# Distribution-Free Testing of Linear Functions on R^n

arXiv: https://arxiv.org/abs/1909.03391 · HTML: https://mlsanity.com/paper/1909.03391/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Sep 2019.

## Built on by (most cited first, 2 of 2)

- [Superpolynomial Lower Bounds for Decision Tree Learning and Testing](https://mlsanity.com/paper/2210.06375/) (2022)
- [Aligning Model Properties via Conformal Risk Control](https://mlsanity.com/paper/2406.18777/) (2024)

## Abstract (arXiv)

We study the problem of testing whether a function f:R^n->R is linear (i.e., both additive and homogeneous) in the distribution-free property testing model, where the distance between functions is measured with respect to an unknown probability distribution over R. We show that, given query access to f, sampling access to the unknown distribution as well as the standard Gaussian, and eps>0, we can distinguish additive functions from functions that are eps-far from additive functions with O((1/eps)log(1/eps)) queries, independent of n. Furthermore, under the assumption that f is a continuous function, the additivity tester can be extended to a distribution-free tester for linearity using the same number of queries. On the other hand, we show that if we are only allowed to get values of f on sampled points, then any distribution-free tester requires Omega(n) samples, even if the underlying distribution is the standard Gaussian.