Skip to contents

GPCA Metrics: Building M and A

Source: vignettes/gpca-metrics.Rmd
gpca-metrics.Rmd

This vignette collects practical recipes for row and column metrics, plus notes on SPD remedies and the experimental gpca_mle() learner.

Why metrics

Metrics encode weighting and correlation. The row metric M changes how observations are compared. The column metric A changes how variables are compared. Both must be symmetric positive definite (SPD) for GPCA. In return, you can bake known structure – temporal smoothness, spatial proximity, group membership, heteroscedastic noise – straight into the decomposition.

Heteroscedastic diagonals

The simplest non-trivial metric is a diagonal that down-weights noisy rows or columns:

set.seed(42)
n <- 60; p <- 20
X <- matrix(rnorm(n * p), n, p)

col_noise_sd <- runif(p, 0.5, 2)
A <- Diagonal(x = 1 / col_noise_sd^2)
row_noise_sd <- runif(n, 0.7, 1.3)
M <- Diagonal(x = 1 / row_noise_sd^2)

fit <- genpca(X, M = M, A = A, ncomp = 3,
              preproc = multivarious::center())
fit$sdev
#> [1] 14.69818 11.24551 10.96887
Inverse-variance weights on columns (top) and rows (bottom). Noisier dimensions get smaller weights.

Inverse-variance weights on columns (top) and rows (bottom). Noisier dimensions get smaller weights.

Recipes

These three patterns – AR(1) smoothing, spatial RBF kernel, graph Laplacian – cover most structured-noise applications.

AR(1) row metric (smoothness in time) 

rho     <- 0.7
n_t     <- 60
idx     <- 0:(n_t - 1)
Sigma_r <- outer(idx, idx, function(i, j) rho^abs(i - j))
M_ar1   <- solve(Sigma_r + 1e-3 * diag(n_t))

Spatial RBF kernel 

coords <- as.matrix(expand.grid(x = 1:8, y = 1:8))
d2     <- as.matrix(dist(coords))^2
ell    <- 2
K      <- exp(-d2 / (2 * ell^2))
A_rbf  <- solve(K + 1e-3 * diag(nrow(K)))

Graph Laplacian 

W <- bandSparse(30, k = c(-1, 0, 1),
                diagonals = list(rep(0.2, 29),
                                 rep(1, 30),
                                 rep(0.2, 29)))
D     <- Diagonal(x = rowSums(W))
A_lap <- (D - W) + 1e-2 * Diagonal(nrow(W))
Three structured metrics. Off-diagonal banding is what couples nearby rows or variables -- it tells GPCA 'treat these dimensions as related, not independent'.

Three structured metrics. Off-diagonal banding is what couples nearby rows or variables – it tells GPCA ‘treat these dimensions as related, not independent’.

Learning metrics with gpca_mle()

When you suspect structured noise but lack good priors, gpca_mle() learns SPD metrics by alternating GPCA with matrix-normal maximum likelihood:

set.seed(1)
n_m <- 40; p_m <- 10
X_mle <- matrix(rnorm(n_m * p_m), n_m, p_m)
fit_mle <- gpca_mle(X_mle, ncomp = 2, max_iter = 6,
                    lambda = 1e-3, scale_fix = "trace",
                    method = "eigen", verbose = FALSE)
range(diag(as.matrix(fit_mle$M)))
#> [1] 289.4551 459.8213
range(diag(as.matrix(fit_mle$A)))
#> [1]  8.230071 69.330188
Learned row metric M (left) and column metric A (right) on i.i.d. Gaussian data. With identity-noise input the learner stays close to identity, with mild row- and column-specific damping.

Learned row metric M (left) and column metric A (right) on i.i.d. Gaussian data. With identity-noise input the learner stays close to identity, with mild row- and column-specific damping.

Keep lambda non-zero, start with a small ncomp, and watch warnings – they signal a metric was repaired during the iteration.

SPD remedies in practice

  • constraints_remedy = "ridge" (default) adds a diagonal jitter to keep metrics SPD.
  • "clip" truncates tiny negative eigenvalues; "identity" falls back to identity if a metric misbehaves.
  • Scale metrics before use (e.g., divide by mean diagonal) to avoid ill-conditioning.
  • After fitting, check range(eigen(A)$values) on small problems to verify conditioning.

Where next

See vignette("gpca-scale") for backend choices, sparse workflows, and covariance-only GPCA.