Hyperbolic Secant representation of the logistic function: Application to probabilistic Multiple Instance Learning for CT intracranial hemorrhage detection

TL;DR

The paper tackles probabilistic MIL for CT intracranial hemorrhage detection by reformulating GP-based MIL with a logistic likelihood. It introduces a Pólya-Gamma augmented model (PG-VGPMIL) and proves its variational updates are equivalent to the original VGPMIL due to the Hyperbolic Secant representations, motivating a general $\psi$-VGPMIL framework that replaces the GSM density with other options. As a concrete realization, it derives Gamma-VGPMIL (G-VGPMIL), which improves predictive performance and efficiency across multiple datasets, including MNIST MIL, MUSK1/2, RSNA, and CQ500. The work demonstrates that a flexible GSM-based inference framework can yield competitive or superior results while enabling instantiations tailored to data characteristics, and points to future extensions such as multi-class problems and alternative GSM choices. Overall, the proposed approach narrows the gap between weakly supervised MIL methods and fully supervised baselines in medical imaging contexts while offering robust uncertainty quantification.

Abstract

Multiple Instance Learning (MIL) is a weakly supervised paradigm that has been successfully applied to many different scientific areas and is particularly well suited to medical imaging. Probabilistic MIL methods, and more specifically Gaussian Processes (GPs), have achieved excellent results due to their high expressiveness and uncertainty quantification capabilities. One of the most successful GP-based MIL methods, VGPMIL, resorts to a variational bound to handle the intractability of the logistic function. Here, we formulate VGPMIL using Pólya-Gamma random variables. This approach yields the same variational posterior approximations as the original VGPMIL, which is a consequence of the two representations that the Hyperbolic Secant distribution admits. This leads us to propose a general GP-based MIL method that takes different forms by simply leveraging distributions other than the Hyperbolic Secant one. Using the Gamma distribution we arrive at a new approach that obtains competitive or superior predictive performance and efficiency. This is validated in a comprehensive experimental study including one synthetic MIL dataset, two well-known MIL benchmarks, and a real-world medical problem. We expect that this work provides useful ideas beyond MIL that can foster further research in the field.

Figures (12)

• Figure 1: Plot of the functions $\theta_{\textnormal{PG}}(x) = \tan(x/2)/(2x)$ (black) and $\theta_{\textnormal{G}}(x; \alpha, \beta) = \alpha/(\beta + x^2/2)$ for $\alpha = 1$ and $\beta \in \left\{ 1, 2, 4, 8 \right\}$ (blue, orange, green, purple).
• Figure 2: MNIST bags and G-VGPMIL predictions (probability to be positive). Positive instances are highlighted with a red frame.
• Figure 3: Training time vs AUC in the MNIST dataset.
• Figure 4: Bag level and instance level performance in MNIST.
• Figure 5: Training time vs AUC for MUSK1 and MUSK2.