# -*- coding: utf-8 -*- # Authors: Alexandre Gramfort # Mathieu Blondel # Robert Layton # Andreas Mueller # Philippe Gervais # Lars Buitinck # Joel Nothman # License: BSD 3 clause import itertools from functools import partial import warnings import numpy as np from scipy.spatial import distance from scipy.sparse import csr_matrix from scipy.sparse import issparse from ..utils.validation import _num_samples from ..utils import check_array from ..utils import gen_even_slices from ..utils import gen_batches, get_chunk_n_rows from ..utils.extmath import row_norms, safe_sparse_dot from ..preprocessing import normalize from ..utils._joblib import Parallel from ..utils._joblib import delayed from ..utils._joblib import effective_n_jobs from .pairwise_fast import _chi2_kernel_fast, _sparse_manhattan # Utility Functions def _return_float_dtype(X, Y): """ 1. If dtype of X and Y is float32, then dtype float32 is returned. 2. Else dtype float is returned. """ if not issparse(X) and not isinstance(X, np.ndarray): X = np.asarray(X) if Y is None: Y_dtype = X.dtype elif not issparse(Y) and not isinstance(Y, np.ndarray): Y = np.asarray(Y) Y_dtype = Y.dtype else: Y_dtype = Y.dtype if X.dtype == Y_dtype == np.float32: dtype = np.float32 else: dtype = np.float return X, Y, dtype def check_pairwise_arrays(X, Y, precomputed=False, dtype=None): """ Set X and Y appropriately and checks inputs If Y is None, it is set as a pointer to X (i.e. not a copy). If Y is given, this does not happen. All distance metrics should use this function first to assert that the given parameters are correct and safe to use. Specifically, this function first ensures that both X and Y are arrays, then checks that they are at least two dimensional while ensuring that their elements are floats (or dtype if provided). Finally, the function checks that the size of the second dimension of the two arrays is equal, or the equivalent check for a precomputed distance matrix. Parameters ---------- X : {array-like, sparse matrix}, shape (n_samples_a, n_features) Y : {array-like, sparse matrix}, shape (n_samples_b, n_features) precomputed : bool True if X is to be treated as precomputed distances to the samples in Y. dtype : string, type, list of types or None (default=None) Data type required for X and Y. If None, the dtype will be an appropriate float type selected by _return_float_dtype. .. versionadded:: 0.18 Returns ------- safe_X : {array-like, sparse matrix}, shape (n_samples_a, n_features) An array equal to X, guaranteed to be a numpy array. safe_Y : {array-like, sparse matrix}, shape (n_samples_b, n_features) An array equal to Y if Y was not None, guaranteed to be a numpy array. If Y was None, safe_Y will be a pointer to X. """ X, Y, dtype_float = _return_float_dtype(X, Y) warn_on_dtype = dtype is not None estimator = 'check_pairwise_arrays' if dtype is None: dtype = dtype_float if Y is X or Y is None: X = Y = check_array(X, accept_sparse='csr', dtype=dtype, warn_on_dtype=warn_on_dtype, estimator=estimator) else: X = check_array(X, accept_sparse='csr', dtype=dtype, warn_on_dtype=warn_on_dtype, estimator=estimator) Y = check_array(Y, accept_sparse='csr', dtype=dtype, warn_on_dtype=warn_on_dtype, estimator=estimator) if precomputed: if X.shape[1] != Y.shape[0]: raise ValueError("Precomputed metric requires shape " "(n_queries, n_indexed). Got (%d, %d) " "for %d indexed." % (X.shape[0], X.shape[1], Y.shape[0])) elif X.shape[1] != Y.shape[1]: raise ValueError("Incompatible dimension for X and Y matrices: " "X.shape[1] == %d while Y.shape[1] == %d" % ( X.shape[1], Y.shape[1])) return X, Y def check_paired_arrays(X, Y): """ Set X and Y appropriately and checks inputs for paired distances All paired distance metrics should use this function first to assert that the given parameters are correct and safe to use. Specifically, this function first ensures that both X and Y are arrays, then checks that they are at least two dimensional while ensuring that their elements are floats. Finally, the function checks that the size of the dimensions of the two arrays are equal. Parameters ---------- X : {array-like, sparse matrix}, shape (n_samples_a, n_features) Y : {array-like, sparse matrix}, shape (n_samples_b, n_features) Returns ------- safe_X : {array-like, sparse matrix}, shape (n_samples_a, n_features) An array equal to X, guaranteed to be a numpy array. safe_Y : {array-like, sparse matrix}, shape (n_samples_b, n_features) An array equal to Y if Y was not None, guaranteed to be a numpy array. If Y was None, safe_Y will be a pointer to X. """ X, Y = check_pairwise_arrays(X, Y) if X.shape != Y.shape: raise ValueError("X and Y should be of same shape. They were " "respectively %r and %r long." % (X.shape, Y.shape)) return X, Y # Pairwise distances def euclidean_distances(X, Y=None, Y_norm_squared=None, squared=False, X_norm_squared=None): """ Considering the rows of X (and Y=X) as vectors, compute the distance matrix between each pair of vectors. For efficiency reasons, the euclidean distance between a pair of row vector x and y is computed as:: dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y)) This formulation has two advantages over other ways of computing distances. First, it is computationally efficient when dealing with sparse data. Second, if one argument varies but the other remains unchanged, then `dot(x, x)` and/or `dot(y, y)` can be pre-computed. However, this is not the most precise way of doing this computation, and the distance matrix returned by this function may not be exactly symmetric as required by, e.g., ``scipy.spatial.distance`` functions. Read more in the :ref:`User Guide `. Parameters ---------- X : {array-like, sparse matrix}, shape (n_samples_1, n_features) Y : {array-like, sparse matrix}, shape (n_samples_2, n_features) Y_norm_squared : array-like, shape (n_samples_2, ), optional Pre-computed dot-products of vectors in Y (e.g., ``(Y**2).sum(axis=1)``) squared : boolean, optional Return squared Euclidean distances. X_norm_squared : array-like, shape = [n_samples_1], optional Pre-computed dot-products of vectors in X (e.g., ``(X**2).sum(axis=1)``) Returns ------- distances : {array, sparse matrix}, shape (n_samples_1, n_samples_2) Examples -------- >>> from sklearn.metrics.pairwise import euclidean_distances >>> X = [[0, 1], [1, 1]] >>> # distance between rows of X >>> euclidean_distances(X, X) array([[0., 1.], [1., 0.]]) >>> # get distance to origin >>> euclidean_distances(X, [[0, 0]]) array([[1. ], [1.41421356]]) See also -------- paired_distances : distances betweens pairs of elements of X and Y. """ X, Y = check_pairwise_arrays(X, Y) if X_norm_squared is not None: XX = check_array(X_norm_squared) if XX.shape == (1, X.shape[0]): XX = XX.T elif XX.shape != (X.shape[0], 1): raise ValueError( "Incompatible dimensions for X and X_norm_squared") else: XX = row_norms(X, squared=True)[:, np.newaxis] if X is Y: # shortcut in the common case euclidean_distances(X, X) YY = XX.T elif Y_norm_squared is not None: YY = np.atleast_2d(Y_norm_squared) if YY.shape != (1, Y.shape[0]): raise ValueError( "Incompatible dimensions for Y and Y_norm_squared") else: YY = row_norms(Y, squared=True)[np.newaxis, :] distances = safe_sparse_dot(X, Y.T, dense_output=True) distances *= -2 distances += XX distances += YY np.maximum(distances, 0, out=distances) if X is Y: # Ensure that distances between vectors and themselves are set to 0.0. # This may not be the case due to floating point rounding errors. distances.flat[::distances.shape[0] + 1] = 0.0 return distances if squared else np.sqrt(distances, out=distances) def _argmin_min_reduce(dist, start): indices = dist.argmin(axis=1) values = dist[np.arange(dist.shape[0]), indices] return indices, values def pairwise_distances_argmin_min(X, Y, axis=1, metric="euclidean", batch_size=None, metric_kwargs=None): """Compute minimum distances between one point and a set of points. This function computes for each row in X, the index of the row of Y which is closest (according to the specified distance). The minimal distances are also returned. This is mostly equivalent to calling: (pairwise_distances(X, Y=Y, metric=metric).argmin(axis=axis), pairwise_distances(X, Y=Y, metric=metric).min(axis=axis)) but uses much less memory, and is faster for large arrays. Parameters ---------- X : {array-like, sparse matrix}, shape (n_samples1, n_features) Array containing points. Y : {array-like, sparse matrix}, shape (n_samples2, n_features) Arrays containing points. axis : int, optional, default 1 Axis along which the argmin and distances are to be computed. metric : string or callable, default 'euclidean' metric to use for distance computation. Any metric from scikit-learn or scipy.spatial.distance can be used. If metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays as input and return one value indicating the distance between them. This works for Scipy's metrics, but is less efficient than passing the metric name as a string. Distance matrices are not supported. Valid values for metric are: - from scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2', 'manhattan'] - from scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. batch_size : integer .. deprecated:: 0.20 Deprecated for removal in 0.22. Use sklearn.set_config(working_memory=...) instead. metric_kwargs : dict, optional Keyword arguments to pass to specified metric function. Returns ------- argmin : numpy.ndarray Y[argmin[i], :] is the row in Y that is closest to X[i, :]. distances : numpy.ndarray distances[i] is the distance between the i-th row in X and the argmin[i]-th row in Y. See also -------- sklearn.metrics.pairwise_distances sklearn.metrics.pairwise_distances_argmin """ if batch_size is not None: warnings.warn("'batch_size' is ignored. It was deprecated in version " "0.20 and will be removed in version 0.22. " "Use sklearn.set_config(working_memory=...) instead.", DeprecationWarning) X, Y = check_pairwise_arrays(X, Y) if metric_kwargs is None: metric_kwargs = {} if axis == 0: X, Y = Y, X indices, values = zip(*pairwise_distances_chunked( X, Y, reduce_func=_argmin_min_reduce, metric=metric, **metric_kwargs)) indices = np.concatenate(indices) values = np.concatenate(values) return indices, values def pairwise_distances_argmin(X, Y, axis=1, metric="euclidean", batch_size=None, metric_kwargs=None): """Compute minimum distances between one point and a set of points. This function computes for each row in X, the index of the row of Y which is closest (according to the specified distance). This is mostly equivalent to calling: pairwise_distances(X, Y=Y, metric=metric).argmin(axis=axis) but uses much less memory, and is faster for large arrays. This function works with dense 2D arrays only. Parameters ---------- X : array-like Arrays containing points. Respective shapes (n_samples1, n_features) and (n_samples2, n_features) Y : array-like Arrays containing points. Respective shapes (n_samples1, n_features) and (n_samples2, n_features) axis : int, optional, default 1 Axis along which the argmin and distances are to be computed. metric : string or callable metric to use for distance computation. Any metric from scikit-learn or scipy.spatial.distance can be used. If metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays as input and return one value indicating the distance between them. This works for Scipy's metrics, but is less efficient than passing the metric name as a string. Distance matrices are not supported. Valid values for metric are: - from scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2', 'manhattan'] - from scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. batch_size : integer .. deprecated:: 0.20 Deprecated for removal in 0.22. Use sklearn.set_config(working_memory=...) instead. metric_kwargs : dict keyword arguments to pass to specified metric function. Returns ------- argmin : numpy.ndarray Y[argmin[i], :] is the row in Y that is closest to X[i, :]. See also -------- sklearn.metrics.pairwise_distances sklearn.metrics.pairwise_distances_argmin_min """ if metric_kwargs is None: metric_kwargs = {} return pairwise_distances_argmin_min(X, Y, axis, metric, metric_kwargs=metric_kwargs, batch_size=batch_size)[0] def manhattan_distances(X, Y=None, sum_over_features=True, size_threshold=None): """ Compute the L1 distances between the vectors in X and Y. With sum_over_features equal to False it returns the componentwise distances. Read more in the :ref:`User Guide `. Parameters ---------- X : array_like An array with shape (n_samples_X, n_features). Y : array_like, optional An array with shape (n_samples_Y, n_features). sum_over_features : bool, default=True If True the function returns the pairwise distance matrix else it returns the componentwise L1 pairwise-distances. Not supported for sparse matrix inputs. size_threshold : int, default=5e8 Unused parameter. Returns ------- D : array If sum_over_features is False shape is (n_samples_X * n_samples_Y, n_features) and D contains the componentwise L1 pairwise-distances (ie. absolute difference), else shape is (n_samples_X, n_samples_Y) and D contains the pairwise L1 distances. Examples -------- >>> from sklearn.metrics.pairwise import manhattan_distances >>> manhattan_distances([[3]], [[3]])#doctest:+ELLIPSIS array([[0.]]) >>> manhattan_distances([[3]], [[2]])#doctest:+ELLIPSIS array([[1.]]) >>> manhattan_distances([[2]], [[3]])#doctest:+ELLIPSIS array([[1.]]) >>> manhattan_distances([[1, 2], [3, 4]],\ [[1, 2], [0, 3]])#doctest:+ELLIPSIS array([[0., 2.], [4., 4.]]) >>> import numpy as np >>> X = np.ones((1, 2)) >>> y = np.full((2, 2), 2.) >>> manhattan_distances(X, y, sum_over_features=False)#doctest:+ELLIPSIS array([[1., 1.], [1., 1.]]) """ if size_threshold is not None: warnings.warn('Use of the "size_threshold" is deprecated ' 'in 0.19 and it will be removed version ' '0.21 of scikit-learn', DeprecationWarning) X, Y = check_pairwise_arrays(X, Y) if issparse(X) or issparse(Y): if not sum_over_features: raise TypeError("sum_over_features=%r not supported" " for sparse matrices" % sum_over_features) X = csr_matrix(X, copy=False) Y = csr_matrix(Y, copy=False) D = np.zeros((X.shape[0], Y.shape[0])) _sparse_manhattan(X.data, X.indices, X.indptr, Y.data, Y.indices, Y.indptr, X.shape[1], D) return D if sum_over_features: return distance.cdist(X, Y, 'cityblock') D = X[:, np.newaxis, :] - Y[np.newaxis, :, :] D = np.abs(D, D) return D.reshape((-1, X.shape[1])) def cosine_distances(X, Y=None): """Compute cosine distance between samples in X and Y. Cosine distance is defined as 1.0 minus the cosine similarity. Read more in the :ref:`User Guide `. Parameters ---------- X : array_like, sparse matrix with shape (n_samples_X, n_features). Y : array_like, sparse matrix (optional) with shape (n_samples_Y, n_features). Returns ------- distance matrix : array An array with shape (n_samples_X, n_samples_Y). See also -------- sklearn.metrics.pairwise.cosine_similarity scipy.spatial.distance.cosine (dense matrices only) """ # 1.0 - cosine_similarity(X, Y) without copy S = cosine_similarity(X, Y) S *= -1 S += 1 np.clip(S, 0, 2, out=S) if X is Y or Y is None: # Ensure that distances between vectors and themselves are set to 0.0. # This may not be the case due to floating point rounding errors. S[np.diag_indices_from(S)] = 0.0 return S # Paired distances def paired_euclidean_distances(X, Y): """ Computes the paired euclidean distances between X and Y Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Y : array-like, shape (n_samples, n_features) Returns ------- distances : ndarray (n_samples, ) """ X, Y = check_paired_arrays(X, Y) return row_norms(X - Y) def paired_manhattan_distances(X, Y): """Compute the L1 distances between the vectors in X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Y : array-like, shape (n_samples, n_features) Returns ------- distances : ndarray (n_samples, ) """ X, Y = check_paired_arrays(X, Y) diff = X - Y if issparse(diff): diff.data = np.abs(diff.data) return np.squeeze(np.array(diff.sum(axis=1))) else: return np.abs(diff).sum(axis=-1) def paired_cosine_distances(X, Y): """ Computes the paired cosine distances between X and Y Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Y : array-like, shape (n_samples, n_features) Returns ------- distances : ndarray, shape (n_samples, ) Notes ------ The cosine distance is equivalent to the half the squared euclidean distance if each sample is normalized to unit norm """ X, Y = check_paired_arrays(X, Y) return .5 * row_norms(normalize(X) - normalize(Y), squared=True) PAIRED_DISTANCES = { 'cosine': paired_cosine_distances, 'euclidean': paired_euclidean_distances, 'l2': paired_euclidean_distances, 'l1': paired_manhattan_distances, 'manhattan': paired_manhattan_distances, 'cityblock': paired_manhattan_distances} def paired_distances(X, Y, metric="euclidean", **kwds): """ Computes the paired distances between X and Y. Computes the distances between (X[0], Y[0]), (X[1], Y[1]), etc... Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray (n_samples, n_features) Array 1 for distance computation. Y : ndarray (n_samples, n_features) Array 2 for distance computation. metric : string or callable The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options specified in PAIRED_DISTANCES, including "euclidean", "manhattan", or "cosine". Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from X as input and return a value indicating the distance between them. Returns ------- distances : ndarray (n_samples, ) Examples -------- >>> from sklearn.metrics.pairwise import paired_distances >>> X = [[0, 1], [1, 1]] >>> Y = [[0, 1], [2, 1]] >>> paired_distances(X, Y) array([0., 1.]) See also -------- pairwise_distances : Computes the distance between every pair of samples """ if metric in PAIRED_DISTANCES: func = PAIRED_DISTANCES[metric] return func(X, Y) elif callable(metric): # Check the matrix first (it is usually done by the metric) X, Y = check_paired_arrays(X, Y) distances = np.zeros(len(X)) for i in range(len(X)): distances[i] = metric(X[i], Y[i]) return distances else: raise ValueError('Unknown distance %s' % metric) # Kernels def linear_kernel(X, Y=None, dense_output=True): """ Compute the linear kernel between X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array of shape (n_samples_1, n_features) Y : array of shape (n_samples_2, n_features) dense_output : boolean (optional), default True Whether to return dense output even when the input is sparse. If ``False``, the output is sparse if both input arrays are sparse. .. versionadded:: 0.20 Returns ------- Gram matrix : array of shape (n_samples_1, n_samples_2) """ X, Y = check_pairwise_arrays(X, Y) return safe_sparse_dot(X, Y.T, dense_output=dense_output) def polynomial_kernel(X, Y=None, degree=3, gamma=None, coef0=1): """ Compute the polynomial kernel between X and Y:: K(X, Y) = (gamma + coef0)^degree Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_1, n_features) Y : ndarray of shape (n_samples_2, n_features) degree : int, default 3 gamma : float, default None if None, defaults to 1.0 / n_features coef0 : float, default 1 Returns ------- Gram matrix : array of shape (n_samples_1, n_samples_2) """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = safe_sparse_dot(X, Y.T, dense_output=True) K *= gamma K += coef0 K **= degree return K def sigmoid_kernel(X, Y=None, gamma=None, coef0=1): """ Compute the sigmoid kernel between X and Y:: K(X, Y) = tanh(gamma + coef0) Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_1, n_features) Y : ndarray of shape (n_samples_2, n_features) gamma : float, default None If None, defaults to 1.0 / n_features coef0 : float, default 1 Returns ------- Gram matrix : array of shape (n_samples_1, n_samples_2) """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = safe_sparse_dot(X, Y.T, dense_output=True) K *= gamma K += coef0 np.tanh(K, K) # compute tanh in-place return K def rbf_kernel(X, Y=None, gamma=None): """ Compute the rbf (gaussian) kernel between X and Y:: K(x, y) = exp(-gamma ||x-y||^2) for each pair of rows x in X and y in Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) gamma : float, default None If None, defaults to 1.0 / n_features Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = euclidean_distances(X, Y, squared=True) K *= -gamma np.exp(K, K) # exponentiate K in-place return K def laplacian_kernel(X, Y=None, gamma=None): """Compute the laplacian kernel between X and Y. The laplacian kernel is defined as:: K(x, y) = exp(-gamma ||x-y||_1) for each pair of rows x in X and y in Y. Read more in the :ref:`User Guide `. .. versionadded:: 0.17 Parameters ---------- X : array of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) gamma : float, default None If None, defaults to 1.0 / n_features Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = -gamma * manhattan_distances(X, Y) np.exp(K, K) # exponentiate K in-place return K def cosine_similarity(X, Y=None, dense_output=True): """Compute cosine similarity between samples in X and Y. Cosine similarity, or the cosine kernel, computes similarity as the normalized dot product of X and Y: K(X, Y) = / (||X||*||Y||) On L2-normalized data, this function is equivalent to linear_kernel. Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray or sparse array, shape: (n_samples_X, n_features) Input data. Y : ndarray or sparse array, shape: (n_samples_Y, n_features) Input data. If ``None``, the output will be the pairwise similarities between all samples in ``X``. dense_output : boolean (optional), default True Whether to return dense output even when the input is sparse. If ``False``, the output is sparse if both input arrays are sparse. .. versionadded:: 0.17 parameter ``dense_output`` for dense output. Returns ------- kernel matrix : array An array with shape (n_samples_X, n_samples_Y). """ # to avoid recursive import X, Y = check_pairwise_arrays(X, Y) X_normalized = normalize(X, copy=True) if X is Y: Y_normalized = X_normalized else: Y_normalized = normalize(Y, copy=True) K = safe_sparse_dot(X_normalized, Y_normalized.T, dense_output=dense_output) return K def additive_chi2_kernel(X, Y=None): """Computes the additive chi-squared kernel between observations in X and Y The chi-squared kernel is computed between each pair of rows in X and Y. X and Y have to be non-negative. This kernel is most commonly applied to histograms. The chi-squared kernel is given by:: k(x, y) = -Sum [(x - y)^2 / (x + y)] It can be interpreted as a weighted difference per entry. Read more in the :ref:`User Guide `. Notes ----- As the negative of a distance, this kernel is only conditionally positive definite. Parameters ---------- X : array-like of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) References ---------- * Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C. Local features and kernels for classification of texture and object categories: A comprehensive study International Journal of Computer Vision 2007 https://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf See also -------- chi2_kernel : The exponentiated version of the kernel, which is usually preferable. sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation to this kernel. """ if issparse(X) or issparse(Y): raise ValueError("additive_chi2 does not support sparse matrices.") X, Y = check_pairwise_arrays(X, Y) if (X < 0).any(): raise ValueError("X contains negative values.") if Y is not X and (Y < 0).any(): raise ValueError("Y contains negative values.") result = np.zeros((X.shape[0], Y.shape[0]), dtype=X.dtype) _chi2_kernel_fast(X, Y, result) return result def chi2_kernel(X, Y=None, gamma=1.): """Computes the exponential chi-squared kernel X and Y. The chi-squared kernel is computed between each pair of rows in X and Y. X and Y have to be non-negative. This kernel is most commonly applied to histograms. The chi-squared kernel is given by:: k(x, y) = exp(-gamma Sum [(x - y)^2 / (x + y)]) It can be interpreted as a weighted difference per entry. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) gamma : float, default=1. Scaling parameter of the chi2 kernel. Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) References ---------- * Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C. Local features and kernels for classification of texture and object categories: A comprehensive study International Journal of Computer Vision 2007 https://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf See also -------- additive_chi2_kernel : The additive version of this kernel sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation to the additive version of this kernel. """ K = additive_chi2_kernel(X, Y) K *= gamma return np.exp(K, K) # Helper functions - distance PAIRWISE_DISTANCE_FUNCTIONS = { # If updating this dictionary, update the doc in both distance_metrics() # and also in pairwise_distances()! 'cityblock': manhattan_distances, 'cosine': cosine_distances, 'euclidean': euclidean_distances, 'l2': euclidean_distances, 'l1': manhattan_distances, 'manhattan': manhattan_distances, 'precomputed': None, # HACK: precomputed is always allowed, never called } def distance_metrics(): """Valid metrics for pairwise_distances. This function simply returns the valid pairwise distance metrics. It exists to allow for a description of the mapping for each of the valid strings. The valid distance metrics, and the function they map to, are: ============ ==================================== metric Function ============ ==================================== 'cityblock' metrics.pairwise.manhattan_distances 'cosine' metrics.pairwise.cosine_distances 'euclidean' metrics.pairwise.euclidean_distances 'l1' metrics.pairwise.manhattan_distances 'l2' metrics.pairwise.euclidean_distances 'manhattan' metrics.pairwise.manhattan_distances ============ ==================================== Read more in the :ref:`User Guide `. """ return PAIRWISE_DISTANCE_FUNCTIONS def _parallel_pairwise(X, Y, func, n_jobs, **kwds): """Break the pairwise matrix in n_jobs even slices and compute them in parallel""" if Y is None: Y = X if effective_n_jobs(n_jobs) == 1: return func(X, Y, **kwds) # TODO: in some cases, backend='threading' may be appropriate fd = delayed(func) ret = Parallel(n_jobs=n_jobs, verbose=0)( fd(X, Y[s], **kwds) for s in gen_even_slices(_num_samples(Y), effective_n_jobs(n_jobs))) return np.hstack(ret) def _pairwise_callable(X, Y, metric, **kwds): """Handle the callable case for pairwise_{distances,kernels} """ X, Y = check_pairwise_arrays(X, Y) if X is Y: # Only calculate metric for upper triangle out = np.zeros((X.shape[0], Y.shape[0]), dtype='float') iterator = itertools.combinations(range(X.shape[0]), 2) for i, j in iterator: out[i, j] = metric(X[i], Y[j], **kwds) # Make symmetric # NB: out += out.T will produce incorrect results out = out + out.T # Calculate diagonal # NB: nonzero diagonals are allowed for both metrics and kernels for i in range(X.shape[0]): x = X[i] out[i, i] = metric(x, x, **kwds) else: # Calculate all cells out = np.empty((X.shape[0], Y.shape[0]), dtype='float') iterator = itertools.product(range(X.shape[0]), range(Y.shape[0])) for i, j in iterator: out[i, j] = metric(X[i], Y[j], **kwds) return out _VALID_METRICS = ['euclidean', 'l2', 'l1', 'manhattan', 'cityblock', 'braycurtis', 'canberra', 'chebyshev', 'correlation', 'cosine', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule', "wminkowski"] def _check_chunk_size(reduced, chunk_size): """Checks chunk is a sequence of expected size or a tuple of same """ is_tuple = isinstance(reduced, tuple) if not is_tuple: reduced = (reduced,) if any(isinstance(r, tuple) or not hasattr(r, '__iter__') for r in reduced): raise TypeError('reduce_func returned %r. ' 'Expected sequence(s) of length %d.' % (reduced if is_tuple else reduced[0], chunk_size)) if any(_num_samples(r) != chunk_size for r in reduced): # XXX: we use int(_num_samples...) because sometimes _num_samples # returns a long in Python 2, even for small numbers. actual_size = tuple(int(_num_samples(r)) for r in reduced) raise ValueError('reduce_func returned object of length %s. ' 'Expected same length as input: %d.' % (actual_size if is_tuple else actual_size[0], chunk_size)) def _precompute_metric_params(X, Y, metric=None, **kwds): """Precompute data-derived metric parameters if not provided """ if metric == "seuclidean" and 'V' not in kwds: if X is Y: V = np.var(X, axis=0, ddof=1) else: V = np.var(np.vstack([X, Y]), axis=0, ddof=1) return {'V': V} if metric == "mahalanobis" and 'VI' not in kwds: if X is Y: VI = np.linalg.inv(np.cov(X.T)).T else: VI = np.linalg.inv(np.cov(np.vstack([X, Y]).T)).T return {'VI': VI} return {} def pairwise_distances_chunked(X, Y=None, reduce_func=None, metric='euclidean', n_jobs=None, working_memory=None, **kwds): """Generate a distance matrix chunk by chunk with optional reduction In cases where not all of a pairwise distance matrix needs to be stored at once, this is used to calculate pairwise distances in ``working_memory``-sized chunks. If ``reduce_func`` is given, it is run on each chunk and its return values are concatenated into lists, arrays or sparse matrices. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, [n_samples_a, n_features] otherwise Array of pairwise distances between samples, or a feature array. Y : array [n_samples_b, n_features], optional An optional second feature array. Only allowed if metric != "precomputed". reduce_func : callable, optional The function which is applied on each chunk of the distance matrix, reducing it to needed values. ``reduce_func(D_chunk, start)`` is called repeatedly, where ``D_chunk`` is a contiguous vertical slice of the pairwise distance matrix, starting at row ``start``. It should return an array, a list, or a sparse matrix of length ``D_chunk.shape[0]``, or a tuple of such objects. If None, pairwise_distances_chunked returns a generator of vertical chunks of the distance matrix. metric : string, or callable The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options allowed by scipy.spatial.distance.pdist for its metric parameter, or a metric listed in pairwise.PAIRWISE_DISTANCE_FUNCTIONS. If metric is "precomputed", X is assumed to be a distance matrix. Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from X as input and return a value indicating the distance between them. n_jobs : int or None, optional (default=None) The number of jobs to use for the computation. This works by breaking down the pairwise matrix into n_jobs even slices and computing them in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. working_memory : int, optional The sought maximum memory for temporary distance matrix chunks. When None (default), the value of ``sklearn.get_config()['working_memory']`` is used. `**kwds` : optional keyword parameters Any further parameters are passed directly to the distance function. If using a scipy.spatial.distance metric, the parameters are still metric dependent. See the scipy docs for usage examples. Yields ------ D_chunk : array or sparse matrix A contiguous slice of distance matrix, optionally processed by ``reduce_func``. Examples -------- Without reduce_func: >>> X = np.random.RandomState(0).rand(5, 3) >>> D_chunk = next(pairwise_distances_chunked(X)) >>> D_chunk # doctest: +ELLIPSIS array([[0. ..., 0.29..., 0.41..., 0.19..., 0.57...], [0.29..., 0. ..., 0.57..., 0.41..., 0.76...], [0.41..., 0.57..., 0. ..., 0.44..., 0.90...], [0.19..., 0.41..., 0.44..., 0. ..., 0.51...], [0.57..., 0.76..., 0.90..., 0.51..., 0. ...]]) Retrieve all neighbors and average distance within radius r: >>> r = .2 >>> def reduce_func(D_chunk, start): ... neigh = [np.flatnonzero(d < r) for d in D_chunk] ... avg_dist = (D_chunk * (D_chunk < r)).mean(axis=1) ... return neigh, avg_dist >>> gen = pairwise_distances_chunked(X, reduce_func=reduce_func) >>> neigh, avg_dist = next(gen) >>> neigh [array([0, 3]), array([1]), array([2]), array([0, 3]), array([4])] >>> avg_dist # doctest: +ELLIPSIS array([0.039..., 0. , 0. , 0.039..., 0. ]) Where r is defined per sample, we need to make use of ``start``: >>> r = [.2, .4, .4, .3, .1] >>> def reduce_func(D_chunk, start): ... neigh = [np.flatnonzero(d < r[i]) ... for i, d in enumerate(D_chunk, start)] ... return neigh >>> neigh = next(pairwise_distances_chunked(X, reduce_func=reduce_func)) >>> neigh [array([0, 3]), array([0, 1]), array([2]), array([0, 3]), array([4])] Force row-by-row generation by reducing ``working_memory``: >>> gen = pairwise_distances_chunked(X, reduce_func=reduce_func, ... working_memory=0) >>> next(gen) [array([0, 3])] >>> next(gen) [array([0, 1])] """ n_samples_X = _num_samples(X) if metric == 'precomputed': slices = (slice(0, n_samples_X),) else: if Y is None: Y = X # We get as many rows as possible within our working_memory budget to # store len(Y) distances in each row of output. # # Note: # - this will get at least 1 row, even if 1 row of distances will # exceed working_memory. # - this does not account for any temporary memory usage while # calculating distances (e.g. difference of vectors in manhattan # distance. chunk_n_rows = get_chunk_n_rows(row_bytes=8 * _num_samples(Y), max_n_rows=n_samples_X, working_memory=working_memory) slices = gen_batches(n_samples_X, chunk_n_rows) # precompute data-derived metric params params = _precompute_metric_params(X, Y, metric=metric, **kwds) kwds.update(**params) for sl in slices: if sl.start == 0 and sl.stop == n_samples_X: X_chunk = X # enable optimised paths for X is Y else: X_chunk = X[sl] D_chunk = pairwise_distances(X_chunk, Y, metric=metric, n_jobs=n_jobs, **kwds) if ((X is Y or Y is None) and PAIRWISE_DISTANCE_FUNCTIONS.get(metric, None) is euclidean_distances): # zeroing diagonal, taking care of aliases of "euclidean", # i.e. "l2" D_chunk.flat[sl.start::_num_samples(X) + 1] = 0 if reduce_func is not None: chunk_size = D_chunk.shape[0] D_chunk = reduce_func(D_chunk, sl.start) _check_chunk_size(D_chunk, chunk_size) yield D_chunk def pairwise_distances(X, Y=None, metric="euclidean", n_jobs=None, **kwds): """ Compute the distance matrix from a vector array X and optional Y. This method takes either a vector array or a distance matrix, and returns a distance matrix. If the input is a vector array, the distances are computed. If the input is a distances matrix, it is returned instead. This method provides a safe way to take a distance matrix as input, while preserving compatibility with many other algorithms that take a vector array. If Y is given (default is None), then the returned matrix is the pairwise distance between the arrays from both X and Y. Valid values for metric are: - From scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2', 'manhattan']. These metrics support sparse matrix inputs. - From scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. These metrics do not support sparse matrix inputs. Note that in the case of 'cityblock', 'cosine' and 'euclidean' (which are valid scipy.spatial.distance metrics), the scikit-learn implementation will be used, which is faster and has support for sparse matrices (except for 'cityblock'). For a verbose description of the metrics from scikit-learn, see the __doc__ of the sklearn.pairwise.distance_metrics function. Read more in the :ref:`User Guide `. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \ [n_samples_a, n_features] otherwise Array of pairwise distances between samples, or a feature array. Y : array [n_samples_b, n_features], optional An optional second feature array. Only allowed if metric != "precomputed". metric : string, or callable The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options allowed by scipy.spatial.distance.pdist for its metric parameter, or a metric listed in pairwise.PAIRWISE_DISTANCE_FUNCTIONS. If metric is "precomputed", X is assumed to be a distance matrix. Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from X as input and return a value indicating the distance between them. n_jobs : int or None, optional (default=None) The number of jobs to use for the computation. This works by breaking down the pairwise matrix into n_jobs even slices and computing them in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. **kwds : optional keyword parameters Any further parameters are passed directly to the distance function. If using a scipy.spatial.distance metric, the parameters are still metric dependent. See the scipy docs for usage examples. Returns ------- D : array [n_samples_a, n_samples_a] or [n_samples_a, n_samples_b] A distance matrix D such that D_{i, j} is the distance between the ith and jth vectors of the given matrix X, if Y is None. If Y is not None, then D_{i, j} is the distance between the ith array from X and the jth array from Y. See also -------- pairwise_distances_chunked : performs the same calculation as this function, but returns a generator of chunks of the distance matrix, in order to limit memory usage. paired_distances : Computes the distances between corresponding elements of two arrays """ if (metric not in _VALID_METRICS and not callable(metric) and metric != "precomputed"): raise ValueError("Unknown metric %s. " "Valid metrics are %s, or 'precomputed', or a " "callable" % (metric, _VALID_METRICS)) if metric == "precomputed": X, _ = check_pairwise_arrays(X, Y, precomputed=True) return X elif metric in PAIRWISE_DISTANCE_FUNCTIONS: func = PAIRWISE_DISTANCE_FUNCTIONS[metric] elif callable(metric): func = partial(_pairwise_callable, metric=metric, **kwds) else: if issparse(X) or issparse(Y): raise TypeError("scipy distance metrics do not" " support sparse matrices.") dtype = bool if metric in PAIRWISE_BOOLEAN_FUNCTIONS else None X, Y = check_pairwise_arrays(X, Y, dtype=dtype) # precompute data-derived metric params params = _precompute_metric_params(X, Y, metric=metric, **kwds) kwds.update(**params) if effective_n_jobs(n_jobs) == 1 and X is Y: return distance.squareform(distance.pdist(X, metric=metric, **kwds)) func = partial(distance.cdist, metric=metric, **kwds) return _parallel_pairwise(X, Y, func, n_jobs, **kwds) # These distances recquire boolean arrays, when using scipy.spatial.distance PAIRWISE_BOOLEAN_FUNCTIONS = [ 'dice', 'jaccard', 'kulsinski', 'matching', 'rogerstanimoto', 'russellrao', 'sokalmichener', 'sokalsneath', 'yule', ] # Helper functions - distance PAIRWISE_KERNEL_FUNCTIONS = { # If updating this dictionary, update the doc in both distance_metrics() # and also in pairwise_distances()! 'additive_chi2': additive_chi2_kernel, 'chi2': chi2_kernel, 'linear': linear_kernel, 'polynomial': polynomial_kernel, 'poly': polynomial_kernel, 'rbf': rbf_kernel, 'laplacian': laplacian_kernel, 'sigmoid': sigmoid_kernel, 'cosine': cosine_similarity, } def kernel_metrics(): """ Valid metrics for pairwise_kernels This function simply returns the valid pairwise distance metrics. It exists, however, to allow for a verbose description of the mapping for each of the valid strings. The valid distance metrics, and the function they map to, are: =============== ======================================== metric Function =============== ======================================== 'additive_chi2' sklearn.pairwise.additive_chi2_kernel 'chi2' sklearn.pairwise.chi2_kernel 'linear' sklearn.pairwise.linear_kernel 'poly' sklearn.pairwise.polynomial_kernel 'polynomial' sklearn.pairwise.polynomial_kernel 'rbf' sklearn.pairwise.rbf_kernel 'laplacian' sklearn.pairwise.laplacian_kernel 'sigmoid' sklearn.pairwise.sigmoid_kernel 'cosine' sklearn.pairwise.cosine_similarity =============== ======================================== Read more in the :ref:`User Guide `. """ return PAIRWISE_KERNEL_FUNCTIONS KERNEL_PARAMS = { "additive_chi2": (), "chi2": frozenset(["gamma"]), "cosine": (), "linear": (), "poly": frozenset(["gamma", "degree", "coef0"]), "polynomial": frozenset(["gamma", "degree", "coef0"]), "rbf": frozenset(["gamma"]), "laplacian": frozenset(["gamma"]), "sigmoid": frozenset(["gamma", "coef0"]), } def pairwise_kernels(X, Y=None, metric="linear", filter_params=False, n_jobs=None, **kwds): """Compute the kernel between arrays X and optional array Y. This method takes either a vector array or a kernel matrix, and returns a kernel matrix. If the input is a vector array, the kernels are computed. If the input is a kernel matrix, it is returned instead. This method provides a safe way to take a kernel matrix as input, while preserving compatibility with many other algorithms that take a vector array. If Y is given (default is None), then the returned matrix is the pairwise kernel between the arrays from both X and Y. Valid values for metric are:: ['rbf', 'sigmoid', 'polynomial', 'poly', 'linear', 'cosine'] Read more in the :ref:`User Guide `. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \ [n_samples_a, n_features] otherwise Array of pairwise kernels between samples, or a feature array. Y : array [n_samples_b, n_features] A second feature array only if X has shape [n_samples_a, n_features]. metric : string, or callable The metric to use when calculating kernel between instances in a feature array. If metric is a string, it must be one of the metrics in pairwise.PAIRWISE_KERNEL_FUNCTIONS. If metric is "precomputed", X is assumed to be a kernel matrix. Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from X as input and return a value indicating the distance between them. filter_params : boolean Whether to filter invalid parameters or not. n_jobs : int or None, optional (default=None) The number of jobs to use for the computation. This works by breaking down the pairwise matrix into n_jobs even slices and computing them in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. **kwds : optional keyword parameters Any further parameters are passed directly to the kernel function. Returns ------- K : array [n_samples_a, n_samples_a] or [n_samples_a, n_samples_b] A kernel matrix K such that K_{i, j} is the kernel between the ith and jth vectors of the given matrix X, if Y is None. If Y is not None, then K_{i, j} is the kernel between the ith array from X and the jth array from Y. Notes ----- If metric is 'precomputed', Y is ignored and X is returned. """ # import GPKernel locally to prevent circular imports from ..gaussian_process.kernels import Kernel as GPKernel if metric == "precomputed": X, _ = check_pairwise_arrays(X, Y, precomputed=True) return X elif isinstance(metric, GPKernel): func = metric.__call__ elif metric in PAIRWISE_KERNEL_FUNCTIONS: if filter_params: kwds = dict((k, kwds[k]) for k in kwds if k in KERNEL_PARAMS[metric]) func = PAIRWISE_KERNEL_FUNCTIONS[metric] elif callable(metric): func = partial(_pairwise_callable, metric=metric, **kwds) else: raise ValueError("Unknown kernel %r" % metric) return _parallel_pairwise(X, Y, func, n_jobs, **kwds)