Approximator: CUR

[nathanielpritchard]

Description

This pull request creates the CUR approximator feature requrested in https://github.com/numlinalg/RLinearAlgebra.jl/issues/147. It contains implementation for the CUR
approximator and the CURCore structures. The CUR approximator is a global structure which is implemented to allow matrix multiplications and eventually back solve operations, the CURCore is how the CUR is fine tuned.

As an example of how this differentiation is necessary, for a selection of rows and columns the core matrix can be formed in at least two different ways. One is the optimal way where if $C$ is the select columns and $R$ is the selected rows from $A$ the core matrix $U = $C^\dagger A R^\dagger$. As this core requires additional accesses to the matrix, that are suboptimal we can also define $U = A[I, J]$ where $I$ are the row indices that make up $R$ and $J$ are the row indices that make up $C$.

This PR features three new files

• cur.jl which includes the structures for the cur decomposition and the functions complete_approximator and mul!.
• CURCore.jl defines abstract types corresponding to the CUR core matrix, as well as a CURCoreAdjoint structure. The functions include basic mul! functionality similar to Compressors.jl, the size function, as well as the revenant completion and update functions (although for compilation reasons these functions are actually placed in cur.jl.
• cross_approximation.jl, features the implementation of the cross approximation CUR core, as well as the complete function and the rapproximate! function. rapproximate is implemented to correspond to the CURCore and not the CUR because the ordering of selection varies based on the CURCore that is used. So it makes sense that the implementation would align with that and not the CUR data structure.

How has this been tested

] activate .
test

Types of changes

• CI
• Docs
• Feature
• Fix
• Performance
• Refactor
• Style
• Test
• Other (use sparingly):

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[dmaldona]

couple of comments