A package for efficiently computing with symmetric extended generator representable semiseparable matrices and a variant thereof. In short this means matrices of the form
K = tril(U*V^T) + triu(V*U^T,1)
as well as
K = tril(U*V^T) + triu(V*U^T,1) + diag(d)
All implemented algorithms (multiplication, Cholesky factorization, forward/backward substitution as well as various traces and determinants) run linear in time and memory w.r.t. to the number of data points n
.
A more in-depth descriptions of the algorithms can be found in [1] or here.
Adding the package can be done through
(@v1.5) pkg> add https://github.com/mipals/SymEGRSSMatrices.jl
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> using SymEGRSSMatrices
julia> import SymEGRSSMatrices: spline_kernel
julia> U, V = spline_kernel(Vector(0.1:0.01:1)', 2); # Creating input such that K is PD
julia> K = SymEGRSSMatrix(U,V); # Symmetric generator representable semiseparable matrix
julia> x = ones(size(K,1)); # Test vector
We can now compute products with K
and K'
. The result are the same as K
is symmetric.
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 SymEGRSSMatrix
structure we can efficiently compute the Cholesky factorization as
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(np)
storage.
A struct for the dealing with symmetric matrices of the form, K = tril(UV') + triu(VU',1) + diag(d)
called SymEGRQSMatrix
is also implemented. The usage is similar to that of SymEGRSSMatrix
and can be created as follows
julia> U, V = spline_kernel(Vector(0.1:0.01:1)', 2); # Creating input such that K is PD
julia> K = SymEGRQSMatrix(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)
[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.