Learning from N-Tuple Data with M Positive Instances: Unbiased Risk Estimation and Theoretical Guarantees
This addresses a problem in weakly supervised learning for scenarios like image classification with region proposals, offering a theoretically grounded method to exploit count-only supervision, though it is incremental as it builds on existing URE frameworks.
The paper tackles weakly supervised learning where each training example is an n-tuple with exactly m positives, but only the count m is observed, by deriving an unbiased risk estimator (URE) and proving theoretical guarantees. It shows the approach outperforms baselines on benchmarks, with favorable precision-recall and F1 trade-offs, and remains robust under class imbalance and diverse tuple configurations.
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.