Minimal Sufficient Representations for Self-interpretable Deep Neural Networks

1. Problem Statement

Deep Neural Networks (DNNs) achieve state-of-the-art predictive performance but suffer from a lack of interpretability and statistical rigor. Key challenges include:

• Overparameterization: DNNs contain far more parameters than necessary, leading to unstable estimation, inflated variance, and computational inefficiency.
• Black Box Nature: The intrinsic dimensionality of the data driving predictions is unknown, and irrelevant features often entangle with informative ones.
• Limitations of Existing Interpretability Methods:
  • Post-hoc methods (e.g., LIME, SHAP) provide local approximations that may be inconsistent or misleading and do not guide the training process.
  • Self-interpretable methods often sacrifice predictive accuracy for interpretability or rely on strict sparsity assumptions (selecting only a few strong variables), failing to capture joint/synergistic effects of weak variables.
  • Lack of Statistical Inference: Existing DNN methods generally lack rigorous hypothesis testing procedures for variables or learned representations, limiting their use in high-stakes fields like medicine.

Core Question: Can we recover the minimal sufficient representations in DNNs to embed interpretability and statistical rigor without degrading predictive accuracy?

2. Methodology: The DeepIn Framework

The authors propose DeepIn, a self-interpretable framework that integrates a learnable linear transformation matrix into standard DNNs.

Model Architecture

The model is defined as:
$$Y \sim g(BX)$$
Where:

• $X \in \mathbb{R}^d$ is the high-dimensional input.
• $B \in \mathbb{R}^{d \times d}$ is a learnable matrix.
• $g(\cdot)$ is a Deep Neural Network (approximated using RePU activation functions, i.e., $\sigma(x) = \{max(x,0)\}^p$ with $p \ge 2$, to facilitate derivative estimation).
• Flexibility: Unlike classical index models that pre-specify a low dimension, $B$ can be up to $d \times d$. If $B=I$, the model reduces to a standard DNN ($Y \sim g(X)$), preserving full expressive capacity.

Estimation and Optimization

The estimator $\hat{\mu} = (\hat{\theta}, \hat{B})$ is obtained by minimizing a regularized loss function:
$$\hat{\mu} = \arg\min_{\mu} \left[ \frac{1}{n}\sum_{i=1}^n L(g_\theta(BX_i), Y_i) + \lambda_1 \rho_1(B) + \lambda_2 \rho_2(B) + \lambda_3 \rho_3(\theta) \right]$$

The penalties serve three distinct purposes:

1. Row Penalty ($\rho_1$): Group Lasso on rows of $B$. Zeroing out rows reduces the effective dimension of the representation ($BX$), identifying the Minimal Representation Dimension (MRD).
2. Column Penalty ($\rho_2$): Group Lasso on columns of $B$. Zeroing out columns performs Variable Selection, excluding irrelevant input variables.
3. Network Penalty ($\rho_3$): An architecture-aware penalty on the DNN weights ($\theta$) that compresses the network structure (depth and width) to match the learned representation dimension, reducing statistical error.

Optimization Algorithm

Due to the non-differentiability of Lasso penalties at zero, the authors employ a subgradient descent algorithm combined with parameter truncation. Thresholds ($\tau_1, \tau_2, \tau_3$) are used to explicitly zero out rows, columns, and weights with small norms during training. A sequential grid search is used to tune hyperparameters.

3. Key Contributions

A. Theoretical Guarantees

• Selection Consistency: The authors prove that the estimator consistently identifies the true minimal representation dimension ($d_0$), the set of important variables ($s_0$), and the minimal network size ($S_{net}^0$) as $n \to \infty$, even in high-dimensional settings where $d$ grows exponentially with $n$.
• Optimal Non-Asymptotic Error Bounds: They derive error bounds of the order $\hat{d}^{\beta \vee (1+\lfloor\beta\rfloor)} (n/\log^c n)^{-2\beta/(\hat{d}+2\beta)}$. This demonstrates that the error rate adapts to the learned intrinsic dimension $\hat{d}$ rather than the ambient dimension $d$, significantly improving generalization.
• Asymptotic Normality: The paper establishes the asymptotic normality of the representation matrix estimator $\hat{B}$ and the functional asymptotic normality of $g$, forming the basis for statistical inference.

B. Statistical Inference (Hypothesis Testing)

Building on the asymptotic normality, the authors develop rigorous testing procedures:

1. Variable Significance: Tests $H_0: \|B_{0, [j]}\|_2 = 0$ to determine if specific input variables are significant.
2. Representation Significance: Tests the joint significance of learned representations (combinations of variables) using a split-sample approach to construct test statistics that follow a standard normal distribution under the null.

C. Empirical Performance

• Superior Accuracy: DeepIn consistently outperforms vanilla DNNs, multi-index models, and other interpretable methods (DFS, LassoNet, GCRNet) in both regression and classification tasks.
• Interpretability: The method automatically uncovers human-interpretable patterns (e.g., specific strokes in digits, garment regions in fashion) without post-hoc analysis.

4. Experimental Results

Simulation Studies

• Settings: Tested on additive/interactive regression and classification models with varying correlation structures.
• Performance: DeepIn achieved the lowest Predictive Error (PE) and Mean Squared Error (MSE).
  • It correctly identified the true dimension (e.g., ~5 or ~10) while competitors often retained the full dimension or failed to adapt.
  • It demonstrated superior Variable Selection with higher True Positive Rates (TPR) and significantly lower False Positive Rates (FPR) compared to DFS, LassoNet, and GCRNet.
• Hypothesis Testing: The proposed tests achieved near-perfect power for informative variables and maintained the nominal significance level (Type I error) for noise variables.

Real-World Benchmarks

1. ADNI (Alzheimer's Disease):
   • Task: Predict cognitive decline (MMSE score) from 48,157 gene expression profiles.
   • Result: DeepIn reduced prediction error by 30.26% compared to DFS. It identified 64 significant genes, many of which (e.g., GBA, EIF2AK3) are biologically validated as relevant to AD pathogenesis.
2. BlogFeedback:
   • Task: Predict blog comment counts from text features.
   • Result: Reduced error by 19.65% compared to DFS, using the fewest dimensions and variables.
3. MNIST & FashionMNIST:
   • Task: Image classification.
   • Result: DeepIn achieved the highest accuracy (e.g., 97.30% on MNIST vs. 95.83% for DNN).
   • Interpretability: Visualizations showed the model focusing on discriminative pixels (e.g., the "neck" and "bottom stroke" for digit 5 vs. the "loop" for digit 6), proving it learns meaningful features rather than noise.

5. Significance and Impact

• Bridging the Gap: DeepIn successfully bridges the gap between the high predictive power of deep learning and the need for statistical rigor and interpretability.
• Beyond Sparsity: By modeling joint effects through the matrix $B$, it overcomes the limitations of methods that assume only a few strong signals exist, making it suitable for complex biological and social data.
• Automatic Architecture Design: The framework automatically determines the optimal network size and variable subset, removing the need for manual hyperparameter tuning of architecture.
• Statistical Validity: The provision of formal hypothesis testing allows deep learning models to be deployed in regulated, high-stakes domains (medicine, policy) where evidence-based decision-making is required.

In conclusion, DeepIn demonstrates that interpretability and statistical rigor can be embedded directly into deep architectures by recovering minimal sufficient representations, leading to models that are not only more accurate and stable but also scientifically explainable.