https://github.com/mipals/symsemiseparablematrices.jl
A package for efficiently computing with symmetric extended generator representable semiseparable matrices (a type of rank structured matrix).
https://github.com/mipals/symsemiseparablematrices.jl
julia linear-algebra rank-structured-matrices
Last synced: 3 months ago
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A package for efficiently computing with symmetric extended generator representable semiseparable matrices (a type of rank structured matrix).
- Host: GitHub
- URL: https://github.com/mipals/symsemiseparablematrices.jl
- Owner: mipals
- License: mit
- Created: 2019-10-23T07:08:57.000Z (over 5 years ago)
- Default Branch: main
- Last Pushed: 2022-06-22T05:56:19.000Z (almost 3 years ago)
- Last Synced: 2025-02-23T06:45:09.784Z (3 months ago)
- Topics: julia, linear-algebra, rank-structured-matrices
- Language: Julia
- Homepage:
- Size: 149 KB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 1
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# SymSemiseparableMatrices.jl
[](https://github.com/mipals/SymSemiseparableMatrices.jl/actions/workflows/CI.yml)
[](https://codecov.io/gh/mipals/SymSemiseparableMatrices.jl)
## Description
A package for efficiently computing with symmetric extended generator representable semiseparable matrices (EGRSS Matrices) and a varient with added diagonal terms. In short this means matrices of the form
$$
K = \text{\textbf{tril}}(UV^\top) + \text{\textbf{triu}}\left(VU^\top,1\right), \quad U,V\in\mathbb{R}^{p\times n}
$$
$$
K = \text{\textbf{tril}}(UV^\top) + \text{\textbf{triu}}\left(VU^\top,1\right) + \text{\textbf{diag}}(d), \quad U,V\in\mathbb{R}^{p\times n},\ d\in\mathbb{R}^n
$$
All implemented algorithms (multiplication, Cholesky factorization, forward/backward substitution as well as various traces and determinants) scales with $O(p^kn)$. Since $p \ll n$ this result in very scalable computations.
A more in-depth descriptions of the algorithms can be found in [1] or [here](https://github.com/mipals/SmoothingSplines.jl/blob/master/mt_mikkel_paltorp.pdf).
## Usage
Adding the package can be done through
```
(@v1.7) pkg> add SymSemiseparableMatrices
```
First we need to create generators U and V that represent the symmetric matrix, ```K = tril(UV') + triu(VU',1)``` as well a test vector ```x```.
```julia
julia> using SymSemiseparableMatrices
julia> import SymSemiseparableMatrices: spline_kernel
julia> U, V = spline_kernel(Vector(0.1:0.01:1)', 2); # Creating input such that K is PD
julia> K = SymSemiseparableMatrix(U,V); # Symmetric generator representable semiseparable matrix
julia> x = ones(size(K)); # Test vector
```
We can now compute products with ```K``` and ```K'```. The result are the same, since ```K``` is symmetric.
```julia
julia> K*x
91×1 Array{Float64,2}:
0.23508333333333334
0.28261583333333334
0.3341535
0.3896073333333333
0.44888933333333336
0.5119124999999999
⋮
11.977057499999997
12.146079333333331
12.31510733333333
12.484138499999995
12.65317083333333
julia> K'*x
91×1 Array{Float64,2}:
0.23508333333333334
0.28261583333333334
0.3341535
0.3896073333333333
0.44888933333333336
0.5119124999999999
⋮
11.977057499999997
12.146079333333331
12.31510733333333
12.484138499999995
12.65317083333333
```
Furthermore from the ```SymSemiseparableMatrix``` structure we can efficiently compute the Cholesky factorization as
```julia
julia> L = cholesky(K); # Computing the Cholesky factorization of K
julia> K*(L'\(L\x))
91×1 Array{Float64,2}:
1.0000000000000036
0.9999999999999982
0.9999999999999956
0.9999999999999944
0.9999999999999951
0.999999999999995
⋮
0.9999999999996279
0.9999999999996153
0.9999999999996028
0.9999999999995898
0.9999999999995764
```
Now ```L``` represents a Cholesky factorization with of form ```L = tril(UW')```, requiring only $O(pn)$ storage.
A struct for the dealing with symmetric matrices of the form, ```K = tril(UV') + triu(VU',1) + diag(d)``` called ```DiaSymSemiseparableMatrix``` is also implemented. The usage is similar to that of ```DiaSymSemiseparableMatrix``` and can be created as follows
```julia
julia> U, V = spline_kernel(Vector(0.1:0.01:1)', 2); # Creating input such that K is PD
julia> K = DiaSymSemiseparableMatrix(U,V,rand(size(U,2)); # Symmetric EGRSS matrix + diagonal
```
The Cholesky factorization of this matrix can be computed using ```cholesky```. Note however here that ```L``` represents a matrix of the form ```L = tril(UW',-1) + diag(c)```
## Benchmarks
### Computing Cholesky factorization of ```K = tril(UV') + triu(VU',1)```
### Computing Cholesky factorization of ```K = tril(UV') + triu(VU',1) + diag(d)```
### Solving linear systems using a Cholesky factorization with the form ```L = tril(UW')```
## References
[1] M. S. Andersen and T. Chen, “Smoothing Splines and Rank Structured Matrices: Revisiting the Spline Kernel,” SIAM Journal on Matrix Analysis and Applications, 2020.
[2] J. Keiner. "Fast Polynomial Transforms." Logos Verlag Berlin, 2011.