Technical Summary: Input-Dependent Fisher Information for Local Sensitivity Analysis of Medical Image Classifiers

Problem Statement

Deep neural networks have achieved high performance in medical image classification but often operate as "black boxes." While post-hoc interpretation methods (e.g., Grad-CAM, Integrated Gradients) are widely used to visualize regions associated with model outputs, they often rely on heuristic quantities that vary across architectures and implementations. Crucially, these attribution maps do not always provide a principled characterization of how a classifier's predictive distribution responds to input perturbations. There is a need for a model-intrinsic tool to analyze local decision sensitivity without requiring architectural redesign or additional supervision.

Methodology

The authors propose a local sensitivity analysis framework based on the Input-Dependent Fisher Information Matrix (iFIM). This approach quantifies the local sensitivity of a classifier's predictive distribution to infinitesimal perturbations of the input image.

Core Framework

1. Definition of iFIM: For a trained classifier $f_\theta$ and input image $x$, the iFIM $F(x)$ is defined as the expected outer product of the gradients of the log-predictive distribution with respect to the input:
   $$F(x) = \mathbb{E}_{y \sim p_\theta(\cdot|x)} [\nabla_x \log p_\theta(y|x) \nabla_x \log p_\theta(y|x)^\top]$$
   This matrix characterizes how the predictive distribution changes locally around $x$.

2. Gram-Matrix Factorization: Directly constructing the full $d \times d$ Fisher matrix (where $d$ is the number of pixels) is computationally infeasible for image-scale inputs. The authors utilize a Gram-matrix formulation to recover the nonzero eigenspectrum efficiently. By defining a matrix $G(x)$ containing weighted class gradients, the iFIM is expressed as $F(x) = G(x)G(x)^\top$. The nonzero eigenvalues of the much smaller $C \times C$ Gram matrix $K(x) = G(x)^\top G(x)$ (where $C$ is the number of classes) coincide with those of $F(x)$. This allows for tractable computation of the leading eigenvectors.

3. Image Decomposition: The input image is decomposed into two orthogonal components based on the spectral structure of the iFIM:

   • Task-Sensitive Component ($x_{sens}$): The projection of the image onto the subspace spanned by the leading eigenvectors of the iFIM (associated with the largest eigenvalues). This component corresponds to directions of high local predictive sensitivity.
   • Task-Insensitive Component ($x_{comp}$): The orthogonal complement, containing lower-sensitivity directions and nullspace components.

   The decomposition is defined as $x = x_{sens} + x_{comp}$. The authors emphasize that these are projections onto sensitivity directions, not necessarily semantic segmentations of anatomy.

Key Contributions

• Formulation: An information-theoretic framework using the iFIM to characterize the local sensitivity of a trained medical image classifier's predictive distribution to input perturbations.
• Computational Efficiency: A Gram-matrix factorization method that recovers the nonzero iFIM eigenspectrum without constructing the full image-dimensional Fisher matrix, enabling application to high-dimensional image data.
• Decomposition Definition: A projection-based decomposition of input images into high-sensitivity and orthogonal low-sensitivity components, with clarified interpretation for low-rank settings (e.g., binary classification).
• Evaluation: Comprehensive evaluation across controlled and clinical medical image classification tasks using perturbation-based analyses and comparisons across multiple architectures.

Experimental Results

The framework was evaluated on three tasks:

1. Retinal OCT: Three-class classification (Normal, Drusen, CNV).
2. Chest X-ray: Cardiomegaly detection.
3. Stylized MRI: Signal-known-statistically/background-known-statistically (SKS/BKS) task.

Findings:

• Visualization: In controlled MRI settings, the task-sensitive component concentrated around known signal regions. In clinical tasks (OCT and X-ray), the task-sensitive maps aligned with pathology-bearing regions (e.g., retinal layers for Drusen/CNV, cardiac silhouette for cardiomegaly), while the insensitive component captured background variations.
• Architecture Dependence: The spatial distribution of the high-sensitivity component varied between VGG16 and ResNet18 architectures, even when both achieved similar predictive performance. This suggests the iFIM reveals architecture-specific local decision geometries.
• Perturbation Analysis:
  • Contrastive Perturbation: When pixels with high magnitude in the task-sensitive iFIM-map were perturbed (replaced with noise), the classifier's F1 score degraded significantly more than when pixels from the task-insensitive map were perturbed. The iFIM-based method achieved a higher performance degradation ratio compared to other attribution methods (DeepSHAP, Grad-CAM variants, etc.).
  • Adversarial Robustness: The task-sensitive iFIM-map exhibited the lowest correlation between clean and adversarially perturbed inputs (approaching zero for OCT and X-ray tasks), indicating that the representation is highly sensitive to decision-altering perturbations.

Significance and Claims

The paper posits that the iFIM framework serves as a principled tool for analyzing local decision sensitivity in medical image classifiers. Unlike conventional attribution heatmaps that offer pixel-wise importance scores, the iFIM provides a model-intrinsic, geometric description of how perturbations near a given input influence the predictive distribution.

The authors claim that:

1. High-sensitivity iFIM components are more strongly coupled to changes in predictive confidence and classification performance than low-sensitivity components.
2. The framework complements existing attribution-based interpretability methods rather than replacing them, offering a different perspective focused on the local sensitivity structure of the predictive distribution.
3. The method allows for the analysis of standard trained classifiers without modifying the model architecture.

The authors remain modest regarding semantic or causal claims, noting that while the decomposition aligns with known signal structures in controlled settings and pathology in clinical settings, strong claims about semantic or causal task relevance would require validation against independent reference standards (e.g., expert annotations). The primary contribution is the characterization of local predictive sensitivity.