SymmetricTensors.jl

SymmetricTensors.jl provides the SymmetricTensors{T, N} type used to store fully symmetric tensors in more efficient way, without most of redundant repetitions. It uses blocks of Array{T, N} stored in Union{Array{Float,N}, Nothing} structure. Repeated blocks are replaced by Void. The module introduces SymmetricTensors{T, N} type and some basic operations on this type. As of 01/01/2017 @kdomino is the lead maintainer of this package.

Installation

Within Julia, just use run

pkg> add SymmetricTensors

to install the files. Julia 1.0 or later is required.

Constructor

julia> data = ones(4,4);


julia> SymmetricTensor{Float64,2}(Union{Nothing, Array{Float64,2}}[[1.0 1.0; 1.0 1.0] [1.0 1.0; 1.0 1.0]; nothing [1.0 1.0; 1.0 1.0]], 2, 2, 4, true)

Converting

From Array{T, N} to SymmetricTensors{T, N}

julia> SymmetricTensors(data::Array{T, N}, bls::Int = 2)

where bls is the size of a block. It is a parameter affecting the compuational speed of cumulants. The block size must fulfill bls ∈ {1, 2,..., dats} where dats = size(data, 1) = ... = size(data, N) otherwise error is risen.

julia> data = ones(4,4);


julia> convert(SymmetricTensor, data, 2)
SymmetricTensor{Float64,2}(Union{Nothing, Array{Float64,2}}[[1.0 1.0; 1.0 1.0] [1.0 1.0; 1.0 1.0]; nothing [1.0 1.0; 1.0 1.0]], 2, 2, 4, true)

From SymmetricTensors{T, N} to Array{T, N}

julia> Array(st::SymmetricTensors{T, N})

Wrong block size:

julia> SymmetricTensor(ones(4,4), 5)
ERROR: DimensionMismatch("bad block size 5 > 4")

Fields

• frame::ArrayNArrays{T,N}: stores data, where ArrayNArrays{T,N} = Array{Union{Array{T, N}, Nothing}, N}
• bls::Int: size of a block,
• bln::Int: number of blocks,
• dats::Int: size of data,
• sqr::Bool: shows if the last block is squared.

Operations

Elementwise addition: +, - is supported between many SymmetricTensors{T, N} while elementwise substraction: - between two SymmetricTensors{T, N}. Addition substraction multiplication and division +, -, *, / is supported between SymmetricTensors{T, N} and a number.

julia> x = SymmetricTensor(ones(4,4));

julia> y = SymmetricTensor(2*ones(4,4));

julia> x+y
SymmetricTensor{Float64,2}(Union{Nothing, Array{Float64,2}}[[3.0 3.0; 3.0 3.0] [3.0 3.0; 3.0 3.0]; #undef [3.0 3.0; 3.0 3.0]], 2, 2, 4, true)

julia> x*10
SymmetricTensor{Float64,2}(Union{Nothing, Array{Float64,2}}[[10.0 10.0; 10.0 10.0] [10.0 10.0; 10.0 10.0]; #undef [10.0 10.0; 10.0 10.0]], 2, 2, 4, true)

The function diag returns a Vector{T}, of all super-diagonal elements of a SymmetricTensor.

julia> data = ones(5,5,5,5);

julia> st = SymmetricTensor(data);

julia> diag(st)
5-element Array{Float64,1}:
 1.0
 1.0
 1.0
 1.0
 1.0

Random Symmetric Tensor generation

To generate random Symmetric Tensor with random elements of typer T form a uniform distribution on [0,1) use rand(SymmetricTensor{T, N}, n::Int, b::Int = 2). Here n denotes data size and b denotes block size.

julia> using Random

julia> Random.seed!(42)

julia> rand(SymmetricTensor{Float64, 2}, 2)
SymmetricTensor{Float64,2}(Union{Nothing, Array{Float64,2}}[[0.533183 0.454029; 0.454029 0.0176868]], 2, 1, 2, true)

getindex and setindex!

julia> using Random

julia> Random.seed!(42)

julia> st = rand(SymmetricTensor{Float64, 2}, 2)
SymmetricTensor{Float64,2}(Union{Nothing, Array{Float64,2}}[[0.533183 0.454029; 0.454029 0.0176868]], 2, 1, 2, true)

julia> st[1,2]
0.4540291355871424

julia> st[2,1]
0.4540291355871424

setindex!(st::SymmetricTensor, x::Float, mulind::Int...) changes all symmetric tensor's elements indexed by mulind to x.

julia> st[1,2] = 10.

julia> convert(Array, st)
2×2 Array{Float64,2}:
  0.533183  10.0      
 10.0        0.0176868

Auxiliary function

julia> unfold(data::Array{T,N}, mode::Int)

unfolds data in a given mode

julia> a = reshape(collect(1.:8.), (2,2,2))
2×2×2 Array{Float64,3}:
[:, :, 1] =
 1.0  3.0
 2.0  4.0

[:, :, 2] =
 5.0  7.0
 6.0  8.0

julia> unfold(a, 1)
2×4 Array{Float64,2}:
 1.0  3.0  5.0  7.0
 2.0  4.0  6.0  8.0

julia> unfold(a, 2)
2×4 Array{Float64,2}:
 1.0  2.0  5.0  6.0
 3.0  4.0  7.0  8.0

julia> unfold(a, 3)
2×4 Array{Float64,2}:
 1.0  2.0  3.0  4.0
 5.0  6.0  7.0  8.0

Block structure

The block usage is motivated by the paper M. D. Schatz, T. M. Low, R. A. van de Geijn, and T. G. Kolda, "Exploiting symmetry in tensors for high performance: Multiplication with symmetric tensors", SIAM Journal on Scientific Computing, 36 (2014), pp. C453–C479 https://doi.org/10.1137/130907215. There only the meaningful part of the symmetric tensor is stored in blocks to decrease the memory and computational overhead. The selection of the optimal block size is not straight forward, however in most cases concerning cumulants one can use 2 or 3.

This project was partially financed by the National Science Centre, Poland – project number 2014/15/B/ST6/05204.