On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction

The evolution of nonlinear SDR has been driven by the need to extract meaningful low-dimensional features from complex high-dimensional data. Early methods like SIR and SAVE provided linear solutions but struggled with nonlinear dependencies. Recent developments include kernel-based approaches and deep learning models, which improved flexibility but often lacked theoretical guarantees such as consistency and exhaustiveness. Deep generative models like diffusion and flow-based methods have shown remarkable generative performance, yet their integration into SDR frameworks remains limited. The challenge is to develop methods that can fully recover the relevant information with rigorous statistical guarantees, especially in high-dimensional and non-Euclidean contexts. This paper addresses this gap by proposing a generative, velocity-field-based approach grounded in modern deep learning techniques.

The core challenge in nonlinear SDR is ensuring that the estimated low-dimensional transformation captures all the information relevant to the response, i.e., exhaustiveness, while maintaining statistical consistency. Existing methods often only guarantee unbiasedness or partial coverage, risking information loss or redundancy. Achieving full recovery at the sample level is particularly difficult due to the complexity of conditional distributions in high dimensions. Moreover, extending these guarantees to responses in non-Euclidean spaces adds further difficulty. The key bottleneck is to design an estimator that can leverage the full distributional information without succumbing to curse-of-dimensionality issues or model misspecification. The paper aims to overcome these obstacles by integrating deep generative models with velocity field modeling within the CSI framework.

The main innovations include: 1) embedding the conditional distribution learning into a velocity field framework, which simplifies the complex distributional structure; 2) establishing the nested structure of velocity fields that directly encode the sufficient transformation; 3) proving the theoretical properties of the estimator, including consistency and exhaustiveness, at both sample and population levels; 4) extending the approach to non-Euclidean responses via ensemble strategies and kernel methods, greatly broadening its scope. These innovations collectively enable a robust, theoretically grounded approach to nonlinear SDR, capable of handling high-dimensional, complex, and non-standard data types.

• �� Model the conditional distribution Y|X using a velocity field derived from stochastic interpolation, enabling a dynamic representation of the distribution; • Use neural networks to approximate the velocity field, ensuring flexibility and expressive power; • Construct an interpolation path between a Gaussian noise and the response, defining a time-dependent velocity field; • Minimize a loss function that aligns the estimated velocity field with the true conditional distribution, jointly estimating the transformation R and the generative model parameters; • Leverage the nested structure of the velocity field to directly recover the sufficient transformation R, ensuring full information coverage; • Extend the framework to non-Euclidean responses by projecting responses into Euclidean spaces via ensemble kernels, then applying the same velocity field estimation; • Theoretically prove the estimator’s consistency and exhaustiveness, validating its effectiveness in high-dimensional and complex data scenarios.

Simulated datasets with known nonlinear relationships and real-world datasets such as spatial spatial data and graph-structured data were used to evaluate performance. Baselines included GSIR, GSAVE, GMDDNet, and DDR. Metrics focused on estimation error, information coverage, and predictive accuracy. Hyperparameters for neural networks were tuned via cross-validation. Ablation studies examined the impact of the nested velocity structure and ensemble techniques. Results consistently showed that GenSDR achieved full recovery of the central σ-field, with errors below 0.05 in simulations and accuracy improvements over 20% in real data. The convergence rate aligned with theoretical predictions, confirming the method’s statistical guarantees.

GenSDR demonstrated superior performance in recovering the central σ-field, with sample errors decreasing at the rate of n^{-1/2}. In high-dimensional nonlinear models, it reduced estimation errors by approximately 30% compared to GSIR and GSAVE. Its extension to non-Euclidean responses maintained stable accuracy, outperforming baseline methods by 15-20%. The experiments validated the theoretical claims of consistency and exhaustiveness, showing that the velocity field approach effectively captures all relevant information even in complex, multimodal, and high-dimensional settings. These results establish GenSDR as a robust and scalable tool for nonlinear dimension reduction.

Applicable to spatial data analysis, graph neural networks, and manifold learning, where responses may reside in complex metric spaces. The method requires responses with conditional densities and smooth velocity fields, making it suitable for high-dimensional feature extraction, predictive modeling, and data visualization in scientific and industrial contexts. Its ability to handle non-Euclidean responses makes it valuable for spatial-temporal modeling, social network analysis, and biological data interpretation, facilitating more accurate and interpretable models.

Dependence on the smoothness and uniqueness of the velocity field may limit performance in highly irregular or multimodal distributions. Neural network training in high dimensions is computationally intensive, requiring significant resources. Extension to extremely complex non-Euclidean spaces may face challenges in maintaining robustness and generalization, especially with limited data. Future work should focus on improving computational efficiency, robustness to model misspecification, and extending theoretical guarantees to broader classes of distributions and data structures.