Constructive approximate transport maps with normalizing flows (2412.19366v2)

Abstract: We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows. Specifically, we construct time-dependent controls $\theta=(w, a, b)$ in the vector field $x\mapsto w(a^\top x + b)+$ to approximately transport a known base density $\rho{\mathrm{B}}$ to a target density $\rho_$. The approximation error is measured in relative entropy, and $\theta$ are constructed piecewise constant, with bounds on the number of switches being provided. Our main result relies on an assumption on the relative tail decay of $\rho_$ and $\rho_{\mathrm{B}}$, and provides hints on characterizing the reachable space of the continuity equation in relative entropy.

• The paper constructs time-dependent controls using normalizing flows to approximate the transport map between a base and a target density.
• A key theorem establishes conditions for achieving arbitrarily small approximation error, rigorously quantified using relative entropy.
• This research offers practical applications in machine learning, such as data augmentation and probabilistic inference, potentially enhancing large-scale AI pipelines.

Constructive Approximate Transport Maps with Normalizing Flows

This paper, authored by Antonio Álvarez-López, Borjan Geshkovski, and Domènec Ruiz-Balet, addresses the problem of approximate controllability in the context of the continuity equation, with a focus on constructing transport maps using normalizing flows. The work explores the use of time-dependent controls to transform a known base density into a target density, utilizing a framework previously established for neural ordinary differential equations (neural ODEs). This approach essentially treats discrete layers within neural networks as a continuous temporal variable, aligning with methodologies for density estimation.

Summary of Contributions

The primary contribution of this research lies in the construction of time-dependent controls, denoted as $\theta=(w, a, b)$, formulated to approximately map a base density $\rho_{\mathrm{B}}$ to a desired target density $\rho_*$. Significantly, the approximation error is quantified through relative entropy, and the controls are constructed to be piecewise constant, with bounds provided on the number of temporal switches. The research hinges on an assumption regarding the relative tail decay between the target and base densities.

A key result of the paper is a theorem that establishes conditions under which approximate transportation with an arbitrarily small error can be achieved. The authors exploit the Csizár-Kullback-Pinsker inequality to bridge the topological gap between $L^1$ approximations and relative entropy, allowing for stronger implications in terms of regularity and control.

Numerical Results and Claims

The paper provides rigorous conditions under which the proposed transport mapping framework correctly approximates the target density. It specifies both upper and lower bounds for the control parameters involved, ensuring the robustness of the proposed methods within prescribed error tolerances. The work emphasizes that the number of switches—comparable to transformations in neural network layers—is inherent to the dimensional complexity of the problem, scaling exponentially with increasing dimensions.

Implications and Future Directions

Practically, this research offers promising applications in machine learning, particularly in scenarios demanding efficient mass transport strategies, such as data augmentation and probabilistic inference. Theoretically, the insights could stimulate further exploration into parameter-efficient control strategies in high-dimensional stochastic processes. Upcoming developments might involve extending this framework to accommodate more complex models involving non-Gaussian densities and non-linear vector fields, which could enrich the understanding of dynamical systems beyond current formulations.

The speculative projection of future AI developments, informed by this paper, includes the potential integration of transport map constructions into large-scale machine learning pipelines, enhancing their adaptability and performance in real-world applications.

Conclusion

The work presented in this paper advances the understanding of approximate controllability in normalizing flows, contributing a methodology that melds theoretical rigor with practical advantages for scalable model deployment. The exploration of the interplay between neural ODEs and density estimation underscores a versatile approach adaptable to various domains within AI—a promising foundation for ongoing and future research.