8.15.2. sklearn.manifold.Isomap

class sklearn.manifold.Isomap(n_neighbors=5, out_dim=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.

out_dim : 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. Note that arpack can handle both dense and sparse data efficiently

‘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 algorithm with Fibonacci Heaps

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

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

Notes

References:

[1] Tenenbaum, J.B.; De Silva, V.; & Langford, J.C. A global geometric

framework for nonlinear dimensionality reduction. Science 290 (5500)

Attributes

embedding_	array-like, shape (n_samples, out_dim)	Stores the embedding vectors
kernel_pca_	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

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.
reconstruction_error()	Compute the reconstruction error for the embedding.
set_params(**params)	Set the parameters of the estimator.
transform(X)	Transform X.

__init__(n_neighbors=5, out_dim=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, cKDTree, NearestNeighbors}

Sample data, shape = (n_samples, n_features), in the form of a numpy array, sparse 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, cKDTree} :

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, out_dim) :

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 the 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, out_dim) :