Add relativistic kinetic

Project.toml

@@ -4,6 +4,7 @@ version = "0.8.0"
 
 [deps]
 AbstractMCMC = "80f14c24-f653-4e6a-9b94-39d6b0f70001"
+AdaptiveRejectionSampling = "c75e803d-635f-53bd-ab7d-544e482d8c75"
 ArgCheck = "dce04be8-c92d-5529-be00-80e4d2c0e197"
 DocStringExtensions = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
@@ -33,6 +34,7 @@ AdvancedHMCOrdinaryDiffEqExt = "OrdinaryDiffEq"
 [compat]
 ADTypes = "1"
 AbstractMCMC = "5.6"
+AdaptiveRejectionSampling = "0.2.1"
 ArgCheck = "1, 2"
 ComponentArrays = "0.15"
 CUDA = "3, 4, 5"

src/AdvancedHMC.jl

@@ -21,6 +21,8 @@ using AbstractMCMC: AbstractMCMC, LogDensityModel
 
 import StatsBase: sample
 
+using AdaptiveRejectionSampling: RejectionSampler, run_sampler!
+
 const DEFAULT_FLOAT_TYPE = typeof(float(0))
 
 include("utilities.jl")
@@ -63,6 +65,11 @@ export Trajectory,
     MultinomialTS,
     find_good_stepsize
 
+include("relativistic/hamiltonian.jl")
+export RelativisticKinetic, DimensionwiseRelativisticKinetic
+
+include("relativistic/metric.jl")
+
 # Useful defaults
 
 @deprecate find_good_eps find_good_stepsize

src/relativistic/hamiltonian.jl

@@ -0,0 +1,71 @@
+abstract type AbstractRelativisticKinetic{T} <: AbstractKinetic end
+
+struct RelativisticKinetic{T} <: AbstractRelativisticKinetic{T}
+    "Mass"
+    m::T
+    "Speed of light"
+    c::T
+end
+
+function relativistic_mass(kinetic::RelativisticKinetic, r, r′=r)
+    return kinetic.m * sqrt(dot(r, r′) / (kinetic.m^2 * kinetic.c^2) + 1)
+end
+function relativistic_energy(kinetic::RelativisticKinetic, r, r′=r)
+    return sum(kinetic.c^2 * relativistic_mass(kinetic, r, r′))
+end
+
+struct DimensionwiseRelativisticKinetic{T} <: AbstractRelativisticKinetic{T}
+    "Mass"
+    m::T
+    "Speed of light"
+    c::T
+end
+
+function relativistic_mass(kinetic::DimensionwiseRelativisticKinetic, r, r′=r)
+    return kinetic.m .* sqrt.(r .* r′ ./ (kinetic.m .^ 2 .* kinetic.c .^ 2) .+ 1)
+end
+function relativistic_energy(kinetic::DimensionwiseRelativisticKinetic, r, r′=r)
+    return sum(kinetic.c .^ 2 .* relativistic_mass(kinetic, r, r′))
+end
+
+function ∂H∂r(
+    h::Hamiltonian{<:UnitEuclideanMetric,<:AbstractRelativisticKinetic}, r::AbstractVecOrMat
+)
+    mass = relativistic_mass(h.kinetic, r)
+    return r ./ mass
+end
+function ∂H∂r(
+    h::Hamiltonian{<:DiagEuclideanMetric,<:AbstractRelativisticKinetic}, r::AbstractVecOrMat
+)
+    r = h.metric.sqrtM⁻¹ .* r
+    mass = relativistic_mass(h.kinetic, r)
+    red_term = r ./ mass # red part of (15)
+    return h.metric.sqrtM⁻¹ .* red_term # (15)
+end
+function ∂H∂r(
+    h::Hamiltonian{<:DenseEuclideanMetric,<:AbstractRelativisticKinetic},
+    r::AbstractVecOrMat,
+)
+    r = h.metric.cholM⁻¹ * r
+    mass = relativistic_mass(h.kinetic, r)
+    red_term = r ./ mass
+    return h.metric.cholM⁻¹' * red_term
+end
+
+function neg_energy(
+    h::Hamiltonian{<:UnitEuclideanMetric,<:AbstractRelativisticKinetic}, r::T, θ::T
+) where {T<:AbstractVector}
+    return -relativistic_energy(h.kinetic, r)
+end
+function neg_energy(
+    h::Hamiltonian{<:DiagEuclideanMetric,<:AbstractRelativisticKinetic}, r::T, θ::T
+) where {T<:AbstractVector}
+    r = h.metric.sqrtM⁻¹ .* r
+    return -relativistic_energy(h.kinetic, r)
+end
+function neg_energy(
+    h::Hamiltonian{<:DenseEuclideanMetric,<:AbstractRelativisticKinetic}, r::T, θ::T
+) where {T<:AbstractVector}
+    r = h.metric.cholM⁻¹ * r
+    return -relativistic_energy(h.kinetic, r)
+end

src/relativistic/metric.jl

@@ -0,0 +1,75 @@
+function rand_angles(rng::AbstractRNG, dim)
+    return rand(rng, dim - 1) .* vcat(fill(π, dim - 2), 2 * π)
+end
+
+"Special case of `polar2spherical` with dimension equal 2"
+polar2cartesian(θ, d) = d * [cos(θ), sin(θ)]
+
+# ref: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
+function polar2spherical(θs, d)
+    cos_lst, sin_lst = cos.(θs), sin.(θs)
+    suffixed_cos_lst = vcat(cos_lst, 1) # [cos(θ[1]), cos(θ[2]), ..., cos(θ[d-1]), 1]
+    prefixed_cumprod_sin_lst = vcat(1, cumprod(sin_lst)) # [1, sin(θ[1]), sin(θ[1]) * sin(θ[2]), ..., sin(θ[1]) * ... * sin(θ[d-1])]
+    return d * prefixed_cumprod_sin_lst .* suffixed_cos_lst
+end
+
+momentum_mode(m, c) = sqrt((1 / c^2 + sqrt(1 / c^2 + 4 * m^2)) / 2) # mode of the momentum distribution
+
+function rand_momentum(
+    rng::AbstractRNG,
+    metric::UnitEuclideanMetric{T},
+    kinetic::RelativisticKinetic{T},
+    ::AbstractVecOrMat,
+) where {T}
+    densityfunc = x -> exp(-relativistic_energy(kinetic, [x])) * x
+    mm = momentum_mode(kinetic.m, kinetic.c)
+    sampler = RejectionSampler(densityfunc, (0.0, Inf), (mm / 2, mm * 2); max_segments=5)
+    sz = size(metric)
+    θs = rand_angles(rng, prod(sz))
+    d = only(run_sampler!(rng, sampler, 1))
+    r = polar2spherical(θs, d * rand(rng, [-1, +1])) # TODO Double check if +/- is needed
+    r = reshape(r, sz)
+    return r
+end
+
+# TODO Support AbstractVector{<:AbstractRNG}
+# FIXME Unit-test this using slice sampler or HMC sampler
+function rand_momentum(
+    rng::AbstractRNG,
+    metric::UnitEuclideanMetric{T},
+    kinetic::DimensionwiseRelativisticKinetic{T},
+    ::AbstractVecOrMat,
+) where {T}
+    h_temp = Hamiltonian(metric, kinetic, identity, identity)
+    densityfunc = x -> exp(neg_energy(h_temp, [x], [x]))
+    sampler = RejectionSampler(densityfunc, (-Inf, Inf); max_segments=5)
+    sz = size(metric)
+    r = run_sampler!(rng, sampler, prod(sz))
+    r = reshape(r, sz)
+    return r
+end
+
+# TODO Support AbstractVector{<:AbstractRNG}
+function rand_momentum(
+    rng::AbstractRNG,
+    metric::DiagEuclideanMetric{T},
+    kinetic::AbstractRelativisticKinetic{T},
+    θ::AbstractVecOrMat,
+) where {T}
+    r = rand_momentum(rng, UnitEuclideanMetric(size(metric)), kinetic, θ)
+    # p' = A p where A = sqrtM
+    r ./= metric.sqrtM⁻¹
+    return r
+end
+# TODO Support AbstractVector{<:AbstractRNG}
+function rand_momentum(
+    rng::AbstractRNG,
+    metric::DenseEuclideanMetric{T},
+    kinetic::AbstractRelativisticKinetic{T},
+    θ::AbstractVecOrMat,
+) where {T}
+    r = rand_momentum(rng, UnitEuclideanMetric(size(metric)), kinetic, θ)
+    # p' = A p where A = cholM
+    ldiv!(metric.cholM⁻¹, r)
+    return r
+end

test/relativistic.jl

@@ -0,0 +1,26 @@
+using ReTest, Random, AdvancedHMC
+
+@testset "Relativistic kinetic construction" begin
+    f = x -> dot(x, x)
+    g = x -> 2x
+    metric = UnitEuclideanMetric(10)
+    h = Hamiltonian(metric, RelativisticKinetic(1.0, 1.0), f, g)
+    @test h.kinetic isa RelativisticKinetic
+end
+
+@testset "Sampling with relativistic kinetic" begin
+    n_samples = 2_000
+    rng = MersenneTwister(1110)
+    initial_θ = rand(D)
+    metric = DiagEuclideanMetric(D)
+    for kineticT in [RelativisticKinetic, DimensionwiseRelativisticKinetic]
+        kinetic = kineticT(1.0, 1.0)
+        h = Hamiltonian(metric, kinetic, ℓπ, ∂ℓπ∂θ)
+        initial_ϵ = find_good_stepsize(h, initial_θ)
+        integrator = Leapfrog(initial_ϵ)
+        kernel = HMCKernel(Trajectory{EndPointTS}(integrator, FixedNSteps(8)))
+        samples, stats = sample(rng, h, kernel, initial_θ, n_samples; progress=true)
+        @test length(samples) == n_samples
+        @test length(stats) == n_samples
+    end
+end

test/runtests.jl

@@ -32,6 +32,7 @@ if GROUP == "All" || GROUP == "AdvancedHMC"
     include("abstractmcmc.jl")
     include("mcmcchains.jl")
     include("constructors.jl")
+    include("relativistic.jl")
 
     Comonicon.@main function runtests(patterns...; dry::Bool=false)
         return retest(patterns...; dry=dry, verbose=Inf)