Classical MDS on shortest-path distances through a nearest-neighbour graph,
approximating geodesics on the manifold. On a Swiss roll two points on
facing sheets are close in space but far along the surface, and only the
graph distance knows the difference.
Usage
morie_esl_isomap(X, k = 2, neighbors = 5)
Arguments
- X
Data, n by p.
- k
Embedding dimension.
- neighbors
Neighbourhood size.
Value
List with embedding, eigenvalues, geodesic,
residual_variance.
Details
The neighbourhood size is the weak spot: too small and the graph
disconnects (the embedding is then undefined, and that is raised rather than
filled in); too large and it short-circuits across a fold, silently
reintroducing the Euclidean distance the method exists to avoid.
References
Tenenbaum, J. B., de Silva, V., & Langford, J. C. (2000).
Science 290(5500), 2319-2323.
Examples
set.seed(1)
tt <- runif(200, 1.5 * pi, 4.5 * pi)
X <- cbind(tt * cos(tt), runif(200, 0, 10), tt * sin(tt))
abs(stats::cor(morie_esl_isomap(X, k = 2, neighbors = 8)$embedding[, 1], tt)) > 0.9
#> [1] FALSE
