I’ve been reading about PPCA, and this post summarizes my understanding of it. I took a lot of this from Pattern Recognition and Machine Learning by Bishop.
The model behind the algorithm is quite simple. We’ve got \(n\) observations of a random variable \(X\) that takes values in \(\mathbb R^m\). We describe a latent representation \(Z\) that has dimension \(m\) or lower as follows. I’ll assume that \(X\) has a zero mean.
\[X | Z \sim \mathcal N(WZ, \sigma^2 I)\] \[Z \sim \mathcal N (0, \mathcal I)\]We can marginalize out \(Z\) out:
\[X \sim \mathcal N(0, WW^T + \sigma^2 I) \;\;\;*\]Tipping & Bishop (1999) showed that the maximum likelihood solution for W is achieved at:
\[W_{ML} = U (L - \sigma^2 I)^{1/2} R\]where \(L\) is a matrix of (the largest) eigenvalues, \(U\) is a matrix of corresponding eigenvectors and \(R\) is an arbitrary orthogonal matrix.
One could work back to \(X\) using the latent variables using:
\[M = W^T W + \sigma^2 I\] \[Z | X \sim \mathcal N(M^{-1} W^T X, \sigma^{-2} M)\]Here’s some Stan code to reproduce the maximum likelihood solution to \(W\). We recover the correct solution up to rotations, as expected.
Stan code for PCA
library(rstan)
library(data.table)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())
m = 2; n = 100
Z = matrix(rnorm(m*n), n, m)
W = matrix(rnorm(m*m), m, m)
X = Z %*% W
pca = prcomp(X, center = F, scale = F)
model_string = "
data {
int m;
int n;
matrix[n, m] X;
}
parameters {
matrix[m, m] W;
}
model {
vector[m] mu;
matrix[m, m] L;
mu = rep_vector(0, m);
L = cholesky_decompose(W * W');
for(i in 1:m) {
W[, i] ~ normal(0, 2);
}
for(i in 1:n) {
X[i, ] ~ multi_normal_cholesky(mu, L);
}
}
"
model = stan_model(model_code = model_string)
data = list(m = m, n = n, X = X)
# optim_lik = function() optimizing(model, data = data)$par
samples = sampling(model, data = data, chains = 2, iter = 1000)
ml_samples = matrix(extract(samples)$W, 1000, 4)
ml_samples = as.data.table(ml_samples)
names(ml_samples) = c('W11', 'W21', 'W12', 'W22')
eig = eigen(cov(X))
U = eig$vectors
L = eig$values
W_no_R = U %*% diag(sqrt(L))
simulate_rotation = function() {
theta = runif(1, 0, 2*pi)
R = matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, 2)
W_ML = W_no_R %*% R
return(W_ML)
}
manual_calc_samples = t(replicate(1000, as.numeric(simulate_rotation())))
manual_calc_samples = as.data.table(manual_calc_samples)
names(manual_calc_samples) = c('MW11', 'MW21', 'MW12', 'MW22')
plot_frame = cbind(ml_samples, manual_calc_samples)
ggplot(plot_frame) +
geom_hex(aes(W11, W21, fill = 'Stan_W[, 1]'), alpha = 0.3, bins = 100) +
geom_hex(aes(W12, W22, fill = 'Stan_W[, 2]'), alpha = 0.3, bins = 100) +
geom_hex(aes(MW11, MW21, fill = 'ML_W[, 1]'), alpha = 0.3, bins = 100) +
geom_hex(aes(MW12, MW22, fill = 'ML_W[, 2]'), alpha = 0.3, bins = 100) +
xlim(-3, 3) + ylim(-3, 3) + labs(x = 'x', y = 'y') +
scale_fill_brewer(palette = 'Spectral') + theme_void()
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