High-dimensional datasets often reside on a low-dimensional geometrical manifold. Manifold learning algorithms aim to retrieve this underlying structure by mapping the data into lower dimensions while minimizing some measure of local (and possibly global) distortion incurred by the map. Bottom-up  approaches address this problem by first constructing low-distortion low-dimensional local views of the data and then integrating them together to obtain a global embedding.

In our work, we investigate the following questions:

1. How to obtain low-distortion low-dimensional local views of high-dimensional data that are robust to noise.

2. How to integrate these local views in an efficient manner to produce a low-dimensional global embedding with distortion guarantees.

3. How does the distortion incurred in the low-dimensional embedding impacts the performance of the downstream tasks.