## Repairing and Denoising Scattered Data for the Reconstruction of Manifolds Embedded in High Dimensions

- Nov. 26, 2018
- 4 p.m.
- LeConte 317R

## Abstract

High dimensional data is increasingly available in many fields, and the problem of extracting valuable information from such data is of primal interest. A common assumption is that high dimensional data is an embedding of a low dimensional manifold. Often, the data suffers from presence of noise, outliers, and non-uniform sampling (which may result in 'holes' in the manifold). Standard approximation tools fail to address those problems - even in low dimensions. In our research, we propose to reconstruct the manifold's geometry by extending the Locally Optimal Projection operator (LOP) algorithm to the high dimensional case. Additionally, we utilize our framework to address other challenges rising in high dimensional data: a) calculation of k-multivariate L1-medians; b) smooth manifold repairing; c) up/down data sampling. We will demonstrate the effectiveness of our framework by considering noisy data from manifolds of 2-10 dimensions embedded in $\mathbb{R}^{60}$ (Figure 1 in the attachment).

Joint work with: David Levin.