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paper_plots.jl
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paper_plots.jl
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include("dynsys.jl")
using Combinatorics
using FileIO
using JLD2
using PyPlot
using Random
function diag_tensor()
T = zeros(Float64, 3, 3, 3)
T[1, 1, 1] = 5
T[2, 2, 2] = 2
T[3, 3, 3] = 1
return T
end
function diag_tensor3_evecs(T::Array{Float64,3})
a, b, c = T[1,1,1], T[2,2,2], T[3,3,3]
evecs = []
push!(evecs, [1.0, 0.0, 0.0])
push!(evecs, [0.0, 1.0, 0.0])
push!(evecs, [0.0, 0.0, 1.0])
push!(evecs, normalize!([b / a, 1.0, 0.0]))
push!(evecs, normalize!([b / c, 0.0, 1.0]))
push!(evecs, normalize!([0.0, c / b, 1.0]))
push!(evecs, normalize!([c / a, c / b, 1.0]))
return evecs
end
function S3_surface()
x1s, x2s = Float64[], Float64[]
for r in range(0.02, 1.0, length=15)
nsamp = max(convert(Int64, round(175 * r)), 4)
for θ in range(0.0, 2 * π, length=nsamp)
x1 = r * cos(θ)
x2 = r * sin(θ)
if x1^2 + x2^2 <= 1.0001
push!(x1s, x1); push!(x2s, x2)
end
end
end
return x1s, x2s
end
function stability1()
eval_map = closest_in_angle([0.0, 0.0, 1.0])
T = diag_tensor()
x1s, x2s = S3_surface()
us, vs = Float64[], Float64[]
for (x1, x2) in zip(x1s, x2s)
x3 = sqrt(abs.(1 - x1^2 - x2^2))
x = [x1, x2, x3]
g = eval_map(collapse(T, x)) - x
push!(us, g[1]); push!(vs, g[2])
end
close()
quiver(x1s, x2s, us, vs, headwidth=2, headlength=3, minshaft=2)
gca()[:set_aspect](aspect=1.0)
# Plot trajectories
function plot_traj(integrator, xs::Vector{Float64}, maxiter::Int64)
push!(xs, 1.0 - sqrt(xs[1]^2 + xs[2]^2))
eval_hist, evec_hist, conv =
TZE_dynsys(T, eval_map, integrator,
x0=xs, maxiter=maxiter)
t1 = vec(evec_hist[1,:])
t2 = vec(evec_hist[2,:])
plot(t1, t2, color="black", lw=0.75, alpha=0.5)
ax = gca()
ax[:scatter](t1, t2, c=collect(1:length(t1)) / length(t1), cmap="hot", s=2)
end
Random.seed!(12345)
xshifts = 0.02 * [ 1, 1, -1, 1, 1, -1, 1]
yshifts = 0.02 * [-1, 1, 1, 1, 1, 1, -1]
for (xs, ys, evec) in zip(xshifts, yshifts, diag_tensor3_evecs(T))
plot_traj(forward_euler(0.01), [evec[1] + xs, evec[2] + ys], 350)
end
# Plot eigenvectors
for evec in diag_tensor3_evecs(T)
scatter([evec[1]], [evec[2]], marker="x", s=100, color="#377eb8")
end
fsz = 20
xlabel(L"$x_1$", fontsize=fsz+2)
ylabel(L"$x_2$", fontsize=fsz+2)
title("Closest to [0, 0, 1]", fontsize=fsz)
ax = gca()
ax[:set_xlim](-1.1, 1.1)
ax[:set_ylim](-1.1, 1.1)
ax[:tick_params]("both", labelsize=fsz-2, length=5, width=1.5)
tight_layout()
savefig("closest_angle.eps")
end
function stability2(normalize=false)
eval_map = smallest_algebraic()
T = diag_tensor()
x1s, x2s = S3_surface()
us, vs = Float64[], Float64[]
for (x1, x2) in zip(x1s, x2s)
x3 = sqrt(abs.(1 - x1^2 - x2^2))
x = [x1, x2, x3]
g = eval_map(collapse(T, x)) - x
push!(us, g[1]); push!(vs, g[2])
end
close()
quiver(x1s, x2s, us, vs, headwidth=2, headlength=3, minshaft=2)
gca()[:set_aspect](aspect=1.0)
# example trajectory
function plot_traj(integrator, xs::Vector{Float64}, maxiter::Int64)
push!(xs, 1.0 - sqrt(xs[1]^2 + xs[2]^2))
eval_hist, evec_hist, conv =
TZE_dynsys(T, eval_map, integrator,
x0=xs, maxiter=maxiter, normalize=normalize)
t1 = vec(evec_hist[1,:])
t2 = vec(evec_hist[2,:])
plot(t1, t2, color="black", lw=0.75, alpha=0.5)
ax = gca()
ax[:scatter](t1, t2, c=collect(1:length(t1)) / length(t1), cmap="hot", s=2)
end
xshifts = 0.01 * [ 1, 1, -1, 1, 1, -1, 1]
yshifts = 0.01 * [-1, 1, 1, 1, -1, 1, -1]
for (xs, ys, evec) in zip(xshifts, yshifts, diag_tensor3_evecs(T))
if normalize; plot_traj(forward_euler(0.01), [evec[1] + xs, evec[2] + ys], 300)
else plot_traj(forward_euler(0.01), [evec[1] + xs, evec[2] + ys], 100)
end
end
# Plot eigenvectors
for evec in diag_tensor3_evecs(T)
scatter([evec[1]], [evec[2]], marker="x", s=100, color="#377eb8")
end
fsz = 20
xlabel(L"$x_1$", fontsize=fsz+2)
ylabel(L"$x_2$", fontsize=fsz+2)
title("Smallest algebraic", fontsize=fsz)
ax = gca()
ax[:set_xlim](-1.1, 1.1)
ax[:set_ylim](-1.1, 1.1)
ax[:tick_params]("both", labelsize=fsz-2, length=5, width=1.5)
tight_layout()
show()
if normalize; savefig("smallest_algebraic_norm.eps")
else savefig("smallest_algebraic.eps")
end
end
#stability2()
# Tensor in Example 3.6 from Kolda and Mayo. "Shifted power method for computing
# tensor eigenpairs." SIMAX, 2011.
function T_36()
function symtensor(T::Array{Float64})
S = zeros(Float64, size(T)...)
d = ndims(T)
for p in permutations(1:d)
S += permutedims(T,p)
end
return S
end
T = zeros(Float64, 3,3,3)
T[1,2,3] = -0.1790
T[2,3,3] = 0.1773 / 2
T[1,1,2] = 0.0516 / 2
T[1,1,3] = -0.0954 /2
T[1,2,2] = -0.1958 /2
T[1,3,3] = -0.2676 /2
T[2,2,2] = 0.3251 /6
T[2,2,3] = 0.2513 /2
T[3,3,3] = 0.0338 /6
T = symtensor(T)
T[1,1,1] = -0.1281
T[2,2,2] = 0.3251
T[3,3,3] = 0.0338
return T
end
function example_36(eigenvalue::Float64)
Random.seed!(1)
T = T_36()
maxiter = 20
all_quotients = []
Λ = kth_smallest_algebraic(2)
FE = forward_euler(0.5)
for i in 1:100
x0=randn(Float64, size(T)[1])
tol = -1.0 # negative to do maximum number of iterations
quotients, xhist = TZE_dynsys(T, Λ, FE, x0=x0, tol=tol, maxiter=maxiter)
push!(all_quotients, quotients)
end
close()
figure()
fsz = 24
xlabel("Iteration", fontsize=fsz)
ylabel("Rayleigh quotient", fontsize=fsz)
if eigenvalue == 0.0018
plot(all_quotients[1], marker="o", lw=3)
ylim(-0.004, 0.0025)
title("V5 (λ = 0.0018)", fontsize=fsz)
elseif eigenvalue == 0.0033
plot(all_quotients[4], marker="o", lw=3)
ylim(-0.09, 0.01)
title("V5 (λ = 0.0033)", fontsize=fsz)
elseif eigenvalue == 0.2294
plot(all_quotients[5], marker="o", lw=3)
ylim(-0.1, 0.25)
title("V5 (λ = 0.2294)", fontsize=fsz)
else
error("Unkown eigenvalue")
end
ax = gca()
ax[:tick_params]("both", labelsize=fsz, length=5, width=1.5)
tight_layout()
savefig(string("ex36-V5-", "$(eigenvalue)"[3:end], ".eps"))
end
# Tensor in Example 4.11 from Cui, Dai, and Nie. "All real eigenvalues of
# symmetric tensors.", SIMAX, 2014.
function T_411(n::Int64)
A = zeros(Float64, n, n, n)
for i in 1:n, j in 1:n, k in 1:n
A[i, j, k] = (-1)^i / i + (-1)^j / j + (-1)^k / k
end
return A
end
function example_411(eigenvalue::Float64, eval_map)
T = T_411(5)
maxiter = 20
tol = -1.0 # negative to run all of the iterations
FE = forward_euler(0.5)
close()
figure()
fsz = 24
xlabel("Iteration", fontsize=fsz)
ylabel("Rayleigh quotient", fontsize=fsz)
x0 = normalize(ones(Float64, size(T)[1]))
figname = ""
if eigenvalue == 9.9779
quotients, xhist = TZE_dynsys(T, eval_map(), FE, x0=x0, tol=tol, maxiter=maxiter)
plot(-quotients, marker="o", lw=3)
title("V1 (λ = 9.9779)", fontsize=fsz)
ylim(5, 10.5)
figname = "ex411-V1.eps"
elseif eigenvalue == 0.0000
quotients, xhist = TZE_dynsys(T, eval_map(), FE, x0=x0, tol=tol, maxiter=maxiter)
plot(quotients, marker="o", lw=3)
title("V2 (λ = 0.0000)", fontsize=fsz)
ylim(-5.5, 0.5)
figname = "ex411-V2.eps"
elseif eigenvalue == 4.2876
if eval_map == largest_algebraic
quotients, xhist = TZE_dynsys(T, eval_map(), FE, x0=x0, tol=tol, maxiter=maxiter)
plot(quotients, marker="o", lw=3)
title("V3 (λ = 4.2876)", fontsize=fsz)
ylim(-6, 5)
figname = "ex411-V3.eps"
end
if map == largest_magnitude
Random.seed!(123456)
x0 = normalize(randn(Float64, size(T)[1]))
quotients, xhist = TZE_dynsys(T, eval_map(), FE, x0=x0, tol=tol, maxiter=maxiter)
@show minimum(quotients), maximum(quotients)
plot(quotients, marker="o", lw=3)
title("V1 (λ = 4.2876)", fontsize=fsz)
ylim(2.5, 4.5)
figname = "ex411-V1-2.eps"
end
else
error("Unkown eigenvalue")
end
ax = gca()
ax[:tick_params]("both", labelsize=fsz, length=5, width=1.5)
tight_layout()
savefig(figname)
end
function scalability(order::Int64)
function get_time(method::String, dim::Int64)
endstr = "evals-$order-$dim.jld2"
if method == "DS"; return load("results/DS-$(endstr)")["time"]; end
if method == "SDP"; return load("SDP/results/SDP-$(endstr)")["time"]; end
if method == "SS-HOPM"; return load("SS-HOPM/results/SS-HOPM-$(endstr)")["time"]; end
end
function get_times(method::String, dims::UnitRange{Int64})
if method == "DS"; return [get_time("DS", dim) for dim in dims]; end
if method == "SDP"; return [get_time("SDP", dim) for dim in dims]; end
if method == "SS-HOPM"; return [get_time("SS-HOPM", dim) for dim in dims]; end
error("Unknown method $method")
end
ds_dims = 5:15
sshopm_dims = 5:15
sdp_dims = 5:15
if order == 5; sdp_dims = 5:10; end
ds_times = get_times("DS", ds_dims)
sshopm_times = get_times("SS-HOPM", sshopm_dims)
sdp_times = get_times("SDP", sdp_dims)
close()
fsz = 24
semilogy(collect(ds_dims), ds_times, lw=2.5, ls="--", marker="s", ms=8, label="DS")
semilogy(collect(sshopm_dims), sshopm_times, lw=2.5, ls=":", marker="x", ms=8, label="SS-HOPM")
semilogy(collect(sdp_dims), sdp_times, lw=2.5, ls="-", marker="o", ms=8, label="SDP")
xlabel("Dimension", fontsize=fsz)
ylabel("Running time (seconds)", fontsize=fsz)
if order == 3
legend(fontsize=fsz-4, loc="upper left", frameon=false,
labelspacing=0.25)
end
title("Order-$(order) tensors", fontsize=fsz)
ax = gca()
ax[:tick_params]("both", labelsize=fsz-4, length=6, width=1.5)
ax[:tick_params]("both", which="minor", length=2, width=1)
tight_layout()
savefig("scalability-$(order).eps")
end
function unique_evals(order::Int64)
function get_evals(method::String, dim::Int64)
endstr = "evals-$order-$dim.jld2"
evals = Float64[]
if method == "DS"; evals = load("results/DS-$(endstr)")["evals"]
elseif method == "SDP"; evals = load("SDP/results/SDP-$(endstr)")["evals"]
elseif method == "SS-HOPM"; evals = load("SS-HOPM/results/SS-HOPM-$(endstr)")["evals"]
else error("Unknown method $method");
end
evals = vec(evals)
if length(evals) == 0; return evals; end
# With odd-order tensors, we have sign ambiguity
if order % 2 == 1; evals = abs.(evals); end
sort!(evals)
# Get everything within tolerance 1e-4 (tolerance used by all methods)
tol = 1e-4
unique_evals = [evals[1]]
for eval in evals
if eval > unique_evals[end] + 1e-4
push!(unique_evals, eval)
end
end
return unique_evals
end
@show get_evals("SDP", 7)
@show get_evals("DS", 7)
@show get_evals("SS-HOPM", 7)
end
;