Manifold Sculpting

Overview

An implementation of Manifold Sculpting algorithm as introduced in Iterative Non-linear Dimensionality Reduction by Manifold Sculpting (Gashler et al., 2007).

Manifold Sculpting is a nonlinear dimensionality reduction technique that preserves distances and angles between neighbor points, by using the loss

$$ L(p_i) = \sum_{j = 1}^{k} \omega_i \left(\left(\frac{d_{ij}^0 - d_{ij}}{2 d_{avg}}\right)^2 + \left(\frac{\theta_{ij}^0 - \theta_{ij}}{\pi}\right)^2\right) $$

The algorithm is:

1. Compute relationship between neighbors
2. Optionally perform PCA
3. Until the stopping criterion is met
4. Scale up dimensions to preserve by σ
5. Scale down dimensions to scale by σ
6. Adjust preserved dimensions by shifting the points
7. Drop dimensions to scale

Usage

The required Python packages are listed in requirements.txt, use

pip install -r requirements.txt

to install them.
All the codes can be compiled using the makefile command make <command> from the base directory. The avalilable commands are:

• config: create th Json file holding the necessary paths
• gendata: generate datasets of required shapes
• run: run the manifold sculpting algorithm and save the results
• gif: create a gif using the checkpoints saved while running the manifold sculpting algorithm.

Results

To validate the dataset and test its efficiency againts the theorethical result and against other methods.

Pro

• Accurate even with a small density of points (250 in the picture)
• Number of neighbors is not that influential on the result

Cons

• Slow
• Many hyperparameters to choose
• Must be rotated to align with the XY plane

The full process can be appreciated in the following gif: