def ler(X, Y, n_components=2, affinity='nearest_neighbors',
n_neighbors=None, gamma=None, mu=1.0, y_gamma=None,
eigen_solver='auto', tol=1e-6, max_iter=100,
random_state=None):
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1)
nbrs.fit(X)
X = nbrs._fit_X
Nx, d_in = X.shape
Ny = Y.shape[0]
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if Nx != Ny:
raise ValueError("X and Y must have same number of points")
if affinity == 'nearest_neighbors':
if n_neighbors >= Nx:
raise ValueError("n_neighbors must be less than number of points")
if n_neighbors == None or n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
elif affinity == 'rbf':
if gamma != None and gamma <= 0:
raise ValueError("n_neighbors must be positive")
else:
raise ValueError("affinity must be 'nearest_neighbors' or 'rbf' must be positive")
if Y.ndim == 1:
Y = Y[:, None]
if y_gamma is None:
dists = pairwise_distances(Y)
y_gamma = 1.0 / median(dists)
if affinity == 'nearest_neighbors':
affinity = kneighbors_graph(X, n_neighbors, include_self=True)
else:
if gamma == None:
dists = pairwise_distances(X)
gamma = 1.0 / median(dists)
affinity = kneighbors_graph(X, n_neighbors, mode='distance', include_self=True)
affinity.data = exp(-gamma * affinity.data ** 2)
K = rbf_kernel(Y, gamma=y_gamma)
lap = laplacian(affinity, normed=True)
lapK = laplacian(K, normed=True)
embedding, _ = null_space(lap + mu * lapK, n_components,
k_skip=1, eigen_solver=eigen_solver,
tol=tol, max_iter=max_iter,
random_state=random_state)
return embedding
def ler(X, Y, n_components=2, affinity='nearest_neighbors',
n_neighbors=None, gamma=None, mu=1.0, y_gamma=None,
eigen_solver='auto', tol=1e-6, max_iter=100,
random_state=None):
"""
Laplacian Eigenmaps for Regression (LER)
Parameters
----------
X : ndarray, 2-dimensional
The data matrix, shape (num_points, num_dims)
Y : ndarray, 1 or 2-dimensional
The response matrix, shape (num_points, num_responses).
n_components : int
Number of dimensions for embedding. Default is 2.
affinity : string or callable, default : "nearest_neighbors"
How to construct the affinity matrix.
- 'nearest_neighbors' : construct affinity matrix by knn graph
- 'rbf' : construct affinity matrix by rbf kernel
n_neighbors : int, optional, default=None
Number of neighbors for kNN graph construction on X.
gamma : float, optional, default=None
Scaling factor for RBF kernel on X.
mu : float, optional, default=1.0
Influence of the Y-similarity penalty.
y_gamma : float, optional
Scaling factor for RBF kernel on Y.
Defaults to the inverse of the median distance between rows of Y.
Returns
-------
embedding : ndarray, 2-dimensional
The embedding of X, shape (num_points, n_components)
"""
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1)
nbrs.fit(X)
X = nbrs._fit_X
Nx, d_in = X.shape
Ny = Y.shape[0]
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if Nx != Ny:
raise ValueError("X and Y must have same number of points")
if affinity == 'nearest_neighbors':
if n_neighbors >= Nx:
raise ValueError("n_neighbors must be less than number of points")
if n_neighbors == None or n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
elif affinity == 'rbf':
if gamma != None and gamma <= 0:
raise ValueError("n_neighbors must be positive")
else:
raise ValueError("affinity must be 'nearest_neighbors' or 'rbf' must be positive")
if Y.ndim == 1:
Y = Y[:, None]
if y_gamma is None:
dists = pairwise_distances(Y)
y_gamma = 1.0 / median(dists)
if affinity == 'nearest_neighbors':
affinity = kneighbors_graph(X, n_neighbors, include_self=True)
else:
if gamma == None:
dists = pairwise_distances(X)
gamma = 1.0 / median(dists)
affinity = kneighbors_graph(X, n_neighbors, mode='distance', include_self=True)
affinity.data = exp(-gamma * affinity.data ** 2)
K = rbf_kernel(Y, gamma=y_gamma)
lap = laplacian(affinity, normed=True)
lapK = laplacian(K, normed=True)
embedding, _ = null_space(lap + mu * lapK, n_components,
k_skip=1, eigen_solver=eigen_solver,
tol=tol, max_iter=max_iter,
random_state=random_state)
return embedding
def ler(X,
Y,
n_components=2,
affinity='nearest_neighbors',
n_neighbors=None,
gamma=None,
mu=1.0,
y_gamma=None,
eigen_solver='auto',
tol=1e-6,
max_iter=100,
random_state=None):
"""
Laplacian Eigenmaps for Regression (LER)
Parameters
----------
X : ndarray, 2-dimensional
The data matrix, shape (num_points, num_dims)
Y : ndarray, 1 or 2-dimensional
The response matrix, shape (num_points, num_responses).
Y[0:] is assumed to provide responses for X[:num_response_points]
n_components : int
Number of dimensions for embedding. Default is 2.
affinity : string or callable, default : "nearest_neighbors"
How to construct the affinity matrix.
- 'nearest_neighbors' : construct affinity matrix by knn graph
- 'rbf' : construct affinity matrix by rbf kernel
n_neighbors : int, optional, default=None
Number of neighbors for kNN graph construction on X.
gamma : float, optional, default=None
Scaling factor for RBF kernel on X.
mu : float, optional, default=1.0
Influence of the Y-similarity penalty.
y_gamma : float, optional
Scaling factor for RBF kernel on Y.
Defaults to the inverse of the median distance between rows of Y.
Returns
-------
embedding : ndarray, 2-dimensional
The embedding of X, shape (num_points, n_components)
"""
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1)
nbrs.fit(X)
X = nbrs._fit_X
Nx, d_in = X.shape
Ny = Y.shape[0]
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if Nx < Ny:
raise ValueError("X should have at least as many points as Y")
if affinity == 'nearest_neighbors':
if n_neighbors >= Nx:
raise ValueError("n_neighbors must be less than number of points")
if n_neighbors is None or n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
elif affinity == 'rbf':
if gamma is not None and gamma <= 0:
raise ValueError("n_neighbors must be positive")
else:
raise ValueError("affinity must be 'nearest_neighbors' or 'rbf' must" +
" be positive")
if Y.ndim == 1:
Y = Y[:, None]
if y_gamma is None:
dists = pairwise_distances(Y)
y_gamma = 1.0 / median(dists)
if affinity == 'nearest_neighbors':
affinity = kneighbors_graph(X, n_neighbors, include_self=True)
else:
if gamma is None:
dists = pairwise_distances(X)
gamma = 1.0 / median(dists)
affinity = kneighbors_graph(X,
n_neighbors,
mode='distance',
include_self=True)
affinity.data = exp(-gamma * affinity.data**2)
K = rbf_kernel(Y, gamma=y_gamma)
lap = laplacian(affinity, normed=True)
lapK = laplacian(K, normed=True)
if Nx > Ny:
# zeros = csr_matrix((Nx-Ny,Nx-Ny),dtype=lap.dtype)
# lapK = bmat([[lapK, None], [None, zeros]])
ones = csr_matrix(np.ones((Nx - Ny, Nx - Ny)), dtype=lap.dtype)
lapK = bmat([[lapK, None], [None, ones]])
embedding, _ = null_space(lap + mu * lapK,
n_components,
k_skip=1,
eigen_solver=eigen_solver,
tol=tol,
max_iter=max_iter,
random_state=random_state)
return embedding
def locally_linear_embedding(
X, n_neighbors, n_components, reg=1e-3, eigen_solver='auto', tol=1e-6,
max_iter=100, method='standard', hessian_tol=1E-4, modified_tol=1E-12,
random_state=None, n_jobs=None):
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
if method not in ('standard', 'hessian', 'modified', 'ltsa'):
raise ValueError("unrecognized method '%s'" % method)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1, n_jobs=n_jobs)
nbrs.fit(X)
X = nbrs._fit_X
N, d_in = X.shape
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if n_neighbors >= N:
raise ValueError(
"Expected n_neighbors <= n_samples, "
" but n_samples = %d, n_neighbors = %d" %
(N, n_neighbors)
)
if n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
M_sparse = (eigen_solver != 'dense')
if method == 'standard':
W = barycenter_kneighbors_graph(nbrs, n_neighbors=n_neighbors, reg=reg, n_jobs=1)
if M_sparse:
M = eye(*W.shape, format=W.format) - W
M = (M.T * M).tocsr()
else:
M = (W.T * W - W.T - W).toarray()
M.flat[::M.shape[0] + 1] += 1 # W = W - I = W - I
elif method == 'hessian':
dp = n_components * (n_components + 1) // 2
if n_neighbors <= n_components + dp:
raise ValueError("for method='hessian', n_neighbors must be "
"greater than "
"[n_components * (n_components + 3) / 2]")
neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
return_distance=False)
neighbors = neighbors[:, 1:]
Yi = np.empty((n_neighbors, 1 + n_components + dp), dtype=np.float64)
Yi[:, 0] = 1
M = np.zeros((N, N), dtype=np.float64)
use_svd = (n_neighbors > d_in)
for i in range(N):
Gi = X[neighbors[i]]
Gi -= Gi.mean(0)
# build Hessian estimator
if use_svd:
U = svd(Gi, full_matrices=0)[0]
else:
Ci = np.dot(Gi, Gi.T)
U = eigh(Ci)[1][:, ::-1]
Yi[:, 1:1 + n_components] = U[:, :n_components]
j = 1 + n_components
for k in range(n_components):
Yi[:, j:j + n_components - k] = (U[:, k:k + 1] *
U[:, k:n_components])
j += n_components - k
Q, R = qr(Yi)
w = Q[:, n_components + 1:]
S = w.sum(0)
S[np.where(abs(S) < hessian_tol)] = 1
w /= S
nbrs_x, nbrs_y = np.meshgrid(neighbors[i], neighbors[i])
M[nbrs_x, nbrs_y] += np.dot(w, w.T)
if M_sparse:
M = csr_matrix(M)
elif method == 'modified':
if n_neighbors < n_components:
raise ValueError("modified LLE requires "
"n_neighbors >= n_components")
neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
return_distance=False)
neighbors = neighbors[:, 1:]
# find the eigenvectors and eigenvalues of each local covariance
# matrix. We want V[i] to be a [n_neighbors x n_neighbors] matrix,
# where the columns are eigenvectors
V = np.zeros((N, n_neighbors, n_neighbors))
nev = min(d_in, n_neighbors)
evals = np.zeros([N, nev])
# choose the most efficient way to find the eigenvectors
use_svd = (n_neighbors > d_in)
if use_svd:
for i in range(N):
X_nbrs = X[neighbors[i]] - X[i]
V[i], evals[i], _ = svd(X_nbrs,
full_matrices=True)
evals **= 2
else:
for i in range(N):
X_nbrs = X[neighbors[i]] - X[i]
C_nbrs = np.dot(X_nbrs, X_nbrs.T)
evi, vi = eigh(C_nbrs)
evals[i] = evi[::-1]
V[i] = vi[:, ::-1]
# find regularized weights: this is like normal LLE.
# because we've already computed the SVD of each covariance matrix,
# it's faster to use this rather than np.linalg.solve
reg = 1E-3 * evals.sum(1)
tmp = np.dot(V.transpose(0, 2, 1), np.ones(n_neighbors))
tmp[:, :nev] /= evals + reg[:, None]
tmp[:, nev:] /= reg[:, None]
w_reg = np.zeros((N, n_neighbors))
for i in range(N):
w_reg[i] = np.dot(V[i], tmp[i])
w_reg /= w_reg.sum(1)[:, None]
# calculate eta: the median of the ratio of small to large eigenvalues
# across the points. This is used to determine s_i, below
rho = evals[:, n_components:].sum(1) / evals[:, :n_components].sum(1)
eta = np.median(rho)
# find s_i, the size of the "almost null space" for each point:
# this is the size of the largest set of eigenvalues
# such that Sum[v; v in set]/Sum[v; v not in set] < eta
s_range = np.zeros(N, dtype=int)
evals_cumsum = stable_cumsum(evals, 1)
eta_range = evals_cumsum[:, -1:] / evals_cumsum[:, :-1] - 1
for i in range(N):
s_range[i] = np.searchsorted(eta_range[i, ::-1], eta)
s_range += n_neighbors - nev # number of zero eigenvalues
# Now calculate M.
# This is the [N x N] matrix whose null space is the desired embedding
M = np.zeros((N, N), dtype=np.float64)
for i in range(N):
s_i = s_range[i]
# select bottom s_i eigenvectors and calculate alpha
Vi = V[i, :, n_neighbors - s_i:]
alpha_i = np.linalg.norm(Vi.sum(0)) / np.sqrt(s_i)
# compute Householder matrix which satisfies
# Hi*Vi.T*ones(n_neighbors) = alpha_i*ones(s)
# using prescription from paper
h = np.full(s_i, alpha_i) - np.dot(Vi.T, np.ones(n_neighbors))
norm_h = np.linalg.norm(h)
if norm_h < modified_tol:
h *= 0
else:
h /= norm_h
# Householder matrix is
# >> Hi = np.identity(s_i) - 2*np.outer(h,h)
# Then the weight matrix is
# >> Wi = np.dot(Vi,Hi) + (1-alpha_i) * w_reg[i,:,None]
# We do this much more efficiently:
Wi = (Vi - 2 * np.outer(np.dot(Vi, h), h) +
(1 - alpha_i) * w_reg[i, :, None])
# Update M as follows:
# >> W_hat = np.zeros( (N,s_i) )
# >> W_hat[neighbors[i],:] = Wi
# >> W_hat[i] -= 1
# >> M += np.dot(W_hat,W_hat.T)
# We can do this much more efficiently:
nbrs_x, nbrs_y = np.meshgrid(neighbors[i], neighbors[i])
M[nbrs_x, nbrs_y] += np.dot(Wi, Wi.T)
Wi_sum1 = Wi.sum(1)
M[i, neighbors[i]] -= Wi_sum1
M[neighbors[i], i] -= Wi_sum1
M[i, i] += s_i
if M_sparse:
M = csr_matrix(M)
elif method == 'ltsa':
neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
return_distance=False)
neighbors = neighbors[:, 1:]
M = np.zeros((N, N))
use_svd = (n_neighbors > d_in)
for i in range(N):
Xi = X[neighbors[i]]
Xi -= Xi.mean(0)
# compute n_components largest eigenvalues of Xi * Xi^T
if use_svd:
v = svd(Xi, full_matrices=True)[0]
else:
Ci = np.dot(Xi, Xi.T)
v = eigh(Ci)[1][:, ::-1]
Gi = np.zeros((n_neighbors, n_components + 1))
Gi[:, 1:] = v[:, :n_components]
Gi[:, 0] = 1. / np.sqrt(n_neighbors)
GiGiT = np.dot(Gi, Gi.T)
nbrs_x, nbrs_y = np.meshgrid(neighbors[i], neighbors[i])
M[nbrs_x, nbrs_y] -= GiGiT
M[neighbors[i], neighbors[i]] += 1
return W
a,b = null_space(M, n_components, k_skip=1, eigen_solver=eigen_solver,
tol=tol, max_iter=max_iter, random_state=random_state)
def ller(X,
Y,
n_neighbors,
n_components,
mu=0.5,
gamma=None,
reg=1e-3,
eigen_solver='auto',
tol=1e-6,
max_iter=100,
random_state=None):
"""
Locally Linear Embedding for Regression (LLER)
Parameters
----------
X : ndarray, 2-dimensional
The data matrix, shape (num_data_points, num_dims)
Y : ndarray, 1 or 2-dimensional
The response matrix, shape (num_response_points, num_responses).
Y[0:] is assumed to provide responses for X[:num_response_points]
n_neighbors : int
Number of neighbors for kNN graph construction.
n_components : int
Number of dimensions for embedding.
mu : float, optional
Influence of the Y-similarity penalty.
gamma : float, optional
Scaling factor for RBF kernel on Y.
Defaults to the inverse of the median distance between rows of Y.
Returns
-------
embedding : ndarray, 2-dimensional
The embedding of X, shape (num_points, n_components)
lle_error : float
The embedding error of X (for a fixed reconstruction matrix W)
ller_error : float
The embedding error of X that takes Y into account.
"""
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
if Y.ndim == 1:
Y = Y[:, None]
if gamma is None:
dists = pairwise_distances(Y)
gamma = 1.0 / np.median(dists)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1)
nbrs.fit(X)
X = nbrs._fit_X
Nx, d_in = X.shape
Ny = Y.shape[0]
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if n_neighbors >= Nx:
raise ValueError("n_neighbors must be less than number of points")
if n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
if Nx < Ny:
raise ValueError("X should have at least as many points as Y")
M_sparse = (eigen_solver != 'dense')
W = barycenter_kneighbors_graph(nbrs, n_neighbors=n_neighbors, reg=reg)
if M_sparse:
M = speye(*W.shape, format=W.format) - W
M = (M.T * M).tocsr()
else:
M = (W.T * W - W.T - W).toarray()
M.flat[::M.shape[0] + 1] += 1
P = rbf_kernel(Y, gamma=gamma)
L = laplacian(P, normed=False)
M /= np.abs(M).max() # optional scaling step
L /= np.abs(L).max()
if Nx > Ny:
# zeros = csr_matrix((Nx-Ny,Nx-Ny),dtype=M.dtype)
# L = bmat([[L, None], [None, zeros]])
ones = csr_matrix(np.ones((Nx - Ny, Nx - Ny)), dtype=M.dtype)
L = bmat([[L, None], [None, ones]])
omega = M + mu * L
embedding, lle_error = null_space(omega,
n_components,
k_skip=1,
eigen_solver=eigen_solver,
tol=tol,
max_iter=max_iter,
random_state=random_state)
ller_error = np.trace(embedding.T.dot(L).dot(embedding))
return embedding, lle_error, ller_error
def ller(X, Y, n_neighbors, n_components, mu=0.5, gamma=None,
reg=1e-3,eigen_solver='auto', tol=1e-6, max_iter=100,
random_state=None):
"""
Locally Linear Embedding for Regression (LLER)
Parameters
----------
X : ndarray, 2-dimensional
The data matrix, shape (num_data_points, num_dims)
Y : ndarray, 1 or 2-dimensional
The response matrix, shape (num_response_points, num_responses).
Y[0:] is assumed to provide responses for X[:num_response_points]
n_neighbors : int
Number of neighbors for kNN graph construction.
n_components : int
Number of dimensions for embedding.
mu : float, optional
Influence of the Y-similarity penalty.
gamma : float, optional
Scaling factor for RBF kernel on Y.
Defaults to the inverse of the median distance between rows of Y.
Returns
-------
embedding : ndarray, 2-dimensional
The embedding of X, shape (num_points, n_components)
lle_error : float
The embedding error of X (for a fixed reconstruction matrix W)
ller_error : float
The embedding error of X that takes Y into account.
"""
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
if Y.ndim == 1:
Y = Y[:, None]
if gamma is None:
dists = pairwise_distances(Y)
gamma = 1.0 / np.median(dists)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1)
nbrs.fit(X)
X = nbrs._fit_X
Nx, d_in = X.shape
Ny = Y.shape[0]
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if n_neighbors >= Nx:
raise ValueError("n_neighbors must be less than number of points")
if n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
if Nx < Ny:
raise ValueError("X should have at least as many points as Y")
M_sparse = (eigen_solver != 'dense')
W = barycenter_kneighbors_graph(
nbrs, n_neighbors=n_neighbors, reg=reg)
if M_sparse:
M = speye(*W.shape, format=W.format) - W
M = (M.T * M).tocsr()
else:
M = (W.T * W - W.T - W).toarray()
M.flat[::M.shape[0] + 1] += 1
P = rbf_kernel(Y, gamma=gamma)
L = laplacian(P, normed=False)
M /= np.abs(M).max() # optional scaling step
L /= np.abs(L).max()
if Nx > Ny:
# zeros = csr_matrix((Nx-Ny,Nx-Ny),dtype=M.dtype)
# L = bmat([[L, None], [None, zeros]])
ones = csr_matrix(np.ones((Nx-Ny,Nx-Ny)),dtype=M.dtype)
L = bmat([[L, None], [None, ones]])
omega = M + mu * L
embedding, lle_error = null_space(omega, n_components, k_skip=1,
eigen_solver=eigen_solver, tol=tol,
max_iter=max_iter,
random_state=random_state)
ller_error = np.trace(embedding.T.dot(L).dot(embedding))
return embedding, lle_error, ller_error