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# Author: Leland McInnes <leland.mcinnes@gmail.com>
#
# License: BSD 3 clause
from __future__ import print_function
from collections import deque, namedtuple
from warnings import warn
import numpy as np
import numba
from umap.sparse import arr_unique, sparse_mul, sparse_diff, sparse_sum
from umap.utils import tau_rand_int, norm
import scipy.sparse
import locale
locale.setlocale(locale.LC_NUMERIC, "C")
# Used for a floating point "nearly zero" comparison
EPS = 1e-8
RandomProjectionTreeNode = namedtuple(
"RandomProjectionTreeNode",
["indices", "is_leaf", "hyperplane", "offset", "left_child", "right_child"],
)
FlatTree = namedtuple("FlatTree", ["hyperplanes", "offsets", "children", "indices"])
@numba.njit(fastmath=True)
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
@numba.njit(fastmath=True, nogil=True)
def euclidean_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 euclidean 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]
# Compute the normal vector to the hyperplane (the vector between
# the two points) and the offset from the origin
hyperplane_offset = 0.0
hyperplane_vector = np.empty(dim, dtype=np.float32)
for d in range(dim):
hyperplane_vector[d] = data[left, d] - data[right, d]
hyperplane_offset -= (
hyperplane_vector[d] * (data[left, d] + data[right, d]) / 2.0
)
# 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
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, hyperplane_offset
@numba.njit(fastmath=True)
def sparse_angular_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]]
left_norm = norm(left_data)
right_norm = norm(right_data)
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)
normalized_left_data = left_data / left_norm
normalized_right_data = right_data / right_norm
hyperplane_inds, hyperplane_data = sparse_diff(
left_inds, normalized_left_data, right_inds, normalized_right_data
)
hyperplane_norm = norm(hyperplane_data)
if abs(hyperplane_norm) < EPS:
hyperplane_norm = 1.0
for d in range(hyperplane_data.shape[0]):
hyperplane_data[d] = hyperplane_data[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
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, None
@numba.njit(fastmath=True)
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 make_euclidean_tree(data, indices, rng_state, leaf_size=30):
if indices.shape[0] > leaf_size:
(
left_indices,
right_indices,
hyperplane,
offset,
) = euclidean_random_projection_split(data, indices, rng_state)
left_node = make_euclidean_tree(data, left_indices, rng_state, leaf_size)
right_node = make_euclidean_tree(data, right_indices, rng_state, leaf_size)
node = RandomProjectionTreeNode(
None, False, hyperplane, offset, left_node, right_node
)
else:
node = RandomProjectionTreeNode(indices, True, None, None, None, None)
return node
def make_angular_tree(data, indices, rng_state, leaf_size=30):
if indices.shape[0] > leaf_size:
(
left_indices,
right_indices,
hyperplane,
offset,
) = angular_random_projection_split(data, indices, rng_state)
left_node = make_angular_tree(data, left_indices, rng_state, leaf_size)
right_node = make_angular_tree(data, right_indices, rng_state, leaf_size)
node = RandomProjectionTreeNode(
None, False, hyperplane, offset, left_node, right_node
)
else:
node = RandomProjectionTreeNode(indices, True, None, None, None, None)
return node
def make_sparse_euclidean_tree(inds, indptr, data, indices, rng_state, leaf_size=30):
if indices.shape[0] > leaf_size:
(
left_indices,
right_indices,
hyperplane,
offset,
) = sparse_euclidean_random_projection_split(
inds, indptr, data, indices, rng_state
)
left_node = make_sparse_euclidean_tree(
inds, indptr, data, left_indices, rng_state, leaf_size
)
right_node = make_sparse_euclidean_tree(
inds, indptr, data, right_indices, rng_state, leaf_size
)
node = RandomProjectionTreeNode(
None, False, hyperplane, offset, left_node, right_node
)
else:
node = RandomProjectionTreeNode(indices, True, None, None, None, None)
return node
def make_sparse_angular_tree(inds, indptr, data, indices, rng_state, leaf_size=30):
if indices.shape[0] > leaf_size:
(
left_indices,
right_indices,
hyperplane,
offset,
) = sparse_angular_random_projection_split(
inds, indptr, data, indices, rng_state
)
left_node = make_sparse_angular_tree(
inds, indptr, data, left_indices, rng_state, leaf_size
)
right_node = make_sparse_angular_tree(
inds, indptr, data, right_indices, rng_state, leaf_size
)
node = RandomProjectionTreeNode(
None, False, hyperplane, offset, left_node, right_node
)
else:
node = RandomProjectionTreeNode(indices, True, None, None, None, None)
return node
def make_tree(data, rng_state, leaf_size=30, angular=False):
"""Construct a random projection tree based on ``data`` with leaves
of size at most ``leaf_size``.
Parameters
----------
data: array of shape (n_samples, n_features)
The original data to be split
rng_state: array of int64, shape (3,)
The internal state of the rng
leaf_size: int (optional, default 30)
The maximum size of any leaf node in the tree. Any node in the tree
with more than ``leaf_size`` will be split further to create child
nodes.
angular: bool (optional, default False)
Whether to use cosine/angular distance to create splits in the tree,
or euclidean distance.
Returns
-------
node: RandomProjectionTreeNode
A random projection tree node which links to its child nodes. This
provides the full tree below the returned node.
"""
is_sparse = scipy.sparse.isspmatrix_csr(data)
indices = np.arange(data.shape[0])
# Make a tree recursively until we get below the leaf size
if is_sparse:
inds = data.indices
indptr = data.indptr
spdata = data.data
if angular:
return make_sparse_angular_tree(
inds, indptr, spdata, indices, rng_state, leaf_size
)
else:
return make_sparse_euclidean_tree(
inds, indptr, spdata, indices, rng_state, leaf_size
)
else:
if angular:
return make_angular_tree(data, indices, rng_state, leaf_size)
else:
return make_euclidean_tree(data, indices, rng_state, leaf_size)
def num_nodes(tree):
"""Determine the number of nodes in a tree"""
if tree.is_leaf:
return 1
else:
return 1 + num_nodes(tree.left_child) + num_nodes(tree.right_child)
def num_leaves(tree):
"""Determine the number of leaves in a tree"""
if tree.is_leaf:
return 1
else:
return num_leaves(tree.left_child) + num_leaves(tree.right_child)
def max_sparse_hyperplane_size(tree):
"""Determine the most number on non zeros in a hyperplane entry"""
if tree.is_leaf:
return 0
else:
return max(
tree.hyperplane.shape[1],
max_sparse_hyperplane_size(tree.left_child),
max_sparse_hyperplane_size(tree.right_child),
)
def recursive_flatten(
tree, hyperplanes, offsets, children, indices, node_num, leaf_num
):
if tree.is_leaf:
children[node_num, 0] = -leaf_num
indices[leaf_num, : tree.indices.shape[0]] = tree.indices
leaf_num += 1
return node_num, leaf_num
else:
if len(tree.hyperplane.shape) > 1:
# sparse case
hyperplanes[node_num][:, : tree.hyperplane.shape[1]] = tree.hyperplane
else:
hyperplanes[node_num] = tree.hyperplane
offsets[node_num] = tree.offset
children[node_num, 0] = node_num + 1
old_node_num = node_num
node_num, leaf_num = recursive_flatten(
tree.left_child,
hyperplanes,
offsets,
children,
indices,
node_num + 1,
leaf_num,
)
children[old_node_num, 1] = node_num + 1
node_num, leaf_num = recursive_flatten(
tree.right_child,
hyperplanes,
offsets,
children,
indices,
node_num + 1,
leaf_num,
)
return node_num, leaf_num
def flatten_tree(tree, leaf_size):
n_nodes = num_nodes(tree)
n_leaves = num_leaves(tree)
if len(tree.hyperplane.shape) > 1:
# sparse case
max_hyperplane_nnz = max_sparse_hyperplane_size(tree)
hyperplanes = np.zeros(
(n_nodes, tree.hyperplane.shape[0], max_hyperplane_nnz), dtype=np.float32
)
else:
hyperplanes = np.zeros((n_nodes, tree.hyperplane.shape[0]), dtype=np.float32)
offsets = np.zeros(n_nodes, dtype=np.float32)
children = -1 * np.ones((n_nodes, 2), dtype=np.int64)
indices = -1 * np.ones((n_leaves, leaf_size), dtype=np.int64)
recursive_flatten(tree, hyperplanes, offsets, children, indices, 0, 0)
return FlatTree(hyperplanes, offsets, children, indices)
@numba.njit()
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
@numba.njit()
def search_flat_tree(point, hyperplanes, offsets, children, indices, rng_state):
node = 0
while children[node, 0] > 0:
side = select_side(hyperplanes[node], offsets[node], point, rng_state)
if side == 0:
node = children[node, 0]
else:
node = children[node, 1]
return indices[-children[node, 0]]
@numba.njit()
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
@numba.njit()
def search_sparse_flat_tree(
point_inds, point_data, hyperplanes, offsets, children, indices, rng_state
):
node = 0
while children[node, 0] > 0:
side = sparse_select_side(
hyperplanes[node], offsets[node], point_inds, point_data, rng_state
)
if side == 0:
node = children[node, 0]
else:
node = children[node, 1]
return indices[-children[node, 0]]
def make_forest(data, n_neighbors, n_trees, rng_state, angular=False):
"""Build a random projection forest with ``n_trees``.
Parameters
----------
data
n_neighbors
n_trees
rng_state
angular
Returns
-------
forest: list
A list of random projection trees.
"""
result = []
leaf_size = max(10, n_neighbors)
try:
result = [
flatten_tree(make_tree(data, rng_state, leaf_size, angular), leaf_size)
for i in range(n_trees)
]
except (RuntimeError, RecursionError, SystemError):
warn(
"Random Projection forest initialisation failed due to recursion"
"limit being reached. Something is a little strange with your "
"data, and this may take longer than normal to compute."
)
return result
def rptree_leaf_array(rp_forest):
"""Generate an array of sets of candidate nearest neighbors by
constructing a random projection forest and taking the leaves of all the
trees. Any given tree has leaves that are a set of potential nearest
neighbors. Given enough trees the set of all such leaves gives a good
likelihood of getting a good set of nearest neighbors in composite. Since
such a random projection forest is inexpensive to compute, this can be a
useful means of seeding other nearest neighbor algorithms.
Parameters
----------
data: array of shape (n_samples, n_features)
The data for which to generate nearest neighbor approximations.
n_neighbors: int
The number of nearest neighbors to attempt to approximate.
rng_state: array of int64, shape (3,)
The internal state of the rng
n_trees: int (optional, default 10)
The number of trees to build in the forest construction.
angular: bool (optional, default False)
Whether to use angular/cosine distance for random projection tree
construction.
Returns
-------
leaf_array: array of shape (n_leaves, max(10, n_neighbors))
Each row of leaf array is a list of indices found in a given leaf.
Since not all leaves are the same size the arrays are padded out with -1
to ensure we can return a single ndarray.
"""
if len(rp_forest) > 0:
leaf_array = np.vstack([tree.indices for tree in rp_forest])
else:
leaf_array = np.array([[-1]])
return leaf_array
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