# Sum-of-Squares Polynomial Flow

arXiv: https://arxiv.org/abs/1905.02325 · HTML: https://mlsanity.com/paper/1905.02325/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2019.

## Built on by (most cited first, 30 of 62)

- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Flow-based Kernel Prior with Application to Blind Super-Resolution](https://mlsanity.com/paper/2103.15977/) (2021)
- [Hierarchical Conditional Flow: A Unified Framework for Image Super-Resolution and Image Rescaling](https://mlsanity.com/paper/2108.05301/) (2021)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [CRoSS: Diffusion Model Makes Controllable, Robust and Secure Image Steganography](https://mlsanity.com/paper/2305.16936/) (2023)
- [Indeterminacy in Generative Models: Characterization and Strong Identifiability](https://mlsanity.com/paper/2206.00801/) (2022)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [Multivariate Quantile Function Forecaster](https://mlsanity.com/paper/2202.11316/) (2022)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows](https://mlsanity.com/paper/2402.06578/) (2024)
- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Learning non-Gaussian graphical models via Hessian scores and triangular transport](https://mlsanity.com/paper/2101.03093/) (2021)
- [Autoregressive Quantile Flows for Predictive Uncertainty Estimation](https://mlsanity.com/paper/2112.04643/) (2021)
- [Sparse Flows: Pruning Continuous-depth Models](https://mlsanity.com/paper/2106.12718/) (2021)
- [Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows](https://mlsanity.com/paper/2110.04227/) (2021)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Flexible Tails for Normalizing Flows](https://mlsanity.com/paper/2406.16971/) (2024)
- [Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flows](https://mlsanity.com/paper/2205.07918/) (2022)
- [An Empirical Study: Extensive Deep Temporal Point Process](https://mlsanity.com/paper/2110.09823/) (2021)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Universal Approximation of Residual Flows in Maximum Mean Discrepancy](https://mlsanity.com/paper/2103.05793/) (2021)
- [Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flows](https://mlsanity.com/paper/2107.02951/) (2021)
- [Transformer Neural Autoregressive Flows](https://mlsanity.com/paper/2401.01855/) (2024)
- [Robust normalizing flows using Bernstein-type polynomials](https://mlsanity.com/paper/2102.03509/) (2021)
- [Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows](https://mlsanity.com/paper/2204.13939/) (2022)
- [Whitening Convergence Rate of Coupling-based Normalizing Flows](https://mlsanity.com/paper/2210.14032/) (2022)
- [Unsupervised tree boosting for learning probability distributions](https://mlsanity.com/paper/2101.11083/) (2021)
- [AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows](https://mlsanity.com/paper/2206.02102/) (2022)

## 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.