Spectral Graph Alignment with grasp

Overview

grasp() aligns two graph domains by computing spectral embeddings, building multi-scale diffusion descriptors, and solving for a functional map plus a node assignment. The method is unsupervised—the latent classes are only consulted in this vignette for diagnostic plots.

We work with the two graph domains provided by alignment_benchmark so the results are directly comparable to the cone_align, parrot, and kema vignettes.

Setup

library(manifoldalign)
library(multidesign)
library(multivarious)
library(tibble)
library(dplyr)
library(ggplot2)
library(scales)

Loading the Benchmark Pair

alignment_benchmark <- manifoldalign::alignment_benchmark
pair_domains <- alignment_benchmark$domains[1:2]

pair_multidesign <- lapply(pair_domains, function(dom) {
  multidesign::multidesign(dom$x, dom$design)
})

pair_names <- names(pair_multidesign)
node_counts <- vapply(pair_multidesign, function(dom) nrow(dom$x), integer(1))

hd_pair <- multidesign::hyperdesign(pair_multidesign)

Running `grasp`

grasp_internal <- grasp(
  hd_pair,
  ncomp = 20,
  q_descriptors = 80,
  sigma = 0.8,
  lambda = 0.1,
  solver = "linear"
)

str(grasp_internal$assignment)
#>  int [1:80] 79 66 29 53 37 13 7 61 17 65 ...

Assignment Diagnostics

The assignment component maps nodes in the second domain to nodes in the first (reference) domain. We compare the assigned pairs with the underlying class labels to quantify alignment quality—remember that the optimiser itself never sees these labels.

assignment <- grasp_internal$assignment
ref_design <- pair_multidesign[[1]]$design
cmp_design <- pair_multidesign[[2]]$design

class_accuracy <- mean(ref_design$condition == cmp_design$condition[assignment])
identity_fraction <- mean(assignment == seq_along(assignment))

tibble(
  metric = c("Class agreement", "Exact identity"),
  value = c(class_accuracy, identity_fraction)
)
#> # A tibble: 2 × 2
#>   metric           value
#>   <chr>            <dbl>
#> 1 Class agreement 0.475 
#> 2 Exact identity  0.0125

n_ref <- node_counts[1]
embed1 <- grasp_internal$s[seq_len(n_ref), , drop = FALSE]
embed2 <- grasp_internal$s[n_ref + seq_len(n_ref), , drop = FALSE]
final_err <- sqrt(mean(rowSums((embed1 - embed2)^2)))

tibble(
  metric = "Post-alignment RMS error",
  value = final_err
)
#> # A tibble: 1 × 2
#>   metric                   value
#>   <chr>                    <dbl>
#> 1 Post-alignment RMS error  1.28

Visualising the Aligned Embeddings

We stack the aligned embeddings for both domains (as returned by grasp()) and colour points by their latent class.

embed1 <- grasp_internal$s[seq_len(n_ref), , drop = FALSE]
embed2 <- grasp_internal$s[n_ref + seq_len(n_ref), , drop = FALSE]

score_tbl <- as_tibble(rbind(embed1, embed2), .name_repair = "minimal")
colnames(score_tbl) <- paste0("comp", seq_len(ncol(score_tbl)))

scores <- score_tbl %>%
  mutate(
    sample = seq_len(nrow(.)),
    domain = rep(pair_names, each = nrow(embed1)),
    condition = c(ref_design$condition, cmp_design$condition[assignment])
  )

 ggplot(scores, aes(x = comp1, y = comp2, colour = condition, shape = domain)) +
  geom_point(size = 2, alpha = 0.85) +
  labs(title = "GRASP alignment: first two components",
       x = "Component 1", y = "Component 2") +
  theme_minimal()

Summary

grasp() performs spectral alignment without supervision; class labels are only used here for evaluation.
The post-alignment RMS error can be used as a quick sanity check that the returned embeddings are in a comparable coordinate system.
Because all alignment vignettes reuse the same alignment_benchmark domains, you can compare GRASP, CONE-Align, PARROT, GPCA, and KEMA directly.