Python implementation for optimization with Low-Rank Inducing Norms

## Project description

# 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.

## Literature:

### Low-rank inducing norms:

- Rank Reduction with Convex Constraints
- Low-rank Inducing Norms with Optimality Interpretations
- Low-rank Optimization with Convex Constraints
- The Use of the r* Heuristic in Covariance Completion Problems
- On optimal low-rank approximation of non-negative matirces

### 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:

- Exact Matrix Completion
- 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)[0]
# Non-convex Douglas-Rachford splitting:
M = drcomplete(N,Index,r,p,solver = 'NDR')[0]
```

#### 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)[0]
# Non-convex Douglas-Rachford splitting:
M = drhankelapprox(H,r,solver = 'NDR')[0]
```

### 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)[0]
```

#### 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)[0]
```

#### 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)
```

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