def select_side(hyperplane, offset, point, rng_state):
margin = offset
for d in range(point.shape[0]):
margin += hyperplane[d] * point[d]
if abs(margin) < EPS:
side = abs(tau_rand_int(rng_state)) % 2
if side == 0:
return 0
else:
return 1
elif margin > 0:
return 0
else:
return 1
def sparse_select_side(hyperplane, offset, point_inds, point_data, rng_state):
margin = offset
hyperplane_inds = arr_unique(hyperplane[0])
hyperplane_data = hyperplane[1, : hyperplane_inds.shape[0]]
aux_inds, aux_data = sparse_mul(
hyperplane_inds, hyperplane_data, point_inds, point_data
)
for d in range(aux_data.shape[0]):
margin += aux_data[d]
if margin == 0:
side = abs(tau_rand_int(rng_state)) % 2
if side == 0:
return 0
else:
return 1
elif margin > 0:
return 0
else:
return 1
def _optimize_layout_euclidean_single_epoch(
head_embedding,
tail_embedding,
head,
tail,
n_vertices,
epochs_per_sample,
a,
b,
rng_state,
gamma,
dim,
move_other,
alpha,
epochs_per_negative_sample,
epoch_of_next_negative_sample,
epoch_of_next_sample,
n,
):
for i in numba.prange(epochs_per_sample.shape[0]):
if epoch_of_next_sample[i] <= n:
j = head[i]
k = tail[i]
current = head_embedding[j]
other = tail_embedding[k]
dist_squared = rdist(current, other)
if dist_squared > 0.0:
grad_coeff = -2.0 * a * b * pow(dist_squared, b - 1.0)
grad_coeff /= a * pow(dist_squared, b) + 1.0
else:
grad_coeff = 0.0
for d in range(dim):
grad_d = clip(grad_coeff * (current[d] - other[d]))
current[d] += grad_d * alpha
if move_other:
other[d] += -grad_d * alpha
epoch_of_next_sample[i] += epochs_per_sample[i]
n_neg_samples = int(
(n - epoch_of_next_negative_sample[i]) / epochs_per_negative_sample[i]
)
for p in range(n_neg_samples):
k = tau_rand_int(rng_state) % n_vertices
other = tail_embedding[k]
dist_squared = rdist(current, other)
if dist_squared > 0.0:
grad_coeff = 2.0 * gamma * b
grad_coeff /= (0.001 + dist_squared) * (
a * pow(dist_squared, b) + 1
)
elif j == k:
continue
else:
grad_coeff = 0.0
for d in range(dim):
if grad_coeff > 0.0:
grad_d = clip(grad_coeff * (current[d] - other[d]))
else:
grad_d = 4.0
current[d] += grad_d * alpha
epoch_of_next_negative_sample[i] += (
n_neg_samples * epochs_per_negative_sample[i]
)
def optimize_layout_inverse(
head_embedding,
tail_embedding,
head,
tail,
weight,
sigmas,
rhos,
n_epochs,
n_vertices,
epochs_per_sample,
a,
b,
rng_state,
gamma=1.0,
initial_alpha=1.0,
negative_sample_rate=5.0,
output_metric=dist.euclidean,
output_metric_kwds=(),
verbose=False,
):
"""Improve an embedding using stochastic gradient descent to minimize the
fuzzy set cross entropy between the 1-skeletons of the high dimensional
and low dimensional fuzzy simplicial sets. In practice this is done by
sampling edges based on their membership strength (with the (1-p) terms
coming from negative sampling similar to word2vec).
Parameters
----------
head_embedding: array of shape (n_samples, n_components)
The initial embedding to be improved by SGD.
tail_embedding: array of shape (source_samples, n_components)
The reference embedding of embedded points. If not embedding new
previously unseen points with respect to an existing embedding this
is simply the head_embedding (again); otherwise it provides the
existing embedding to embed with respect to.
head: array of shape (n_1_simplices)
The indices of the heads of 1-simplices with non-zero membership.
tail: array of shape (n_1_simplices)
The indices of the tails of 1-simplices with non-zero membership.
weight: array of shape (n_1_simplices)
The membership weights of the 1-simplices.
n_epochs: int
The number of training epochs to use in optimization.
n_vertices: int
The number of vertices (0-simplices) in the dataset.
epochs_per_sample: array of shape (n_1_simplices)
A float value of the number of epochs per 1-simplex. 1-simplices with
weaker membership strength will have more epochs between being sampled.
a: float
Parameter of differentiable approximation of right adjoint functor
b: float
Parameter of differentiable approximation of right adjoint functor
rng_state: array of int64, shape (3,)
The internal state of the rng
gamma: float (optional, default 1.0)
Weight to apply to negative samples.
initial_alpha: float (optional, default 1.0)
Initial learning rate for the SGD.
negative_sample_rate: int (optional, default 5)
Number of negative samples to use per positive sample.
verbose: bool (optional, default False)
Whether to report information on the current progress of the algorithm.
Returns
-------
embedding: array of shape (n_samples, n_components)
The optimized embedding.
"""
dim = head_embedding.shape[1]
move_other = head_embedding.shape[0] == tail_embedding.shape[0]
alpha = initial_alpha
epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
epoch_of_next_sample = epochs_per_sample.copy()
for n in range(n_epochs):
for i in range(epochs_per_sample.shape[0]):
if epoch_of_next_sample[i] <= n:
j = head[i]
k = tail[i]
current = head_embedding[j]
other = tail_embedding[k]
dist_output, grad_dist_output = output_metric(
current, other, *output_metric_kwds
)
w_l = weight[i]
grad_coeff = -(1 / (w_l * sigmas[k] + 1e-6))
for d in range(dim):
grad_d = clip(grad_coeff * grad_dist_output[d])
current[d] += grad_d * alpha
if move_other:
other[d] += -grad_d * alpha
epoch_of_next_sample[i] += epochs_per_sample[i]
n_neg_samples = int(
(n - epoch_of_next_negative_sample[i])
/ epochs_per_negative_sample[i]
)
for p in range(n_neg_samples):
k = tau_rand_int(rng_state) % n_vertices
other = tail_embedding[k]
dist_output, grad_dist_output = output_metric(
current, other, *output_metric_kwds
)
# w_l = 0.0 # for negative samples, the edge does not exist
w_h = np.exp(-max(dist_output - rhos[k], 1e-6) / (sigmas[k] + 1e-6))
grad_coeff = -gamma * ((0 - w_h) / ((1 - w_h) * sigmas[k] + 1e-6))
for d in range(dim):
grad_d = clip(grad_coeff * grad_dist_output[d])
current[d] += grad_d * alpha
epoch_of_next_negative_sample[i] += (
n_neg_samples * epochs_per_negative_sample[i]
)
alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
if verbose and n % int(n_epochs / 10) == 0:
print("\tcompleted ", n, " / ", n_epochs, "epochs")
return head_embedding
def sparse_euclidean_random_projection_split(inds, indptr, data, indices, rng_state):
"""Given a set of ``indices`` for data points from a sparse data set
presented in csr sparse format as inds, indptr and data, create
a random hyperplane to split the data, returning two arrays indices
that fall on either side of the hyperplane. This is the basis for a
random projection tree, which simply uses this splitting recursively.
This particular split uses cosine distance to determine the hyperplane
and which side each data sample falls on.
Parameters
----------
inds: array
CSR format index array of the matrix
indptr: array
CSR format index pointer array of the matrix
data: array
CSR format data array of the matrix
indices: array of shape (tree_node_size,)
The indices of the elements in the ``data`` array that are to
be split in the current operation.
rng_state: array of int64, shape (3,)
The internal state of the rng
Returns
-------
indices_left: array
The elements of ``indices`` that fall on the "left" side of the
random hyperplane.
indices_right: array
The elements of ``indices`` that fall on the "left" side of the
random hyperplane.
"""
# Select two random points, set the hyperplane between them
left_index = tau_rand_int(rng_state) % indices.shape[0]
right_index = tau_rand_int(rng_state) % indices.shape[0]
right_index += left_index == right_index
right_index = right_index % indices.shape[0]
left = indices[left_index]
right = indices[right_index]
left_inds = inds[indptr[left] : indptr[left + 1]]
left_data = data[indptr[left] : indptr[left + 1]]
right_inds = inds[indptr[right] : indptr[right + 1]]
right_data = data[indptr[right] : indptr[right + 1]]
# Compute the normal vector to the hyperplane (the vector between
# the two points) and the offset from the origin
hyperplane_offset = 0.0
hyperplane_inds, hyperplane_data = sparse_diff(
left_inds, left_data, right_inds, right_data
)
offset_inds, offset_data = sparse_sum(left_inds, left_data, right_inds, right_data)
offset_data = offset_data / 2.0
offset_inds, offset_data = sparse_mul(
hyperplane_inds, hyperplane_data, offset_inds, offset_data
)
for d in range(offset_data.shape[0]):
hyperplane_offset -= offset_data[d]
# For each point compute the margin (project into normal vector, add offset)
# If we are on lower side of the hyperplane put in one pile, otherwise
# put it in the other pile (if we hit hyperplane on the nose, flip a coin)
n_left = 0
n_right = 0
side = np.empty(indices.shape[0], np.int8)
for i in range(indices.shape[0]):
margin = hyperplane_offset
i_inds = inds[indptr[indices[i]] : indptr[indices[i] + 1]]
i_data = data[indptr[indices[i]] : indptr[indices[i] + 1]]
mul_inds, mul_data = sparse_mul(
hyperplane_inds, hyperplane_data, i_inds, i_data
)
for d in range(mul_data.shape[0]):
margin += mul_data[d]
if abs(margin) < EPS:
side[i] = abs(tau_rand_int(rng_state)) % 2
if side[i] == 0:
n_left += 1
else:
n_right += 1
elif margin > 0:
side[i] = 0
n_left += 1
else:
side[i] = 1
n_right += 1
# Now that we have the counts allocate arrays
indices_left = np.empty(n_left, dtype=np.int64)
indices_right = np.empty(n_right, dtype=np.int64)
# Populate the arrays with indices according to which side they fell on
n_left = 0
n_right = 0
for i in range(side.shape[0]):
if side[i] == 0:
indices_left[n_left] = indices[i]
n_left += 1
else:
indices_right[n_right] = indices[i]
n_right += 1
hyperplane = np.vstack((hyperplane_inds, hyperplane_data))
return indices_left, indices_right, hyperplane, hyperplane_offset
def angular_random_projection_split(data, indices, rng_state):
"""Given a set of ``indices`` for data points from ``data``, create
a random hyperplane to split the data, returning two arrays indices
that fall on either side of the hyperplane. This is the basis for a
random projection tree, which simply uses this splitting recursively.
This particular split uses cosine distance to determine the hyperplane
and which side each data sample falls on.
Parameters
----------
data: array of shape (n_samples, n_features)
The original data to be split
indices: array of shape (tree_node_size,)
The indices of the elements in the ``data`` array that are to
be split in the current operation.
rng_state: array of int64, shape (3,)
The internal state of the rng
Returns
-------
indices_left: array
The elements of ``indices`` that fall on the "left" side of the
random hyperplane.
indices_right: array
The elements of ``indices`` that fall on the "left" side of the
random hyperplane.
"""
dim = data.shape[1]
# Select two random points, set the hyperplane between them
left_index = tau_rand_int(rng_state) % indices.shape[0]
right_index = tau_rand_int(rng_state) % indices.shape[0]
right_index += left_index == right_index
right_index = right_index % indices.shape[0]
left = indices[left_index]
right = indices[right_index]
left_norm = norm(data[left])
right_norm = norm(data[right])
if abs(left_norm) < EPS:
left_norm = 1.0
if abs(right_norm) < EPS:
right_norm = 1.0
# Compute the normal vector to the hyperplane (the vector between
# the two points)
hyperplane_vector = np.empty(dim, dtype=np.float32)
for d in range(dim):
hyperplane_vector[d] = (data[left, d] / left_norm) - (
data[right, d] / right_norm
)
hyperplane_norm = norm(hyperplane_vector)
if abs(hyperplane_norm) < EPS:
hyperplane_norm = 1.0
for d in range(dim):
hyperplane_vector[d] = hyperplane_vector[d] / hyperplane_norm
# For each point compute the margin (project into normal vector)
# If we are on lower side of the hyperplane put in one pile, otherwise
# put it in the other pile (if we hit hyperplane on the nose, flip a coin)
n_left = 0
n_right = 0
side = np.empty(indices.shape[0], np.int8)
for i in range(indices.shape[0]):
margin = 0.0
for d in range(dim):
margin += hyperplane_vector[d] * data[indices[i], d]
if abs(margin) < EPS:
side[i] = abs(tau_rand_int(rng_state)) % 2
if side[i] == 0:
n_left += 1
else:
n_right += 1
elif margin > 0:
side[i] = 0
n_left += 1
else:
side[i] = 1
n_right += 1
# Now that we have the counts allocate arrays
indices_left = np.empty(n_left, dtype=np.int64)
indices_right = np.empty(n_right, dtype=np.int64)
# Populate the arrays with indices according to which side they fell on
n_left = 0
n_right = 0
for i in range(side.shape[0]):
if side[i] == 0:
indices_left[n_left] = indices[i]
n_left += 1
else:
indices_right[n_right] = indices[i]
n_right += 1
return indices_left, indices_right, hyperplane_vector, None