mlsanityData to 13 Oct 2026

Sum-of-Squares Polynomial Flow

arXiv 1905.02325 ↗First posted 7 May 2019cs.LG stat.MLcited by 62

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Abstract (arXiv)

Triangular map is a recent construct in probability theory that allows one to transform any source probability density function to any target density function. Based on triangular maps, we propose a general framework for high-dimensional density estimation, by specifying one-dimensional transformations (equivalently conditional densities) and appropriate conditioner networks. This framework (a) reveals the commonalities and differences of existing autoregressive and flow based methods, (b) allows a unified understanding of the limitations and representation power of these recent approaches and, (c) motivates us to uncover a new Sum-of-Squares (SOS) flow that is interpretable, universal, and easy to train. We perform several synthetic experiments on various density geometries to demonstrate the benefits (and short-comings) of such transformations. SOS flows achieve competitive results in simulations and several real-world datasets.

Key facts. Cited by 62 papers in the map (data through 13 Oct 2026). First posted on arXiv 7 May 2019.

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Citing papers per year: 2021: 23; 2022: 19; 2023: 4; 2024: 12; 2025: 3; 2026: 1.

62 papers.

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  1. 2021Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Modelscited by 252
  2. 2021Flow-based Kernel Prior with Application to Blind Super-Resolutioncited by 74
  3. 2021Hierarchical Conditional Flow: A Unified Framework for Image Super-Resolution and Image Rescalingcited by 73
  4. 2021Smooth Normalizing Flowscited by 53
  5. 2023CRoSS: Diffusion Model Makes Controllable, Robust and Secure Image Steganographycited by 50
  6. 2022Indeterminacy in Generative Models: Characterization and Strong Identifiabilitycited by 33
  7. 2022Universal approximation property of invertible neural networkscited by 30
  8. 2024Sampling in Unit Time with Kernel Fisher-Rao Flowcited by 27
  9. 2023Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inferencecited by 20
  10. 2022Multivariate Quantile Function Forecastercited by 19
  11. 2022On minimax density estimation via measure transportcited by 18
  12. 2024On the Universality of Volume-Preserving and Coupling-Based Normalizing Flowscited by 18
38 more
  1. 2022Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularizationcited by 17
  2. 2021Learning non-Gaussian graphical models via Hessian scores and triangular transportcited by 16
  3. 2021Autoregressive Quantile Flows for Predictive Uncertainty Estimationcited by 15
  4. 2021Sparse Flows: Pruning Continuous-depth Modelscited by 13
  5. 2021Universal Joint Approximation of Manifolds and Densities by Simple Injective Flowscited by 13
  6. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 13
  7. 2024Flexible Tails for Normalizing Flowscited by 11
  8. 2021An Empirical Study: Extensive Deep Temporal Point Processcited by 10
  9. 2022VQ-Flows: Vector Quantized Local Normalizing Flowscited by 10
  10. 2022Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flowscited by 10
  11. 2021Universal Approximation of Residual Flows in Maximum Mean Discrepancycited by 7
  12. 2021Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flowscited by 7
  13. 2024Transformer Neural Autoregressive Flowscited by 7
  14. 2021Robust normalizing flows using Bernstein-type polynomialscited by 6
  15. 2022Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flowscited by 6
  16. 2022Whitening Convergence Rate of Coupling-based Normalizing Flowscited by 6
  17. 2021Unsupervised tree boosting for learning probability distributionscited by 5
  18. 2022AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flowscited by 5
  19. 2024Fast and Unified Path Gradient Estimators for Normalizing Flowscited by 5
  20. 2021Manifold Density Estimation via Generalized Dequantizationcited by 4
  21. 2021Subspace Detours Meet Gromov-Wassersteincited by 4
  22. 2024Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equationcited by 4
  23. 2024Efficient Distribution Matching of Representations via Noise-Injected Deep InfoMaxcited by 4
  24. 2025A friendly introduction to triangular transportcited by 4
  25. 2022Bernstein Flows for Flexible Posteriors in Variational Bayescited by 3
  26. 2024Distribution Guided Active Feature Acquisitioncited by 3
  27. 2021Abelian Neural Networkscited by 2
  28. 2022Universality of parametric Coupling Flows over parametric diffeomorphismscited by 2
  29. 2022Neural Inverse Transform Samplercited by 2
  30. 2023On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networkscited by 2
  31. 2024ISR: Invertible Symbolic Regressioncited by 2
  32. 2021Generalization of the Change of Variables Formula with Applications to Residual Flowscited by 1
  33. 2021Incremental Non-Gaussian Inference for SLAM Using Normalizing Flowscited by 1
  34. 2021Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regressioncited by 1
  35. 2021ELF: Exact-Lipschitz Based Universal Density Approximator Flowcited by 1
  36. 2022Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transportcited by 1
  37. 2022Variational Monte Carlo Approach to Partial Differential Equations with Neural Networkscited by 1
  38. 2022Beyond Hawkes: Neural Multi-event Forecasting on Spatio-temporal Point Processescited by 1

50 most cited of 62 All 62 (JSON)