Generation Properties of Stochastic Interpolation under Finite Training Set

Researchers derive closed-form expressions for optimal velocity fields in stochastic interpolation generative models trained on finite datasets, demonstrating that deterministic processes exactly recover training samples while stochastic processes add Gaussian noise. The work formalizes underfitting and overfitting for generative models, showing that estimation errors produce convex combinations of training samples with mixed noise corruption.

This theoretical paper advances the mathematical foundations of generative modeling by rigorously analyzing stochastic interpolation under realistic finite training conditions. Rather than assuming infinite data, the authors characterize how models behave with limited samples—a critical practical concern that bridges theory and implementation. The closed-form derivations provide interpretability into why deterministic processes memorize training data while stochastic variants add controlled noise, offering insights into the fundamental trade-off between fidelity and generalization.

Stochastic interpolation represents an emerging framework competing with diffusion models and other generative approaches. By establishing formal definitions of underfitting and overfitting specific to generative models, this work fills a conceptual gap in the literature. Previous analyses often borrowed terminology from discriminative learning without accounting for generative-specific dynamics. The finding that estimation errors produce convex combinations of training samples corrupted by mixed noise types explains why real-world generative models sometimes collapse toward training data or produce artifacts.

For practitioners building generative systems, these theoretical guarantees inform architecture and training choices. Understanding exactly how finite sample sizes degrade generation quality enables better regularization strategies and helps predict when additional training data provides meaningful improvements. The experimental validation on both generation and downstream classification tasks demonstrates practical relevance beyond theoretical constructs.

Future work should extend these analyses to high-dimensional settings and examine computational implications of the derived velocity fields. The framework's ability to characterize both deterministic and stochastic generation modes positions it as a valuable tool for optimizing generative model design and deployment.

• →Stochastic interpolation generates closed-form expressions for optimal velocity fields under finite training data constraints.
• →Deterministic processes exactly memorize training samples while stochastic variants add controlled Gaussian noise.
• →Formal definitions of underfitting and overfitting for generative models reveal they produce convex combinations of training data with mixed noise.
• →Theoretical analysis validated experimentally on generation and classification tasks demonstrates practical applicability.
• →Framework provides interpretability into generative model behavior that informs regularization and training strategies.