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Gaussian Process Latent Variable Model

In a previous tutorial, we have discussed latent variable models, in particular probabilistic principal component analysis (pPCA). Here, we show how we can extend the mapping provided by pPCA to non-linear mappings between input and output. For more details about the Gaussian Process Latent Variable Model (GPLVM), we refer the reader to the original publication and a further extension.

In short, the GPVLM is a dimensionality reduction technique that allows us to embed a high-dimensional dataset in a lower-dimensional embedding. Importantly, it provides the advantage that the linear mappings from the embedded space can be non-linearised through the use of Gaussian Processes.

Let's start by loading some dependencies.

using Turing
using AbstractGPs, Random

using LinearAlgebra
using VegaLite, DataFrames, StatsPlots, StatsBase
using RDatasets


We demonstrate the GPLVM with a very small dataset: Fisher's Iris data set. This is mostly for reasons of run time, so the tutorial can be run quickly. As you will see, one of the major drawbacks of using GPs is their speed, although this is an active area of research. We will briefly touch on some ways to speed things up at the end of this tutorial. We transform the original data with non-linear operations in order to demonstrate the power of GPs to work on non-linear relationships, while keeping the problem reasonably small.

data = dataset("datasets", "iris")
species = data[!, "Species"]
index = shuffle(1:150)
# we extract the four measured quantities,
# so the dimension of the data is only d=4 for this toy example
dat = Matrix(data[index, 1:4])
labels = data[index, "Species"]

# non-linearize data to demonstrate ability of GPs to deal with non-linearity
dat[:, 1] = 0.5 * dat[:, 1] .^ 2 + 0.1 * dat[:, 1] .^ 3
dat[:, 2] = dat[:, 2] .^ 3 + 0.2 * dat[:, 2] .^ 4
dat[:, 3] = 0.1 * exp.(dat[:, 3]) - 0.2 * dat[:, 3] .^ 2
dat[:, 4] = 0.5 * log.(dat[:, 4]) .^ 2 + 0.01 * dat[:, 3] .^ 5

# normalize data
dt = fit(ZScoreTransform, dat; dims=1);
StatsBase.transform!(dt, dat);

We will start out by demonstrating the basic similarity between pPCA (see the tutorial on this topic) and the GPLVM model. Indeed, pPCA is basically equivalent to running the GPLVM model with an automatic relevance determination (ARD) linear kernel.

First, we re-introduce the pPCA model (see the tutorial on pPCA for details)

@model function pPCA(x, ::Type{TV}=Array{Float64}) where {TV}
    # Dimensionality of the problem.
    N, D = size(x)
    # latent variable z
    z ~ filldist(Normal(), D, N)
    # weights/loadings W
    w ~ filldist(Normal(), D, D)
    mu = (w * z)'
    for d in 1:D
        x[:, d] ~ MvNormal(mu[:, d], I)

We define two different kernels, a simple linear kernel with an Automatic Relevance Determination transform and a squared exponential kernel.

linear_kernel(α) = LinearKernel()  ARDTransform(α)
sekernel(α, σ) = σ * SqExponentialKernel()  ARDTransform(α);

And here is the GPLVM model. We create separate models for the two types of kernel.

@model function GPLVM_linear(Y, K=4, ::Type{T}=Float64) where {T}

    # Dimensionality of the problem.
    N, D = size(Y)
    # K is the dimension of the latent space
    @assert K <= D
    noise = 1e-3

    # Priors
    α ~ MvLogNormal(MvNormal(zeros(K), I))
    Z ~ filldist(Normal(), K, N)
    mu ~ filldist(Normal(), N)

    kernel = linear_kernel(α)

    gp = GP(mu, kernel)
    cv = cov(gp(ColVecs(Z), noise))
    return Y ~ filldist(MvNormal(mu, cv), D)

@model function GPLVM(Y, K=4, ::Type{T}=Float64) where {T}

    # Dimensionality of the problem.
    N, D = size(Y)
    # K is the dimension of the latent space
    @assert K <= D
    noise = 1e-3

    # Priors
    α ~ MvLogNormal(MvNormal(zeros(K), I))
    σ ~ LogNormal(0.0, 1.0)
    Z ~ filldist(Normal(), K, N)
    mu ~ filldist(Normal(), N)

    kernel = sekernel(α, σ)

    gp = GP(mu, kernel)
    cv = cov(gp(ColVecs(Z), noise))
    return Y ~ filldist(MvNormal(mu, cv), D)
# Standard GPs don't scale very well in n, so we use a small subsample for the purpose of this tutorial
n_data = 40
# number of features to use from dataset
n_features = 4
# latent dimension for GP case
ndim = 4;
ppca = pPCA(dat[1:n_data, 1:n_features])
chain_ppca = sample(ppca, NUTS(), 1000);
# we extract the posterior mean estimates of the parameters from the chain
w = reshape(mean(group(chain_ppca, :w))[:, 2], (n_features, n_features))
z = reshape(mean(group(chain_ppca, :z))[:, 2], (n_features, n_data))
X = w * z

df_pre = DataFrame(z', :auto)
rename!(df_pre, Symbol.(["z" * string(i) for i in collect(1:n_features)]))
df_pre[!, :type] = labels[1:n_data]
p_ppca = @vlplot(:point, x = :z1, y = :z2, color = "type:n")(df_pre)

We can see that the pPCA fails to distinguish the groups. In particular, the setosa species is not clearly separated from versicolor and virginica. This is due to the non-linearities that we introduced, as without them the two groups can be clearly distinguished using pPCA (see the pPCA tutorial).

Let's try the same with our linear kernel GPLVM model.

gplvm_linear = GPLVM_linear(dat[1:n_data, 1:n_features], ndim)

chain_linear = sample(gplvm_linear, NUTS(), 500)
# we extract the posterior mean estimates of the parameters from the chain
z_mean = reshape(mean(group(chain_linear, :Z))[:, 2], (n_features, n_data))
alpha_mean = mean(group(chain_linear, :α))[:, 2]
4-element Vector{Float64}:
df_gplvm_linear = DataFrame(z_mean', :auto)
rename!(df_gplvm_linear, Symbol.(["z" * string(i) for i in collect(1:ndim)]))
df_gplvm_linear[!, :sample] = 1:n_data
df_gplvm_linear[!, :labels] = labels[1:n_data]
alpha_indices = sortperm(alpha_mean; rev=true)[1:2]
df_gplvm_linear[!, :ard1] = z_mean[alpha_indices[1], :]
df_gplvm_linear[!, :ard2] = z_mean[alpha_indices[2], :]

p_linear = @vlplot(:point, x = :ard1, y = :ard2, color = "labels:n")(df_gplvm_linear)
[3, 1]

We can see that similar to the pPCA case, the linear kernel GPLVM fails to distinguish between the two groups (setosa on the one hand, and virginica and verticolor on the other).

Finally, we demonstrate that by changing the kernel to a non-linear function, we are able to separate the data again.

gplvm = GPLVM(dat[1:n_data, 1:n_features], ndim)

chain_gplvm = sample(gplvm, NUTS(), 500)
# we extract the posterior mean estimates of the parameters from the chain
z_mean = reshape(mean(group(chain_gplvm, :Z))[:, 2], (ndim, n_data))
alpha_mean = mean(group(chain_gplvm, :α))[:, 2]
4-element Vector{Float64}:
df_gplvm = DataFrame(z_mean', :auto)
rename!(df_gplvm, Symbol.(["z" * string(i) for i in collect(1:ndim)]))
df_gplvm[!, :sample] = 1:n_data
df_gplvm[!, :labels] = labels[1:n_data]
alpha_indices = sortperm(alpha_mean; rev=true)[1:2]
df_gplvm[!, :ard1] = z_mean[alpha_indices[1], :]
df_gplvm[!, :ard2] = z_mean[alpha_indices[2], :]

p_gplvm = @vlplot(:point, x = :ard1, y = :ard2, color = "labels:n")(df_gplvm)
[2, 1]

Now, the split between the two groups is visible again.

Speeding up inference

Gaussian processes tend to be slow, as they naively require operations in the order of $O(n^3)$. Here, we demonstrate a simple speedup using the Stheno library. Speeding up Gaussian process inference is an active area of research.

using Stheno
@model function GPLVM_sparse(Y, K, ::Type{T}=Float64) where {T}

    # Dimensionality of the problem.
    N, D = size(Y)
    # dimension of latent space
    @assert K <= D
    # number of inducing points
    n_inducing = 25
    noise = 1e-3

    # Priors
    α ~ MvLogNormal(MvNormal(zeros(K), I))
    σ ~ LogNormal(1.0, 1.0)
    Z ~ filldist(Normal(), K, N)
    mu ~ filldist(Normal(), N)

    kernel = σ * SqExponentialKernel()  ARDTransform(α)

    ## Standard
    # gpc = GPC()
    # f = atomic(GP(kernel), gpc)
    # gp = f(ColVecs(Z), noise)
    # Y ~ filldist(gp, D)

    ## SPARSE GP
    #  xu = reshape(repeat(locations, K), :, K) # inducing points
    #  xu = reshape(repeat(collect(range(-2.0, 2.0; length=20)), K), :, K) # inducing points
    lbound = minimum(Y) + 1e-6
    ubound = maximum(Y) - 1e-6
    #  locations ~ filldist(Uniform(lbound, ubound), n_inducing)
    #  locations = [-2., -1.5 -1., -0.5, -0.25, 0.25, 0.5, 1., 2.]
    #  locations = collect(LinRange(lbound, ubound, n_inducing))
    locations = quantile(vec(Y), LinRange(0.01, 0.99, n_inducing))
    xu = reshape(locations, 1, :)
    gp = atomic(GP(kernel), GPC())
    fobs = gp(ColVecs(Z), noise)
    finducing = gp(xu, 1e-12)
    sfgp = SparseFiniteGP(fobs, finducing)
    cv = cov(sfgp.fobs)
    return Y ~ filldist(MvNormal(mu, cv), D)
GPLVM_sparse (generic function with 3 methods)
n_data = 50
gplvm_sparse = GPLVM_sparse(dat[1:n_data, :], ndim)

chain_gplvm_sparse = sample(gplvm_sparse, NUTS(), 500)
# we extract the posterior mean estimates of the parameters from the chain
z_mean = reshape(mean(group(chain_gplvm_sparse, :Z))[:, 2], (ndim, n_data))
alpha_mean = mean(group(chain_gplvm_sparse, :α))[:, 2]
4-element Vector{Float64}:
df_gplvm_sparse = DataFrame(z_mean', :auto)
rename!(df_gplvm_sparse, Symbol.(["z" * string(i) for i in collect(1:ndim)]))
df_gplvm_sparse[!, :sample] = 1:n_data
df_gplvm_sparse[!, :labels] = labels[1:n_data]
alpha_indices = sortperm(alpha_mean; rev=true)[1:2]
df_gplvm_sparse[!, :ard1] = z_mean[alpha_indices[1], :]
df_gplvm_sparse[!, :ard2] = z_mean[alpha_indices[2], :]
p_sparse = @vlplot(:point, x = :ard1, y = :ard2, color = "labels:n")(df_gplvm_sparse)

Comparing the runtime, between the two versions, we can observe a clear speed-up with the sparse version.


These tutorials are a part of the TuringTutorials repository, found at:

To locally run this tutorial, do the following commands:

using TuringTutorials
TuringTutorials.weave("12-gaussian-process", "12_gaussian-process.jmd")

Computer Information:

Julia Version 1.6.6
Commit b8708f954a (2022-03-28 07:17 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/7aa0085e-79a4-45f3-a5bd-9743c91cf3da

Package Information:

      Status `/cache/build/default-amdci4-6/julialang/turingtutorials/tutorials/12-gaussian-process/Project.toml`
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  [91d4177d] Opus_jll v1.3.2+0
  [2f80f16e] PCRE_jll v8.44.0+0
  [30392449] Pixman_jll v0.40.1+0
  [ea2cea3b] Qt5Base_jll v5.15.3+1
  [f50d1b31] Rmath_jll v0.3.0+0
  [a2964d1f] Wayland_jll v1.19.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.9.12+0
  [aed1982a] XSLT_jll v1.1.34+0
  [4f6342f7] Xorg_libX11_jll v1.6.9+4
  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [3161d3a3] Zstd_jll v1.5.2+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [05823500] OpenLibm_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll