240 lines
7.0 KiB
Julia
240 lines
7.0 KiB
Julia
using JLD
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using JuMP
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using SCS
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using GroupRings
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using PropertyT
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import Nemo: Group, GroupElem
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using ArgParse
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include("OrbitDecomposition.jl")
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immutable ProblemData{T}
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name::String
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Us::Vector{SparseMatrixCSC{Float64,Int}}
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Ps::Vector{Array{JuMP.Variable,2}}
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cnstr::Vector{T}
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laplacian::SparseVector{Float64}
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laplacianSq::SparseVector{Float64}
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dims::Vector{Int}
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end
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function sparsify!{T}(U::AbstractArray{T}, eps=eps(T))
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# n = rank(U)
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U[abs.(U) .< eps] = zero(T)
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# @assert rank(U) == n
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return sparse(U)
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end
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sparsify{T}(U::AbstractArray{T}, eps=eps(T)) = sparsify!(deepcopy(U), eps)
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small_to_zero!{T}(A::AbstractArray{T}, eps=eps(T)) = A[abs(A) .< eps] = zero(T)
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function init_model(Uπs)
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m = JuMP.Model();
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l = size(Uπs,1)
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P = Vector{Array{JuMP.Variable,2}}(l)
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for k in 1:l
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s = size(Uπs[k],2)
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P[k] = JuMP.@variable(m, P[k][i=1:s, j=1:s])
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JuMP.@SDconstraint(m, P[k] >= 0.0)
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end
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JuMP.@variable(m, λ >= 0.0)
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JuMP.@objective(m, Max, λ)
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return m, P
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end
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function init_ProblemData(name::String)
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splap = load(joinpath(name, "delta.jld"), "Δ");
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pm = load(joinpath(name, "pm.jld"), "pm");
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cnstr = PropertyT.constraints_from_pm(pm);
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splap² = GroupRings.mul(splap, splap, pm);
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Uπs = load(joinpath(name, "U_pis.jld"), "Uπs");
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Uπs = sparsify.(Uπs);
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#dimensions of the corresponding πs:
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dims = load(joinpath(name, "U_pis.jld"), "dims")
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m, P = init_model(Uπs);
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orbits = load(joinpath(name, "orbits.jld"), "orbits");
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n = size(Uπs[1],1)
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orb_spcnstrm = [orbit_constraint(cnstr[collect(orb)], n) for orb in orbits]
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orb_splap = orbit_spvector(splap, orbits)
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orb_splap² = orbit_spvector(splap², orbits)
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orb_SOutF4data = ProblemData(name, Uπs, P, orb_spcnstrm, orb_splap, orb_splap², dims);
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return m, orb_SOutF4data
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end
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function transform{T}(U::AbstractArray{T,2}, V::AbstractArray{T,2}, eps=eps(T))
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w = U'*V*U
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sparsify!(w, eps)
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dropzeros!(w)
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return w
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end
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A(data::ProblemData, π, t) = data.dims[π]*transform(data.Us[π], data.cnstr[t])
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function constrLHS(m::JuMP.Model, data::ProblemData, t)
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l = endof(data.Us)
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lhs = @expression(m, sum(vecdot(A(data, π, t), data.Ps[π]) for π in 1:l))
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return lhs
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end
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function addconstraints!(m::JuMP.Model, data::ProblemData, l::Int=length(data.cnstr); var::Symbol = :λ)
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λ = getvariable(m, var)
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for t in 1:l
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d, d² = data.laplacian[t], data.laplacianSq[t]
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lhs = constrLHS(m, data, t)
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if lhs == zero(lhs)
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if d == 0 && d² == 0
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info("Detected empty constraint")
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continue
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else
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warn("Adding unsatisfiable constraint!")
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end
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end
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JuMP.@constraint(m, lhs == d² - λ*d)
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end
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end
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function reconstructP(m::JuMP.Model, data::ProblemData)
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computedPs = [getvalue(P) for P in data.Ps]
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return sum(data.dims[π]*data.Us[π]*computedPs[π]*data.Us[π]' for π in 1:endof(data.Ps))
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end
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function cpuinfo_physicalcores()
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maxcore = -1
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for line in eachline("/proc/cpuinfo")
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if startswith(line, "core id")
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maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
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end
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end
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maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
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return maxcore + 1
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end
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function parse_commandline()
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s = ArgParseSettings()
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@add_arg_table s begin
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"--tol"
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help = "set numerical tolerance for the SDP solver (default: 1e-5)"
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arg_type = Float64
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default = 1e-5
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"--iterations"
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help = "set maximal number of iterations for the SDP solver (default: 20000)"
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arg_type = Int
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default = 20000
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"--upper-bound"
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help = "Set an upper bound for the spectral gap (default: Inf)"
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arg_type = Float64
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default = Inf
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"--cpus"
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help = "Set number of cpus used by solver (default: auto)"
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arg_type = Int
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required = false
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"-N"
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help = "Consider automorphisms of free group on N generators (default: N=2)"
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arg_type = Int
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default = 2
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"--radius"
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help = "Find the decomposition over B_r(e,S)"
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arg_type = Int
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default = 2
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end
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return parse_args(s)
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end
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function create_SDP_problem(name::String; upper_bound=Inf)
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info(PropertyT.logger, "Loading data....")
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t = @timed SDP_problem, orb_data = init_ProblemData(name);
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info(PropertyT.logger, PropertyT.timed_msg(t))
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if upper_bound < Inf
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λ = JuMP.getvariable(SDP_problem, :λ)
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JuMP.@constraint(SDP_problem, λ <= upper_bound)
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end
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info(PropertyT.logger, "Adding constraints... ")
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t = @timed addconstraints!(SDP_problem, orb_data)
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info(PropertyT.logger, PropertyT.timed_msg(t))
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return SDP_problem, orb_data
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end
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function λandP(m::JuMP.Model, data::ProblemData)
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info(PropertyT.logger, "Solving SDP problem...")
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varλ = JuMP.getvariable(m, :λ)
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varP = data.Ps
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λ, P = PropertyT.λandP(data.name, m, varλ, varP)
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recP = reconstructP(m, data)
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fname = PropertyT.λSDPfilenames(data.name)[2]
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save(fname, "origP", P, "P", recP)
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return λ, recP
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end
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function init_orbit_data{T<:Nemo.GroupElem}(logger, name::String, G::Nemo.Group,
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S::Vector{T}, AutS::Nemo.Group; radius=2)
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ex(fname) = isfile(joinpath(name, fname))
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files_exists = ex.(["delta.jld", "pm.jld", "U_pis.jld", "orbits.jld"])
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if !all(files_exists)
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compute_orbit_data(logger, name, SOutFN, S, AutS, radius=radius)
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end
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return 0
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end
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function orbit_check_propertyT(logger, dirname, G, S, AutS, solver, upper_bound)
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init_orbit_data(logger, dirname, G, S, AutS; radius=2)
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Δ = PropertyT.ΔandSDPconstraints(name, G)[1]
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fnames = PropertyT.λSDPfilenames(dirname)
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if all(isfile.(fnames))
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λ, P = PropertyT.λandP(dirname)
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else
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info(logger, "Creating SDP problem...")
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SDP_problem, orb_data = create_SDP_problem(dirname, upper_bound=upper_bound)
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JuMP.setsolver(SDP_problem, solver)
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λ, P = λandP(SDP_problem, orb_data)
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end
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info(PropertyT.logger, "λ = $λ")
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info(PropertyT.logger, "sum(P) = $(sum(P))")
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info(PropertyT.logger, "maximum(P) = $(maximum(P))")
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info(PropertyT.logger, "minimum(P) = $(minimum(P))")
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if λ > 0
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sgap = PropertyT.check_distance_to_positive_cone(Δ, λ, P, tol=tol, rational=false, len=2*radius)
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if isa(sgap, Interval)
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sgap = sgap.lo
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end
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if sgap > 0
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info(logger, "λ ≥ $(Float64(trunc(sgap,12)))")
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Kazhdan_κ = PropertyT.Kazhdan_from_sgap(sgap, length(S))
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Kazhdan_κ = Float64(trunc(Kazhdan_κ, 12))
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info(logger, "κ($name, S) ≥ $Kazhdan_κ: Group HAS property (T)!")
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return true
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else
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sgap = Float64(trunc(sgap, 12))
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info(logger, "λ($name, S) ≥ $sgap: Group may NOT HAVE property (T)!")
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return false
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end
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end
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info(logger, "κ($name, S) ≥ $λ < 0: Tells us nothing about property (T)")
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return false
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end
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