Source code for umap.umap_

# Author: Leland McInnes <leland.mcinnes@gmail.com>
#
# License: BSD 3 clause
from __future__ import print_function
from warnings import warn
import time

from scipy.optimize import curve_fit
from sklearn.base import BaseEstimator
from sklearn.utils import check_random_state, check_array
from sklearn.metrics import pairwise_distances
from sklearn.preprocessing import normalize
from sklearn.neighbors import KDTree

try:
    import joblib
except ImportError:
    # sklearn.externals.joblib is deprecated in 0.21, will be removed in 0.23
    from sklearn.externals import joblib

import numpy as np
import scipy.sparse
import scipy.sparse.csgraph
import numba

import umap.distances as dist

import umap.sparse as sparse

from umap.utils import (
    tau_rand_int,
    deheap_sort,
    submatrix,
    ts,
    fast_knn_indices,
)
from umap.rp_tree import rptree_leaf_array, make_forest
from umap.nndescent import (
    make_nn_descent,
    make_initialisations,
    make_initialized_nnd_search,
    initialise_search,
)
from umap.spectral import spectral_layout

import locale

locale.setlocale(locale.LC_NUMERIC, "C")

INT32_MIN = np.iinfo(np.int32).min + 1
INT32_MAX = np.iinfo(np.int32).max - 1

SMOOTH_K_TOLERANCE = 1e-5
MIN_K_DIST_SCALE = 1e-3
NPY_INFINITY = np.inf


[docs]@numba.njit( fastmath=True ) # benchmarking `parallel=True` shows it to *decrease* performance def smooth_knn_dist( distances, k, n_iter=64, local_connectivity=1.0, bandwidth=1.0, ): """Compute a continuous version of the distance to the kth nearest neighbor. That is, this is similar to knn-distance but allows continuous k values rather than requiring an integral k. In essence we are simply computing the distance such that the cardinality of fuzzy set we generate is k. Parameters ---------- distances: array of shape (n_samples, n_neighbors) Distances to nearest neighbors for each samples. Each row should be a sorted list of distances to a given samples nearest neighbors. k: float The number of nearest neighbors to approximate for. n_iter: int (optional, default 64) We need to binary search for the correct distance value. This is the max number of iterations to use in such a search. local_connectivity: int (optional, default 1) The local connectivity required -- i.e. the number of nearest neighbors that should be assumed to be connected at a local level. The higher this value the more connected the manifold becomes locally. In practice this should be not more than the local intrinsic dimension of the manifold. bandwidth: float (optional, default 1) The target bandwidth of the kernel, larger values will produce larger return values. Returns ------- knn_dist: array of shape (n_samples,) The distance to kth nearest neighbor, as suitably approximated. nn_dist: array of shape (n_samples,) The distance to the 1st nearest neighbor for each point. """ target = np.log2(k) * bandwidth rho = np.zeros(distances.shape[0]) result = np.zeros(distances.shape[0]) mean_distances = np.mean(distances) for i in range(distances.shape[0]): lo = 0.0 hi = NPY_INFINITY mid = 1.0 # TODO: This is very inefficient, but will do for now. FIXME ith_distances = distances[i] non_zero_dists = ith_distances[ith_distances > 0.0] if non_zero_dists.shape[0] >= local_connectivity: index = int(np.floor(local_connectivity)) interpolation = local_connectivity - index if index > 0: rho[i] = non_zero_dists[index - 1] if interpolation > SMOOTH_K_TOLERANCE: rho[i] += interpolation * ( non_zero_dists[index] - non_zero_dists[index - 1] ) else: rho[i] = interpolation * non_zero_dists[0] elif non_zero_dists.shape[0] > 0: rho[i] = np.max(non_zero_dists) for n in range(n_iter): psum = 0.0 for j in range(1, distances.shape[1]): d = distances[i, j] - rho[i] if d > 0: psum += np.exp(-(d / mid)) else: psum += 1.0 if np.fabs(psum - target) < SMOOTH_K_TOLERANCE: break if psum > target: hi = mid mid = (lo + hi) / 2.0 else: lo = mid if hi == NPY_INFINITY: mid *= 2 else: mid = (lo + hi) / 2.0 result[i] = mid # TODO: This is very inefficient, but will do for now. FIXME if rho[i] > 0.0: mean_ith_distances = np.mean(ith_distances) if ( result[i] < MIN_K_DIST_SCALE * mean_ith_distances ): result[i] = ( MIN_K_DIST_SCALE * mean_ith_distances ) else: if ( result[i] < MIN_K_DIST_SCALE * mean_distances ): result[i] = ( MIN_K_DIST_SCALE * mean_distances ) return result, rho
[docs]def nearest_neighbors( X, n_neighbors, metric, metric_kwds, angular, random_state, verbose=False, ): """Compute the ``n_neighbors`` nearest points for each data point in ``X`` under ``metric``. This may be exact, but more likely is approximated via nearest neighbor descent. Parameters ---------- X: array of shape (n_samples, n_features) The input data to compute the k-neighbor graph of. n_neighbors: int The number of nearest neighbors to compute for each sample in ``X``. metric: string or callable The metric to use for the computation. metric_kwds: dict Any arguments to pass to the metric computation function. angular: bool Whether to use angular rp trees in NN approximation. random_state: np.random state The random state to use for approximate NN computations. verbose: bool Whether to print status data during the computation. Returns ------- knn_indices: array of shape (n_samples, n_neighbors) The indices on the ``n_neighbors`` closest points in the dataset. knn_dists: array of shape (n_samples, n_neighbors) The distances to the ``n_neighbors`` closest points in the dataset. """ if verbose: print(ts(), "Finding Nearest Neighbors") if metric == "precomputed": # Note that this does not support sparse distance matrices yet ... # Compute indices of n nearest neighbors knn_indices = fast_knn_indices(X, n_neighbors) # Compute the nearest neighbor distances # (equivalent to np.sort(X)[:,:n_neighbors]) knn_dists = X[ np.arange(X.shape[0])[:, None], knn_indices ].copy() rp_forest = [] else: if callable(metric): distance_func = metric elif metric in dist.named_distances: distance_func = dist.named_distances[metric] else: raise ValueError( "Metric is neither callable, " + "nor a recognised string" ) if metric in ( "cosine", "correlation", "dice", "jaccard", ): angular = True rng_state = random_state.randint( INT32_MIN, INT32_MAX, 3 ).astype(np.int64) if scipy.sparse.isspmatrix_csr(X): if metric in sparse.sparse_named_distances: distance_func = sparse.sparse_named_distances[ metric ] if metric in sparse.sparse_need_n_features: metric_kwds["n_features"] = X.shape[1] else: raise ValueError( "Metric {} not supported for sparse " + "data".format(metric) ) metric_nn_descent = sparse.make_sparse_nn_descent( distance_func, tuple(metric_kwds.values()) ) # TODO: Hacked values for now n_trees = 5 + int( round((X.shape[0]) ** 0.5 / 20.0) ) n_iters = max( 5, int(round(np.log2(X.shape[0]))) ) if verbose: print( ts(), "Building RP forest with", str(n_trees), "trees", ) rp_forest = make_forest( X, n_neighbors, n_trees, rng_state, angular ) leaf_array = rptree_leaf_array(rp_forest) if verbose: print( ts(), "NN descent for", str(n_iters), "iterations", ) knn_indices, knn_dists = metric_nn_descent( X.indices, X.indptr, X.data, X.shape[0], n_neighbors, rng_state, max_candidates=60, rp_tree_init=True, leaf_array=leaf_array, n_iters=n_iters, verbose=verbose, ) else: metric_nn_descent = make_nn_descent( distance_func, tuple(metric_kwds.values()) ) # TODO: Hacked values for now n_trees = 5 + int( round((X.shape[0]) ** 0.5 / 20.0) ) n_iters = max( 5, int(round(np.log2(X.shape[0]))) ) if verbose: print( ts(), "Building RP forest with", str(n_trees), "trees", ) rp_forest = make_forest( X, n_neighbors, n_trees, rng_state, angular ) leaf_array = rptree_leaf_array(rp_forest) if verbose: print( ts(), "NN descent for", str(n_iters), "iterations", ) knn_indices, knn_dists = metric_nn_descent( X, n_neighbors, rng_state, max_candidates=60, rp_tree_init=True, leaf_array=leaf_array, n_iters=n_iters, verbose=verbose, ) if np.any(knn_indices < 0): warn( "Failed to correctly find n_neighbors for some samples." "Results may be less than ideal. Try re-running with" "different parameters." ) if verbose: print(ts(), "Finished Nearest Neighbor Search") return knn_indices, knn_dists, rp_forest
[docs]@numba.njit(parallel=True, fastmath=True) def compute_membership_strengths( knn_indices, knn_dists, sigmas, rhos ): """Construct the membership strength data for the 1-skeleton of each local fuzzy simplicial set -- this is formed as a sparse matrix where each row is a local fuzzy simplicial set, with a membership strength for the 1-simplex to each other data point. Parameters ---------- knn_indices: array of shape (n_samples, n_neighbors) The indices on the ``n_neighbors`` closest points in the dataset. knn_dists: array of shape (n_samples, n_neighbors) The distances to the ``n_neighbors`` closest points in the dataset. sigmas: array of shape(n_samples) The normalization factor derived from the metric tensor approximation. rhos: array of shape(n_samples) The local connectivity adjustment. Returns ------- rows: array of shape (n_samples * n_neighbors) Row data for the resulting sparse matrix (coo format) cols: array of shape (n_samples * n_neighbors) Column data for the resulting sparse matrix (coo format) vals: array of shape (n_samples * n_neighbors) Entries for the resulting sparse matrix (coo format) """ n_samples = knn_indices.shape[0] n_neighbors = knn_indices.shape[1] rows = np.zeros(knn_indices.size, dtype=np.int64) cols = np.zeros(knn_indices.size, dtype=np.int64) vals = np.zeros(knn_indices.size, dtype=np.float64) for i in range(n_samples): for j in range(n_neighbors): if knn_indices[i, j] == -1: continue # We didn't get the full knn for i if knn_indices[i, j] == i: val = 0.0 elif knn_dists[i, j] - rhos[i] <= 0.0: val = 1.0 else: val = np.exp( -( (knn_dists[i, j] - rhos[i]) / (sigmas[i]) ) ) rows[i * n_neighbors + j] = i cols[i * n_neighbors + j] = knn_indices[i, j] vals[i * n_neighbors + j] = val return rows, cols, vals
[docs]def fuzzy_simplicial_set( X, n_neighbors, random_state, metric, metric_kwds={}, knn_indices=None, knn_dists=None, angular=False, set_op_mix_ratio=1.0, local_connectivity=1.0, verbose=False, ): """Given a set of data X, a neighborhood size, and a measure of distance compute the fuzzy simplicial set (here represented as a fuzzy graph in the form of a sparse matrix) associated to the data. This is done by locally approximating geodesic distance at each point, creating a fuzzy simplicial set for each such point, and then combining all the local fuzzy simplicial sets into a global one via a fuzzy union. Parameters ---------- X: array of shape (n_samples, n_features) The data to be modelled as a fuzzy simplicial set. n_neighbors: int The number of neighbors to use to approximate geodesic distance. Larger numbers induce more global estimates of the manifold that can miss finer detail, while smaller values will focus on fine manifold structure to the detriment of the larger picture. random_state: numpy RandomState or equivalent A state capable being used as a numpy random state. metric: string or function (optional, default 'euclidean') The metric to use to compute distances in high dimensional space. If a string is passed it must match a valid predefined metric. If a general metric is required a function that takes two 1d arrays and returns a float can be provided. For performance purposes it is required that this be a numba jit'd function. Valid string metrics include: * euclidean (or l2) * manhattan (or l1) * cityblock * braycurtis * canberra * chebyshev * correlation * cosine * dice * hamming * jaccard * kulsinski * mahalanobis * matching * minkowski * rogerstanimoto * russellrao * seuclidean * sokalmichener * sokalsneath * sqeuclidean * yule * wminkowski Metrics that take arguments (such as minkowski, mahalanobis etc.) can have arguments passed via the metric_kwds dictionary. At this time care must be taken and dictionary elements must be ordered appropriately; this will hopefully be fixed in the future. metric_kwds: dict (optional, default {}) Arguments to pass on to the metric, such as the ``p`` value for Minkowski distance. knn_indices: array of shape (n_samples, n_neighbors) (optional) If the k-nearest neighbors of each point has already been calculated you can pass them in here to save computation time. This should be an array with the indices of the k-nearest neighbors as a row for each data point. knn_dists: array of shape (n_samples, n_neighbors) (optional) If the k-nearest neighbors of each point has already been calculated you can pass them in here to save computation time. This should be an array with the distances of the k-nearest neighbors as a row for each data point. angular: bool (optional, default False) Whether to use angular/cosine distance for the random projection forest for seeding NN-descent to determine approximate nearest neighbors. set_op_mix_ratio: float (optional, default 1.0) Interpolate between (fuzzy) union and intersection as the set operation used to combine local fuzzy simplicial sets to obtain a global fuzzy simplicial sets. Both fuzzy set operations use the product t-norm. The value of this parameter should be between 0.0 and 1.0; a value of 1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy intersection. local_connectivity: int (optional, default 1) The local connectivity required -- i.e. the number of nearest neighbors that should be assumed to be connected at a local level. The higher this value the more connected the manifold becomes locally. In practice this should be not more than the local intrinsic dimension of the manifold. verbose: bool (optional, default False) Whether to report information on the current progress of the algorithm. Returns ------- fuzzy_simplicial_set: coo_matrix A fuzzy simplicial set represented as a sparse matrix. The (i, j) entry of the matrix represents the membership strength of the 1-simplex between the ith and jth sample points. """ if knn_indices is None or knn_dists is None: knn_indices, knn_dists, _ = nearest_neighbors( X, n_neighbors, metric, metric_kwds, angular, random_state, verbose=verbose, ) sigmas, rhos = smooth_knn_dist( knn_dists, n_neighbors, local_connectivity=local_connectivity, ) rows, cols, vals = compute_membership_strengths( knn_indices, knn_dists, sigmas, rhos ) result = scipy.sparse.coo_matrix( (vals, (rows, cols)), shape=(X.shape[0], X.shape[0]) ) result.eliminate_zeros() transpose = result.transpose() prod_matrix = result.multiply(transpose) result = ( set_op_mix_ratio * (result + transpose - prod_matrix) + (1.0 - set_op_mix_ratio) * prod_matrix ) result.eliminate_zeros() return result
[docs]@numba.njit() def fast_intersection( rows, cols, values, target, unknown_dist=1.0, far_dist=5.0, ): """Under the assumption of categorical distance for the intersecting simplicial set perform a fast intersection. Parameters ---------- rows: array An array of the row of each non-zero in the sparse matrix representation. cols: array An array of the column of each non-zero in the sparse matrix representation. values: array An array of the value of each non-zero in the sparse matrix representation. target: array of shape (n_samples) The categorical labels to use in the intersection. unknown_dist: float (optional, default 1.0) The distance an unknown label (-1) is assumed to be from any point. far_dist float (optional, default 5.0) The distance between unmatched labels. Returns ------- None """ for nz in range(rows.shape[0]): i = rows[nz] j = cols[nz] if (target[i] == -1) or (target[j] == -1): values[nz] *= np.exp(-unknown_dist) elif target[i] != target[j]: values[nz] *= np.exp(-far_dist) return
[docs]def reset_local_connectivity(simplicial_set): """Reset the local connectivity requirement -- each data sample should have complete confidence in at least one 1-simplex in the simplicial set. We can enforce this by locally rescaling confidences, and then remerging the different local simplicial sets together. Parameters ---------- simplicial_set: sparse matrix The simplicial set for which to recalculate with respect to local connectivity. Returns ------- simplicial_set: sparse_matrix The recalculated simplicial set, now with the local connectivity assumption restored. """ simplicial_set = normalize(simplicial_set, norm="max") transpose = simplicial_set.transpose() prod_matrix = simplicial_set.multiply(transpose) simplicial_set = ( simplicial_set + transpose - prod_matrix ) simplicial_set.eliminate_zeros() return simplicial_set
[docs]def categorical_simplicial_set_intersection( simplicial_set, target, unknown_dist=1.0, far_dist=5.0 ): """Combine a fuzzy simplicial set with another fuzzy simplicial set generated from categorical data using categorical distances. The target data is assumed to be categorical label data (a vector of labels), and this will update the fuzzy simplicial set to respect that label data. TODO: optional category cardinality based weighting of distance Parameters ---------- simplicial_set: sparse matrix The input fuzzy simplicial set. target: array of shape (n_samples) The categorical labels to use in the intersection. unknown_dist: float (optional, default 1.0) The distance an unknown label (-1) is assumed to be from any point. far_dist float (optional, default 5.0) The distance between unmatched labels. Returns ------- simplicial_set: sparse matrix The resulting intersected fuzzy simplicial set. """ simplicial_set = simplicial_set.tocoo() fast_intersection( simplicial_set.row, simplicial_set.col, simplicial_set.data, target, unknown_dist, far_dist, ) simplicial_set.eliminate_zeros() return reset_local_connectivity(simplicial_set)
def general_simplicial_set_intersection( simplicial_set1, simplicial_set2, weight ): result = (simplicial_set1 + simplicial_set2).tocoo() left = simplicial_set1.tocsr() right = simplicial_set2.tocsr() sparse.general_sset_intersection( left.indptr, left.indices, left.data, right.indptr, right.indices, right.data, result.row, result.col, result.data, weight, ) return result
[docs]def make_epochs_per_sample(weights, n_epochs): """Given a set of weights and number of epochs generate the number of epochs per sample for each weight. Parameters ---------- weights: array of shape (n_1_simplices) The weights ofhow much we wish to sample each 1-simplex. n_epochs: int The total number of epochs we want to train for. Returns ------- An array of number of epochs per sample, one for each 1-simplex. """ result = -1.0 * np.ones( weights.shape[0], dtype=np.float64 ) n_samples = n_epochs * (weights / weights.max()) result[n_samples > 0] = ( float(n_epochs) / n_samples[n_samples > 0] ) return result
[docs]@numba.njit() def clip(val): """Standard clamping of a value into a fixed range (in this case -4.0 to 4.0) Parameters ---------- val: float The value to be clamped. Returns ------- The clamped value, now fixed to be in the range -4.0 to 4.0. """ if val > 4.0: return 4.0 elif val < -4.0: return -4.0 else: return val
[docs]@numba.njit("f4(f4[:],f4[:])", fastmath=True) def rdist(x, y): """Reduced Euclidean distance. Parameters ---------- x: array of shape (embedding_dim,) y: array of shape (embedding_dim,) Returns ------- The squared euclidean distance between x and y """ result = 0.0 for i in range(x.shape[0]): result += (x[i] - y[i]) ** 2 return result
[docs]@numba.njit(fastmath=True, parallel=True) def optimize_layout( head_embedding, tail_embedding, head, tail, n_epochs, n_vertices, epochs_per_sample, a, b, rng_state, gamma=1.0, initial_alpha=1.0, negative_sample_rate=5.0, 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. 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_samples: 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_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] ) 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
[docs]def simplicial_set_embedding( data, graph, n_components, initial_alpha, a, b, gamma, negative_sample_rate, n_epochs, init, random_state, metric, metric_kwds, verbose, ): """Perform a fuzzy simplicial set embedding, using a specified initialisation method and then minimizing the fuzzy set cross entropy between the 1-skeletons of the high and low dimensional fuzzy simplicial sets. Parameters ---------- data: array of shape (n_samples, n_features) The source data to be embedded by UMAP. graph: sparse matrix The 1-skeleton of the high dimensional fuzzy simplicial set as represented by a graph for which we require a sparse matrix for the (weighted) adjacency matrix. n_components: int The dimensionality of the euclidean space into which to embed the data. initial_alpha: float Initial learning rate for the SGD. a: float Parameter of differentiable approximation of right adjoint functor b: float Parameter of differentiable approximation of right adjoint functor gamma: float Weight to apply to negative samples. negative_sample_rate: int (optional, default 5) The number of negative samples to select per positive sample in the optimization process. Increasing this value will result in greater repulsive force being applied, greater optimization cost, but slightly more accuracy. n_epochs: int (optional, default 0) The number of training epochs to be used in optimizing the low dimensional embedding. Larger values result in more accurate embeddings. If 0 is specified a value will be selected based on the size of the input dataset (200 for large datasets, 500 for small). init: string How to initialize the low dimensional embedding. Options are: * 'spectral': use a spectral embedding of the fuzzy 1-skeleton * 'random': assign initial embedding positions at random. * A numpy array of initial embedding positions. random_state: numpy RandomState or equivalent A state capable being used as a numpy random state. metric: string The metric used to measure distance in high dimensional space; used if multiple connected components need to be layed out. metric_kwds: dict Key word arguments to be passed to the metric function; used if multiple connected components need to be layed out. 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 of ``graph`` into an ``n_components`` dimensional euclidean space. """ graph = graph.tocoo() graph.sum_duplicates() n_vertices = graph.shape[1] if n_epochs <= 0: # For smaller datasets we can use more epochs if graph.shape[0] <= 10000: n_epochs = 500 else: n_epochs = 200 graph.data[ graph.data < (graph.data.max() / float(n_epochs)) ] = 0.0 graph.eliminate_zeros() if isinstance(init, str) and init == "random": embedding = random_state.uniform( low=-10.0, high=10.0, size=(graph.shape[0], n_components), ).astype(np.float32) elif isinstance(init, str) and init == "spectral": # We add a little noise to avoid local minima for optimization to come initialisation = spectral_layout( data, graph, n_components, random_state, metric=metric, metric_kwds=metric_kwds, ) expansion = 10.0 / np.abs(initialisation).max() embedding = (initialisation * expansion).astype( np.float32 ) + random_state.normal( scale=0.0001, size=[graph.shape[0], n_components], ).astype( np.float32 ) else: init_data = np.array(init) if len(init_data.shape) == 2: if ( np.unique(init_data, axis=0).shape[0] < init_data.shape[0] ): tree = KDTree(init_data) dist, ind = tree.query(init_data, k=2) nndist = np.mean(dist[:, 1]) embedding = init_data + random_state.normal( scale=0.001 * nndist, size=init_data.shape, ).astype(np.float32) else: embedding = init_data epochs_per_sample = make_epochs_per_sample( graph.data, n_epochs ) head = graph.row tail = graph.col rng_state = random_state.randint( INT32_MIN, INT32_MAX, 3 ).astype(np.int64) embedding = optimize_layout( embedding, embedding, head, tail, n_epochs, n_vertices, epochs_per_sample, a, b, rng_state, gamma, initial_alpha, negative_sample_rate, verbose=verbose, ) return embedding
[docs]@numba.njit() def init_transform(indices, weights, embedding): """Given indices and weights and an original embeddings initialize the positions of new points relative to the indices and weights (of their neighbors in the source data). Parameters ---------- indices: array of shape (n_new_samples, n_neighbors) The indices of the neighbors of each new sample weights: array of shape (n_new_samples, n_neighbors) The membership strengths of associated 1-simplices for each of the new samples. embedding: array of shape (n_samples, dim) The original embedding of the source data. Returns ------- new_embedding: array of shape (n_new_samples, dim) An initial embedding of the new sample points. """ result = np.zeros( (indices.shape[0], embedding.shape[1]), dtype=np.float32, ) for i in range(indices.shape[0]): for j in range(indices.shape[1]): for d in range(embedding.shape[1]): result[i, d] += ( weights[i, j] * embedding[indices[i, j], d] ) return result
[docs]def find_ab_params(spread, min_dist): """Fit a, b params for the differentiable curve used in lower dimensional fuzzy simplicial complex construction. We want the smooth curve (from a pre-defined family with simple gradient) that best matches an offset exponential decay. """ def curve(x, a, b): return 1.0 / (1.0 + a * x ** (2 * b)) xv = np.linspace(0, spread * 3, 300) yv = np.zeros(xv.shape) yv[xv < min_dist] = 1.0 yv[xv >= min_dist] = np.exp( -(xv[xv >= min_dist] - min_dist) / spread ) params, covar = curve_fit(curve, xv, yv) return params[0], params[1]
[docs]class UMAP(BaseEstimator): """Uniform Manifold Approximation and Projection Finds a low dimensional embedding of the data that approximates an underlying manifold. Parameters ---------- n_neighbors: float (optional, default 15) The size of local neighborhood (in terms of number of neighboring sample points) used for manifold approximation. Larger values result in more global views of the manifold, while smaller values result in more local data being preserved. In general values should be in the range 2 to 100. n_components: int (optional, default 2) The dimension of the space to embed into. This defaults to 2 to provide easy visualization, but can reasonably be set to any integer value in the range 2 to 100. metric: string or function (optional, default 'euclidean') The metric to use to compute distances in high dimensional space. If a string is passed it must match a valid predefined metric. If a general metric is required a function that takes two 1d arrays and returns a float can be provided. For performance purposes it is required that this be a numba jit'd function. Valid string metrics include: * euclidean * manhattan * chebyshev * minkowski * canberra * braycurtis * mahalanobis * wminkowski * seuclidean * cosine * correlation * haversine * hamming * jaccard * dice * russelrao * kulsinski * rogerstanimoto * sokalmichener * sokalsneath * yule Metrics that take arguments (such as minkowski, mahalanobis etc.) can have arguments passed via the metric_kwds dictionary. At this time care must be taken and dictionary elements must be ordered appropriately; this will hopefully be fixed in the future. n_epochs: int (optional, default None) The number of training epochs to be used in optimizing the low dimensional embedding. Larger values result in more accurate embeddings. If None is specified a value will be selected based on the size of the input dataset (200 for large datasets, 500 for small). learning_rate: float (optional, default 1.0) The initial learning rate for the embedding optimization. init: string (optional, default 'spectral') How to initialize the low dimensional embedding. Options are: * 'spectral': use a spectral embedding of the fuzzy 1-skeleton * 'random': assign initial embedding positions at random. * A numpy array of initial embedding positions. min_dist: float (optional, default 0.1) The effective minimum distance between embedded points. Smaller values will result in a more clustered/clumped embedding where nearby points on the manifold are drawn closer together, while larger values will result on a more even dispersal of points. The value should be set relative to the ``spread`` value, which determines the scale at which embedded points will be spread out. spread: float (optional, default 1.0) The effective scale of embedded points. In combination with ``min_dist`` this determines how clustered/clumped the embedded points are. set_op_mix_ratio: float (optional, default 1.0) Interpolate between (fuzzy) union and intersection as the set operation used to combine local fuzzy simplicial sets to obtain a global fuzzy simplicial sets. Both fuzzy set operations use the product t-norm. The value of this parameter should be between 0.0 and 1.0; a value of 1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy intersection. local_connectivity: int (optional, default 1) The local connectivity required -- i.e. the number of nearest neighbors that should be assumed to be connected at a local level. The higher this value the more connected the manifold becomes locally. In practice this should be not more than the local intrinsic dimension of the manifold. repulsion_strength: float (optional, default 1.0) Weighting applied to negative samples in low dimensional embedding optimization. Values higher than one will result in greater weight being given to negative samples. negative_sample_rate: int (optional, default 5) The number of negative samples to select per positive sample in the optimization process. Increasing this value will result in greater repulsive force being applied, greater optimization cost, but slightly more accuracy. transform_queue_size: float (optional, default 4.0) For transform operations (embedding new points using a trained model_ this will control how aggressively to search for nearest neighbors. Larger values will result in slower performance but more accurate nearest neighbor evaluation. a: float (optional, default None) More specific parameters controlling the embedding. If None these values are set automatically as determined by ``min_dist`` and ``spread``. b: float (optional, default None) More specific parameters controlling the embedding. If None these values are set automatically as determined by ``min_dist`` and ``spread``. random_state: int, RandomState instance or None, optional (default: None) If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. metric_kwds: dict (optional, default None) Arguments to pass on to the metric, such as the ``p`` value for Minkowski distance. If None then no arguments are passed on. angular_rp_forest: bool (optional, default False) Whether to use an angular random projection forest to initialise the approximate nearest neighbor search. This can be faster, but is mostly on useful for metric that use an angular style distance such as cosine, correlation etc. In the case of those metrics angular forests will be chosen automatically. target_n_neighbors: int (optional, default -1) The number of nearest neighbors to use to construct the target simplcial set. If set to -1 use the ``n_neighbors`` value. target_metric: string or callable (optional, default 'categorical') The metric used to measure distance for a target array is using supervised dimension reduction. By default this is 'categorical' which will measure distance in terms of whether categories match or are different. Furthermore, if semi-supervised is required target values of -1 will be trated as unlabelled under the 'categorical' metric. If the target array takes continuous values (e.g. for a regression problem) then metric of 'l1' or 'l2' is probably more appropriate. target_metric_kwds: dict (optional, default None) Keyword argument to pass to the target metric when performing supervised dimension reduction. If None then no arguments are passed on. target_weight: float (optional, default 0.5) weighting factor between data topology and target topology. A value of 0.0 weights entirely on data, a value of 1.0 weights entirely on target. The default of 0.5 balances the weighting equally between data and target. transform_seed: int (optional, default 42) Random seed used for the stochastic aspects of the transform operation. This ensures consistency in transform operations. verbose: bool (optional, default False) Controls verbosity of logging. """ def __init__( self, n_neighbors=15, n_components=2, metric="euclidean", n_epochs=None, learning_rate=1.0, init="spectral", min_dist=0.1, spread=1.0, set_op_mix_ratio=1.0, local_connectivity=1.0, repulsion_strength=1.0, negative_sample_rate=5, transform_queue_size=4.0, a=None, b=None, random_state=None, metric_kwds=None, angular_rp_forest=False, target_n_neighbors=-1, target_metric="categorical", target_metric_kwds=None, target_weight=0.5, transform_seed=42, verbose=False, ): self.n_neighbors = n_neighbors self.metric = metric self.metric_kwds = metric_kwds self.n_epochs = n_epochs self.init = init self.n_components = n_components self.repulsion_strength = repulsion_strength self.learning_rate = learning_rate self.spread = spread self.min_dist = min_dist self.set_op_mix_ratio = set_op_mix_ratio self.local_connectivity = local_connectivity self.negative_sample_rate = negative_sample_rate self.random_state = random_state self.angular_rp_forest = angular_rp_forest self.transform_queue_size = transform_queue_size self.target_n_neighbors = target_n_neighbors self.target_metric = target_metric self.target_metric_kwds = target_metric_kwds self.target_weight = target_weight self.transform_seed = transform_seed self.verbose = verbose self.a = a self.b = b def _validate_parameters(self): if ( self.set_op_mix_ratio < 0.0 or self.set_op_mix_ratio > 1.0 ): raise ValueError( "set_op_mix_ratio must be between 0.0 and 1.0" ) if self.repulsion_strength < 0.0: raise ValueError( "repulsion_strength cannot be negative" ) if self.min_dist > self.spread: raise ValueError( "min_dist must be less than or equal to spread" ) if self.min_dist < 0.0: raise ValueError( "min_dist must be greater than 0.0" ) if not isinstance( self.init, str ) and not isinstance(self.init, np.ndarray): raise ValueError( "init must be a string or ndarray" ) if isinstance(self.init, str) and self.init not in ( "spectral", "random", ): raise ValueError( 'string init values must be "spectral" or "random"' ) if ( isinstance(self.init, np.ndarray) and self.init.shape[1] != self.n_components ): raise ValueError( "init ndarray must match n_components value" ) if not isinstance( self.metric, str ) and not callable(self.metric): raise ValueError( "metric must be string or callable" ) if self.negative_sample_rate < 0: raise ValueError( "negative sample rate must be positive" ) if self._initial_alpha < 0.0: raise ValueError( "learning_rate must be positive" ) if self.n_neighbors < 2: raise ValueError( "n_neighbors must be greater than 2" ) if ( self.target_n_neighbors < 2 and self.target_n_neighbors != -1 ): raise ValueError( "target_n_neighbors must be greater than 2" ) if not isinstance(self.n_components, int): raise ValueError("n_components must be an int") if self.n_components < 1: raise ValueError( "n_components must be greater than 0" ) if self.n_epochs is not None and ( self.n_epochs <= 10 or not isinstance(self.n_epochs, int) ): raise ValueError( "n_epochs must be a positive integer " "larger than 10" )
[docs] def fit(self, X, y=None): """Fit X into an embedded space. Optionally use y for supervised dimension reduction. Parameters ---------- X : array, shape (n_samples, n_features) or (n_samples, n_samples) If the metric is 'precomputed' X must be a square distance matrix. Otherwise it contains a sample per row. If the method is 'exact', X may be a sparse matrix of type 'csr', 'csc' or 'coo'. y : array, shape (n_samples) A target array for supervised dimension reduction. How this is handled is determined by parameters UMAP was instantiated with. The relevant attributes are ``target_metric`` and ``target_metric_kwds``. """ X = check_array( X, dtype=np.float32, accept_sparse="csr" ) self._raw_data = X # Handle all the optional arguments, setting default if self.a is None or self.b is None: self._a, self._b = find_ab_params( self.spread, self.min_dist ) else: self._a = self.a self._b = self.b if self.metric_kwds is not None: self._metric_kwds = self.metric_kwds else: self._metric_kwds = {} if self.target_metric_kwds is not None: self._target_metric_kwds = ( self.target_metric_kwds ) else: self._target_metric_kwds = {} if isinstance(self.init, np.ndarray): init = check_array( self.init, dtype=np.float32, accept_sparse=False, ) else: init = self.init self._initial_alpha = self.learning_rate self._validate_parameters() if self.verbose: print(str(self)) # Error check n_neighbors based on data size if X.shape[0] <= self.n_neighbors: if X.shape[0] == 1: self.embedding_ = np.zeros( (1, self.n_components) ) # needed to sklearn comparability return self warn( "n_neighbors is larger than the dataset size; truncating to " "X.shape[0] - 1" ) self._n_neighbors = X.shape[0] - 1 else: self._n_neighbors = self.n_neighbors if scipy.sparse.isspmatrix_csr(X): if not X.has_sorted_indices: X.sort_indices() self._sparse_data = True else: self._sparse_data = False random_state = check_random_state(self.random_state) if self.verbose: print("Construct fuzzy simplicial set") # Handle small cases efficiently by computing all distances if X.shape[0] < 4096: self._small_data = True dmat = pairwise_distances( X, metric=self.metric, **self._metric_kwds ) self.graph_ = fuzzy_simplicial_set( dmat, self._n_neighbors, random_state, "precomputed", self._metric_kwds, None, None, self.angular_rp_forest, self.set_op_mix_ratio, self.local_connectivity, self.verbose, ) else: self._small_data = False # Standard case ( self._knn_indices, self._knn_dists, self._rp_forest, ) = nearest_neighbors( X, self._n_neighbors, self.metric, self._metric_kwds, self.angular_rp_forest, random_state, self.verbose, ) self.graph_ = fuzzy_simplicial_set( X, self.n_neighbors, random_state, self.metric, self._metric_kwds, self._knn_indices, self._knn_dists, self.angular_rp_forest, self.set_op_mix_ratio, self.local_connectivity, self.verbose, ) self._search_graph = scipy.sparse.lil_matrix( (X.shape[0], X.shape[0]), dtype=np.int8 ) self._search_graph.rows = self._knn_indices self._search_graph.data = ( self._knn_dists != 0 ).astype(np.int8) self._search_graph = self._search_graph.maximum( self._search_graph.transpose() ).tocsr() if callable(self.metric): self._distance_func = self.metric elif self.metric in dist.named_distances: self._distance_func = dist.named_distances[ self.metric ] elif self.metric == "precomputed": warn( "Using precomputed metric; transform will be unavailable for new data" ) else: raise ValueError( "Metric is neither callable, " + "nor a recognised string" ) if self.metric != "precomputed": self._dist_args = tuple( self._metric_kwds.values() ) self._random_init, self._tree_init = make_initialisations( self._distance_func, self._dist_args ) self._search = make_initialized_nnd_search( self._distance_func, self._dist_args ) if y is not None: if len(X) != len(y): raise ValueError( "Length of x = {len_x}, length of y = {len_y}, while it must be equal.".format( len_x=len(X), len_y=len(y) ) ) y_ = check_array(y, ensure_2d=False) if self.target_metric == "categorical": if self.target_weight < 1.0: far_dist = 2.5 * ( 1.0 / (1.0 - self.target_weight) ) else: far_dist = 1.0e12 self.graph_ = categorical_simplicial_set_intersection( self.graph_, y_, far_dist=far_dist ) else: if self.target_n_neighbors == -1: target_n_neighbors = self._n_neighbors else: target_n_neighbors = ( self.target_n_neighbors ) # Handle the small case as precomputed as before if y.shape[0] < 4096: ydmat = pairwise_distances( y_[np.newaxis, :].T, metric=self.target_metric, **self._target_metric_kwds ) target_graph = fuzzy_simplicial_set( ydmat, target_n_neighbors, random_state, "precomputed", self._target_metric_kwds, None, None, False, 1.0, 1.0, False, ) else: # Standard case target_graph = fuzzy_simplicial_set( y_[np.newaxis, :].T, target_n_neighbors, random_state, self.target_metric, self._target_metric_kwds, None, None, False, 1.0, 1.0, False, ) # product = self.graph_.multiply(target_graph) # # self.graph_ = 0.99 * product + 0.01 * (self.graph_ + # # target_graph - # # product) # self.graph_ = product self.graph_ = general_simplicial_set_intersection( self.graph_, target_graph, self.target_weight, ) self.graph_ = reset_local_connectivity( self.graph_ ) if self.n_epochs is None: n_epochs = 0 else: n_epochs = self.n_epochs if self.verbose: print(ts(), "Construct embedding") self.embedding_ = simplicial_set_embedding( self._raw_data, self.graph_, self.n_components, self._initial_alpha, self._a, self._b, self.repulsion_strength, self.negative_sample_rate, n_epochs, init, random_state, self.metric, self._metric_kwds, self.verbose, ) if self.verbose: print(ts() + " Finished embedding") self._input_hash = joblib.hash(self._raw_data) return self
[docs] def fit_transform(self, X, y=None): """Fit X into an embedded space and return that transformed output. Parameters ---------- X : array, shape (n_samples, n_features) or (n_samples, n_samples) If the metric is 'precomputed' X must be a square distance matrix. Otherwise it contains a sample per row. y : array, shape (n_samples) A target array for supervised dimension reduction. How this is handled is determined by parameters UMAP was instantiated with. The relevant attributes are ``target_metric`` and ``target_metric_kwds``. Returns ------- X_new : array, shape (n_samples, n_components) Embedding of the training data in low-dimensional space. """ self.fit(X, y) return self.embedding_
[docs] def transform(self, X): """Transform X into the existing embedded space and return that transformed output. Parameters ---------- X : array, shape (n_samples, n_features) New data to be transformed. Returns ------- X_new : array, shape (n_samples, n_components) Embedding of the new data in low-dimensional space. """ # If we fit just a single instance then error if self.embedding_.shape[0] == 1: raise ValueError( "Transform unavailable when model was fit with" "only a single data sample." ) # If we just have the original input then short circuit things X = check_array( X, dtype=np.float32, accept_sparse="csr" ) x_hash = joblib.hash(X) if x_hash == self._input_hash: return self.embedding_ if self._sparse_data: raise ValueError( "Transform not available for sparse input." ) elif self.metric == "precomputed": raise ValueError( "Transform of new data not available for " "precomputed metric." ) X = check_array(X, dtype=np.float32, order="C") random_state = check_random_state( self.transform_seed ) rng_state = random_state.randint( INT32_MIN, INT32_MAX, 3 ).astype(np.int64) if self._small_data: dmat = pairwise_distances( X, self._raw_data, metric=self.metric, **self._metric_kwds ) indices = np.argpartition( dmat, self._n_neighbors )[:, : self._n_neighbors] dmat_shortened = submatrix( dmat, indices, self._n_neighbors ) indices_sorted = np.argsort(dmat_shortened) indices = submatrix( indices, indices_sorted, self._n_neighbors ) dists = submatrix( dmat_shortened, indices_sorted, self._n_neighbors, ) else: init = initialise_search( self._rp_forest, self._raw_data, X, int( self._n_neighbors * self.transform_queue_size ), self._random_init, self._tree_init, rng_state, ) result = self._search( self._raw_data, self._search_graph.indptr, self._search_graph.indices, init, X, ) indices, dists = deheap_sort(result) indices = indices[:, : self._n_neighbors] dists = dists[:, : self._n_neighbors] adjusted_local_connectivity = max( 0, self.local_connectivity - 1.0 ) sigmas, rhos = smooth_knn_dist( dists, self._n_neighbors, local_connectivity=adjusted_local_connectivity, ) rows, cols, vals = compute_membership_strengths( indices, dists, sigmas, rhos ) graph = scipy.sparse.coo_matrix( (vals, (rows, cols)), shape=(X.shape[0], self._raw_data.shape[0]), ) # This was a very specially constructed graph with constant degree. # That lets us do fancy unpacking by reshaping the csr matrix indices # and data. Doing so relies on the constant degree assumption! csr_graph = normalize(graph.tocsr(), norm="l1") inds = csr_graph.indices.reshape( X.shape[0], self._n_neighbors ) weights = csr_graph.data.reshape( X.shape[0], self._n_neighbors ) embedding = init_transform( inds, weights, self.embedding_ ) if self.n_epochs is None: # For smaller datasets we can use more epochs if graph.shape[0] <= 10000: n_epochs = 100 else: n_epochs = 30 else: n_epochs = self.n_epochs // 3.0 graph.data[ graph.data < (graph.data.max() / float(n_epochs)) ] = 0.0 graph.eliminate_zeros() epochs_per_sample = make_epochs_per_sample( graph.data, n_epochs ) head = graph.row tail = graph.col embedding = optimize_layout( embedding, self.embedding_.astype( np.float32, copy=True ), # Fixes #179 & #217 head, tail, n_epochs, graph.shape[1], epochs_per_sample, self._a, self._b, rng_state, self.repulsion_strength, self._initial_alpha, self.negative_sample_rate, verbose=self.verbose, ) return embedding