sklearn.manifold.Isomap

class sklearn.manifold.Isomap(n_neighbors=5, n_components=2, eigen_solver='auto', tol=0, max_iter=None, path_method='auto', neighbors_algorithm='auto')

Isomap Embedding

Non-linear dimensionality reduction through Isometric Mapping

Parameters:

n_neighbors : integer

number of neighbors to consider for each point.

n_components : integer

number of coordinates for the manifold

eigen_solver : [‘auto’|’arpack’|’dense’]

‘auto’ : Attempt to choose the most efficient solver for the given problem.

‘arpack’ : Use Arnoldi decomposition to find the eigenvalues and eigenvectors.

‘dense’ : Use a direct solver (i.e. LAPACK) for the eigenvalue decomposition.

tol : float

Convergence tolerance passed to arpack or lobpcg. not used if eigen_solver == ‘dense’.

max_iter : integer

Maximum number of iterations for the arpack solver. not used if eigen_solver == ‘dense’.

path_method : string [‘auto’|’FW’|’D’]

Method to use in finding shortest path.

‘auto’ : attempt to choose the best algorithm automatically.

‘FW’ : Floyd-Warshall algorithm.

‘D’ : Dijkstra’s algorithm.

neighbors_algorithm : string [‘auto’|’brute’|’kd_tree’|’ball_tree’]

Algorithm to use for nearest neighbors search, passed to neighbors.NearestNeighbors instance.

Attributes:

embedding_ : array-like, shape (n_samples, n_components)

Stores the embedding vectors.

kernel_pca_ : object

KernelPCA object used to implement the embedding.

training_data_ : array-like, shape (n_samples, n_features)

Stores the training data.

nbrs_ : sklearn.neighbors.NearestNeighbors instance

Stores nearest neighbors instance, including BallTree or KDtree if applicable.

dist_matrix_ : array-like, shape (n_samples, n_samples)

Stores the geodesic distance matrix of training data.

References

[R146]

Tenenbaum, J.B.; De Silva, V.; & Langford, J.C. A global geometric framework for nonlinear dimensionality reduction. Science 290 (5500)

Methods

fit(X[, y])	Compute the embedding vectors for data X
fit_transform(X[, y])	Fit the model from data in X and transform X.
get_params([deep])	Get parameters for this estimator.
reconstruction_error()	Compute the reconstruction error for the embedding.
set_params(**params)	Set the parameters of this estimator.
transform(X)	Transform X.

__init__(n_neighbors=5, n_components=2, eigen_solver='auto', tol=0, max_iter=None, path_method='auto', neighbors_algorithm='auto')

fit(X, y=None)

Compute the embedding vectors for data X

Parameters:

X : {array-like, sparse matrix, BallTree, KDTree, NearestNeighbors}

Sample data, shape = (n_samples, n_features), in the form of a numpy array, precomputed tree, or NearestNeighbors object.

Returns:

self : returns an instance of self.

fit_transform(X, y=None)

Fit the model from data in X and transform X.

Parameters:

X: {array-like, sparse matrix, BallTree, KDTree} :

Training vector, where n_samples in the number of samples and n_features is the number of features.

Returns:

X_new: array-like, shape (n_samples, n_components) :

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deep: boolean, optional :

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

params : mapping of string to any

Parameter names mapped to their values.

reconstruction_error()

Compute the reconstruction error for the embedding.

Returns:reconstruction_error : float

Notes

The cost function of an isomap embedding is

E = frobenius_norm[K(D) - K(D_fit)] / n_samples

Where D is the matrix of distances for the input data X, D_fit is the matrix of distances for the output embedding X_fit, and K is the isomap kernel:

K(D) = -0.5 * (I - 1/n_samples) * D^2 * (I - 1/n_samples)

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The former have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns:self :

transform(X)

Transform X.

This is implemented by linking the points X into the graph of geodesic distances of the training data. First the n_neighbors nearest neighbors of X are found in the training data, and from these the shortest geodesic distances from each point in X to each point in the training data are computed in order to construct the kernel. The embedding of X is the projection of this kernel onto the embedding vectors of the training set.

Parameters:X: array-like, shape (n_samples, n_features) : Returns:X_new: array-like, shape (n_samples, n_components) :

Examples using sklearn.manifold.Isomap

Comparison of Manifold Learning methods

Manifold Learning methods on a severed sphere

Manifold learning on handwritten digits: Locally Linear Embedding, Isomap...