use multi-line @info where possible
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@ -140,49 +140,57 @@ end
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###############################################################################
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function distance_to_positive_cone(Δ::GroupRingElem, λ, Q; R::Int=2)
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@info("------------------------------------------------------------")
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@info("Checking in floating-point arithmetic...")
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@info("λ = $λ")
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separator = "-"^76
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info_strs = [separator,
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"Checking in floating-point arithmetic...",
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"λ = $λ"]
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@info(join(info_strs, "\n"))
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eoi = Δ^2-λ*Δ
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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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@info("ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ≈ $(@sprintf("%.10f", aug(residual)))")
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L1_norm = norm(residual,1)
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@info("‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ≈ $(@sprintf("%.10f", L1_norm))")
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distance = λ - 2.0^(2ceil(log2(R)))*L1_norm
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@info("Floating point distance (to positive cone) ≈")
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@info("$(@sprintf("%.10f", distance))")
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info_strs = ["Numerical metrics:",
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"ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ≈ $(@sprintf("%.10f", aug(residual)))",
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"‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ≈ $(@sprintf("%.10f", L1_norm))",
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"Floating point distance (to positive cone) ≈",
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"$(@sprintf("%.10f", distance))"]
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@info(join(info_strs, "\n"))
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if distance ≤ 0
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return distance
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end
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@info("-"^76)
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@info("Checking in interval arithmetic...")
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λ = @interval(λ)
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@info("λ ∈ $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 = Δ^2 - λ*Δ
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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... ")
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@info((check ? "They do." : "FAILED!"))
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info_strs = ["Checking that sum of every column contains 0.0...",
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(check ? "DONE!" : "FAILED!")]
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@info(join(info_strs, "\n"))
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check || @warn("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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@info("ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ∈ $(aug(residual))")
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L1_norm = norm(residual,1)
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@info("‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ∈ $(L1_norm)")
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distance = λ - 2.0^(2ceil(log2(R)))*L1_norm
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@info("Interval distance (to positive cone) ∈")
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@info("$(distance)")
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@info("-"^76)
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info_strs = ["Numerical metrics:",
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"ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ∈ $(aug(residual))",
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"‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ∈ $(L1_norm)",
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"Interval distance (to positive cone) ∈",
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"$(distance)",
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separator]
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@info(join(info_strs, "\n"))
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return distance.lo
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end
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@ -195,8 +203,27 @@ end
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Kazhdan(λ::Number, N::Integer) = sqrt(2*λ/N)
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function interpret_results(sett::Settings, sgap::Number)
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function check_property_T(sett::Settings)
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print_summary(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 print_summary(sett::Settings)
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separator = "="^76
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info_strs = [separator,
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"Running tests for $(sett.name):",
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"Upper bound for λ: $(sett.upper_bound), on radius $(sett.radius).",
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"Warmstart: $(sett.warmstart)",
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"Results will be stored in ./$(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, separator)
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@info(join(info_strs, "\n"))
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end
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function interpret_results(sett::Settings, sgap::Number)
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if sgap > 0
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Kazhdan_κ = Kazhdan(sgap, length(sett.S))
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if Kazhdan_κ > 0
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@ -204,19 +231,18 @@ function interpret_results(sett::Settings, sgap::Number)
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return true
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end
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end
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@info("λ($(sett.name), S) ≥ $sgap < 0: Tells us nothing about property (T)")
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info_strs = ["The certified lower bound on the spectral gap is negative:",
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"λ($(sett.name), S) ≥ 0.0 > $sgap",
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"This tells us nothing about property (T)"]
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@info(join(info_strs, "\n"))
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return false
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end
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function check_property_T(sett::Settings)
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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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@info("="^76)
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@info("Running tests for $(sett.name):")
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@info("Upper bound for λ: $(sett.upper_bound), on radius $(sett.radius).")
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@info("Solver is $(sett.solver)")
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@info("Warmstart: $(sett.warmstart)")
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@info("="^76)
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if isfile(filename(sett,:Δ))
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# cached
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@ -240,17 +266,18 @@ function check_property_T(sett::Settings)
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@warn("Solver did not produce a valid solution!")
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end
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end
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@info("λ = $λ")
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@info("sum(P) = $(sum(P))")
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@info("maximum(P) = $(maximum(P))")
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@info("minimum(P) = $(minimum(P))")
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info_strs = ["λ = $λ",
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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( (P.+ P')./2 ))
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sgap = distance_to_positive_cone(Δ, λ, Q, R=sett.radius)
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return interpret_results(sett, sgap)
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certified_sgap = distance_to_positive_cone(Δ, λ, Q, R=sett.radius)
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return certified_sgap
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end
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@ -17,20 +17,25 @@ function OrbitData(RG::GroupRing, autS::Group, verbose=true)
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@assert sum(length(o) for o in orbs) == length(RG.basis)
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verbose && @info("The action has $(length(orbs)) orbits")
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verbose && @info("Projections in the Group Ring of AutS = $autS")
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verbose && @info("Finding projections in the Group Ring of $(autS)")
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@time autS_mps = Projections.rankOne_projections(GroupRing(autS, collect(autS)))
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verbose && @info("AutS-action matrix representatives")
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verbose && @info("Finding AutS-action matrix representation")
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@time preps = perm_reps(autS, RG.basis[1:size(RG.pm,1)], RG.basis_dict)
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@time mreps = matrix_reps(preps)
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verbose && @info("Projection matrices Uπs")
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verbose && @info("Computing the projection matrices Uπs")
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@time Uπs = [orthSVD(matrix_repr(p, mreps)) for p in autS_mps]
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multiplicities = size.(Uπs,2)
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verbose && @info("multiplicities = $multiplicities")
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dimensions = [Int(p[autS()]*Int(order(autS))) for p in autS_mps]
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verbose && @info("dimensions = $dimensions")
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if verbose
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info_strs = ["",
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lpad("multiplicities", 14) * " =" * join(lpad.(multiplicities, 4), ""),
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lpad("dimensions", 14) * " =" * join(lpad.(dimensions, 4), "")
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]
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@info(join(info_strs, "\n"))
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end
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@assert dot(multiplicities, dimensions) == size(RG.pm,1)
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return OrbitData(orbs, preps, Uπs, dimensions)
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