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reorganize/rewrite tests!
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@ -1,51 +1,203 @@
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function check_positivity(elt, unit; upper_bound=Inf, halfradius=2, optimizer)
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@time sos_problem =
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PropertyT.sos_problem_primal(elt, unit, upper_bound=upper_bound)
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status, _ = PropertyT.solve(sos_problem, optimizer)
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P = JuMP.value.(sos_problem[:P])
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Q = real.(sqrt(P))
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certified, λ_cert = PropertyT.certify_solution(
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elt,
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unit,
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JuMP.objective_value(sos_problem),
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Q,
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halfradius=halfradius,
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)
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return status, certified, λ_cert
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end
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@testset "1703.09680 Examples" begin
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@testset "1703.09680 Examples" begin
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@testset "SL(2,Z)" begin
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@testset "SL(2,Z)" begin
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N = 2
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N = 2
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G = MatrixAlgebra(zz, N)
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G = MatrixGroups.SpecialLinearGroup{N}(Int8)
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S = PropertyT.generating_set(G)
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RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
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rm("SL($N,Z)", recursive=true, force=true)
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Δ = let RG = RG, S = S
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sett = PropertyT.Settings("SL($N,Z)", G, S, with_SCS(20000, accel=20); upper_bound=0.1)
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RG(length(S)) - sum(RG(s) for s in S)
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@info sett
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λ = PropertyT.spectral_gap(sett)
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@test λ < 0.0
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@test PropertyT.interpret_results(sett, λ) == false
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end
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end
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@testset "SL(3,Z)" begin
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elt = Δ^2
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unit = Δ
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ub = 0.1
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status, certified, λ = check_positivity(
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elt,
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unit,
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upper_bound=ub,
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halfradius=2,
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optimizer=scs_optimizer(
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eps=1e-10,
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max_iters=5_000,
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accel=50,
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alpha=1.9,
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)
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)
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@test status == JuMP.ALMOST_OPTIMAL
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@test !certified
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@test λ < 0
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end
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@testset "SL(3,F₅)" begin
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N = 3
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N = 3
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G = MatrixAlgebra(zz, N)
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G = MatrixGroups.SpecialLinearGroup{N}(SymbolicWedderburn.Characters.FiniteFields.GF{5})
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S = PropertyT.generating_set(G)
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RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
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rm("SL($N,Z)", recursive=true, force=true)
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Δ = let RG = RG, S = S
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sett = PropertyT.Settings("SL($N,Z)", G, S, with_SCS(1000, accel=20); upper_bound=0.1)
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RG(length(S)) - sum(RG(s) for s in S)
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end
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@info sett
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elt = Δ^2
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unit = Δ
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ub = 1.01 # 1.5
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λ = PropertyT.spectral_gap(sett)
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status, certified, λ = check_positivity(
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@test λ > 0.099
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elt,
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@test PropertyT.interpret_results(sett, λ) == true
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unit,
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upper_bound=ub,
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halfradius=2,
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optimizer=scs_optimizer(
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eps=1e-10,
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max_iters=5_000,
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accel=50,
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alpha=1.9,
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)
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)
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@test PropertyT.check_property_T(sett) == true #second run should be fast
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@test status == JuMP.OPTIMAL
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@test certified
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@test λ > 1
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end
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end
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@testset "SAut(F₂)" begin
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@testset "SAut(F₂)" begin
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N = 2
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N = 2
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G = SAut(FreeGroup(N))
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G = SpecialAutomorphismGroup(FreeGroup(N))
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S = PropertyT.generating_set(G)
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RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
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rm("SAut(F$N)", recursive=true, force=true)
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Δ = let RG = RG, S = S
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sett = PropertyT.Settings("SAut(F$N)", G, S, with_SCS(10000);
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RG(length(S)) - sum(RG(s) for s in S)
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upper_bound=0.15)
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end
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@info sett
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elt = Δ^2
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unit = Δ
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ub = 0.1
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λ = PropertyT.spectral_gap(sett)
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status, certified, λ = check_positivity(
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@test λ < 0.0
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elt,
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@test PropertyT.interpret_results(sett, λ) == false
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unit,
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upper_bound=0.1,
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halfradius=2,
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optimizer=scs_optimizer(
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eps=1e-10,
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max_iters=5_000,
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accel=50,
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alpha=1.9,
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)
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)
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@test status == JuMP.ALMOST_OPTIMAL
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@test λ < 0
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@test !certified
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end
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@testset "SL(3,Z) has (T)" begin
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n = 3
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SL = MatrixGroups.SpecialLinearGroup{n}(Int8)
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RSL, S, sizes = PropertyT.group_algebra(SL, halfradius=2, twisted=true)
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Δ = RSL(length(S)) - sum(RSL(s) for s in S)
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@testset "basic formulation" begin
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elt = Δ^2
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unit = Δ
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ub = 0.1
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opt_problem = PropertyT.sos_problem_primal(
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elt,
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unit,
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upper_bound=ub,
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augmented=false,
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)
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status, _ = PropertyT.solve(
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opt_problem,
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cosmo_optimizer(
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eps=1e-10,
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max_iters=10_000,
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accel=0,
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alpha=1.5,
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),
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)
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@test status == JuMP.OPTIMAL
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λ = JuMP.value(opt_problem[:λ])
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@test λ > 0.09
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Q = real.(sqrt(JuMP.value.(opt_problem[:P])))
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certified, λ_cert = PropertyT.certify_solution(
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elt,
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unit,
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λ,
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Q,
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halfradius=2,
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augmented=false,
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)
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@test certified
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@test isapprox(λ_cert, λ, rtol=1e-5)
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end
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@testset "augmented formulation" begin
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elt = Δ^2
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unit = Δ
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ub = 0.20001 # Inf
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opt_problem = PropertyT.sos_problem_primal(
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elt,
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unit,
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upper_bound=ub,
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augmented=true,
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)
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status, _ = PropertyT.solve(
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opt_problem,
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scs_optimizer(
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eps=1e-10,
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max_iters=10_000,
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accel=-10,
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alpha=1.5,
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),
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)
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@test status == JuMP.OPTIMAL
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λ = JuMP.value(opt_problem[:λ])
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Q = real.(sqrt(JuMP.value.(opt_problem[:P])))
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certified, λ_cert = PropertyT.certify_solution(
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elt,
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unit,
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λ,
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Q,
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halfradius=2,
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augmented=true,
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)
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@test certified
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@test isapprox(λ_cert, λ, rtol=1e-5)
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@test λ_cert > 2 // 10
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end
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end
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end
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end
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end
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@ -1,104 +1,225 @@
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function check_positivity(elt, unit, wd; upper_bound=Inf, halfradius=2, optimizer)
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@assert aug(elt) == aug(unit) == 0
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@time sos_problem, Ps =
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PropertyT.sos_problem_primal(elt, unit, wd, upper_bound=upper_bound)
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@time status, _ = PropertyT.solve(sos_problem, optimizer)
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Q = let Ps = Ps
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flPs = [real.(sqrt(JuMP.value.(P))) for P in Ps]
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PropertyT.reconstruct(flPs, wd)
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end
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λ = JuMP.value(sos_problem[:λ])
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sos = let RG = parent(elt), Q = Q
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z = zeros(eltype(Q), length(basis(RG)))
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res = AlgebraElement(z, RG)
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cnstrs = PropertyT.constraints(basis(RG), RG.mstructure, augmented=true)
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PropertyT._cnstr_sos!(res, Q, cnstrs)
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end
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residual = elt - λ * unit - sos
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λ_fl = PropertyT.sufficient_λ(residual, λ, halfradius=2)
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λ_fl < 0 && return status, false, λ_fl
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sos = let RG = parent(elt), Q = [PropertyT.IntervalArithmetic.@interval(q) for q in Q]
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z = zeros(eltype(Q), length(basis(RG)))
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res = AlgebraElement(z, RG)
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cnstrs = PropertyT.constraints(basis(RG), RG.mstructure, augmented=true)
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PropertyT._cnstr_sos!(res, Q, cnstrs)
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end
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λ_int = PropertyT.IntervalArithmetic.@interval(λ)
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residual_int = elt - λ_int * unit - sos
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λ_int = PropertyT.sufficient_λ(residual_int, λ_int, halfradius=2)
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return status, λ_int > 0, PropertyT.IntervalArithmetic.inf(λ_int)
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end
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@testset "1712.07167 Examples" begin
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@testset "1712.07167 Examples" begin
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@testset "SL(3,Z)" begin
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N = 3
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G = MatrixAlgebra(zz, N)
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S = PropertyT.generating_set(G)
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autS = WreathProduct(SymmetricGroup(2), SymmetricGroup(N))
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NAME = "SL($N,Z)_orbit"
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rm(NAME, recursive=true, force=true)
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sett = PropertyT.Settings(NAME, G, S, autS, with_SCS(1000, accel=20);
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upper_bound=0.27, force_compute=false)
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@info sett
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λ = PropertyT.spectral_gap(sett)
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@test λ < 0.0
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@test PropertyT.interpret_results(sett, λ) == false
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# second run just checks the solution due to force_compute=false above
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@test λ == PropertyT.spectral_gap(sett)
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@test PropertyT.check_property_T(sett) == false
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sett = PropertyT.Settings(NAME, G, S, autS, with_SCS(4000, accel=20);
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upper_bound=0.27, force_compute=true)
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@info sett
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λ = PropertyT.spectral_gap(sett)
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@test λ > 0.269999
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@test PropertyT.interpret_results(sett, λ) == true
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# this should be very fast due to warmstarting:
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@test λ ≈ PropertyT.spectral_gap(sett) atol=1e-5
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@test PropertyT.check_property_T(sett) == true
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##########
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# Symmetrizing by SymmetricGroup(3):
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sett = PropertyT.Settings(NAME, G, S, SymmetricGroup(N), with_SCS(4000, accel=20, warm_start=false);
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upper_bound=0.27, force_compute=true)
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@info sett
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λ = PropertyT.spectral_gap(sett)
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@test λ > 0.269999
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@test PropertyT.interpret_results(sett, λ) == true
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end
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@testset "SL(4,Z)" begin
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N = 4
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G = MatrixAlgebra(zz, N)
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S = PropertyT.generating_set(G)
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autS = WreathProduct(SymmetricGroup(2), SymmetricGroup(N))
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NAME = "SL($N,Z)_orbit"
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rm(NAME, recursive=true, force=true)
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sett = PropertyT.Settings(NAME, G, S, autS, with_SCS(1000, accel=20);
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upper_bound=1.3, force_compute=false)
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@info sett
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λ = PropertyT.spectral_gap(sett)
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@test λ < 0.0
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@test PropertyT.interpret_results(sett, λ) == false
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# second run just checks the solution due to force_compute=false above
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@test λ == PropertyT.spectral_gap(sett)
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@test PropertyT.check_property_T(sett) == false
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sett = PropertyT.Settings(NAME, G, S, autS, with_SCS(7000, accel=20, warm_start=true);
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upper_bound=1.3, force_compute=true)
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@info sett
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@time λ = PropertyT.spectral_gap(sett)
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@test λ > 1.2999
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@test PropertyT.interpret_results(sett, λ) == true
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# this should be very fast due to warmstarting:
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@time @test λ ≈ PropertyT.spectral_gap(sett) atol=1e-5
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@time @test PropertyT.check_property_T(sett) == true
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end
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@testset "SAut(F₃)" begin
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@testset "SAut(F₃)" begin
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N = 3
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N = 3
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G = SAut(FreeGroup(N))
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G = SpecialAutomorphismGroup(FreeGroup(N))
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S = PropertyT.generating_set(G)
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RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
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autS = WreathProduct(SymmetricGroup(2), SymmetricGroup(N))
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NAME = "SAut(F$N)_orbit"
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P = PermGroup(perm"(1,2)", Perm(circshift(1:N, -1)))
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Σ = PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
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act = PropertyT.action_by_conjugation(G, Σ)
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wd = WedderburnDecomposition(
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Float64,
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Σ,
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act,
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basis(RG),
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StarAlgebras.Basis{UInt16}(@view basis(RG)[1:sizes[2]]),
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)
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rm(NAME, recursive=true, force=true)
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Δ = let RG = RG, S = S
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RG(length(S)) - sum(RG(s) for s in S)
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end
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sett = PropertyT.Settings(NAME, G, S, autS, with_SCS(1000);
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elt = Δ^2
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upper_bound=0.15)
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unit = Δ
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ub = Inf
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@info sett
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status, certified, λ_cert = check_positivity(
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elt,
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unit,
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wd,
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upper_bound=ub,
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halfradius=2,
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optimizer=cosmo_optimizer(
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eps=1e-7,
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max_iters=10_000,
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accel=50,
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|
alpha=1.9,
|
||||||
|
),
|
||||||
|
)
|
||||||
|
|
||||||
@test PropertyT.check_property_T(sett) == false
|
@test status == JuMP.OPTIMAL
|
||||||
|
@test !certified
|
||||||
|
@test λ_cert < 0
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "SL(3,Z) has (T)" begin
|
||||||
|
n = 3
|
||||||
|
|
||||||
|
SL = MatrixGroups.SpecialLinearGroup{n}(Int8)
|
||||||
|
RSL, S, sizes = PropertyT.group_algebra(SL, halfradius=2, twisted=true)
|
||||||
|
|
||||||
|
Δ = RSL(length(S)) - sum(RSL(s) for s in S)
|
||||||
|
|
||||||
|
@testset "Wedderburn formulation" begin
|
||||||
|
P = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
|
Σ = PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
|
act = PropertyT.action_by_conjugation(SL, Σ)
|
||||||
|
wd = WedderburnDecomposition(
|
||||||
|
Rational{Int},
|
||||||
|
Σ,
|
||||||
|
act,
|
||||||
|
basis(RSL),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSL)[1:sizes[2]]),
|
||||||
|
)
|
||||||
|
|
||||||
|
elt = Δ^2
|
||||||
|
unit = Δ
|
||||||
|
ub = 0.2801
|
||||||
|
|
||||||
|
@test_throws ErrorException PropertyT.sos_problem_primal(
|
||||||
|
elt,
|
||||||
|
unit,
|
||||||
|
wd,
|
||||||
|
upper_bound=ub,
|
||||||
|
augmented=false,
|
||||||
|
)
|
||||||
|
|
||||||
|
wdfl = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Float64,
|
||||||
|
Σ,
|
||||||
|
act,
|
||||||
|
basis(RSL),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSL)[1:sizes[2]]),
|
||||||
|
)
|
||||||
|
|
||||||
|
model, varP = PropertyT.sos_problem_primal(
|
||||||
|
elt,
|
||||||
|
unit,
|
||||||
|
wdfl,
|
||||||
|
upper_bound=ub,
|
||||||
|
augmented=false,
|
||||||
|
)
|
||||||
|
|
||||||
|
status, warm = PropertyT.solve(
|
||||||
|
model,
|
||||||
|
scs_optimizer(
|
||||||
|
eps=1e-10,
|
||||||
|
max_iters=20_000,
|
||||||
|
accel=-20,
|
||||||
|
alpha=1.2,
|
||||||
|
),
|
||||||
|
)
|
||||||
|
|
||||||
|
Q = @time let varP = varP
|
||||||
|
Qs = map(varP) do P
|
||||||
|
real.(sqrt(JuMP.value.(P)))
|
||||||
|
end
|
||||||
|
PropertyT.reconstruct(Qs, wdfl)
|
||||||
|
end
|
||||||
|
λ = JuMP.value(model[:λ])
|
||||||
|
|
||||||
|
sos = PropertyT.compute_sos(parent(elt), Q; augmented=false)
|
||||||
|
|
||||||
|
certified, λ_cert = PropertyT.certify_solution(
|
||||||
|
elt,
|
||||||
|
unit,
|
||||||
|
λ,
|
||||||
|
Q,
|
||||||
|
halfradius=2,
|
||||||
|
augmented=false,
|
||||||
|
)
|
||||||
|
|
||||||
|
@test certified
|
||||||
|
@test λ_cert >= 28 // 100
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "augmented Wedderburn formulation" begin
|
||||||
|
elt = Δ^2
|
||||||
|
unit = Δ
|
||||||
|
ub = Inf
|
||||||
|
|
||||||
|
P = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
|
Σ = PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
|
act = PropertyT.action_by_conjugation(SL, Σ)
|
||||||
|
|
||||||
|
wdfl = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Float64,
|
||||||
|
Σ,
|
||||||
|
act,
|
||||||
|
basis(RSL),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSL)[1:sizes[2]]),
|
||||||
|
)
|
||||||
|
|
||||||
|
opt_problem, varP = PropertyT.sos_problem_primal(
|
||||||
|
elt,
|
||||||
|
unit,
|
||||||
|
wdfl,
|
||||||
|
upper_bound=ub,
|
||||||
|
# augmented = true # since both elt and unit are augmented
|
||||||
|
)
|
||||||
|
|
||||||
|
status, _ = PropertyT.solve(
|
||||||
|
opt_problem,
|
||||||
|
scs_optimizer(
|
||||||
|
eps=1e-8,
|
||||||
|
max_iters=20_000,
|
||||||
|
accel=0,
|
||||||
|
alpha=1.9,
|
||||||
|
),
|
||||||
|
)
|
||||||
|
|
||||||
|
@test status == JuMP.OPTIMAL
|
||||||
|
|
||||||
|
Q = @time let varP = varP
|
||||||
|
Qs = map(varP) do P
|
||||||
|
real.(sqrt(JuMP.value.(P)))
|
||||||
|
end
|
||||||
|
PropertyT.reconstruct(Qs, wdfl)
|
||||||
|
end
|
||||||
|
|
||||||
|
certified, λ_cert = PropertyT.certify_solution(
|
||||||
|
elt,
|
||||||
|
unit,
|
||||||
|
JuMP.objective_value(opt_problem),
|
||||||
|
Q,
|
||||||
|
halfradius=2,
|
||||||
|
# augmented = true # since both elt and unit are augmented
|
||||||
|
)
|
||||||
|
|
||||||
|
@test certified
|
||||||
|
@test λ_cert > 28 // 100
|
||||||
|
end
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|||||||
@ -1,3 +1,5 @@
|
|||||||
|
using SparseArrays
|
||||||
|
|
||||||
@testset "Sq, Adj, Op" begin
|
@testset "Sq, Adj, Op" begin
|
||||||
function isconstant_on_orbit(v, orb)
|
function isconstant_on_orbit(v, orb)
|
||||||
isempty(orb) && return true
|
isempty(orb) && return true
|
||||||
@ -6,240 +8,237 @@
|
|||||||
end
|
end
|
||||||
|
|
||||||
@testset "unit tests" begin
|
@testset "unit tests" begin
|
||||||
for N in [3,4]
|
|
||||||
M = MatrixAlgebra(zz, N)
|
|
||||||
|
|
||||||
@test PropertyT.EltaryMat(M, 1, 2) isa MatAlgElem
|
|
||||||
e12 = PropertyT.EltaryMat(M, 1, 2)
|
|
||||||
@test e12[1,2] == 1
|
|
||||||
@test inv(e12)[1,2] == -1
|
|
||||||
|
|
||||||
S = PropertyT.generating_set(M)
|
|
||||||
@test e12 ∈ S
|
|
||||||
|
|
||||||
@test length(PropertyT.generating_set(M)) == 2N*(N-1)
|
|
||||||
@test all(inv(s) ∈ S for s in S)
|
|
||||||
|
|
||||||
A = SAut(FreeGroup(N))
|
|
||||||
@test length(PropertyT.generating_set(A)) == 4N*(N-1)
|
|
||||||
S = PropertyT.generating_set(A)
|
|
||||||
@test all(inv(s) ∈ S for s in S)
|
|
||||||
end
|
|
||||||
|
|
||||||
@test PropertyT.isopposite(perm"(1,2,3)(4)", perm"(1,4,2)")
|
@test PropertyT.isopposite(perm"(1,2,3)(4)", perm"(1,4,2)")
|
||||||
@test PropertyT.isadjacent(perm"(1,2,3)", perm"(1,2)(3)")
|
@test PropertyT.isadjacent(perm"(1,2,3)", perm"(1,2)(3)")
|
||||||
|
|
||||||
@test !PropertyT.isopposite(perm"(1,2,3)", perm"(1,2)(3)")
|
@test !PropertyT.isopposite(perm"(1,2,3)", perm"(1,2)(3)")
|
||||||
@test !PropertyT.isadjacent(perm"(1,4)", perm"(2,3)(4)")
|
@test !PropertyT.isadjacent(perm"(1,4)", perm"(2,3)(4)")
|
||||||
|
|
||||||
@test isconstant_on_orbit([1,1,1,2,2], [2,3])
|
@test isconstant_on_orbit([1, 1, 1, 2, 2], [2, 3])
|
||||||
@test !isconstant_on_orbit([1,1,1,2,2], [2,3,4])
|
@test !isconstant_on_orbit([1, 1, 1, 2, 2], [2, 3, 4])
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "Sq, Adj, Op" begin
|
@testset "Sq, Adj, Op in SL(4,Z)" begin
|
||||||
|
|
||||||
N = 4
|
N = 4
|
||||||
M = MatrixAlgebra(zz, N)
|
G = MatrixGroups.SpecialLinearGroup{N}(Int8)
|
||||||
S = PropertyT.generating_set(M)
|
|
||||||
Δ = PropertyT.Laplacian(S, 2)
|
|
||||||
RG = parent(Δ)
|
|
||||||
|
|
||||||
autS = WreathProduct(SymmetricGroup(2), SymmetricGroup(N))
|
RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
|
||||||
orbits = PropertyT.orbit_decomposition(autS, RG.basis)
|
|
||||||
|
|
||||||
@test PropertyT.Sq(RG) isa GroupRingElem
|
Δ = let RG = RG, S = S
|
||||||
sq = PropertyT.Sq(RG)
|
RG(length(S)) - sum(RG(s) for s in S)
|
||||||
@test all(isconstant_on_orbit(sq, orb) for orb in orbits)
|
end
|
||||||
|
|
||||||
@test PropertyT.Adj(RG) isa GroupRingElem
|
P = PermGroup(perm"(1,2)", Perm(circshift(1:N, -1)))
|
||||||
adj = PropertyT.Adj(RG)
|
Σ = PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
@test all(isconstant_on_orbit(adj, orb) for orb in orbits)
|
act = PropertyT.action_by_conjugation(G, Σ)
|
||||||
|
|
||||||
@test PropertyT.Op(RG) isa GroupRingElem
|
wd = WedderburnDecomposition(
|
||||||
op = PropertyT.Op(RG)
|
Float64,
|
||||||
@test all(isconstant_on_orbit(op, orb) for orb in orbits)
|
Σ,
|
||||||
|
act,
|
||||||
|
basis(RG),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RG)[1:sizes[2]]),
|
||||||
|
)
|
||||||
|
ivs = SymbolicWedderburn.invariant_vectors(wd)
|
||||||
|
|
||||||
sq, adj, op = PropertyT.SqAdjOp(RG, N)
|
sq, adj, op = PropertyT.SqAdjOp(RG, N)
|
||||||
@test sq == PropertyT.Sq(RG)
|
|
||||||
@test adj == PropertyT.Adj(RG)
|
|
||||||
@test op == PropertyT.Op(RG)
|
|
||||||
|
|
||||||
e = one(M)
|
@test all(
|
||||||
g = PropertyT.EltaryMat(M, 1,2)
|
isconstant_on_orbit(sq, SparseArrays.nonzeroinds(iv)) for iv in ivs
|
||||||
h = PropertyT.EltaryMat(M, 1,3)
|
)
|
||||||
k = PropertyT.EltaryMat(M, 3,4)
|
|
||||||
|
|
||||||
edges = N*(N-1)÷2
|
@test all(
|
||||||
@test sq[e] == 20*edges
|
isconstant_on_orbit(adj, SparseArrays.nonzeroinds(iv)) for iv in ivs
|
||||||
|
)
|
||||||
|
|
||||||
|
@test all(
|
||||||
|
isconstant_on_orbit(op, SparseArrays.nonzeroinds(iv)) for iv in ivs
|
||||||
|
)
|
||||||
|
|
||||||
|
e = one(G)
|
||||||
|
g = G([alphabet(G)[MatrixGroups.ElementaryMatrix{N}(1, 2, Int8(1))]])
|
||||||
|
h = G([alphabet(G)[MatrixGroups.ElementaryMatrix{N}(1, 3, Int8(1))]])
|
||||||
|
k = G([alphabet(G)[MatrixGroups.ElementaryMatrix{N}(3, 4, Int8(1))]])
|
||||||
|
|
||||||
|
@test sq[e] == 120
|
||||||
@test sq[g] == sq[h] == -8
|
@test sq[g] == sq[h] == -8
|
||||||
@test sq[g^2] == sq[h^2] == 1
|
@test sq[g^2] == sq[h^2] == 1
|
||||||
@test sq[g*h] == sq[h*g] == 0
|
@test sq[g*h] == sq[h*g] == 0
|
||||||
|
|
||||||
# @test adj[e] == ...
|
@test adj[e] == 384
|
||||||
@test adj[g] == adj[h] # == ...
|
@test adj[g] == adj[h] == -32
|
||||||
@test adj[g^2] == adj[h^2] == 0
|
@test adj[g^2] == adj[h^2] == 0
|
||||||
@test adj[g*h] == adj[h*g] # == ...
|
@test adj[g*h] == adj[h*g] == 2
|
||||||
|
@test adj[k*h] == adj[h*k] == 1
|
||||||
|
|
||||||
|
@test op[e] == 96
|
||||||
# @test op[e] == ...
|
@test op[g] == op[h] == -8
|
||||||
@test op[g] == op[h] # == ...
|
|
||||||
@test op[g^2] == op[h^2] == 0
|
@test op[g^2] == op[h^2] == 0
|
||||||
@test op[g*h] == op[h*g] == 0
|
@test op[g*h] == op[h*g] == 0
|
||||||
@test op[g*k] == op[k*g] # == ...
|
@test op[g*k] == op[k*g] == 2
|
||||||
@test op[h*k] == op[k*h] == 0
|
@test op[h*k] == op[k*h] == 0
|
||||||
end
|
end
|
||||||
|
|
||||||
|
@testset "SAut(F₃)" begin
|
||||||
|
n = 3
|
||||||
|
G = SpecialAutomorphismGroup(FreeGroup(n))
|
||||||
|
RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
|
||||||
|
sq, adj, op = PropertyT.SqAdjOp(RG, n)
|
||||||
|
|
||||||
|
@test sq(one(G)) == 216
|
||||||
|
@test all(sq(g) == -16 for g in gens(G))
|
||||||
|
@test adj(one(G)) == 384
|
||||||
|
@test all(adj(g) == -32 for g in gens(G))
|
||||||
|
@test iszero(op)
|
||||||
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
|
|
||||||
@testset "1812.03456 examples" begin
|
@testset "1812.03456 examples" begin
|
||||||
|
|
||||||
function SOS_residual(x::GroupRingElem, Q::Matrix)
|
|
||||||
RG = parent(x)
|
|
||||||
@time sos = PropertyT.compute_SOS(RG, Q);
|
|
||||||
return x - sos
|
|
||||||
end
|
|
||||||
|
|
||||||
function check_positivity(elt, Δ, orbit_data, upper_bound, warm=nothing; with_solver=with_SCS(20_000, accel=10))
|
|
||||||
SDP_problem, varP = PropertyT.SOS_problem_primal(elt, Δ, orbit_data; upper_bound=upper_bound)
|
|
||||||
|
|
||||||
status, warm = PropertyT.solve(SDP_problem, with_solver, warm);
|
|
||||||
Base.Libc.flush_cstdio()
|
|
||||||
@info "Optimization status:" status
|
|
||||||
|
|
||||||
λ = value(SDP_problem[:λ])
|
|
||||||
Ps = [value.(P) for P in varP]
|
|
||||||
|
|
||||||
Qs = real.(sqrt.(Ps));
|
|
||||||
Q = PropertyT.reconstruct(Qs, orbit_data);
|
|
||||||
|
|
||||||
b = SOS_residual(elt - λ*Δ, Q)
|
|
||||||
return b, λ, warm
|
|
||||||
end
|
|
||||||
|
|
||||||
@testset "SL(3,Z)" begin
|
@testset "SL(3,Z)" begin
|
||||||
N = 3
|
n = 3
|
||||||
halfradius = 2
|
|
||||||
M = MatrixAlgebra(zz, N)
|
G = MatrixGroups.SpecialLinearGroup{n}(Int8)
|
||||||
S = PropertyT.generating_set(M)
|
RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
|
||||||
Δ = PropertyT.Laplacian(S, halfradius)
|
|
||||||
RG = parent(Δ)
|
Δ = RG(length(S)) - sum(RG(s) for s in S)
|
||||||
orbit_data = PropertyT.BlockDecomposition(RG, WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
|
|
||||||
orbit_data = PropertyT.decimate(orbit_data);
|
P = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
|
Σ = PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
|
act = PropertyT.action_by_conjugation(G, Σ)
|
||||||
|
|
||||||
|
wd = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Float64,
|
||||||
|
Σ,
|
||||||
|
act,
|
||||||
|
basis(RG),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RG)[1:sizes[2]]),
|
||||||
|
)
|
||||||
|
|
||||||
|
sq, adj, op = PropertyT.SqAdjOp(RG, n)
|
||||||
|
|
||||||
@testset "Sq₃ is SOS" begin
|
@testset "Sq₃ is SOS" begin
|
||||||
elt = PropertyT.Sq(RG)
|
elt = sq
|
||||||
UB = 0.05 # 0.105?
|
UB = Inf # λ ≈ 0.1040844
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
status, certified, λ_cert = check_positivity(
|
||||||
Base.Libc.flush_cstdio()
|
elt,
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
Δ,
|
||||||
|
wd,
|
||||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
upper_bound=UB,
|
||||||
|
halfradius=2,
|
||||||
@test 2^2*norm(residual, 1) < 2λ/100
|
optimizer=cosmo_optimizer(accel=50, alpha=1.9)
|
||||||
|
)
|
||||||
|
@test status == JuMP.OPTIMAL
|
||||||
|
@test certified
|
||||||
|
@test λ_cert > 104 // 1000
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "Adj₃ is SOS" begin
|
@testset "Adj₃ is SOS" begin
|
||||||
elt = PropertyT.Adj(RG)
|
elt = adj
|
||||||
UB = 0.1 # 0.157?
|
UB = Inf # λ ≈ 0.15858018
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
status, certified, λ_cert = check_positivity(
|
||||||
Base.Libc.flush_cstdio()
|
elt,
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
Δ,
|
||||||
|
wd,
|
||||||
@test 2^2*norm(residual, 1) < λ
|
upper_bound=UB,
|
||||||
@test 2^2*norm(residual, 1) < λ/100
|
halfradius=2,
|
||||||
|
optimizer=cosmo_optimizer(accel=50, alpha=1.9)
|
||||||
|
)
|
||||||
|
@test status == JuMP.OPTIMAL
|
||||||
|
@test certified
|
||||||
|
@test λ_cert > 1585 // 10000
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "Op₃ is empty, so can not be certified" begin
|
@testset "Op₃ is empty, so can not be certified" begin
|
||||||
elt = PropertyT.Op(RG)
|
elt = op
|
||||||
UB = Inf
|
UB = Inf
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
status, certified, λ_cert = check_positivity(
|
||||||
Base.Libc.flush_cstdio()
|
elt,
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
Δ,
|
||||||
|
wd,
|
||||||
@test 2^2*norm(residual, 1) > λ
|
upper_bound=UB,
|
||||||
|
halfradius=2,
|
||||||
|
optimizer=cosmo_optimizer(accel=50, alpha=1.9)
|
||||||
|
)
|
||||||
|
@test status == JuMP.OPTIMAL
|
||||||
|
@test !certified
|
||||||
|
@test λ_cert < 0
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "SL(4,Z)" begin
|
@testset "SL(4,Z)" begin
|
||||||
N = 4
|
n = 4
|
||||||
halfradius = 2
|
|
||||||
M = MatrixAlgebra(zz, N)
|
|
||||||
S = PropertyT.generating_set(M)
|
|
||||||
Δ = PropertyT.Laplacian(S, halfradius)
|
|
||||||
RG = parent(Δ)
|
|
||||||
orbit_data = PropertyT.BlockDecomposition(RG, WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
|
|
||||||
orbit_data = PropertyT.decimate(orbit_data);
|
|
||||||
|
|
||||||
@testset "Sq₄ is SOS" begin
|
G = MatrixGroups.SpecialLinearGroup{n}(Int8)
|
||||||
elt = PropertyT.Sq(RG)
|
RG, S, sizes = PropertyT.group_algebra(G, halfradius=2, twisted=true)
|
||||||
UB = 0.2 # 0.3172
|
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
Δ = RG(length(S)) - sum(RG(s) for s in S)
|
||||||
Base.Libc.flush_cstdio()
|
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
|
||||||
|
|
||||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
P = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
@test 2^2*norm(residual, 1) < λ/100
|
Σ = PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
|
act = PropertyT.action_by_conjugation(G, Σ)
|
||||||
|
|
||||||
|
wd = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Float64,
|
||||||
|
Σ,
|
||||||
|
act,
|
||||||
|
basis(RG),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RG)[1:sizes[2]]),
|
||||||
|
)
|
||||||
|
|
||||||
|
sq, adj, op = PropertyT.SqAdjOp(RG, n)
|
||||||
|
|
||||||
|
@testset "Sq is SOS" begin
|
||||||
|
elt = sq
|
||||||
|
UB = Inf # λ ≈ 0.31670
|
||||||
|
|
||||||
|
status, certified, λ_cert = check_positivity(
|
||||||
|
elt,
|
||||||
|
Δ,
|
||||||
|
wd,
|
||||||
|
upper_bound=UB,
|
||||||
|
halfradius=2,
|
||||||
|
optimizer=cosmo_optimizer(accel=50, alpha=1.9)
|
||||||
|
)
|
||||||
|
@test status == JuMP.OPTIMAL
|
||||||
|
@test certified
|
||||||
|
@test λ_cert > 316 // 1000
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "Adj₄ is SOS" begin
|
@testset "Adj is SOS" begin
|
||||||
elt = PropertyT.Adj(RG)
|
elt = adj
|
||||||
UB = 0.3 # 0.5459?
|
UB = 0.541 # λ ≈ 0.545710
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
status, certified, λ_cert = check_positivity(
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
elt,
|
||||||
|
Δ,
|
||||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
wd,
|
||||||
@test 2^2*norm(residual, 1) < λ/100
|
upper_bound=UB,
|
||||||
|
halfradius=2,
|
||||||
|
optimizer=cosmo_optimizer(accel=50, alpha=1.9)
|
||||||
|
)
|
||||||
|
@test status == JuMP.OPTIMAL
|
||||||
|
@test certified
|
||||||
|
@test λ_cert > 54 // 100
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "we can't cerify that Op₄ SOS" begin
|
@testset "Op is a sum of squares, but not an order unit" begin
|
||||||
elt = PropertyT.Op(RG)
|
elt = op
|
||||||
UB = 2.0
|
UB = Inf
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB,
|
status, certified, λ_cert = check_positivity(
|
||||||
with_solver=with_SCS(20_000, accel=10, eps=2e-10))
|
elt,
|
||||||
Base.Libc.flush_cstdio()
|
Δ,
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
wd,
|
||||||
|
upper_bound=UB,
|
||||||
@test 2^2*norm(residual, 1) > λ # i.e. we can't certify positivity
|
halfradius=2,
|
||||||
end
|
optimizer=cosmo_optimizer(accel=50, alpha=1.9)
|
||||||
|
)
|
||||||
@testset "Adj₄ + Op₄ is SOS" begin
|
@test status == JuMP.OPTIMAL
|
||||||
elt = PropertyT.Adj(RG) + PropertyT.Op(RG)
|
@test !certified
|
||||||
UB = 0.6 # 0.82005
|
@test -1e-2 < λ_cert < 0
|
||||||
|
|
||||||
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
|
|
||||||
Base.Libc.flush_cstdio()
|
|
||||||
@info "obtained λ and residual" λ norm(residual, 1)
|
|
||||||
|
|
||||||
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
|
||||||
@test 2^2*norm(residual, 1) < λ/100
|
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
# @testset "Adj₄ + 100 Op₄ ∈ ISAut(F₄) is SOS" begin
|
|
||||||
# N = 4
|
|
||||||
# halfradius = 2
|
|
||||||
# M = SAut(FreeGroup(N))
|
|
||||||
# S = PropertyT.generating_set(M)
|
|
||||||
# Δ = PropertyT.Laplacian(S, halfradius)
|
|
||||||
# RG = parent(Δ)
|
|
||||||
# orbit_data = PropertyT.BlockDecomposition(RG, WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
|
|
||||||
# orbit_data = PropertyT.decimate(orbit_data);
|
|
||||||
#
|
|
||||||
# @time elt = PropertyT.Adj(RG) + 100PropertyT.Op(RG)
|
|
||||||
# UB = 0.05
|
|
||||||
#
|
|
||||||
# warm = nothing
|
|
||||||
#
|
|
||||||
# residual, λ, warm = check_positivity(elt, Δ, orbit_data, UB, warm, with_solver=with_SCS(20_000, accel=10))
|
|
||||||
# @info "obtained λ and residual" λ norm(residual, 1)
|
|
||||||
#
|
|
||||||
# @test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
|
|
||||||
# @test 2^2*norm(residual, 1) < λ/100
|
|
||||||
# end
|
|
||||||
end
|
end
|
||||||
|
|||||||
255
test/actions.jl
255
test/actions.jl
@ -1,101 +1,210 @@
|
|||||||
@testset "actions on Group[Rings]" begin
|
function test_action(basis, group, act)
|
||||||
Eij = PropertyT.EltaryMat
|
action = SymbolicWedderburn.action
|
||||||
ssgs(M::MatAlgebra, i, j) = (S = [Eij(M, i, j), Eij(M, j, i)];
|
return @testset "action definition" begin
|
||||||
S = unique([S; inv.(S)]); S)
|
@test all(basis) do b
|
||||||
|
e = one(group)
|
||||||
function ssgs(A::AutGroup, i, j)
|
action(act, e, b) == b
|
||||||
rmuls = [Groups.transvection_R(i,j), Groups.transvection_R(j,i)]
|
|
||||||
lmuls = [Groups.transvection_L(i,j), Groups.transvection_L(j,i)]
|
|
||||||
gen_set = A.([rmuls; lmuls])
|
|
||||||
return unique([gen_set; inv.(gen_set)])
|
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "actions on SL(3,Z) and its group ring" begin
|
a = let a = rand(basis)
|
||||||
N = 3
|
while isone(a)
|
||||||
halfradius = 2
|
a = rand(basis)
|
||||||
M = MatrixAlgebra(zz, N)
|
|
||||||
S = PropertyT.generating_set(M)
|
|
||||||
E_R, sizes = Groups.wlmetric_ball(S, one(M), radius=2halfradius);
|
|
||||||
|
|
||||||
rdict = GroupRings.reverse_dict(E_R)
|
|
||||||
pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
|
|
||||||
RG = GroupRing(M, E_R, rdict, pm)
|
|
||||||
|
|
||||||
@testset "correctness of actions" begin
|
|
||||||
Δ = length(S)*RG(1) - sum(RG(s) for s in S)
|
|
||||||
@test Δ == PropertyT.spLaplacian(RG, S)
|
|
||||||
|
|
||||||
elt = S[5]
|
|
||||||
x = RG(1) - RG(elt)
|
|
||||||
elt2 = E_R[rand(sizes[1]:sizes[2])]
|
|
||||||
y = 2RG(elt2) - RG(elt)
|
|
||||||
|
|
||||||
for G in [SymmetricGroup(N), WreathProduct(SymmetricGroup(2), SymmetricGroup(N))]
|
|
||||||
@test all(one(M)^g == one(M) for g in G)
|
|
||||||
@test all(rdict[m^g] <= sizes[1] for g in G for m in S)
|
|
||||||
@test all(m^g*n^g == (m*n)^g for g in G for m in S for n in S)
|
|
||||||
|
|
||||||
@test all(Δ^g == Δ for g in G)
|
|
||||||
@test all(x^g == RG(1) - RG(elt^g) for g in G)
|
|
||||||
|
|
||||||
@test all(2RG(elt2^g) - RG(elt^g) == y^g for g in G)
|
|
||||||
end
|
end
|
||||||
|
@assert !isone(a)
|
||||||
|
a
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "small Laplacians" begin
|
g, h = let g_h = rand(group, 2)
|
||||||
for (i,j) in PropertyT.indexing(N)
|
while any(isone, g_h)
|
||||||
Sij = ssgs(M, i,j)
|
g_h = rand(group, 2)
|
||||||
Δij= PropertyT.spLaplacian(RG, Sij)
|
end
|
||||||
|
@assert all(!isone, g_h)
|
||||||
|
g_h
|
||||||
|
end
|
||||||
|
|
||||||
@test all(Δij^p == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in SymmetricGroup(N))
|
action = SymbolicWedderburn.action
|
||||||
|
@test action(act, g, a) in basis
|
||||||
|
@test action(act, h, a) in basis
|
||||||
|
@test action(act, h, action(act, g, a)) == action(act, g * h, a)
|
||||||
|
|
||||||
@test all(Δij^g == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
|
@test all([(g, h) for g in group for h in group]) do (g, h)
|
||||||
|
x = action(act, h, action(act, g, a))
|
||||||
|
y = action(act, g * h, a)
|
||||||
|
x == y
|
||||||
|
end
|
||||||
|
|
||||||
|
if act isa SymbolicWedderburn.ByPermutations
|
||||||
|
@test all(basis) do b
|
||||||
|
action(act, g, b) ∈ basis && action(act, h, b) ∈ basis
|
||||||
|
end
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
@testset "actions on SAut(F_3) and its group ring" begin
|
## Testing
|
||||||
N = 3
|
|
||||||
halfradius = 2
|
|
||||||
M = SAut(FreeGroup(N))
|
|
||||||
S = PropertyT.generating_set(M)
|
|
||||||
E_R, sizes = Groups.wlmetric_ball(S, one(M), radius=2halfradius);
|
|
||||||
|
|
||||||
rdict = GroupRings.reverse_dict(E_R)
|
@testset "Actions on SL(3,ℤ)" begin
|
||||||
pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
|
|
||||||
RG = GroupRing(M, E_R, rdict, pm)
|
|
||||||
|
|
||||||
|
n = 3
|
||||||
|
|
||||||
@testset "correctness of actions" begin
|
SL = MatrixGroups.SpecialLinearGroup{n}(Int8)
|
||||||
|
RSL, S, sizes = PropertyT.group_algebra(SL, halfradius=2, twisted=true)
|
||||||
|
|
||||||
Δ = length(S)*RG(1) - sum(RG(s) for s in S)
|
@testset "Permutation action" begin
|
||||||
@test Δ == PropertyT.spLaplacian(RG, S)
|
|
||||||
|
|
||||||
elt = S[5]
|
Γ = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
x = RG(1) - RG(elt)
|
ΓpA = PropertyT.action_by_conjugation(SL, Γ)
|
||||||
elt2 = E_R[rand(sizes[1]:sizes[2])]
|
|
||||||
y = 2RG(elt2) - RG(elt)
|
|
||||||
|
|
||||||
for G in [SymmetricGroup(N), WreathProduct(SymmetricGroup(2), SymmetricGroup(N))]
|
test_action(basis(RSL), Γ, ΓpA)
|
||||||
@test all(one(M)^g == one(M) for g in G)
|
|
||||||
@test all(rdict[m^g] <= sizes[1] for g in G for m in S)
|
|
||||||
@test all(m^g*n^g == (m*n)^g for g in G for m in S for n in S)
|
|
||||||
|
|
||||||
@test all(Δ^g == Δ for g in G)
|
@testset "mps is successful" begin
|
||||||
@test all(x^g == RG(1) - RG(elt^g) for g in G)
|
|
||||||
|
|
||||||
@test all(2RG(elt2^g) - RG(elt^g) == y^g for g in G)
|
charsΓ =
|
||||||
|
SymbolicWedderburn.Character{
|
||||||
|
Rational{Int},
|
||||||
|
}.(SymbolicWedderburn.irreducible_characters(Γ))
|
||||||
|
|
||||||
|
RΓ = SymbolicWedderburn._group_algebra(Γ)
|
||||||
|
|
||||||
|
@time mps, simple =
|
||||||
|
SymbolicWedderburn.minimal_projection_system(charsΓ, RΓ)
|
||||||
|
@test all(simple)
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "Wedderburn decomposition" begin
|
||||||
|
wd = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Rational{Int},
|
||||||
|
Γ,
|
||||||
|
ΓpA,
|
||||||
|
basis(RSL),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSL)[1:sizes[2]])
|
||||||
|
)
|
||||||
|
|
||||||
|
@test length(invariant_vectors(wd)) == 918
|
||||||
|
@test SymbolicWedderburn.size.(direct_summands(wd), 1) == [40, 23, 18]
|
||||||
|
@test all(issimple, direct_summands(wd))
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
for (i,j) in PropertyT.indexing(N)
|
@testset "Wreath action" begin
|
||||||
Sij = ssgs(M, i,j)
|
|
||||||
Δij= PropertyT.spLaplacian(RG, Sij)
|
|
||||||
|
|
||||||
@test all(Δij^p == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in SymmetricGroup(N))
|
Γ = let P = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
|
PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
|
end
|
||||||
|
|
||||||
@test all(Δij^g == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
|
ΓpA = PropertyT.action_by_conjugation(SL, Γ)
|
||||||
|
|
||||||
|
test_action(basis(RSL), Γ, ΓpA)
|
||||||
|
|
||||||
|
@testset "mps is successful" begin
|
||||||
|
|
||||||
|
charsΓ =
|
||||||
|
SymbolicWedderburn.Character{
|
||||||
|
Rational{Int},
|
||||||
|
}.(SymbolicWedderburn.irreducible_characters(Γ))
|
||||||
|
|
||||||
|
RΓ = SymbolicWedderburn._group_algebra(Γ)
|
||||||
|
|
||||||
|
@time mps, simple =
|
||||||
|
SymbolicWedderburn.minimal_projection_system(charsΓ, RΓ)
|
||||||
|
@test all(simple)
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "Wedderburn decomposition" begin
|
||||||
|
wd = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Rational{Int},
|
||||||
|
Γ,
|
||||||
|
ΓpA,
|
||||||
|
basis(RSL),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSL)[1:sizes[2]])
|
||||||
|
)
|
||||||
|
|
||||||
|
@test length(invariant_vectors(wd)) == 247
|
||||||
|
@test SymbolicWedderburn.size.(direct_summands(wd), 1) == [14, 9, 6, 14, 12]
|
||||||
|
@test all(issimple, direct_summands(wd))
|
||||||
|
end
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
|
@testset "Actions on SAut(F4)" begin
|
||||||
|
|
||||||
|
n = 4
|
||||||
|
|
||||||
|
SAutFn = SpecialAutomorphismGroup(FreeGroup(n))
|
||||||
|
RSAutFn, S, sizes = PropertyT.group_algebra(SAutFn, halfradius=1, twisted=true)
|
||||||
|
|
||||||
|
@testset "Permutation action" begin
|
||||||
|
|
||||||
|
Γ = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
|
ΓpA = PropertyT.action_by_conjugation(SAutFn, Γ)
|
||||||
|
|
||||||
|
test_action(basis(RSAutFn), Γ, ΓpA)
|
||||||
|
|
||||||
|
@testset "mps is successful" begin
|
||||||
|
|
||||||
|
charsΓ =
|
||||||
|
SymbolicWedderburn.Character{
|
||||||
|
Rational{Int},
|
||||||
|
}.(SymbolicWedderburn.irreducible_characters(Γ))
|
||||||
|
|
||||||
|
RΓ = SymbolicWedderburn._group_algebra(Γ)
|
||||||
|
|
||||||
|
@time mps, simple =
|
||||||
|
SymbolicWedderburn.minimal_projection_system(charsΓ, RΓ)
|
||||||
|
@test all(simple)
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "Wedderburn decomposition" begin
|
||||||
|
wd = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Rational{Int},
|
||||||
|
Γ,
|
||||||
|
ΓpA,
|
||||||
|
basis(RSAutFn),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSAutFn)[1:sizes[1]])
|
||||||
|
)
|
||||||
|
|
||||||
|
@test length(invariant_vectors(wd)) == 93
|
||||||
|
@test SymbolicWedderburn.size.(direct_summands(wd), 1) == [4, 8, 5, 4]
|
||||||
|
@test all(issimple, direct_summands(wd))
|
||||||
|
end
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "Wreath action" begin
|
||||||
|
|
||||||
|
Γ = let P = PermGroup(perm"(1,2)", Perm(circshift(1:n, -1)))
|
||||||
|
PropertyT.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
|
||||||
|
end
|
||||||
|
|
||||||
|
ΓpA = PropertyT.action_by_conjugation(SAutFn, Γ)
|
||||||
|
|
||||||
|
test_action(basis(RSAutFn), Γ, ΓpA)
|
||||||
|
|
||||||
|
@testset "mps is successful" begin
|
||||||
|
|
||||||
|
charsΓ =
|
||||||
|
SymbolicWedderburn.Character{
|
||||||
|
Rational{Int},
|
||||||
|
}.(SymbolicWedderburn.irreducible_characters(Γ))
|
||||||
|
|
||||||
|
RΓ = SymbolicWedderburn._group_algebra(Γ)
|
||||||
|
|
||||||
|
@time mps, simple =
|
||||||
|
SymbolicWedderburn.minimal_projection_system(charsΓ, RΓ)
|
||||||
|
@test all(simple)
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "Wedderburn decomposition" begin
|
||||||
|
wd = SymbolicWedderburn.WedderburnDecomposition(
|
||||||
|
Rational{Int},
|
||||||
|
Γ,
|
||||||
|
ΓpA,
|
||||||
|
basis(RSAutFn),
|
||||||
|
StarAlgebras.Basis{UInt16}(@view basis(RSAutFn)[1:sizes[1]])
|
||||||
|
)
|
||||||
|
|
||||||
|
@test length(invariant_vectors(wd)) == 18
|
||||||
|
@test SymbolicWedderburn.size.(direct_summands(wd), 1) == [1, 1, 2, 2, 1, 2, 2, 1]
|
||||||
|
@test all(issimple, direct_summands(wd))
|
||||||
|
end
|
||||||
|
end
|
||||||
end
|
end
|
||||||
|
|||||||
40
test/graded_adj.jl
Normal file
40
test/graded_adj.jl
Normal file
@ -0,0 +1,40 @@
|
|||||||
|
@testset "Adj for SpN via grading" begin
|
||||||
|
|
||||||
|
genus = 3
|
||||||
|
halfradius = 2
|
||||||
|
|
||||||
|
SpN = MatrixGroups.SymplecticGroup{2genus}(Int8)
|
||||||
|
|
||||||
|
RSpN, S_sp, sizes_sp = PropertyT.group_algebra(SpN, halfradius=halfradius, twisted=true)
|
||||||
|
|
||||||
|
Δ, Δs = let RG = RSpN, S = S_sp, ψ = identity
|
||||||
|
Δ = RG(length(S)) - sum(RG(s) for s in S)
|
||||||
|
Δs = PropertyT.laplacians(
|
||||||
|
RG,
|
||||||
|
S,
|
||||||
|
x -> (gx = PropertyT.grading(ψ(x)); Set([gx, -gx])),
|
||||||
|
)
|
||||||
|
Δ, Δs
|
||||||
|
end
|
||||||
|
|
||||||
|
@testset "Adj correctness: genus=$genus" begin
|
||||||
|
|
||||||
|
all_subtypes = (
|
||||||
|
:A₁, :C₁, Symbol("A₁×A₁"), Symbol("C₁×C₁"), Symbol("A₁×C₁"), :A₂, :C₂
|
||||||
|
)
|
||||||
|
|
||||||
|
@test PropertyT.Adj(Δs, :A₂)[one(SpN)] == 384
|
||||||
|
@test iszero(PropertyT.Adj(Δs, Symbol("A₁×A₁")))
|
||||||
|
@test iszero(PropertyT.Adj(Δs, Symbol("C₁×C₁")))
|
||||||
|
|
||||||
|
@testset "divisibility by 16" begin
|
||||||
|
for subtype in all_subtypes
|
||||||
|
subtype in (:A₁, :C₁) && continue
|
||||||
|
@test isinteger(PropertyT.Adj(Δs, subtype)[one(SpN)] / 16)
|
||||||
|
end
|
||||||
|
end
|
||||||
|
@test sum(PropertyT.Adj(Δs, subtype) for subtype in all_subtypes) == Δ^2
|
||||||
|
end
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
54
test/optimizers.jl
Normal file
54
test/optimizers.jl
Normal file
@ -0,0 +1,54 @@
|
|||||||
|
## Optimizers
|
||||||
|
|
||||||
|
import JuMP
|
||||||
|
import SCS
|
||||||
|
|
||||||
|
function scs_optimizer(;
|
||||||
|
accel=10,
|
||||||
|
alpha=1.5,
|
||||||
|
eps=1e-9,
|
||||||
|
max_iters=100_000,
|
||||||
|
verbose=true,
|
||||||
|
linear_solver=SCS.DirectSolver
|
||||||
|
)
|
||||||
|
return JuMP.optimizer_with_attributes(
|
||||||
|
SCS.Optimizer,
|
||||||
|
"acceleration_lookback" => accel,
|
||||||
|
"acceleration_interval" => 10,
|
||||||
|
"alpha" => alpha,
|
||||||
|
"eps_abs" => eps,
|
||||||
|
"eps_rel" => eps,
|
||||||
|
"linear_solver" => linear_solver,
|
||||||
|
"max_iters" => max_iters,
|
||||||
|
"warm_start" => true,
|
||||||
|
"verbose" => verbose,
|
||||||
|
)
|
||||||
|
end
|
||||||
|
|
||||||
|
import COSMO
|
||||||
|
|
||||||
|
function cosmo_optimizer(;
|
||||||
|
accel=15,
|
||||||
|
alpha=1.6,
|
||||||
|
eps=1e-9,
|
||||||
|
max_iters=100_000,
|
||||||
|
verbose=true,
|
||||||
|
verbose_timing=verbose,
|
||||||
|
decompose=false
|
||||||
|
)
|
||||||
|
return JuMP.optimizer_with_attributes(
|
||||||
|
COSMO.Optimizer,
|
||||||
|
"accelerator" => COSMO.with_options(
|
||||||
|
COSMO.AndersonAccelerator,
|
||||||
|
mem=max(accel, 2)
|
||||||
|
),
|
||||||
|
"alpha" => alpha,
|
||||||
|
"decompose" => decompose,
|
||||||
|
"eps_abs" => eps,
|
||||||
|
"eps_rel" => eps,
|
||||||
|
"max_iter" => max_iters,
|
||||||
|
"verbose" => verbose,
|
||||||
|
"verbose_timing" => verbose_timing,
|
||||||
|
"check_termination" => 200,
|
||||||
|
)
|
||||||
|
end
|
||||||
@ -1,18 +1,27 @@
|
|||||||
using Test
|
using Test
|
||||||
using LinearAlgebra, SparseArrays
|
using LinearAlgebra
|
||||||
using AbstractAlgebra, Groups, GroupRings
|
using SparseArrays
|
||||||
|
BLAS.set_num_threads(1)
|
||||||
|
ENV["OMP_NUM_THREADS"] = 4
|
||||||
|
|
||||||
|
using Groups
|
||||||
|
using Groups.GroupsCore
|
||||||
|
import Groups.MatrixGroups
|
||||||
|
|
||||||
using PropertyT
|
using PropertyT
|
||||||
using JLD
|
using SymbolicWedderburn
|
||||||
|
using SymbolicWedderburn.StarAlgebras
|
||||||
|
using SymbolicWedderburn.PermutationGroups
|
||||||
|
|
||||||
using JuMP, SCS
|
include("optimizers.jl")
|
||||||
|
|
||||||
with_SCS(iters; accel=0, eps=1e-10, warm_start=true) =
|
@testset "PropertyT" begin
|
||||||
with_optimizer(SCS.Optimizer,
|
|
||||||
linear_solver=SCS.DirectSolver, max_iters=iters,
|
|
||||||
acceleration_lookback=accel, eps=eps, warm_start=warm_start)
|
|
||||||
|
|
||||||
include("1703.09680.jl")
|
include("actions.jl")
|
||||||
include("actions.jl")
|
|
||||||
include("1712.07167.jl")
|
include("1703.09680.jl")
|
||||||
include("SOS_correctness.jl")
|
include("1712.07167.jl")
|
||||||
include("1812.03456.jl")
|
include("1812.03456.jl")
|
||||||
|
|
||||||
|
include("graded_adj.jl")
|
||||||
|
end
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user