# Goodness-of-fit tests on manifolds

arXiv: https://arxiv.org/abs/1909.05229 · HTML: https://mlsanity.com/paper/1909.05229/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2019.

## Built on by (most cited first, 1 of 1)

- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)

## Abstract (arXiv)

We develop a general theory for the goodness-of-fit test to non-linear models. In particular, we assume that the observations are noisy samples of a submanifold defined by a yaosufficiently smooth non-linear map. The observation noise is additive Gaussian. Our main result shows that the "residual" of the model fit, by solving a non-linear least-square problem, follows a (possibly noncentral) chi2 distribution. The parameters of the chi2 distribution are related to the model order and dimension of the problem. We further present a method to select the model orders sequentially. We demonstrate the broad application of the general theory in machine learning and signal processing, including determining the rank of low-rank (possibly complex-valued) matrices and tensors from noisy, partial, or indirect observations, determining the number of sources in signal demixing, and potential applications in determining the number of hidden nodes in neural networks.