mlsanityData to 7 Oct 2026

Constructive approximate transport maps with normalizing flows

arXiv 2412.19366 ↗First posted 26 Dec 2024math.OC stat.MLcited by 4

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Abstract (arXiv)

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 θ=(w, a, b) in the vector field xmapsto w(a^top x + b)_+ to approximately transport a known base density ρB to a target density ρ_*. The approximation error is measured in relative entropy, and θ 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 ρ_* and ρB, and provides hints on characterizing the reachable space of the continuity equation in relative entropy.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 25 of its 42 references that are in the map. First posted on arXiv 26 Dec 2024.

Builds on

25 of 42 references.

References · 25

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2022Flow Matching for Generative Modelingcited by 5,426
  3. 2018Neural Ordinary Differential Equationscited by 3,733
  4. 2015Variational Inference with Normalizing Flowscited by 2,161
  5. 2016Variational Inference: A Review for Statisticianscited by 1,466
  6. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  7. 2014NICE: Non-linear Independent Components Estimationcited by 1,052
  8. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  9. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  10. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  11. 2019Deep Learning via Dynamical Systems: An Approximation Perspectivecited by 78
  12. 2021Neural ODE control for classification, approximation and transportcited by 56
13 more
  1. 2016Improving Variational Auto-Encoders using Householder Flowcited by 41
  2. 2019Deep neural networks, generic universal interpolation, and controlled ODEscited by 28
  3. 2024Measure-to-measure interpolation using Transformerscited by 28
  4. 2023Distribution learning via neural differential equations: a nonparametric statistical perspectivecited by 22
  5. 2023Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein spacecited by 20
  6. 2023Interpolation, Approximation and Controllability of Deep Neural Networkscited by 15
  7. 2022Neural ODE Control for Trajectory Approximation of Continuity Equationcited by 8
  8. 2023Control of neural transport for normalizing flowscited by 8
  9. 2024Interplay between depth and width for interpolation in neural ODEscited by 8
  10. 2021Deep Learning Approximation of Diffeomorphisms via Linear-Control Systemscited by 7
  11. 2017Approximate and exact controllability of the continuity equation with a localized vector fieldcited by 4
  12. 2023Normalizing flows as approximations of optimal transport maps via linear-control neural ODEscited by 2
  13. 2024Some Remarks on Controllability of the Liouville Equationcited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 3; 2026: 1.

4 papers.

  1. 2025Continuous Temporal Learning of Probability Distributions via Neural ODEs with Applications in Continuous Glucose Monitoring Datacited by 0
  2. 2025Machine Learning and Control: Foundations, Advances, and Perspectivescited by 0
  3. 2025Convergence, design and training of continuous-time dropout as a random batch methodcited by 0
  4. 2026Constructive interpolation and generalization rates for neural ODEs: a control perspectivecited by 0