Scikit-Learn · Class

MDS

MDS embeds pairwise dissimilarities in a lower-dimensional space using SMACOF. It supports metric and nonmetric variants and can use Euclidean or precomputed dissimilarities.

How it works

MDS uses the SMACOF algorithm to place observations in a lower-dimensional space while preserving their pairwise dissimilarities as well as possible. Metric mode preserves quantitative distances, whereas nonmetric mode uses the rank order of dissimilarities; repeated initializations can improve the selected solution.

Syntax

class sklearn.manifold.MDS(n_components=2, *, metric_mds=True, n_init=1, init='warn', max_iter=300, verbose=0, eps=1e-6, n_jobs=None, random_state=None, dissimilarity='deprecated', metric='euclidean', metric_params=None, normalized_stress='auto')

Parameters

Name	Type	Description
`n_components`	`int, default=2`	Number of dimensions in which to immerse the dissimilarities.
`metric_mds`	`bool, default=True`	Whether to perform metric MDS; when false, perform nonmetric MDS.
`n_init`	`int, default=1`	Number of SMACOF runs with different initializations.
`max_iter`	`int, default=300`	Maximum iterations for one SMACOF run.
`verbose`	`int, default=0`	Verbosity level.
`eps`	`float, default=1e-6`	Relative stress tolerance for convergence.
`n_jobs`	`int or None, default=None`	Number of parallel jobs used for multiple initializations.
`random_state`	`int, RandomState instance or None, default=None`	Controls random number generation.
`dissimilarity`	`{'euclidean', 'precomputed'}`	Dissimilarity measure: Euclidean or precomputed.
`metric_params`	`dict or None, default=None`	Metric configuration passed to distance computation.
`normalized_stress`	`{'auto', True, False}, default='auto'`	Stress normalization behavior.
`metric`	`str, default='euclidean'`	Deprecated compatibility parameter shown by the stable signature.

Returns

The page documents fitted attributes and transformation methods; fit_transform returns the transformed embedding array.

Key points

Use n_components to choose how many coordinates each observation receives in the embedding.
Increase n_init to try multiple starting configurations and reduce dependence on one initialization.
max_iter bounds the optimization work performed by each individual SMACOF run.
eps determines when an optimization run is considered converged based on relative stress improvement.
Set metric_mds to select quantitative-distance preservation or rank-based nonmetric embedding.
n_jobs can control parallel execution of multiple initialization runs.

Common mistakes

In nonmetric mode, zero dissimilarities are treated as missing values, so zero entries should not be used to represent genuine equal distances.
Interpret the stress_ value according to the selected mode: metric MDS defaults to raw stress, while nonmetric MDS defaults to normalized stress.

Example: Embed Iris data with MDS

from sklearn.manifold import MDS
from sklearn.datasets import load_iris

X, _ = load_iris(return_X_y=True)
embedding = MDS(n_components=2, random_state=0)
X_2d = embedding.fit_transform(X)
print(X_2d.shape)

This creates a two-dimensional embedding of the Iris samples using Euclidean metric MDS.

When to use it

Visualization

Explore high-dimensional observations in two or three coordinates.

Precomputed dissimilarities

Represent custom pairwise distances that are not ordinary feature-space Euclidean distances.

Distance-based representation

Recover a low-dimensional arrangement from similarity or distance data.

Interview questions

QWhat problem does MDS solve?

MDS is a manifold-learning estimator for embedding dissimilarities into a chosen number of dimensions. It supports metric and nonmetric modes and optimizes the embedding with SMACOF.

Follow-up: How would you supply distances rather than feature vectors? Set dissimilarity='precomputed' and pass the square dissimilarity matrix to fit or fit_transform.

QWhy might you increase n_init?

n_init controls how many SMACOF runs use different initializations. The estimator can run those initializations in parallel when appropriate, and the resulting solution is selected using stress.

Follow-up: How can reproducibility be improved? Provide a fixed random_state, while remembering that changing initialization can change the optimization result.

QWhat do n_components and dissimilarity control?

n_components specifies the dimensionality of the embedding, while dissimilarity selects either pairwise Euclidean distances computed from the input or distances supplied directly. The default dissimilarity is Euclidean.

Follow-up: When should precomputed distances be used? Use them when the pairwise relationship is not adequately represented by Euclidean distances in the original feature space.

MDS.fit
MDS.fit_transform
MDS.get_params
MDS.set_params
ClassicalMDS

Source: MDS — scikit-learn 1.9.1 documentation. Checked 2026-10-15.

MDS