decompy

decompy is a Python package containing several robust algorithms for matrix decomposition and analysis. The types of algorithms include

• Robust PCA or SVD-based methods

• Matrix completion methods

• Robust matrix or tensor factorization methods.

• Matrix rank estimation methods.

Features

• Data decomposition using various methods

• Support for sparse decomposition, low-rank approximation, and more

• User-friendly API for easy integration into your projects

• Extensive documentation and examples

Installation

You can install decompy using pip:

pip install decompy

Usage

Here’s a simple example demonstrating how to use decompy for data decomposition:

import numpy as np
from decompy.matrix_factorization import RobustSVDDensityPowerDivergence

# Load your data
data = np.arange(100).reshape(20,5).astype(np.float64)

# Perform data decomposition
algo = RobustSVDDensityPowerDivergence(alpha = 0.5)
result = algo.decompose(data)

# Access the decomposed components
U, V = result.singular_vectors(type = "both")
S = result.singular_values()
low_rank_component = U @ S @ V.T
sparse_component = data - low_rank_component

print(low_rank_component)
print(sparse_component)

While the singular values are about 573 and 7.11 for this case (check the S variable), it can get highly affected if you use the simple SVD and change a single entry of the data matrix.

s2 = np.linalg.svd(data, compute_uv = False)
print(np.round(s2, 2))    # estimated by usual SVD
print(np.diag(np.round(S, 2)))    # estimated by robust SVD


data[1, 1] = 10000  # just change a single entry
s3 = np.linalg.svd(data, compute_uv = False)
print(np.round(s3, 2))   # usual SVD shoots up
s4 = algo.decompose(data).singular_values()
print(np.diag(np.round(s4, 2)))

You can find more example notebooks in the examples folder. For more detailed usage instructions, please refer to the documentation.

Contributing

Contributions are welcome! If you find any issues or have suggestions for improvements, please create an issue or submit a pull request on the GitHub repository. For contributing developers, please refer to CONTRIBUTING.md file.

License

This project is licensed under the BSD 3-Clause License.

List of Algorithms available in the decompy library

Matrix Factorization Methods

Currently, there are 19 matrix factorization methods available in decompy, as follows:

• Alternating Direction Method (Yuan and Yang, 2009) - matrix_factorization/adm.py

• Augmented Lagrangian Method (Tang and Nehorai) - matrix_factorization/alm.py

• AS-RPCA: Active Subspace: Towards Scalable Low-Rank Learning (Liu and Yan, 2012) - matrix_factorization/asrpca.py

• Dual RPCA (Lin et al. 2009) - matrix_factorization/dual.py

• Exact Augmented Lagrangian Method (Lin, Chen and Ma, 2010) - matrix_factorization/ealm.py

• Robust PCA using Fast PCP Method (Rodriguez and Wohlberg, 2013) - matrix_factorization/fpcp.py

• Grassmann Average (Hauberg et al. 2014) website - matrix_factorization/ga.py

• Trimmed Grassmann Average (Hauberg et al. 2014) website - matrix_factorization/ga.py (with trim_percent value more than 0)

• Inexact Augmented Lagrangian Method (Lin et al. 2009) website - matrix_factorization/ialm.py

• L1 Filtering (Liu et al. 2011) - matrix_factorization/l1f.py

• Linearized ADM with Adaptive Penalty (Lin et al. 2011) - matrix_factorization/ladmap.py

• MoG-RPCA: Mixture of Gaussians RPCA (Zhao et al. 2014) website - matrix_factorization/mog.py

• Outlier Pursuit Xu et al, 2011 - matrix_factorization/op.py

• Principal Component Pursuit (PCP) Method (Candes et al. 2009) - matrix_factorization/pcp.py

• RegL1-ALM: Robust low-rank matrix approximation with missing data and outliers (Zheng et al. 2012) website - matrix_factorization/regl1alm.py

• Robust SVD using Density Power Divergence (rSVDdpd) Algorithm (Roy et al, 2023) - matrix_factorization/rsvddpd.py

• SVT: Singular Value Thresholding (Cai et al. 2008) website - matrix_factorization/svt.py

• Symmetric Alternating Direction Augmented Lagrangian Method (SADAL) (Goldfarb et al. 2010) - matrix_factorization/sadal.py

• Robust PCA using Variational Bayes method (Babacan et al 2012) - matrix_factorization/vbrpca.py

Methods to be added (Coming soon)

• R2PCP: Riemannian Robust Principal Component Pursuit (Hintermüller and Wu, 2014)

• DECOLOR: Contiguous Outliers in the Low-Rank Representation (Zhou et al. 2011) website1 website2

Rank Estimation Methods

In rankmethods/penalized.py -

• Elbow method

• Akaike’s Information Criterion (AIC) - https://link.springer.com/chapter/10.1007/978-1-4612-1694-0_15

• Bayesian Information Criterion (BIC) - https://doi.org/10.1214/aos/1176344136

• Bai and Ng’s Information Criterion for spatiotemporal decomposition (PC1, PC2, PC3, IC1, IC2, IC3) - https://doi.org/10.1111/1468-0262.00273

• Divergence Information Criterion (DIC) - https://doi.org/10.1080/03610926.2017.1307405

In rankmethods/cvrank.py -

• Gabriel style Cross validation - http://www.numdam.org/item/JSFS_2002__143_3-4_5_0/

• Wold style cross validation separate row and column deletion - https://www.jstor.org/stable/1267581

• Bi-cross validation (Owen and Perry) - https://doi.org/10.1214/08-AOAS227

In rankmethods/bayes.py -

• Bayesian rank estimation method by Hoffman - https://www.jstor.org/stable/27639896