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Python implementation for optimization with Low-Rank Inducing Norms

# LRIPy

Python code for Low-rank optimization by Low-Rank Inducing Norms as well as non-convex Douglas-Rachford.

## Purpose:

Low-rank rank inducing norms and non-convex Proximal Splitting Algoriths attempt to find exact rank/cardinality-r solutions to minimization problems with convex loss functions, i.e., avoiding of regularzation heuristics. LRIPy provides Python implementations for the proximal mappings of the low-rank inducing Frobenius and Spectral norms, as well as, their epi-graph projections and non-convex counter parts.

## Installation

The easiest way to install the package is to run `pip install lripy`. To install the package from source, run `python setup.py install` in the main folder.

## Documentation

In the following it holds that

• for the low-rank inducing Frobenius norm: p = 2
• for the low-rank inducing Spectral norm: p = 'inf'

### Examples

There are two examples in the "example" folder:

1. Exact Matrix Completion
2. Low-rank approximation with Hankel constraint

### Optimization

LRIPy contains Douglas-Rachford splitting implementations for "Exact Matrix Completion" and "Low-rank Hankel Approximation", both with low-rank inducing norms, as well as, non-convex Douglas-Rachford splitting. It is easy to modify these functions for other constraints!

#### Exact Matrix completion

Let N be a matrix and Index be a binary matrix of the same size, where the ones indicate the known entries N. We attempt to find a rank-r completion M:

``````# Import the Douglas-Rachford Completion function:

from lripy import drcomplete

# Low-rank inducing norms with Douglas-Rachford splitting:

M = drcomplete(N,Index,r,p)

# Non-convex Douglas-Rachford splitting:

M = drcomplete(N,Index,r,p,solver = 'NDR')
``````

#### Low-rank Hankel Approximation

Let H be a matrix. We attempt to find a rank-r Hankel approximation M that minimizes the Frobenius norm:

``````# Import the Douglas-Rachford Hankel Approximation function:

from lripy import drhankelapprox

# Low-rank inducing norms with Douglas-Rachford splitting:

M = drhankelapprox(H,r)

# Non-convex Douglas-Rachford splitting:

M = drhankelapprox(H,r,solver = 'NDR')
``````

### Proximal Mappings

LRIPy provides Python implemenations for the proximal mappings to the low-rank inducing Frobenius and Spectral norm as well as their epi-graph projections and non-convex counter parts.

#### Low-rank inducing Spectral and Frobenius norms:

Proximal mapping of the low-rank inducing norms at Z with parameter r and scaling factor gamma:

``````X = proxnormrast(Z,r,p,gamma)
``````

#### Squared Low-rank inducing Spectral and Frobenius norms:

Proximal mapping of the SQUARED low-rank inducing norms at Z with parameter r and scaling factor gamma:

``````X = proxnormrast_square(Z,r,p,gamma)
``````

#### Projection onto the epi-graph of the low-rank inducing norms:

Projection of (Z,zv) on the epi-graph of the low-rank inducing norms with parameter r and scaling factor gamma:

``````X,xv = projrast(Z,zv,r,p,gamma)[0:2]
``````

#### Non-convex proximal mappings for Frobenius and Spectral norm:

Non-convex proximal mapping of at Z with parameter r and scaling factor gamma:

``````X = proxnonconv(Z,r,p,gamma)
``````

#### Non-convex proximal mappings for squared Frobenius and Spectral norm:

Non-convex proximal mapping for the SQUARED norms at Z with parameter r and scaling factor gamma:

``````X = proxnonconv_square(Z,r,p,gamma)
``````

## Project details

This version 0.0.2 0.0.1