Interpretable weakly-supervised learning through kernel density matrices: A digital pathology use case

Classification methods based on deep learning require selecting between fully-supervised or weakly-supervised approaches, each presenting limitations in uncertainty quantification and interpretability. A framework unifying both supervision modes while maintaining quantifiable interpretation metrics remains unexplored. We introduce WiSDoM (Weakly-Supervised Density Matrices), which uses kernel matrices to model probability distributions of input data and their labels. The framework integrates: (1) differentiable kernel density matrices enabling stochastic gradient descent optimization, (2) local-global attention mechanisms for multi-scale feature weighting, (3) data-driven prototype generation through kernel space sampling, and (4) ordinal regression through density matrix operations. WiSDoM was validated through supervised patch classification (κ = 0.896) and weakly-supervised whole-slide classification (κ = 0.930) on histopathology images. WiSDoM generates three quantifiable outputs: posterior probability distributions, variance-based uncertainty maps, and phenotype prototypes. Through validation in a Gleason grading task at a patch and whole-slide level using histopathology images, WiSDoM demonstrated consistent performance across supervision modes (κ > 0.89) and prototype interpretability (0.88 expert agreement). These results show that kernel density matrices can serve as a foundation for classification models requiring both prediction interpretability and uncertainty quantification across supervision modes.