GP Sampling

Project.toml

@@ -9,6 +9,7 @@ Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
 IrrationalConstants = "92d709cd-6900-40b7-9082-c6be49f344b6"
 KernelFunctions = "ec8451be-7e33-11e9-00cf-bbf324bd1392"
+KernelSpectralDensities = "027d52a2-76e5-4228-9bfe-bc7e0f5a8348"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 PDMats = "90014a1f-27ba-587c-ab20-58faa44d9150"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
@@ -24,6 +25,7 @@ Distributions = "0.19, 0.20, 0.21, 0.22, 0.23, 0.24, 0.25"
 FillArrays = "0.7, 0.8, 0.9, 0.10, 0.11, 0.12, 0.13, 1"
 IrrationalConstants = "0.1, 0.2"
 KernelFunctions = "0.9, 0.10"
+KernelSpectralDensities = "0.2.0"
 LinearAlgebra = "1"
 PDMats = "0.11"
 Random = "1"

src/AbstractGPs.jl

@@ -13,6 +13,7 @@ using RecipesBase
 using IrrationalConstants: log2π
 
 using KernelFunctions: ColVecs, RowVecs
+using KernelSpectralDensities
 
 using ChainRulesCore: ChainRulesCore
 
@@ -33,6 +34,7 @@ export rand!,
     posterior,
     update_posterior
 export ColVecs, RowVecs
+export GPSampler, CholeskySampling, Conditional, Independent, RFFSampling, PathwiseSampling
 
 # Various bits of utility functionality.
 include("util/common_covmat_ops.jl")
@@ -56,6 +58,9 @@ include("sparse_approximations.jl")
 # LatentGP and LatentFiniteGP objects to accommodate GPs with non-Gaussian likelihoods.
 include("latent_gp.jl")
 
+# Different sampling methods
+include("sampling.jl")
+
 # Plotting utilities.
 include("util/plotting.jl")
 

src/sampling.jl

@@ -0,0 +1,264 @@
+abstract type AbstractGPSamplingMethod end
+
+SeedableRNG = Union{Xoshiro,MersenneTwister}
+
+_rand(rng, d) = Random.rand(rng, d)
+function _rand(rng::AbstractRNG, ::Type{T}) where {T<:SeedableRNG}
+    return T(Random.rand(rng, 1:typemax(Int)))
+end
+
+# ## Interface
+
+struct GPSample{F,S}
+    fun::F
+    sample::S
+end
+
+(gs::GPSample)(x::AbstractArray) = eval_at(gs.fun, gs.sample, x)
+
+# This may become more challenging once we extend to multi-input GPS
+(gs::GPSample)(x::Number) = only(gs([x]))
+
+"""
+    GPSampler(gp::AbstractGPs.AbstractGP, method::AbstractGPSamplingMethod)
+Creates a sampler for the given `gp` using the specified `method`. 
+
+```jldoctest
+julia> f = GP(Matern32Kernel());
+
+julia> gps = GPSampler(f, CholeskySampling());
+
+julia> rand(gps);
+```   
+"""
+struct GPSampler{F,S} <: Random.Sampler{GPSample}
+    fun::F
+    sampler::S
+
+    # Specify input types here, since it is a "public" interface
+    function GPSampler(gp::AbstractGPs.AbstractGP, method::AbstractGPSamplingMethod)
+        fun, sampler = method(gp)
+        return new{typeof(fun),typeof(sampler)}(fun, sampler)
+    end
+end
+
+# Don't love the deepcopy here
+# issue is "pass by sharing" and the mutable struct in CholeskySampling
+function Random.rand(rng::AbstractRNG, gs::GPSampler)
+    return GPSample(deepcopy(gs.fun), _rand(rng, gs.sampler))
+end
+
+# ## Utils
+
+_get_prior(gp::AbstractGPs.GP) = gp
+_get_prior(pgp::AbstractGPs.PosteriorGP) = pgp.prior
+
+function get_obs_variance(pgp::AbstractGPs.PosteriorGP)
+    σk = pgp.prior.kernel(0, 0)
+    v = diag(pgp.data.C.L * pgp.data.C.U) .- σk
+    return v
+end
+
+function get_target_prior(pgp::AbstractGPs.PosteriorGP)
+    m = pgp.data.δ
+    σ2 = get_obs_variance(pgp)
+    return MvNormal(m, sqrt.(σ2))
+end
+
+#########################
+# Function Space/ Cholesky
+
+"""
+    CholeskySampling(s=Conditional, generator=Xoshiro)
+Sampling by using the standard way, via Cholesky decomposition. 
+Arguments:
+- `s`: Sampling type, either `Conditional` or `Independent`. Default is `Conditional`.
+- `generator`: Random number generator to use in each sample. Default is `Xoshiro`.
+"""
+struct CholeskySampling{M,G} <: AbstractGPSamplingMethod
+    function CholeskySampling(s=Conditional, generator=Xoshiro)
+        return new{s,generator}()
+    end
+end
+
+function (cs::CholeskySampling{M,G})(gp) where {M,G}
+    return M(gp), G
+end
+
+"""
+    Conditional
+Generates a GP sample that conditions function samples on all previous samples.
+"""
+mutable struct Conditional{GPT<:AbstractGPs.AbstractGP}
+    gp::GPT
+end
+
+function Conditional(gp::AbstractGPs.GP)
+    data = (
+        α=Vector{Float64}(undef, 0),
+        C=Cholesky(UpperTriangular(Matrix{Float64}(undef, 0, 0))),
+        x=Vector{Float64}(undef, 0),
+        δ=Vector{Float64}(undef, 0),
+    )
+    pgp = AbstractGPs.PosteriorGP(gp, data)
+    return Conditional(pgp)
+end
+
+function eval_at(s::Conditional, rng, x::AbstractArray)
+    if isassigned(s.gp.data.x, 1)
+        pgp = s.gp
+    else
+        pgp = s.gp.prior
+    end
+    fgp = pgp(x)
+    y = rand(rng, fgp)
+    s.gp = posterior(fgp, y)
+    return y
+end
+
+"""
+    Independent
+Generates a GP sample that samples function samples independent from previous samples.
+"""
+struct Independent{GPT<:AbstractGPs.AbstractGP}
+    gp::GPT
+    function Independent(gp)
+        return new{typeof(gp)}(gp)
+    end
+end
+
+function eval_at(s::Independent, rng, x::AbstractArray)
+    gp = s.gp
+    fgp = gp(x)
+    y = rand(rng, fgp)
+    return y
+end
+
+# ## WeightSpace
+
+# ### Utils
+
+get_weight_distribution(::AbstractGPs.GP, rff) = MvNormal(ones(rff.l))
+
+function get_weight_distribution(pgp::AbstractGPs.PosteriorGP, rff)
+    d = get_target_prior(pgp)
+
+    P = rff.(pgp.data.x)
+    Pt = reduce(hcat, P)
+    Cp = Symmetric(Pt * (d.Σ \ Pt') + I)
+    C = cholesky(Cp)
+
+    μ = C \ (Pt * (d.Σ \ d.μ))
+    Σ = C \ I
+    return MvNormal(μ, Symmetric(Σ))
+end
+
+# ### Main
+
+"""
+    RFFSampling(l::Int, rff_type::Type{<:KernelSpectralDensities.AbstractRFF}=DoubleRFF)
+Sampling by using Random Fourier Features.
+Arguments:
+- `l`: Number of random Fourier features to use.
+- `rff_type`: Type of random Fourier features to use. Default is `DoubleRFF`.
+"""
+struct RFFSampling{RFF,RNG} <: AbstractGPSamplingMethod
+    l::Int
+    rng::RNG
+    function RFFSampling(
+        rng, l; rff_type::Type{<:KernelSpectralDensities.AbstractRFF}=DoubleRFF
+    )
+        return new{rff_type,typeof(rng)}(l, rng)
+    end
+end
+
+function RFFSampling(l; rff_type::Type{<:KernelSpectralDensities.AbstractRFF}=DoubleRFF)
+    return RFFSampling(Random.default_rng(), l; rff_type)
+end
+
+function (rffs::RFFSampling{RFF})(gp) where {RFF}
+    prior = _get_prior(gp)
+    # for now, hardcoding "1", later expand for multi-input
+    S = SpectralDensity(prior.kernel, 1)
+    # ToDo: add rng to RFF
+    rff = RFF(rffs.rng, S, rffs.l)
+
+    ws = get_weight_distribution(gp, rff)
+
+    return rff, ws
+end
+
+function eval_at(rff::KernelSpectralDensities.AbstractRFF, w, x::AbstractArray)
+    return dot.(rff.(x), Ref(w))
+end
+
+# ## PathwiseSampler
+
+# ### utils
+struct KernelBasis{K}
+    ker::K
+    x::AbstractArray
+end
+
+(kb::KernelBasis)(x) = kb.ker.(Ref(x), kb.x)
+
+function update_basis(pgp, cs::CholeskySampling)
+    ker = pgp.prior.kernel
+    x = pgp.data.x
+    return KernelBasis(ker, x)
+end
+
+function update_basis(pgp, rffs::RFFSampling)
+    rff, _ = rffs(pgp)
+
+    return rff
+end
+
+# ### Main
+
+"""
+    PathwiseSampling(l::Int)
+Sampling by using pathwise sampling, which uses RFF sampling for the prior and an update rule
+based on the kernel. Takes as an input the number of random Fourier features `l` to use.
+"""
+struct PathwiseSampling{P,U} <: AbstractGPSamplingMethod
+    prior::P
+    update::U
+end
+
+function PathwiseSampling(l::Int)
+    return PathwiseSampling(RFFSampling(l), CholeskySampling())
+end
+
+struct PathwiseSampler{PS,TS,D}
+    prior_sampler::PS
+    target_sampler::TS
+    data::D
+end
+
+function (ps::PathwiseSampling)(pgp::AbstractGPs.PosteriorGP)
+    upd_fun = update_basis(pgp, ps.update)
+
+    prior = pgp.prior
+    prior_sampler = GPSampler(prior, ps.prior)
+
+    target_dist = get_target_prior(pgp)
+
+    data = (C=pgp.data.C, x=pgp.data.x)
+    return upd_fun, PathwiseSampler(prior_sampler, target_dist, data)
+end
+
+function _rand(rng::AbstractRNG, ps::PathwiseSampler)
+    prior = rand(rng, ps.prior_sampler)
+    f = prior(ps.data.x)
+
+    ts = rand(rng, ps.target_sampler)
+
+    v = ps.data.C \ (ts - f)
+
+    return (prior=prior, v=v)
+end
+
+function eval_at(basis::KernelBasis, s, x::AbstractArray)
+    return s.prior(x) .+ dot.(basis.(x), Ref(s.v))
+end

test/runtests.jl

@@ -77,6 +77,10 @@ include("test_util.jl")
         println(" ")
         @info "Ran latent_gp tests"
 
+        include("sampling.jl")
+        println(" ")
+        @info "Ran sampling tests"
+
         include("deprecations.jl")
         println(" ")
         @info "Ran deprecation tests"