1
0
mirror of https://github.com/kalmarek/PropertyT.jl.git synced 2024-12-24 18:10:29 +01:00

replace BlockDecomposition by SymbolicWedderburn

This commit is contained in:
Marek Kaluba 2022-11-07 15:34:30 +01:00
parent 92d9e468d2
commit 9511e34de4
No known key found for this signature in database
GPG Key ID: 8BF1A3855328FC15
4 changed files with 2 additions and 553 deletions

View File

@ -4,23 +4,16 @@ authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
version = "0.3.2" version = "0.3.2"
[deps] [deps]
AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
Dates = "ade2ca70-3891-5945-98fb-dc099432e06a" Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253" IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
JLD = "4138dd39-2aa7-5051-a626-17a0bb65d9c8"
JuMP = "4076af6c-e467-56ae-b986-b466b2749572" JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13" SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
SymbolicWedderburn = "858aa9a9-4c7c-4c62-b466-2421203962a2"
[compat] [compat]
AbstractAlgebra = "^0.10.0"
GroupRings = "^0.3.2"
Groups = "^0.5.0"
IntervalArithmetic = "^0.16.0" IntervalArithmetic = "^0.16.0"
JLD = "^0.9.0"
JuMP = "^0.20.0" JuMP = "^0.20.0"
SCS = "^0.7.0" SCS = "^0.7.0"
julia = "^1.3.0" julia = "^1.3.0"

View File

@ -1,22 +1,16 @@
__precompile__() __precompile__()
module PropertyT module PropertyT
using AbstractAlgebra
using LinearAlgebra using LinearAlgebra
using SparseArrays using SparseArrays
using Dates using Dates
using Groups using Groups
using GroupRings using SymbolicWedderburn
using JLD
using JuMP using JuMP
import AbstractAlgebra: Group, NCRing
include("laplacians.jl") include("laplacians.jl")
include("RGprojections.jl")
include("blockdecomposition.jl")
include("sos_sdps.jl") include("sos_sdps.jl")
include("checksolution.jl") include("checksolution.jl")

View File

@ -1,251 +0,0 @@
module Projections
using AbstractAlgebra
using Groups
using GroupRings
export PermCharacter, DirectProdCharacter, rankOne_projections
###############################################################################
#
# Characters of Symmetric Group and DirectProduct
#
###############################################################################
abstract type AbstractCharacter end
struct PermCharacter <: AbstractCharacter
p::Generic.Partition
end
struct DirectProdCharacter{N, T<:AbstractCharacter} <: AbstractCharacter
chars::NTuple{N, T}
end
function (chi::DirectProdCharacter)(g::DirectPowerGroupElem)
res = 1
for (χ, elt) in zip(chi.chars, g.elts)
res *= χ(elt)
end
return res
end
function (chi::PermCharacter)(g::Generic.Perm)
R = AbstractAlgebra.partitionseq(chi.p)
p = Partition(Generic.permtype(g))
return Int(Generic.MN1inner(R, p, 1, Generic._charvalsTable))
end
AbstractAlgebra.dim(χ::PermCharacter) = dim(YoungTableau(χ.p))
for T in [PermCharacter, DirectProdCharacter]
@eval begin
function (chi::$T)(X::GroupRingElem)
RG = parent(X)
z = zero(eltype(X))
result = z
for i in 1:length(X.coeffs)
if X.coeffs[i] != z
result += chi(RG.basis[i])*X.coeffs[i]
end
end
return result
end
end
end
characters(G::Generic.SymmetricGroup) = (PermCharacter(p) for p in AllParts(G.n))
function characters(G::DirectPowerGroup{N}) where N
nfold_chars = Iterators.repeated(characters(G.group), N)
return (DirectProdCharacter(idx) for idx in Iterators.product(nfold_chars...))
end
###############################################################################
#
# Projections
#
###############################################################################
function central_projection(RG::GroupRing, chi::AbstractCharacter, T::Type=Rational{Int})
result = RG(zeros(T, length(RG.basis)))
dim = chi(one(RG.group))
ord = Int(order(RG.group))
for g in RG.basis
result[g] = convert(T, (dim//ord)*chi(g))
end
return result
end
function alternating_emb(RG::GroupRing{Gr,T}, V::Vector{T}, S::Type=Rational{Int}) where {Gr<:AbstractAlgebra.AbstractPermutationGroup, T<:GroupElem}
res = RG(S)
for g in V
res[g] += sign(g)
end
return res
end
function idempotents(RG::GroupRing{Generic.SymmetricGroup{S}}, T::Type=Rational{Int}) where S<:Integer
if RG.group.n == 1
return GroupRingElem{T}[one(RG,T)]
elseif RG.group.n == 2
Id = one(RG,T)
transp = RG(perm"(1,2)", T)
return GroupRingElem{T}[1//2*(Id + transp), 1//2*(Id - transp)]
end
projs = Vector{Vector{Generic.Perm{S}}}()
for l in 2:RG.group.n
u = RG.group([circshift([i for i in 1:l], -1); [i for i in l+1:RG.group.n]])
i = 0
while (l-1)*i <= RG.group.n
v = RG.group(circshift(collect(1:RG.group.n), i))
k = inv(v)*u*v
push!(projs, generateGroup([k], RG.group.n))
i += 1
end
end
idems = Vector{GroupRingElem{T}}()
for p in projs
append!(idems, [RG(p, T)//length(p), alternating_emb(RG, p, T)//length(p)])
end
return unique(idems)
end
function rankOne_projection(chi::PermCharacter, idems::Vector{T}) where {T<:GroupRingElem}
RG = parent(first(idems))
S = eltype(first(idems))
ids = [one(RG, S); idems]
zzz = zero(S)
for (i,j,k) in Base.product(ids, ids, ids)
if chi(i) == zzz || chi(j) == zzz || chi(k) == zzz
continue
else
elt = i*j*k
if elt^2 != elt
continue
elseif chi(elt) == one(S)
return elt
# return (i,j,k)
end
end
end
throw("Couldn't find rank-one projection for $chi")
end
function rankOne_projections(RG::GroupRing{<:Generic.SymmetricGroup}, T::Type=Rational{Int})
if RG.group.n == 1
return [GroupRingElem([one(T)], RG)]
end
RGidems = idempotents(RG, T)
min_projs = [central_projection(RG,chi)*rankOne_projection(chi,RGidems) for chi in characters(RG.group)]
return min_projs
end
function ifelsetuple(a,b, k, n)
x = [repeat([a], k); repeat([b], n-k)]
return tuple(x...)
end
function orbit_selector(n::Integer, k::Integer,
chi::AbstractCharacter, psi::AbstractCharacter)
return Projections.DirectProdCharacter(ifelsetuple(chi, psi, k, n))
end
function rankOne_projections(RBn::GroupRing{G}, T::Type=Rational{Int}) where {G<:WreathProduct}
Bn = RBn.group
N = Bn.P.n
# projections as elements of the group rings RSₙ
Sn_rankOnePr = [rankOne_projections(
GroupRing(SymmetricGroup(i), collect(SymmetricGroup(i))))
for i in typeof(N)(1):N]
# embedding into group ring of BN
RN = GroupRing(Bn.N, collect(Bn.N))
sign, id = collect(characters(Bn.N.group))
# Bn.N = (Z/2Z)ⁿ characters corresponding to the first k coordinates:
BnN_orbits = Dict(i => orbit_selector(N, i, sign, id) for i in 0:N)
Q = Dict(i => RBn(g -> Bn(g), central_projection(RN, BnN_orbits[i], T)) for i in 0:N)
Q = Dict(key => GroupRings.dense(val) for (key, val) in Q)
all_projs = [Q[0]*RBn(g->Bn(g), p) for p in Sn_rankOnePr[N]]
r = collect(1:N)
for i in 1:N-1
first_emb = g->Bn(Generic.emb!(one(Bn.P), g, view(r, 1:i)))
last_emb = g->Bn(Generic.emb!(one(Bn.P), g, view(r, (i+1):N)))
Sk_first = (RBn(first_emb, p) for p in Sn_rankOnePr[i])
Sk_last = (RBn(last_emb, p) for p in Sn_rankOnePr[N-i])
append!(all_projs,
[Q[i]*p1*p2 for (p1,p2) in Base.product(Sk_first,Sk_last)])
end
append!(all_projs, [Q[N]*RBn(g->Bn(g), p) for p in Sn_rankOnePr[N]])
return all_projs
end
##############################################################################
#
# General Groups Misc
#
##############################################################################
"""
products(X::Vector{GroupElem}, Y::Vector{GroupElem}[, op=*])
Return a vector of all possible products (or `op(x,y)`), where `x ∈ X` and
`y ∈ Y`. You may change which operation is used by specifying `op` argument.
"""
function products(X::AbstractVector{T}, Y::AbstractVector{T}, op=*) where {T<:GroupElem}
result = Vector{T}()
seen = Set{T}()
sizehint!(result, length(X)*length(Y))
sizehint!(seen, length(X)*length(Y))
for x in X
for y in Y
z = op(x,y)
if !in(z, seen)
push!(seen, z)
push!(result, z)
end
end
end
return result
end
"""
generateGroup(gens::Vector{GroupElem}, r, [op=*])
Produce all elements of a group generated by elements in `gens` in ball of
radius `r` (word-length metric induced by `gens`). The binary operation can
be optionally specified.
"""
function generateGroup(gens::Vector{T}, r, op=*) where {T<:GroupElem}
n = 0
R = 1
elts = [one(first(gens)); gens]
while n length(elts) && R < r
# @show elts
R += 1
n = length(elts)
elts = products(gens, elts, op)
end
return elts
end
end # of module Projections

View File

@ -1,287 +0,0 @@
###############################################################################
#
# BlockDecomposition
#
###############################################################################
struct BlockDecomposition{T<:AbstractArray{Float64, 2}, GEl<:GroupElem, P<:Generic.Perm}
orbits::Vector{Vector{Int}}
preps::Dict{GEl, P}
Uπs::Vector{T}
dims::Vector{Int}
end
function BlockDecomposition(RG::GroupRing, autS::Group; verbose=true)
verbose && @info "Decomposing basis of RG into orbits of" autS
@time orbs = orbit_decomposition(autS, RG.basis, RG.basis_dict)
@assert sum(length(o) for o in orbs) == length(RG.basis)
verbose && @info "The action has $(length(orbs)) orbits"
verbose && @info "Finding projections in the Group Ring of" autS
@time autS_mps = Projections.rankOne_projections(GroupRing(autS, collect(autS)))
verbose && @info "Finding AutS-action matrix representation"
@time preps = perm_reps(autS, RG.basis[1:size(RG.pm,1)], RG.basis_dict)
@time mreps = matrix_reps(preps)
verbose && @info "Computing the projection matrices Uπs"
@time Uπs = [orthSVD(matrix_repr(p, mreps)) for p in autS_mps]
multiplicities = size.(Uπs,2)
dimensions = [Int(p[one(autS)]*Int(order(autS))) for p in autS_mps]
if verbose
info_strs = ["",
lpad("multiplicities", 14) * " =" * join(lpad.(multiplicities, 4), ""),
lpad("dimensions", 14) * " =" * join(lpad.(dimensions, 4), "")
]
@info join(info_strs, "\n")
end
@assert dot(multiplicities, dimensions) == size(RG.pm,1)
return BlockDecomposition(orbs, preps, Uπs, dimensions)
end
function decimate(od::BlockDecomposition; verbose=true)
nzros = [i for i in 1:length(od.Uπs) if !isempty(od.Uπs[i])]
Us = sparsify!.(od.Uπs, eps(Float64) * 1e4, verbose = verbose)[nzros]
#dimensions of the corresponding Uπs:
dims = od.dims[nzros]
return BlockDecomposition(od.orbits, od.preps, Array{Float64}.(Us), dims)
end
function orthSVD(M::AbstractMatrix{T}) where {T<:AbstractFloat}
fact = svd(convert(Matrix{T}, M))
M_rank = sum(fact.S .> maximum(size(M)) * eps(T))
return fact.U[:, 1:M_rank]
end
orbit_decomposition(
G::Group,
E::AbstractVector,
rdict = GroupRings.reverse_dict(E);
op = ^,
) = orbit_decomposition(collect(G), E, rdict; op=op)
function orbit_decomposition(elts::AbstractVector{<:GroupElem}, E::AbstractVector, rdict=GroupRings.reverse_dict(E); op=^)
tovisit = trues(size(E));
orbits = Vector{Vector{Int}}()
orbit = zeros(Int, length(elts))
for i in eachindex(E)
if tovisit[i]
g = E[i]
Threads.@threads for j in eachindex(elts)
orbit[j] = rdict[op(g, elts[j])]
end
tovisit[orbit] .= false
push!(orbits, unique(orbit))
end
end
return orbits
end
###############################################################################
#
# Sparsification
#
###############################################################################
dens(M::SparseMatrixCSC) = nnz(M)/length(M)
dens(M::AbstractArray) = count(!iszero, M)/length(M)
function sparsify!(M::SparseMatrixCSC{Tv,Ti}, tol=eps(Tv); verbose=false) where {Tv,Ti}
densM = dens(M)
droptol!(M, tol)
verbose && @info(
"Sparsified density:",
rpad(densM, 20),
"",
rpad(dens(M), 20),
" ($(nnz(M)) non-zeros)"
)
return M
end
function sparsify!(M::AbstractArray{T}, tol=eps(T); verbose=false) where T
densM = dens(M)
clamp_small!(M, tol)
if verbose
@info("Sparsifying $(size(M))-matrix... \n $(rpad(densM, 20))$(rpad(dens(M),20))), ($(count(!iszero, M)) non-zeros)")
end
return sparse(M)
end
function clamp_small!(M::AbstractArray{T}, tol=eps(T)) where T
for n in eachindex(M)
if abs(M[n]) < tol
M[n] = zero(T)
end
end
return M
end
function sparsify(U::AbstractArray{T}, tol=eps(T); verbose=false) where T
return sparsify!(deepcopy(U), tol, verbose=verbose)
end
###############################################################################
#
# perm-, matrix-, representations
#
###############################################################################
function perm_repr(g::GroupElem, E::Vector, E_dict)
p = Vector{Int}(undef, length(E))
for (i,elt) in enumerate(E)
p[i] = E_dict[elt^g]
end
return p
end
function perm_reps(G::Group, E::Vector, E_rdict=GroupRings.reverse_dict(E))
elts = collect(G)
l = length(elts)
preps = Vector{Generic.Perm}(undef, l)
permG = SymmetricGroup(length(E))
Threads.@threads for i in 1:l
preps[i] = permG(PropertyT.perm_repr(elts[i], E, E_rdict), false)
end
return Dict(elts[i]=>preps[i] for i in 1:l)
end
function matrix_repr(x::GroupRingElem, mreps::Dict)
nzeros = findall(!iszero, x.coeffs)
return sum(x[i].*mreps[parent(x).basis[i]] for i in nzeros)
end
function matrix_reps(preps::Dict{T,Generic.Perm{I}}) where {T<:GroupElem, I<:Integer}
kk = collect(keys(preps))
mreps = Vector{SparseMatrixCSC{Float64, Int}}(undef, length(kk))
Threads.@threads for i in 1:length(kk)
mreps[i] = AbstractAlgebra.matrix_repr(preps[kk[i]])
end
return Dict(kk[i] => mreps[i] for i in 1:length(kk))
end
###############################################################################
#
# actions
#
###############################################################################
function Base.:^(y::GroupRingElem, g::GroupRingElem, op = ^)
res = parent(y)()
for elt in GroupRings.supp(g)
res += g[elt] * ^(y, elt, op)
end
return res
end
function Base.:^(y::GroupRingElem, g::GroupElem, op = ^)
RG = parent(y)
result = zero(RG, eltype(y.coeffs))
for (idx, c) in enumerate(y.coeffs)
if !iszero(c)
result[op(RG.basis[idx], g)] = c
end
end
return result
end
function Base.:^(
y::GroupRingElem{T,<:SparseVector},
g::GroupElem,
op = ^,
) where {T}
RG = parent(y)
index = [RG.basis_dict[op(RG.basis[idx], g)] for idx in y.coeffs.nzind]
result = GroupRingElem(sparsevec(index, y.coeffs.nzval, y.coeffs.n), RG)
return result
end
###############################################################################
#
# perm && WreathProductElems actions: MatAlgElem
#
###############################################################################
function Base.:^(A::MatAlgElem, p::Generic.Perm)
length(p.d) == size(A, 1) == size(A, 2) ||
throw("Can't act via $p on matrix of size $(size(A))")
result = similar(A)
@inbounds for i = 1:size(A, 1)
for j = 1:size(A, 2)
result[p[i], p[j]] = A[i, j] # action by permuting rows and colums/conjugation
end
end
return result
end
function Base.:^(A::MatAlgElem, g::WreathProductElem{N}) where {N}
# @assert N == size(A,1) == size(A,2)
flips = ntuple(i -> (g.n[i].d[1] == 1 && g.n[i].d[2] == 2 ? 1 : -1), N)
result = similar(A)
R = base_ring(parent(A))
tmp = R(1)
@inbounds for i = 1:size(A, 1)
for j = 1:size(A, 2)
x = A[i, j]
if flips[i] * flips[j] == 1
result[g.p[i], g.p[j]] = x
else
result[g.p[i], g.p[j]] = -x
end
end
end
return result
end
###############################################################################
#
# perm && WreathProductElems actions: Automorphism
#
###############################################################################
function Base.:^(a::Automorphism, g::GroupElem)
Ag = parent(a)(g)
Ag_inv = inv(Ag)
res = append!(Ag, a, Ag_inv)
return Groups.freereduce!(res)
end
(A::AutGroup)(p::Generic.Perm) = A(Groups.AutSymbol(p))
function (A::AutGroup)(g::WreathProductElem)
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
parent(g).P.n == length(A.objectGroup.gens) ||
throw("No natural embedding of $(parent(g)) into $A")
elt = one(A)
Id = one(parent(g.n.elts[1]))
for i = 1:length(g.p.d)
if g.n.elts[i] != Id
push!(elt, Groups.flip(i))
end
end
push!(elt, Groups.AutSymbol(g.p))
return elt
end
# fallback:
Base.one(p::Generic.Perm) = Perm(length(p.d))