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geodesic.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra, Printf, Statistics\n",
"using ForwardDiff"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"gmc1 (generic function with 3 methods)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function gmc1(logπ, M, N, T=100, ϵ=0.01)\n",
" X = zeros(2,N)\n",
" \n",
" x = randn(2)\n",
" x ./= norm(x)\n",
" \n",
" tol = sqrt(eps())\n",
"\n",
" for n = 1:N\n",
" x0 = x\n",
" \n",
" Π = I - x*x'\n",
" A = Π * M * Π\n",
"\n",
" Λ,U = eigen(A)\n",
" Λ = [abs(λ) < tol ? 0.0 : λ for λ in Λ]\n",
" invΛ = [abs(λ) < tol ? 0.0 : 1/λ for λ in Λ]\n",
"\n",
" A⁺ = U*Diagonal(invΛ)*U'\n",
" v = U * (sqrt.(invΛ).*randn(2))\n",
"\n",
" logDetA = sum(log(λ) for λ in Λ if abs(λ) >= tol)\n",
" e = -logπ(x) + logDetA + v'*A*v/2\n",
"\n",
" for t = 1:T\n",
" # 5\n",
" v += ϵ/2 * A⁺ * ( ForwardDiff.gradient(logπ, x) - A⁺*(Π*M*x))\n",
" # 6\n",
" vv = U * Diagonal(sqrt.(Λ)) * U' * v\n",
" # 7\n",
" α = norm(vv)\n",
" nvv = vv / α \n",
" x,nvv = cos(α*ϵ)*x+sin(α*ϵ)*nvv, -sin(α*ϵ)*x+cos(α*ϵ)*nvv\n",
" vv = nvv * α\n",
"\n",
" # recompute quantities\n",
" Π = I - x*x'\n",
" A = Π * M * Π\n",
" \n",
" Λ,U = eigen(A)\n",
" Λ = [abs(λ) < tol ? 0.0 : λ for λ in Λ]\n",
" invΛ = [abs(λ) < tol ? 0.0 : 1/λ for λ in Λ]\n",
"\n",
" A⁺ = U*Diagonal(invΛ)*U'\n",
"\n",
" # 8\n",
" v = U * Diagonal(sqrt.(invΛ)) * U' * vv\n",
" # 9\n",
" v += ϵ/2 * A⁺ * ( -A⁺*(Π*M*x))\n",
" end\n",
" logDetA = sum(log(λ) for λ in Λ if abs(λ) >= tol)\n",
" ee = -logπ(x) + logDetA + v'*A*v/2\n",
" acc_prob = exp(e-ee)\n",
" #@printf \"acc prob = %.3f\\n\" min(acc_prob,1)\n",
" if !(rand() < acc_prob)\n",
" x = x0\n",
" end\n",
" X[:,n] = x\n",
" end\n",
" X \n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"gmc2 (generic function with 3 methods)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function gmc2(logπ, M, N, T=100, ϵ=0.01)\n",
" X = zeros(2,N)\n",
" \n",
" x = randn(2)\n",
" x ./= norm(x)\n",
" \n",
" tol = sqrt(eps())\n",
"\n",
" for n = 1:N\n",
" x0 = x\n",
" \n",
" Π = I - x*x'\n",
" A = Π * M * Π\n",
"\n",
" Λ,U = eigen(A)\n",
" Λ = [abs(λ) < tol ? 0.0 : λ for λ in Λ]\n",
" invΛ = [abs(λ) < tol ? 0.0 : 1/λ for λ in Λ]\n",
"\n",
" A⁺ = U*Diagonal(invΛ)*U'\n",
" v = U * (sqrt.(invΛ).*randn(2))\n",
"\n",
" e = -logπ(x) + v'*A*v/2\n",
"\n",
" for t = 1:T\n",
" # 5\n",
" v += ϵ/2 * A⁺ * (ForwardDiff.gradient(logπ, x) + A⁺*(Π*M*x))\n",
" # 6\n",
" vv = U * Diagonal(sqrt.(Λ)) * U' * v\n",
" # 7\n",
" α = norm(vv)\n",
" nvv = vv / α \n",
" x,nvv = cos(α*ϵ)*x+sin(α*ϵ)*nvv, -sin(α*ϵ)*x+cos(α*ϵ)*nvv\n",
" vv = nvv * α\n",
"\n",
" # recompute quantities\n",
" Π = I - x*x'\n",
" A = Π * M * Π\n",
" \n",
" Λ,U = eigen(A)\n",
" Λ = [abs(λ) < tol ? 0.0 : λ for λ in Λ]\n",
" invΛ = [abs(λ) < tol ? 0.0 : 1/λ for λ in Λ]\n",
"\n",
" A⁺ = U*Diagonal(invΛ)*U'\n",
"\n",
" # 8\n",
" v = U * Diagonal(sqrt.(invΛ)) * U' * vv\n",
" # 9\n",
" v += ϵ/2 * A⁺ * ( +A⁺*(Π*M*x))\n",
"\n",
" end\n",
" ee = -logπ(x) + v'*A*v/2\n",
" acc_prob = exp(e-ee)\n",
" #@printf \"acc prob = %.3f\\n\" min(acc_prob,1)\n",
" if !(rand() < acc_prob)\n",
" x = x0\n",
" end\n",
" X[:,n] = x\n",
" end\n",
" X \n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"using Plots"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
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"metadata": {},
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}
],
"source": [
"# Algorithm 1, identity matrix\n",
"X = gmc1(x->dot(x,[1,1]), [1.0 0.0; 0.0 1.0], 10_000)\n",
"A = vec(mapslices(x -> atan(x[1],x[2]), X, dims=1))\n",
"histogram(A, nbins=range(-pi,stop=pi,length=20))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
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"metadata": {},
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}
],
"source": [
"# Algorithm 2, identity matrix\n",
"X = gmc2(x->dot(x,[1,1]), [1.0 0.0; 0.0 1.0], 10_000)\n",
"A = vec(mapslices(x -> atan(x[1],x[2]), X, dims=1))\n",
"histogram(A, nbins=range(-pi,stop=pi,length=20))"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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"metadata": {},
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}
],
"source": [
"# Algorithm 1, non-identity matrix\n",
"X = gmc1(x->dot(x,[1,1]),[1.0 0.0; 0.0 4.0], 10_000)\n",
"A = vec(mapslices(x -> atan(x[1],x[2]), X, dims=1))\n",
"histogram(A, nbins=range(-pi,stop=pi,length=20))"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
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},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Algorithm 2, non-identity matrix\n",
"X = gmc2(x->dot(x,[1,1]),[1.0 0.0; 0.0 16.0], 10_000)\n",
"A = vec(mapslices(x -> atan(x[1],x[2]), X, dims=1))\n",
"histogram(A, nbins=range(-pi,stop=pi,length=20))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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[[Sockets]]
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[[SparseArrays]]
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Project.toml
[deps]
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
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