estimate, learning the desired positive-versus-negative classifier.

Then, the key is to capture the temporal evolution of node pair (term pair) relations from just the positive and unlabeled data. We propose a variational inference model to estimate the positive prior, and incorporate it in the learning of node pairembeddings, which arethenused forlinkprediction.

Weakly supervised learning has emerged as a practical alternative to fully supervised learning when complete and accurate labels are costly or infeasible to acquire. However, many existing methods are tailored to specific supervision patterns -- such as positive-unlabeled (PU), unlabeled-unlabeled (UU), complementary-label (CLL), partial-label (PLL), or similarity-unlabeled annotations -- and rely on post-hoc corrections to mitigate instability induced by indirect supervision. We propose a principled, unified framework that bypasses such post-hoc adjustments by directly formulating a stable surrogate risk grounded in the structure of weakly supervised data. The formulation naturally subsumes diverse settings -- including PU, UU, CLL, PLL, multi-class unlabeled, and tuple-based learning -- under a single optimization objective. We further establish a non-asymptotic generalization bound via Rademacher complexity that clarifies how supervision structure, model capacity, and sample size jointly govern performance. Beyond this, we analyze the effect of class-prior misspecification on the bound, deriving explicit terms that quantify its impact, and we study identifiability, giving sufficient conditions -- most notably via supervision stratification across groups -- under which the target risk is recoverable. Extensive experiments show consistent gains across class priors, dataset scales, and class counts -- without heuristic stabilization -- while exhibiting robustness to overfitting.

Our work is related to one-class classification which is aimed at "describing" the positive class by clustering-related methods, but one-class classification does not have the ability to tune hyper-parameters

Weakly supervised learning often operates with coarse aggregate signals rather than instance labels. We study a setting where each training example is an $n$-tuple containing exactly m positives, while only the count m per tuple is observed. This NTMP (N-tuple with M positives) supervision arises in, e.g., image classification with region proposals and multi-instance measurements. We show that tuple counts admit a trainable unbiased risk estimator (URE) by linking the tuple-generation process to latent instance marginals. Starting from fixed (n,m), we derive a closed-form URE and extend it to variable tuple sizes, variable counts, and their combination. Identification holds whenever the effective mixing rate is separated from the class prior. We establish generalization bounds via Rademacher complexity and prove statistical consistency with standard rates under mild regularity assumptions. To improve finite-sample stability, we introduce simple ReLU corrections to the URE that preserve asymptotic correctness. Across benchmarks converted to NTMP tasks, the approach consistently outperforms representative weak-supervision baselines and yields favorable precision-recall and F1 trade-offs. It remains robust under class-prior imbalance and across diverse tuple configurations, demonstrating that count-only supervision can be exploited effectively through a theoretically grounded and practically stable objective.