Quadrics for Manifold Learning in Feature Space

This repository contains a PyTorch-based implementation of the algorithms and examples from the paper Quadric Hypersurface Intersection for Manifold Learning in Feature Space.

Requirements

You will need Python 3.6+ and the following packages:

• numpy
• pytorch 1.7
• sklearn 0.24.2 if you need kernel models
• matplotlib for plotting in the .ipynb example notebooks
• tqdm
• scipy for the toy_example.ipynb notebook
• pandas for tables

Quadric Models

The folder named quadrics contains an implementation of quadric intersection manifold:

• model.py is the implementation of quadrics as torch.nn.Module,
• quadrics.py is a wrapper with sklearn-like interface (including trainer implementation).

Examples

The folder named examples contains Jupyter notebooks with the following examples.

• toy_example.ipynb: approximating the seam line of a tennis ball using an intersection of two quadrics.
• similarity_robustification.ipynb: demonstration and comparison of different similarity robustification methods.
• outlier_detection.ipynb: demonstration and comparison of different outlier detection methods.

Data

To download the precomputed ArcFace embeddings follow this link. Put the downloaded file into the project folder. The archive contains the embeedings generated by ArcFace for the given datasets:

• MS1M-ArcFace
• megaface
• cplfw
• calfw
• flickr
• cplfw outliers and anime faces

Here is the link for image datasets.

Apart from the standard datasets we used a new hand-crafted embedding dataset cplfw and anime outliers. It was constructed by selecting (manually) the outliers from CPLFW (they can be found in labels/cplfw_outliers_labels.json, notation corresponds to labels/cplfw_labels_full.txt) and mixing them with anime-face-dataset.

Pre-trained quadric models (with parameters from config.json) may be downloaded here.

To extract the archive and to prepare the folders for the models and features calculation execute .sh file:

• ./prepare_folders.sh

Scripts

Use create_models.py script to fit the manifolds. With default parameters it will fit OneClassSVM and PCA.

For fitting quadrics with the distance of order 2 use the following command:

python create_models.py --methods quadrics

For fitting quadrics with the algebraic distance use the following command:

python create_models.py --methods quadrics_algebraic

Warning: quadrics fitting requires a lot of time! Fitting an intersection of 100 quadrics took us 72 hours on a pair of Quadro RTX 8000 GPUs.

Citation

@article{pavutnitskiy2021quadric,
  title={Quadric Hypersurface Intersection for Manifold Learning in Feature Space},
  author={Pavutnitskiy, Fedor and Ivanov, Sergei O and Abramov, Evgeny and Borovitskiy, Viacheslav and Klochkov, Artem and Vialov, Viktor and Zaikovskii, Anatolii and Petiushko, Aleksandr},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  year={2022},
  organization={PMLR}
}