def lmds(matrix, k=2):
# Find Landmark points
N = matrix.shape[0]
landmarks = _getLandmarkPoints(N)
num_landmarks = len(landmarks)
landmark_matrix = matrix[landmarks]
sqdistances = compute_distances(landmark_matrix, landmark_matrix)
# Do normal MDS on landmark points
means = np.mean(sqdistances, axis=1) # this is called mu_n in the paper
global_mean = np.mean(means) # this is called mu in the paper
# this is called B in the paper
distances_balanced = -(sqdistances - means[np.newaxis, :] -
means[:, np.newaxis] + global_mean) / 2
# find the eigenvectors and eigenvalues with our all-purpose hammer
# for the paper, Lambda = lambda, Q = V
Q, Lambda, _ = np.linalg.svd(distances_balanced)
k = min(k, len(Lambda))
mdsarray_sharp = Q[:, :k] # called L^sharp transpose in the paper
mdsarray = multiply(
mdsarray_sharp,
np.sqrt(Lambda)[np.newaxis, :k]) # called L transpose in the paper
# Make Triangulation Object
return LMDSProjection(landmark_matrix, mdsarray_sharp, means)
def lmds(matrix, k=2):
# Find Landmark points
N = matrix.shape[0]
landmarks = _getLandmarkPoints(N)
num_landmarks = len(landmarks)
landmark_matrix = matrix[landmarks]
sqdistances = compute_distances(landmark_matrix, landmark_matrix)
# Do normal MDS on landmark points
means = np.mean(sqdistances, axis=1) # this is called mu_n in the paper
global_mean = np.mean(means) # this is called mu in the paper
# this is called B in the paper
distances_balanced = -(sqdistances - means[np.newaxis, :] - means[:, np.newaxis] + global_mean) / 2
# find the eigenvectors and eigenvalues with our all-purpose hammer
# for the paper, Lambda = lambda, Q = V
Q, Lambda, _ = np.linalg.svd(distances_balanced)
k = min(k, len(Lambda))
mdsarray_sharp = Q[:, :k] # called L^sharp transpose in the paper
mdsarray = multiply(mdsarray_sharp, np.sqrt(Lambda)[np.newaxis, :k]) # called L transpose in the paper
# Make Triangulation Object
return LMDSProjection(landmark_matrix, mdsarray_sharp, means)
def compute_distances(matrix1, matrix2):
nmatrix1 = normalize(matrix1)
nmatrix2 = normalize(matrix2)
distances = np.arccos(np.maximum(-1, np.minimum(1, dot(nmatrix1, nmatrix2.T))))
assert isinstance(distances, DenseMatrix)
assert distances.same_row_labels_as(nmatrix1)
return multiply(distances, distances)
def compute_distances(matrix1, matrix2):
nmatrix1 = normalize(matrix1)
nmatrix2 = normalize(matrix2)
distances = np.arccos(
np.maximum(-1, np.minimum(1, dot(nmatrix1, nmatrix2.T))))
assert isinstance(distances, DenseMatrix)
assert distances.same_row_labels_as(nmatrix1)
return multiply(distances, distances)
