Optimize an edge-KK local repair layout

edge.kk() is the edge-restricted Kamada–Kawai local repair operator for the experimental GMDS layout program. Earlier notes called this operator edge-gKK or edge-isometric GMDS; edge-KK is the preferred name because the objective is restricted to graph edges rather than all graph-geodesic pairs. It minimizes weighted edge-length stress $$ \frac{1}{2}\sum_{(i,j)\in E} k_{ij} \left(\|z_i-z_j\|_2 - s w_{ij}\right)^2 $$ using deterministic gradient descent with Armijo backtracking. The default density.mix.schedule runs a continuation from density-weighted stiffnesses toward uniform stiffnesses.

Usage

edge.kk(
  coords = NULL,
  prepared = NULL,
  edges = NULL,
  n = NULL,
  adj.list = NULL,
  weight.list = NULL,
  edge.weights = NULL,
  dim = 2L,
  init = c("classical_mds", "metric_mds", "weighted_grip", "random"),
  weighted.grip.args = list(),
  stiffness.method = c("density", "uniform", "distance_power"),
  stiffness.transform = c("identity", "sqrt", "log"),
  density.mix.schedule = c(0, 0.25, 0.5, 0.75, 1),
  bandwidth = NULL,
  density.n = 512L,
  distance.power = 0,
  stiffness.floor = 0,
  stiffness.ceiling = Inf,
  scale.mode = c("profiled", "identity", "fixed_initial", "user"),
  scale = NULL,
  max.iter = 50L,
  initial.step = 1,
  step.shrink = 0.5,
  armijo.factor = 1e-04,
  grad.tol = 1e-08,
  min.step = 1e-08,
  edge.length.epsilon = 1e-08,
  distance.floor = 1e-08,
  recenter = TRUE,
  return.trace = TRUE,
  diagnostics = TRUE,
  seed = 1L,
  engine = c("cpp", "R")
)

Arguments

coords: Optional starting coordinates. If omitted, init is used.
prepared: Optional object returned by prepare.edge.kk(), prepare.graph.geodesic.mds() or prepare.geodesic.kk(). Edge-only objects from prepare.edge.kk() report only edge diagnostics; all-pairs GMDS path and chord diagnostics are unavailable.
edges: Two-column integer edge matrix used when prepared is omitted. Supply either edges/edge.weights or adj.list/weight.list, not both. Raw graph inputs cannot be combined with a prepared object.
n: Finite positive integer vertex count. When supplied with a prepared object, it must match the stored graph size.
adj.list: Optional adjacency list used when prepared is omitted.
weight.list: Optional edge-weight list parallel to adj.list.
edge.weights: Optional positive edge weights parallel to edges.
dim: Target embedding dimension.
init: Starting layout used when coords is omitted. "classical_mds" (the default) uses classical.mds(). "metric_mds" uses stress-minimizing metric.mds() with its default native SGD backend. Both require an all-pairs prepared object. "weighted_grip" runs grip() with metric = "edge_length" on the graph edges first, and "random" uses centered Gaussian coordinates.
weighted.grip.args: Named list of additional tuning arguments passed to grip() when init = "weighted_grip". Graph inputs, dim, seed, and metric are supplied by edge.kk() and may not be repeated here.
stiffness.method, stiffness.transform, density.mix.schedule, bandwidth, density.n: Parameters passed to edge.length.density.stiffness().
distance.power, stiffness.floor, stiffness.ceiling: Additional stiffness constructor parameters.
scale.mode: Scale policy for edge targets. "profiled" analytically refits s at every state evaluation, "identity" fixes s = 1, "fixed_initial" fits s once at the first continuation stage, and "user" uses scale.
scale: User scale for scale.mode = "user".
max.iter: Maximum iterations per continuation stage.
initial.step, step.shrink, armijo.factor, grad.tol, min.step: Line-search controls.
edge.length.epsilon: Small stabilizer for fixed-path embedded lengths.
distance.floor: Positive floor for relative residuals.
recenter: If TRUE, recenter the layout after accepted steps.
return.trace: If TRUE, keep per-iteration trace rows and coordinate frames. If FALSE, omit those payloads while retaining compact stage summaries in metadata$stage_summaries.
diagnostics: If TRUE, attach the common GMDS diagnostic panel.
seed: Random seed used for init = "weighted_grip" and init = "random".
engine: Optimizer engine. "cpp" uses the Rcpp backend for the edge-stress loop; "R" uses the reference implementation.

Value

A "grip_gmds_layout" object with method "edge_kk".

Details

For scalable repair from an existing layout, pass coords or an edge-only object from prepare.edge.kk(). When raw graph inputs are supplied, edge.kk() uses edge-only preparation whenever coords are supplied or init is "weighted_grip" or "random". Use init = "weighted_grip" for a scalable weighted-GRIP warm start followed by edge-KK polish. If coords is omitted and either MDS initializer is requested, use an all-pairs prepared object from prepare.graph.geodesic.mds() or prepare.geodesic.kk() instead.

Examples

cycle_edges <- edges.cycle(6)
theta <- seq(0, 2 * pi, length.out = 7)[-7]
initial <- cbind(cos(theta), sin(theta))
prepared <- prepare.edge.kk(cycle_edges, n = 6)
fit <- edge.kk(
  coords = initial, prepared = prepared,
  max.iter = 1, density.n = 32, engine = "R"
)