8.16.3. sklearn.manifold.locally_linear_embedding

sklearn.manifold.locally_linear_embedding(X, n_neighbors, out_dim, reg=0.001, eigen_solver='auto', tol=1e-06, max_iter=100, method='standard', hessian_tol=0.0001, modified_tol=1e-12, random_state=None)

Perform a Locally Linear Embedding analysis on the data.

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.

n_neighbors : integer

number of neighbors to consider for each point.

out_dim : integer

number of coordinates for the manifold.

reg : float

regularization constant, multiplies the trace of the local covariance matrix of the distances.

eigen_solver : string, {‘auto’, ‘arpack’, ‘dense’}

auto : algorithm will attempt to choose the best method for input data

arpack : use arnoldi iteration in shift-invert mode.

For this method, M may be a dense matrix, sparse matrix, or general linear operator.

dense : use standard dense matrix operations for the eigenvalue

decomposition. For this method, M must be an array or matrix type. This method should be avoided for large problems.

tol : float, optional

Tolerance for ‘arpack’ method Not used if eigen_solver==’dense’.

max_iter : integer

maximum number of iterations for the arpack solver.

method : {‘standard’, ‘hessian’, ‘modified’, ‘ltsa’}

standard : use the standard locally linear embedding algorithm.

see reference [R68]

hessian : use the Hessian eigenmap method. This method requires

n_neighbors > out_dim * (1 + (out_dim + 1) / 2. see reference [R69]

modified : use the modified locally linear embedding algorithm.

see reference [R70]

ltsa : use local tangent space alignment algorithm

see reference [R71]

hessian_tol : float, optional

Tolerance for Hessian eigenmapping method. Only used if method == ‘hessian’

modified_tol : float, optional

Tolerance for modified LLE method. Only used if method == ‘modified’

random_state: numpy.RandomState, optional :

The generator used to initialize the centers. Defaults to numpy.random.

Returns :

Y : array-like, shape [n_samples, out_dim]

Embedding vectors.

squared_error : float

Reconstruction error for the embedding vectors. Equivalent to norm(Y - W Y, 'fro')**2, where W are the reconstruction weights.

References

[R68]

(1, 2) Roweis, S. & Saul, L. Nonlinear dimensionality reduction by locally linear embedding. Science 290:2323 (2000).

[R69]

(1, 2) Donoho, D. & Grimes, C. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci U S A. 100:5591 (2003).

[R70]

(1, 2) Zhang, Z. & Wang, J. MLLE: Modified Locally Linear Embedding Using Multiple Weights. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.382

[R71]

(1, 2) Zhang, Z. & Zha, H. Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. Journal of Shanghai Univ. 8:406 (2004)