# Fast Distribution To Real Regression

arXiv: https://arxiv.org/abs/1311.2236 · HTML: https://mlsanity.com/paper/1311.2236/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Nov 2013.

## Built on by (most cited first, 5 of 5)

- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Transparent Single-Cell Set Classification with Kernel Mean Embeddings](https://mlsanity.com/paper/2201.07322/) (2022)
- [Distribution-based Sketching of Single-Cell Samples](https://mlsanity.com/paper/2207.00584/) (2022)
- [Coefficient-based Regularized Distribution Regression](https://mlsanity.com/paper/2208.12427/) (2022)
- [Improved learning theory for kernel distribution regression with two-stage sampling](https://mlsanity.com/paper/2308.14335/) (2023)

## Abstract (arXiv)

We study the problem of distribution to real-value regression, where one aims to regress a mapping f that takes in a distribution input covariate P∈ I (for a non-parametric family of distributions I) and outputs a real-valued response Y=f(P) + ε. This setting was recently studied, and a "Kernel-Kernel" estimator was introduced and shown to have a polynomial rate of convergence. However, evaluating a new prediction with the Kernel-Kernel estimator scales as Omega(N). This causes the difficult situation where a large amount of data may be necessary for a low estimation risk, but the computation cost of estimation becomes infeasible when the data-set is too large. To this end, we propose the Double-Basis estimator, which looks to alleviate this big data problem in two ways: first, the Double-Basis estimator is shown to have a computation complexity that is independent of the number of of instances N when evaluating new predictions after training; secondly, the Double-Basis estimator is shown to have a fast rate of convergence for a general class of mappings f∈F.