remove old laplacians.jl, 1712.* 1812.*
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0e5799862b
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@ -1,329 +0,0 @@
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###############################################################################
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# Settings and filenames
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abstract type Settings end
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struct Naive{El} <: Settings
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name::String
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G::Union{Group, NCRing}
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S::Vector{El}
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halfradius::Int
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upper_bound::Float64
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solver::JuMP.OptimizerFactory
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force_compute::Bool
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end
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struct Symmetrized{El} <: Settings
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name::String
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G::Union{Group, NCRing}
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S::Vector{El}
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autS::Group
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halfradius::Int
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upper_bound::Float64
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solver::JuMP.OptimizerFactory
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force_compute::Bool
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end
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function Settings(name::String,
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G::Union{Group, NCRing}, S::AbstractVector{El}, solver::JuMP.OptimizerFactory;
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halfradius=2, upper_bound=1.0, force_compute=false) where El <: Union{GroupElem, NCRingElem}
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return Naive(name, G, S, halfradius, upper_bound, solver, force_compute)
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end
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function Settings(name::String,
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G::Union{Group, NCRing}, S::AbstractVector{El}, autS::Group, solver::JuMP.OptimizerFactory;
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halfradius=2, upper_bound=1.0, force_compute=false) where El <: Union{GroupElem, NCRingElem}
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return Symmetrized(name, G, S, autS, halfradius, upper_bound, solver, force_compute)
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end
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suffix(s::Settings) = "$(s.upper_bound)"
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prepath(s::Settings) = s.name
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fullpath(s::Settings) = joinpath(prepath(s), suffix(s))
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filename(sett::Settings, s::Symbol; kwargs...) = filename(sett, Val{s}; kwargs...)
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filename(sett::Settings, ::Type{Val{:fulllog}}; kwargs...) =
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filename(fullpath(sett), "full", "log", suffix=Dates.now(); kwargs...)
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filename(sett::Settings, ::Type{Val{:solverlog}}; kwargs...) =
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filename(fullpath(sett), "solver", "log", suffix=Dates.now(); kwargs...)
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filename(sett::Settings, ::Type{Val{:Δ}}; kwargs...) =
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filename(prepath(sett), "delta", "jld"; kwargs...)
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filename(sett::Settings, ::Type{Val{:BlockDecomposition}}; kwargs...) =
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filename(prepath(sett), "BlockDecomposition", "jld"; kwargs...)
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filename(sett::Settings, ::Type{Val{:solution}}; kwargs...) =
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filename(fullpath(sett), "solution", "jld"; kwargs...)
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function filename(sett::Settings, ::Type{Val{:warmstart}}; kwargs...)
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filename(fullpath(sett), "warmstart", "jld"; kwargs...)
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end
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function filename(path::String, name, extension; prefix=nothing, suffix=nothing)
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pre = isnothing(prefix) ? "" : "$(prefix)_"
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suf = isnothing(suffix) ? "" : "_$(suffix)"
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return joinpath(path, "$pre$name$suf.$extension")
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end
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###############################################################################
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# Approximation by SOS (logged & warmstarted)
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function warmstart(sett::Settings)
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warmstart_fname = filename(sett, :warmstart)
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try
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ws = load(warmstart_fname, "warmstart")
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@info "Loaded $warmstart_fname."
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return ws
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catch ex
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@warn "$(ex.msg). Could not provide a warmstart to the solver."
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return nothing
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end
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end
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function approximate_by_SOS(sett::Naive,
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elt::GroupRingElem, orderunit::GroupRingElem;
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solverlog=tempname()*".log")
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isdir(fullpath(sett)) || mkpath(fullpath(sett))
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@info "Creating SDP problem..."
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SDP_problem = SOS_problem_primal(elt, orderunit, upper_bound=sett.upper_bound)
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@info Base.repr(SDP_problem)
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@info "Logging solver's progress into $solverlog"
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ws = warmstart(sett)
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@time status, ws = PropertyT.solve(solverlog, SDP_problem, sett.solver, ws)
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@info "Optimization finished:" status
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P = value.(SDP_problem[:P])
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λ = value(SDP_problem[:λ])
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if any(isnan, P)
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@warn "The solution seems to contain NaNs. Not overriding warmstart.jld"
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else
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save(filename(sett, :warmstart),
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"warmstart", (ws.primal, ws.dual, ws.slack),
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"P", P,
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"λ", λ)
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end
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save(filename(sett, :warmstart, suffix=Dates.now()),
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"warmstart", (ws.primal, ws.dual, ws.slack),
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"P", P,
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"λ", λ)
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return λ, P
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end
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function approximate_by_SOS(sett::Symmetrized,
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elt::GroupRingElem, orderunit::GroupRingElem;
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solverlog=tempname()*".log")
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isdir(fullpath(sett)) || mkpath(fullpath(sett))
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orbit_data = try
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orbit_data = load(filename(sett, :BlockDecomposition), "BlockDecomposition")
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@info "Loaded orbit data."
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orbit_data
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catch ex
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@warn ex.msg
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GroupRings.hasbasis(parent(orderunit)) ||
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throw("You need to define basis of Group Ring to compute orbit decomposition!")
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@info "Computing orbit and Wedderburn decomposition..."
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orbit_data = BlockDecomposition(parent(orderunit), sett.autS)
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save(filename(sett, :BlockDecomposition), "BlockDecomposition", orbit_data)
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orbit_data
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end
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orbit_data = decimate(orbit_data)
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@info "Creating SDP problem..."
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SDP_problem, varP = SOS_problem_primal(elt, orderunit, orbit_data, upper_bound=sett.upper_bound)
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@info Base.repr(SDP_problem)
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@info "Logging solver's progress into $solverlog"
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ws = warmstart(sett)
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@time status, ws = PropertyT.solve(solverlog, SDP_problem, sett.solver, ws)
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@info "Optimization finished:" status
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λ = value(SDP_problem[:λ])
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Ps = [value.(P) for P in varP]
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if any(any(isnan, P) for P in Ps)
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@warn "The solution seems to contain NaNs. Not overriding warmstart.jld"
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else
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save(filename(sett, :warmstart),
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"warmstart", (ws.primal, ws.dual, ws.slack),
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"Ps", Ps,
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"λ", λ)
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end
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save(filename(sett, :warmstart, suffix=Dates.now()),
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"warmstart", (ws.primal, ws.dual, ws.slack),
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"Ps", Ps,
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"λ", λ)
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@info "Reconstructing P..."
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@time P = reconstruct(Ps, orbit_data)
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return λ, P
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end
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###############################################################################
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# Checking solution
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function certify_SOS_decomposition(elt::GroupRingElem, orderunit::GroupRingElem,
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λ::Number, Q::AbstractMatrix; R::Int=2)
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separator = "-"^76
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@info "$separator\nChecking in floating-point arithmetic..." λ
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eoi = elt - λ*orderunit
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@info("Computing sum of squares decomposition...")
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@time residual = eoi - compute_SOS(parent(eoi), augIdproj(Q))
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L1_norm = norm(residual,1)
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floatingpoint_λ = λ - 2.0^(2ceil(log2(R)))*L1_norm
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info_strs = ["Numerical metrics of the obtained SOS:",
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"ɛ(elt - λu - ∑ξᵢ*ξᵢ) ≈ $(aug(residual))",
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"‖elt - λu - ∑ξᵢ*ξᵢ‖₁ ≈ $(L1_norm)",
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"Floating point (NOT certified) λ ≈"]
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@info join(info_strs, "\n") floatingpoint_λ
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if floatingpoint_λ ≤ 0
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return floatingpoint_λ
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end
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λ = @interval(λ)
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info_strs = [separator,
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"Checking in interval arithmetic...",
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"λ ∈ $λ"]
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@info(join(info_strs, "\n"))
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eoi = elt - λ*orderunit
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@info("Projecting columns of Q to the augmentation ideal...")
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@time Q, check = augIdproj(Interval, Q)
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@info "Checking that sum of every column contains 0.0..." check_augmented=check
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check || @error("The following numbers are meaningless!")
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@info("Computing sum of squares decomposition...")
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@time residual = eoi - compute_SOS(parent(eoi), Q)
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L1_norm = norm(residual,1)
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certified_λ = λ - 2.0^(2ceil(log2(R)))*L1_norm
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info_strs = ["Numerical metrics of the obtained SOS:",
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"ɛ(elt - λu - ∑ξᵢ*ξᵢ) ∈ $(aug(residual))",
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"‖elt - λu - ∑ξᵢ*ξᵢ‖₁ ∈ $(L1_norm)",
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"Interval aritmetic (certified) λ ∈"]
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@info join(info_strs, "\n") certified_λ
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return certified_λ
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end
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function spectral_gap(Δ::GroupRingElem, λ::Number, Q::AbstractMatrix; R::Int=2)
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@info "elt = Δ², u = Δ"
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return certify_SOS_decomposition(Δ^2, Δ, λ, Q, R=R)
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end
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###############################################################################
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# Interpreting the numerical results
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Kazhdan_constant(λ::Number, N::Integer) = sqrt(2*λ/N)
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Kazhdan_constant(λ::Interval, N::Integer) = IntervalArithmetic.inf(sqrt(2*λ/N))
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function check_property_T(sett::Settings)
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@info sett
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certified_sgap = spectral_gap(sett)
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return interpret_results(sett, certified_sgap)
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end
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function Base.show(io::IO, sett::Settings)
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info_strs = ["PropertyT Settings:",
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"Group: $(sett.name)",
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"Upper bound for λ: $(sett.upper_bound), on halfradius $(sett.halfradius).",
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"Force computations: $(sett.force_compute);",
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"Results will be stored in ./$(PropertyT.prepath(sett));",
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"Solver: $(typeof(sett.solver()))",
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"Solvers options: "]
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append!(info_strs, [rpad(" $k", 30)* "→ \t$v" for (k,v) in sett.solver().options])
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push!(info_strs, "="^76)
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print(io, join(info_strs, "\n"))
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end
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function interpret_results(name::String, sgap::Number, N::Integer)
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if sgap > 0
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κ = Kazhdan_constant(sgap, N)
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@info "κ($name, S) ≥ $κ: Group HAS property (T)!"
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return true
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end
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info_strs = [
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"The certified lower bound on the spectral gap is negative:",
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"λ($name, S) ≥ 0.0 > $sgap",
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"This tells us nothing about property (T)",
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]
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@info join(info_strs, "\n")
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return false
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end
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interpret_results(sett::Settings, sgap::Number) =
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interpret_results(sett.name, sgap, length(sett.S))
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function spectral_gap(sett::Settings)
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fp = PropertyT.fullpath(sett)
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isdir(fp) || mkpath(fp)
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Δ = try
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Δ = loadGRElem(filename(sett,:Δ), sett.G)
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@info "Loaded precomputed Δ."
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Δ
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catch ex
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@warn ex.msg
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@info "Computing Δ..."
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Δ = Laplacian(sett.S, sett.halfradius)
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saveGRElem(filename(sett, :Δ), Δ)
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Δ
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end
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function compute(sett, Δ)
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@info "Computing λ and P..."
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λ, P = approximate_by_SOS(sett, Δ^2, Δ;
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solverlog=filename(sett, :solverlog))
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save(filename(sett, :solution), "λ", λ, "P", P)
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λ < 0 && @warn "Solver did not produce a valid solution!"
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return λ, P
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end
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if sett.force_compute
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λ, P = compute(sett, Δ)
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else
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λ, P =try
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λ, P = load(filename(sett, :solution), "λ", "P")
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@info "Loaded existing λ and P."
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λ, P
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catch ex
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@warn ex.msg
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compute(sett, Δ)
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end
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end
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info_strs = ["Numerical metrics of matrix solution:",
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"sum(P) = $(sum(P))",
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"maximum(P) = $(maximum(P))",
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"minimum(P) = $(minimum(P))"]
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@info join(info_strs, "\n")
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isapprox(eigvals(P), abs.(eigvals(P))) ||
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@warn "The solution matrix doesn't seem to be positive definite!"
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@time Q = real(sqrt(Symmetric( (P.+ P')./2 )))
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certified_sgap = spectral_gap(Δ, λ, Q, R=sett.halfradius)
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return certified_sgap
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end
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@ -1,26 +0,0 @@
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indexing(n) = [(i,j) for i in 1:n for j in 1:n if i≠j]
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function generating_set(G::AutGroup{N}, n=N) where N
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rmuls = [Groups.transvection_R(i,j) for (i,j) in indexing(n)]
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lmuls = [Groups.transvection_L(i,j) for (i,j) in indexing(n)]
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gen_set = G.([rmuls; lmuls])
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return [gen_set; inv.(gen_set)]
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end
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function EltaryMat(M::MatAlgebra, i::Integer, j::Integer, val=1)
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@assert i ≠ j
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@assert 1 ≤ i ≤ nrows(M)
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@assert 1 ≤ j ≤ ncols(M)
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m = one(M)
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m[i,j] = val
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return m
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end
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function generating_set(M::MatAlgebra, n=nrows(M))
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elts = [EltaryMat(M, i,j) for (i,j) in indexing(n)]
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return elem_type(M)[elts; inv.(elts)]
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end
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include("sqadjop.jl")
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@ -12,7 +12,6 @@ using Groups
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using StarAlgebras
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using SymbolicWedderburn
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include("laplacians.jl")
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include("constraint_matrix.jl")
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include("sos_sdps.jl")
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include("certify.jl")
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@ -1,51 +0,0 @@
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###############################################################################
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#
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# Laplacians
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#
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###############################################################################
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function spLaplacian(RG::GroupRing, S::AbstractVector, T::Type=Float64)
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result = RG(T)
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result[one(RG.group)] = T(length(S))
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for s in S
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result[s] -= one(T)
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end
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return result
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end
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function Laplacian(S::AbstractVector{REl}, halfradius) where REl<:Union{NCRingElem, GroupElem}
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G = parent(first(S))
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@info "Generating metric ball of radius" radius=2halfradius
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@time E_R, sizes = Groups.wlmetric_ball(S, radius=2halfradius)
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@info "Generated balls:" sizes
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@info "Creating product matrix..."
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rdict = GroupRings.reverse_dict(E_R)
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@time pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=true)
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RG = GroupRing(G, E_R, rdict, pm)
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Δ = spLaplacian(RG, S)
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return Δ
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end
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function saveGRElem(fname::String, g::GroupRingElem)
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RG = parent(g)
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JLD.save(fname, "coeffs", g.coeffs, "pm", RG.pm, "G", RG.group)
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end
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function loadGRElem(fname::String, RG::GroupRing)
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coeffs = load(fname, "coeffs")
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return GroupRingElem(coeffs, RG)
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end
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function loadGRElem(fname::String, G::Union{Group, NCRing})
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pm = load(fname, "pm")
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RG = GroupRing(G, pm)
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return loadGRElem(fname, RG)
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end
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function loadGRElem(fname::String)
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pm, G = load(fname, "pm", "G")
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RG = GroupRing(G, pm)
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return loadGRElem(fname, RG)
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end
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