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
---
Caption photos easily. On the left, a road goes through a tunnel. Middle, leaves artistically fall in a hipster photoshoot. Right, in another hipster photoshoot, a lumberjack grasps a handful of pine needles.
This image can also have a caption. It's like magic.

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.

You can also have artistically styled 2/3 + 1/3 images, like these.

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

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