| | |
| | |
Stat |
Members: 3643 Articles: 2'487'895 Articles rated: 2609
29 March 2024 |
|
| | | |
|
Article overview
| |
|
Fitting a manifold of large reach to noisy data | Charles Fefferman
; Sergei Ivanov
; Matti Lassas
; Hariharan Narayanan
; | Date: |
11 Oct 2019 | Abstract: | Let ${mathcal M}subset {mathbb R}^n$ be a $C^2$-smooth compact submanifold
of dimension $d$. Assume that the volume of ${mathcal M}$ is at most $V$ and
the reach (i.e. the normal injectivity radius) of ${mathcal M}$ is greater
than $ au$. Moreover, let $mu$ be a probability measure on ${mathcal M}$
which density on ${mathcal M}$ is a strictly positive Lipschitz-smooth
function. Let $x_jin {mathcal M}$, $j=1,2,dots,N$ be $N$ independent random
samples from distribution $mu$. Also, let $xi_j$, $j=1,2,dots, N$ be
independent random samples from a Gaussian random variable in ${mathbb R}^n$
having covariance $sigma^2I$, where $sigma$ is less than a certain specified
function of $d, V$ and $ au$. We assume that we are given the data points
$y_j=x_j+xi_j,$ $j=1,2,dots,N$, modelling random points of ${mathcal M}$
with measurement noise. We develop an algorithm which produces from these data,
with high probability, a $d$ dimensional submanifold ${mathcal M}_osubset
{mathbb R}^n$ whose Hausdorff distance to ${mathcal M}$ is less than
$Cdsigma^2/ au$ and whose reach is greater than $c{ au}/d^6$ with universal
constants $C,c > 0$. The number $N$ of random samples required depends almost
linearly on $n$, polynomially on $sigma^{-1}$ and exponentially on $d$ | Source: | arXiv, 1910.5084 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
| News, job offers and information for researchers and scientists:
| |