# Euclidean Distance Geometry

Point cloud recovery from partial pairwise distances

In the applied sciences, a common measurement to make is the distance between objects, such as the distance between sensors in a network or the distance between atoms in a protein. A collection of distances contains a lot of powerful information; namely, if in a point cloud of \(n\) objects in \(d\) dimensions I know every distance between pairs of points, I can reconstruct the underlying object (up to translation/rotation.) The problem becomes considerably more interesting with access to only a few distances.

More mathematically, consider a set of vectors \(\{\mathbf{p}_k\}_{k=1}^n \subset \mathbb{R}^d\). From these vectors we can construct a matrix \(\mathbf{P} = [\mathbf{p}_1 ... \mathbf{p}_n]^T\in\mathbb{R}^{n\times d}\). We seek to recover this matrix from entries of the squared distance matrix \(\mathbf{D} = [d_{ij}^2]= \Vert \mathbf{p}_i - \mathbf{p}_j\Vert_2^2\).

```
---
layout: page
title: project
description: a project with a background image
img: /assets/img/12.jpg
---
```

You can also put regular text between your rows of images. Say you wanted to write a little bit about your project before you posted the rest of the images. You describe how you toiled, sweated, *bled* for your project, and then… you reveal its glory in the next row of images.

The code is simple. Just wrap your images with `<div class="col-sm">`

and place them inside `<div class="row">`

(read more about the Bootstrap Grid system). To make images responsive, add `img-fluid`

class to each; for rounded corners and shadows use `rounded`

and `z-depth-1`

classes. Here’s the code for the last row of images above:

```
<div class="row justify-content-sm-center">
<div class="col-sm-8 mt-3 mt-md-0">
{% include figure.html path="assets/img/6.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
</div>
<div class="col-sm-4 mt-3 mt-md-0">
{% include figure.html path="assets/img/11.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
</div>
</div>
```

## References

## 2023

- Riemannian Optimization for Euclidean Distance Geometry
*In OPT-ML*, 2023