Finite-Time Analysis of Discrete-Time Stochastic Interpolants

The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions. However, prior analyses of this framework have primarily focused on the continuous-time setting, assuming perfect solution of the underlying equations. In this work, we present the first discrete-time analysis of the stochastic interpolant framework, where we introduce a innovative discrete-time sampler and derive a finite-time upper bound on its distribution estimation error. Our result provides a novel quantification on how different factors, including the distance between source and target distributions and estimation accuracy, affect the convergence rate and also offers a new principled way to design efficient schedule for convergence acceleration. Finally, numerical experiments are conducted on the discrete-time sampler to corroborate our theoretical findings.

The stochastic interpolant framework is a powerful method used to build models that can generate new data by transforming one type of data into another. It relies on mathematical equations that describe how systems change over time, helping these models learn how to modify data in useful ways. However, most research on these models has focused on continuous-time settings, assuming that the equations can be solved perfectly.In our work, we shift the focus to how these models behave in discrete-time settings, which is more realistic for many practical applications. We introduce a method for sampling data in this discrete-time framework and show how accurately the model can estimate the desired data distribution over time. Our results also explain how factors like the difference between the starting and target data sets influence how quickly the model can reach the desired outcome. Additionally, we provide strategies for speeding up this process.To validate our findings, we conduct experiments that demonstrate how the theory works in practice. This research offers both a deeper understanding of how these models perform in real-world settings and practical methods for improving their efficiency.