linalg.cmds

linalg.cmds(D, d=2, coords=True)

Constructs coordinates from squared distances using Classical Multi-Dimensional Scaling.

CMDS is a coordinatization algorithm that generates d-dimensional coordinates from a Euclidean distance matrix D. Algorithmically, D is converted into a doubly-centered Gram matrix G whose eigen decomposition is used to produce coordinates minimizing a notion of ‘strain’.

Parameters

Name	Type	Description	Default
`D`	ArrayLike	Squared Euclidean distance matrix, or set of (squared) pairwise distances.	required
`d`	int	dimension of the embedding to produce.	`2`
`coords`	bool	whether to return the embedding (default = True), or just return the eigenvectors	`True`

Returns

Type	Description
Union[np.ndarray, tuple]	if `coords = True`, the coordinates from the largest `d` eigenvectors of `G`’s eigendecomposition. Otherwise, the eigenvalues and eigenvectors are returned. See the Examples section for more details.

Notes

PCA is dual to CMDS in the sense that the embedding produced by CMDS on the Euclidean distance matrix from X satisfies the same reconstruction loss as with PCA. In particular, when X comes from Euclidean space, the output of pca(…) will match the output of cmds(…) exactly up to rotation and translation.

See Also

Examples

import numpy as np 
from geomcover.linalg import pca, cmds

## Start with a random set of points in R^3 + its distance matrix
X = np.random.uniform(size=(50,3))
D = np.linalg.norm(X - X[:,np.newaxis], axis=2)

## Note that CMDS takes as input *squared* distances
Y_pca = pca(X, d=2)
Y_mds = cmds(D**2, d=2)

## Get distance matrices for both embeddings
Y_pca_D = np.linalg.norm(Y_pca - Y_pca[:,np.newaxis], axis=2)
Y_mds_D = np.linalg.norm(Y_mds - Y_mds[:,np.newaxis], axis=2)

## Up to rotation and translation, the coordinates are identical
all_close = np.allclose(Y_pca_D, Y_mds_D)
print(f"PCA and MDS coord. distances identical? {all_close}")

PCA and MDS coord. distances identical? True