Colormann

Colormann (Convex Low Rank Matrix Approximation using Nuclear Norm constraint) extracts the shared and distinct hidden components from a noisy data matrix $\mathbf{X}$ using two steps:

1. Decompose $\mathbf{X}$ as the sum of a low-rank component ($\mathbf{Y}$) and a sparse component ($\mathbf{M}$). Colormann provides a choice of several convex algorithms for the low rank matrix approximation (LRMA), using a combination of weighted nuclear norm and $\ell_1$ norm regularization (see below).
2. Decompose $\mathbf{Y}$, $\mathbf{M}$ or their sum ($\mathbf{Y} + \mathbf{M}$) as a product of loadings and factors using singular value decomposition (SVD).

In most cases, the factors obtained from the low-rank component $\mathbf{Y}$ would be the shared hidden component and the factors obtained from the sparse component $\mathbf{M}$ would be the distinct hidden components. However, they may have different interpretations depending on the structure of the input data.

LRMA algorithms

The following algorithms have been implemented:

• Rank minimization with weighted nuclear norm constraint using Frank-Wolfe algorithm.
• Rank minimization with nuclear norm and $\ell_1$ norm constraint using Frank-Wolfe algorithm.
• RobustPCA with inexact augmented Lagrange multiplier algorithm.

Installation

The software can be installed directly from github using pip:

pip install git+https://github.com/banskt/colormann.git

For development, download this repository and install using the -e flag:

git clone https://github.com/banskt/colormann.git colormann # or use the SSH link
cd colormann
pip install -e .

Related algorithms

Also see Sparse PCA, RobustPCA, Weighted PCA, Bayesian PCA.

PCA is not convex and cannot be made so.