GPLVM tutorial

tutorials/12-gaussian-process-latent-variable-model/12_gaussian-process-latent-variable-model.jmd

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+---
+title: Gaussian Process Latent Variable Model
+permalink: /:collection/:name/
+redirect_from: tutorials/12-gaussian-process-latent-variable-model/
+---
+
+# 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](https://jmlr.org/papers/v6/lawrence05a.html) and a [further extension](http://proceedings.mlr.press/v9/titsias10a/titsias10a.pdf).
+
+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.
+```julia
+using Turing
+using AbstractGPs, KernelFunctions, Random, Plots
+
+using LinearAlgebra
+using VegaLite, DataFrames, StatsPlots, StatsBase
+using RDatasets
+
+Random.seed!(1789);
+```
+
+We demonstrate the GPLVM with a very small dataset: [Fisher's Iris data set](https://en.wikipedia.org/wiki/Iris_flower_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.
+
+```julia
+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)
+```julia
+@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)
+  end
+end;
+```
+
+
+We define two different kernels, a simple linear kernel with an Automatic Relevance Determination transform and a
+squared exponential kernel.
+```julia
+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.
+```julia
+@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))
+  Y ~ filldist(MvNormal(mu, cv), D)
+end;
+
+@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))
+  Y ~ filldist(MvNormal(mu, cv), D)
+end;
+```
+
+```julia
+# 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;
+```
+
+```julia
+ppca = pPCA(dat[1:n_data, 1:n_features])
+chain_ppca = sample(ppca, NUTS(), 1000);
+```
+
+```julia
+# 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 = df_pre |>  @vlplot(:point, x=:z1, y=:z2, color="type:n")
+```
+
+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.
+```julia
+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]
+```
+
+```julia
+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]
+println(alpha_indices)
+df_gplvm_linear[!,:ard1] = z_mean[alpha_indices[1], :]
+df_gplvm_linear[!,:ard2] = z_mean[alpha_indices[2], :]
+
+p_linear = df_gplvm_linear |>  @vlplot(:point, x=:ard1, y=:ard2, color="labels:n")
+p_linear
+```
+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.
+```julia
+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]
+```
+
+```julia
+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]
+println(alpha_indices)
+df_gplvm[!,:ard1] = z_mean[alpha_indices[1], :]
+df_gplvm[!,:ard2] = z_mean[alpha_indices[2], :]
+
+p_gplvm = df_gplvm |>  @vlplot(:point, x=:ard1, y=:ard2, color="labels:n")
+p_gplvm
+```
+
+
+```julia; echo=false
+let
+  @assert abs(mean(z_mean[alpha_indices[1], labels[1:n_data] .== "setosa"]) - mean(z_mean[alpha_indices[1], labels[1:n_data] .!= "setosa"])) > 1.4
+end
+```
+
+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.
+
+```julia
+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 = Stheno.wrap(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 = Stheno.wrap(GP(kernel), GPC())
+  fobs = gp(ColVecs(Z), noise)
+  finducing = gp(xu, 1e-12)
+  sfgp = SparseFiniteGP(fobs, finducing)
+  cv = cov(sfgp.fobs)
+  Y ~ filldist(MvNormal(mu, cv), D)
+
+end
+```
+
+```julia
+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]
+```
+
+```julia
+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 = df_gplvm_sparse |>  @vlplot(:point, x=:ard1, y=:ard2, color="labels:n")
+p_sparse
+```
+
+Comparing the runtime, between the two versions, we can observe a clear speed-up with the sparse version.
+
+```julia; echo=false
+let
+  @assert abs(mean(z_mean[alpha_indices[1], labels[1:n_data] .== "setosa"]) - mean(z_mean[alpha_indices[1], labels[1:n_data] .!= "setosa"])) > 1.0
+
+  @assert abs(mean(z_mean[alpha_indices[2], labels[1:n_data] .== "setosa"]) - mean(z_mean[alpha_indices[2], labels[1:n_data] .!= "setosa"])) > 0.1
+end
+```
+
+```julia, echo=false, skip="notebook"
+if isdefined(Main, :TuringTutorials)
+    Main.TuringTutorials.tutorial_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file])
+end
+```