# Quantum Adaptive Fourier Features for Neural Density Estimation

arXiv: https://arxiv.org/abs/2208.00564 · HTML: https://mlsanity.com/paper/2208.00564/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 1 Aug 2022.

## Builds on (most cited first, 14 of 14)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels](https://mlsanity.com/paper/1412.8293/) (2014)
- [Hashing-Based-Estimators for Kernel Density in High Dimensions](https://mlsanity.com/paper/1808.10530/) (2018)
- [Learning with Density Matrices and Random Features](https://mlsanity.com/paper/2102.04394/) (2021)
- [ASKIT: Approximate Skeletonization Kernel-Independent Treecode in High Dimensions](https://mlsanity.com/paper/1410.0260/) (2014)
- [Random Fourier Features via Fast Surrogate Leverage Weighted Sampling](https://mlsanity.com/paper/1911.09158/) (2019)
- [Not-So-Random Features](https://mlsanity.com/paper/1710.10230/) (2017)

## Built on by (most cited first, 6 of 6)

- [AD-DMKDE: Anomaly Detection through Density Matrices and Fourier Features](https://mlsanity.com/paper/2210.14796/) (2022)
- [Fast Kernel Density Estimation with Density Matrices and Random Fourier Features](https://mlsanity.com/paper/2208.01206/) (2022)
- [InQMAD: Incremental Quantum Measurement Anomaly Detection](https://mlsanity.com/paper/2210.05061/) (2022)
- [LEAN-DMKDE: Quantum Latent Density Estimation for Anomaly Detection](https://mlsanity.com/paper/2211.08525/) (2022)
- [MEMO-QCD: Quantum Density Estimation through Memetic Optimisation for Quantum Circuit Design](https://mlsanity.com/paper/2406.08591/) (2024)
- [Latent Anomaly Detection Through Density Matrices](https://mlsanity.com/paper/2408.07623/) (2024)

## Abstract (arXiv)

Density estimation is a fundamental task in statistics and machine learning applications. Kernel density estimation is a powerful tool for non-parametric density estimation in low dimensions; however, its performance is poor in higher dimensions. Moreover, its prediction complexity scale linearly with more training data points. This paper presents a method for neural density estimation that can be seen as a type of kernel density estimation, but without the high prediction computational complexity. The method is based on density matrices, a formalism used in quantum mechanics, and adaptive Fourier features. The method can be trained without optimization, but it could be also integrated with deep learning architectures and trained using gradient descent. Thus, it could be seen as a form of neural density estimation method. The method was evaluated in different synthetic and real datasets, and its performance compared against state-of-the-art neural density estimation methods, obtaining competitive results.