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saschatimme/Voronoi_MLE.ipynb

Created July 30, 2019 12:48
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Voronoi_MLE.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Starting solutions\n",
"\n",
"Here we compute the starting solutions. There should be 39, because that is the MLE degree for this problem."
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5-element Array{Polynomial{true,Int64},1}:\n",
" p0 + p1 + p2 + p3 + p4 + p5 - 1 \n",
" 20p0p2p4 - 10p0p3² - 8p1²p4 + 4p1p2p3 - p2³ \n",
" 100p0p2p5 - 20p0p3p4 - 40p1²p5 + 4p1p2p4 + 2p1p3² - p2²p3\n",
" 100p0p3p5 - 40p0p4² - 20p1p2p5 + 4p1p3p4 + 2p2²p4 - p2p3²\n",
" 20p1p3p5 - 8p1p4² - 10p2²p5 + 4p2p3p4 - p3³ "
]
},
"execution_count": 139,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using HomotopyContinuation, DynamicPolynomials, LinearAlgebra\n",
"p = @polyvar p0 p1 p2 p3 p4 p5\n",
"u = @polyvar u0 u1 u2 u3 u4 u5\n",
"\n",
"f0 = p0+p1+p2+p3+p4+p5-1\n",
"f1 = 20*p0*p2*p4-10*p0*p3^2-8*p1^2*p4+4*p1*p2*p3-p2^3\n",
"f2 = 100*p0*p2*p5-20*p0*p3*p4-40*p1^2*p5+4*p1*p2*p4+2*p1*p3^2-p2^2*p3\n",
"f3 = 100*p0*p3*p5-40*p0*p4^2-20*p1*p2*p5+4*p1*p3*p4+2*p2^2*p4-p2*p3^2\n",
"f4 = 20*p1*p3*p5-8*p1*p4^2-10*p2^2*p5+4*p2*p3*p4-p3^3\n",
"F = [f0, f1, f2, f3, f4]"
]
},
{
"cell_type": "code",
"execution_count": 140,
"metadata": {},
"outputs": [],
"source": [
"@polyvar λ[1:3]\n",
"A = randn(ComplexF64, 3, 5) \n",
"J_AF = differentiate(A * F, p)\n",
"G = [(p .* J_AF') * λ .- u; F];"
]
},
{
"cell_type": "code",
"execution_count": 146,
"metadata": {},
"outputs": [],
"source": [
"# Generate random start solutions"
]
},
{
"cell_type": "code",
"execution_count": 149,
"metadata": {},
"outputs": [],
"source": [
"p₀ = [518/9375, 124/625, 192/625, 168/625, 86/625, 307/9375];"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6-element Array{Complex{Float64},1}:\n",
" -0.007759331588900692 - 0.051702519582698656im\n",
" -0.02901864267019256 - 0.1338686976337791im \n",
" -0.07080646952592631 - 0.2342026009161865im \n",
" -0.012616054415686057 - 0.23526730344954921im \n",
" -0.04455595100361198 - 0.07782179318649807im \n",
" 0.0008519378150538565 - 0.033993931962325565im"
]
},
"execution_count": 141,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# We know that p₀ is a point in our variety, let's use use this to find a solution to `G`\n",
"λ₀ = randn(ComplexF64, 3)\n",
"u₀ = [gᵢ( p=> p₀) for gᵢ in (p .* J_AF')] * λ₀"
]
},
{
"cell_type": "code",
"execution_count": 142,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.1653756241042735e-16"
]
},
"execution_count": 142,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Check that we have indeed constructed solution\n",
"norm([gᵢ(p => p₀, u => u₀, λ => λ₀) for gᵢ in G])"
]
},
{
"cell_type": "code",
"execution_count": 150,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\r",
"\u001b[32mSolutions found: 39 \t Time: 0:00:00\u001b[39m\n"
]
},
{
"data": {
"text/plain": [
"MonodromyResult\n",
"==================================\n",
"• 39 solutions (0 real)\n",
"• return code → success\n",
"• 576 tracked paths\n",
"• seed → 153568\n"
]
},
"execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"res_generic = monodromy_solve(G, [p₀; λ₀], u₀; parameters=[u...], target_solutions_count=39)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Create hexagon and sample from it"
]
},
{
"cell_type": "code",
"execution_count": 152,
"metadata": {},
"outputs": [],
"source": [
"vertices = [\n",
" [0, 651/1625, 0, 30569/58500, 43/2250, 3377/58500],\n",
" [0, 124/375, 88/375, 77/375, 86/375, 0],\n",
" [8288/76875, 0, 3176/5125, 0, 1376/5125, 307/76875],\n",
" [259/1875, 0, 52/125, 91/250, 0, 307/3750],\n",
" [518/76875, 1984/5125, 0, 2779/5125, 0, 4912/76875],\n",
" [2849/29250, 31/1125, 8734/14625, 0, 903/3250, 0]];"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The vertices only span a two-dimensional hexagon. We can check this by checking that these points lie in a 2 dimensional affine space:"
]
},
{
"cell_type": "code",
"execution_count": 156,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2"
]
},
"execution_count": 156,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rank(hcat(vertices[2:end]...) .- vertices[1])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In order to sample from the polygon and to visualize it let's reduce to a two dimensional picture:"
]
},
{
"cell_type": "code",
"execution_count": 162,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6-element Array{Array{Float64,1},1}:\n",
" [0.0, 0.0, 0.0, 0.0, 0.0, 0.0] \n",
" [0.0, -0.0699487, 0.234667, -0.317214, 0.210222, -0.0577265] \n",
" [0.107811, -0.400615, 0.619707, -0.522547, 0.249377, -0.053733] \n",
" [0.138133, -0.400615, 0.416, -0.158547, -0.0191111, 0.0241402] \n",
" [0.00673821, -0.0134934, 0.0, 0.0196969, -0.0191111, 0.00616944]\n",
" [0.0974017, -0.37306, 0.597197, -0.522547, 0.258735, -0.0577265]"
]
},
"execution_count": 162,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# 1) translate to the origin\n",
"P = vertices .- Ref(vertices[1])"
]
},
{
"cell_type": "code",
"execution_count": 302,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×2 Array{Float64,2}:\n",
" -0.114006 0.258541\n",
" 0.423633 -0.580681\n",
" -0.655313 0.211174\n",
" 0.55257 0.4703 \n",
" -0.263705 -0.544105\n",
" 0.0568203 0.18477 "
]
},
"execution_count": 302,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's use a QR decomposition to find a basis of the two dimensional space\n",
"B = qr(hcat(P...), Val(true)).Q[:,1:2]"
]
},
{
"cell_type": "code",
"execution_count": 303,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6-element Array{Array{Float64,1},1}:\n",
" [-0.0, -0.0] \n",
" [-0.417412, -0.184061] \n",
" [-0.945666, 3.4093e-17]\n",
" [-0.539269, 0.296485] \n",
" [0.0097897, 0.0303793] \n",
" [-0.920749, -0.0292758]"
]
},
"execution_count": 303,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# compute coordinates of our points in the new basis B\n",
"coords = Ref(B) .\\ P"
]
},
{
"cell_type": "code",
"execution_count": 304,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Polyhedron DefaultPolyhedron{Float64,Polyhedra.Intersection{Float64,Array{Float64,1},Int64},Polyhedra.Hull{Float64,Array{Float64,1},Int64}}:\n",
"6-element iterator of Array{Float64,1}:\n",
" [-0.0, -0.0]\n",
" [-0.417412, -0.184061]\n",
" [-0.945666, 3.4093e-17]\n",
" [-0.539269, 0.296485]\n",
" [0.0097897, 0.0303793]\n",
" [-0.920749, -0.0292758]"
]
},
"execution_count": 304,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Polyhedra\n",
"hexagon = polyhedron(vrep(coords))"
]
},
{
"cell_type": "code",
"execution_count": 305,
"metadata": {},
"outputs": [
{
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},
"execution_count": 305,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(hexagon)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Sampling"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We sample uniformly from the polygon by using rejection sampling"
]
},
{
"cell_type": "code",
"execution_count": 306,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#171 (generic function with 1 method)"
]
},
"execution_count": 306,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sampler = let\n",
" poly = hexagon\n",
" # sample from box\n",
" x_min, x_max = extrema(first.(coords))\n",
" y_min, y_max = extrema(last.(coords))\n",
" () -> begin\n",
" while true\n",
" x = rand()*(x_max-x_min) + x_min\n",
" y = rand()*(y_max-y_min) + y_min\n",
" if [x,y] ∈ poly\n",
" return [x,y]\n",
" end\n",
" end\n",
" end\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 343,
"metadata": {},
"outputs": [
{
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},
"execution_count": 343,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's sample 1000 points and check whether this looks reasonable\n",
"test_sample = [sampler() for _ in 1:1000]\n",
"plot(hexagon; fill=:transparent)\n",
"scatter!(first.(test_sample), last.(test_sample), color=:indianred, markersize=2.5, markerstrokewidth=0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Computing things"
]
},
{
"cell_type": "code",
"execution_count": 317,
"metadata": {},
"outputs": [],
"source": [
"sample_coords = [sampler() for _ in 1:200]\n",
"sample_points = [vertices[1] + B * v for v in sample_coords];"
]
},
{
"cell_type": "code",
"execution_count": 318,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" -0.46615495777161764 \n",
" 0.008157264312942164"
]
},
"execution_count": 318,
"metadata": {},
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}
],
"source": [
"c₀ = B \\ (p₀ - vertices[1])"
]
},
{
"cell_type": "code",
"execution_count": 342,
"metadata": {},
"outputs": [
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},
"execution_count": 342,
"metadata": {},
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}
],
"source": [
"plot(hexagon; fill=:transparent)\n",
"scatter!(first.(sample_coords), last.(sample_coords), color=:indianred, markersize=2.5, markerstrokewidth=0)\n",
"scatter!(c₀[1:1], c₀[2:2], color=:lightseagreen, markersize=5., markerstrokewidth=0)"
]
},
{
"cell_type": "code",
"execution_count": 320,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"PathTracker"
]
},
"execution_count": 320,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"S = solutions(res_generic)\n",
"tracker = pathtracker(G, S; parameters=[u...], start_parameters=u₀, target_parameters=sample_points[1])"
]
},
{
"cell_type": "code",
"execution_count": 347,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"best_result (generic function with 1 method)"
]
},
"execution_count": 347,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function best_result(result, p)\n",
" argmax = nothing\n",
" max_mle = -Inf\n",
" for r in result\n",
" x = r.solution\n",
" is_real_solution = all(i -> abs(imag(x[i])) < 1e-8, 1:6)\n",
" is_real_solution || continue\n",
" \n",
" pos_solution = all(i -> real(x[i]) ≥ -1e-8, 1:6)\n",
" pos_solution || continue\n",
" \n",
" mle = sum(i -> p[i]*log(real(x[i])), 1:6)\n",
" if mle > max_mle\n",
" max_mle = mle\n",
" argmax = [real(x[i]) for i in 1:6]\n",
" end\n",
" end\n",
" argmax\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 348,
"metadata": {},
"outputs": [],
"source": [
"distances = map(sample_points) do p\n",
" set_parameters!(tracker; target_parameters=p)\n",
" argmax = best_result(solve(tracker, S, threading=false), p)\n",
" norm(argmax - p₀)\n",
"end;"
]
},
{
"cell_type": "code",
"execution_count": 349,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8-element Array{Array{Float64,1},1}:\n",
" [-0.464435, 0.0087003] \n",
" [-0.4624, -0.0353325] \n",
" [-0.477379, -0.00459659]\n",
" [-0.484581, 0.00544768] \n",
" [-0.478832, -0.0427221] \n",
" [-0.444336, -0.0473178] \n",
" [-0.461735, 0.0196813] \n",
" [-0.437166, -0.0306598] "
]
},
"execution_count": 349,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"green = sample_coords[distances .< 1e-8]"
]
},
{
"cell_type": "code",
"execution_count": 350,
"metadata": {},
"outputs": [
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"</svg>\n"
]
},
"execution_count": 350,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(hexagon; fill=:transparent)\n",
"scatter!(first.(sample_coords), last.(sample_coords), color=:indianred, markersize=2.5, markerstrokewidth=0)\n",
"scatter!(first.(green), last.(green), color=:MEDIUMPURPLE, markersize=2.5, markerstrokewidth=0)\n",
"scatter!(c₀[1:1], c₀[2:2], color=:lightseagreen, markersize=5, markerstrokewidth=0)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.1.0",
"language": "julia",
"name": "julia-1.1"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.1.0"
}
},
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"nbformat_minor": 1
}
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