# -*- coding: utf-8 -*-
"""
Various useful functions
"""
# Author: Remi Flamary <remi.flamary@unice.fr>
#
# License: MIT License
from functools import reduce
import time
import numpy as np
from scipy.spatial.distance import cdist
import sys
import warnings
from inspect import signature
from .backend import get_backend, Backend, NumpyBackend, JaxBackend
__time_tic_toc = time.time()
[docs]
def tic():
r"""Python implementation of Matlab tic() function"""
global __time_tic_toc
__time_tic_toc = time.time()
[docs]
def toc(message="Elapsed time : {} s"):
r"""Python implementation of Matlab toc() function"""
t = time.time()
print(message.format(t - __time_tic_toc))
return t - __time_tic_toc
[docs]
def toq():
r"""Python implementation of Julia toc() function"""
t = time.time()
return t - __time_tic_toc
[docs]
def kernel(x1, x2, method="gaussian", sigma=1, **kwargs):
r"""Compute kernel matrix"""
nx = get_backend(x1, x2)
if method.lower() in ["gaussian", "gauss", "rbf"]:
K = nx.exp(-dist(x1, x2) / (2 * sigma**2))
return K
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def laplacian(x):
r"""Compute Laplacian matrix"""
nx = get_backend(x)
L = nx.diag(nx.sum(x, axis=0)) - x
return L
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def list_to_array(*lst, nx=None):
r"""Convert a list if in numpy format"""
lst_not_empty = [a for a in lst if len(a) > 0 and not isinstance(a, list)]
if nx is None: # find backend
if len(lst_not_empty) == 0:
type_as = np.zeros(0)
nx = get_backend(type_as)
else:
nx = get_backend(*lst_not_empty)
type_as = lst_not_empty[0]
else:
if len(lst_not_empty) == 0:
type_as = None
else:
type_as = lst_not_empty[0]
if len(lst) > 1:
return [
nx.from_numpy(np.array(a), type_as=type_as) if isinstance(a, list) else a
for a in lst
]
else:
if isinstance(lst[0], list):
return nx.from_numpy(np.array(lst[0]), type_as=type_as)
else:
return lst[0]
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def proj_simplex(v, z=1):
r"""Compute the closest point (orthogonal projection) on the
generalized `(n-1)`-simplex of a vector :math:`\mathbf{v}` wrt. to the Euclidean
distance, thus solving:
.. math::
\mathcal{P}(w) \in \mathop{\arg \min}_\gamma \| \gamma - \mathbf{v} \|_2
s.t. \ \gamma^T \mathbf{1} = z
\gamma \geq 0
If :math:`\mathbf{v}` is a 2d array, compute all the projections wrt. axis 0
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends.
Parameters
----------
v : {array-like}, shape (n, d)
z : int, optional
'size' of the simplex (each vectors sum to z, 1 by default)
Returns
-------
h : ndarray, shape (`n`, `d`)
Array of projections on the simplex
"""
nx = get_backend(v)
n = v.shape[0]
if v.ndim == 1:
d1 = 1
v = v[:, None]
else:
d1 = 0
d = v.shape[1]
# sort u in ascending order
u = nx.sort(v, axis=0)
# take the descending order
u = nx.flip(u, 0)
cssv = nx.cumsum(u, axis=0) - z
ind = nx.arange(n, type_as=v)[:, None] + 1
cond = u - cssv / ind > 0
rho = nx.sum(cond, 0)
theta = cssv[rho - 1, nx.arange(d)] / rho
w = nx.maximum(v - theta[None, :], nx.zeros(v.shape, type_as=v))
if d1:
return w[:, 0]
else:
return w
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def projection_sparse_simplex(V, max_nz, z=1, axis=None, nx=None):
r"""Projection of :math:`\mathbf{V}` onto the simplex with cardinality constraint (maximum number of non-zero elements) and then scaled by `z`.
.. math::
P\left(\mathbf{V}, max_nz, z\right) = \mathop{\arg \min}_{\substack{\mathbf{y} >= 0 \\ \sum_i \mathbf{y}_i = z} \\ ||p||_0 \le \text{max_nz}} \quad \|\mathbf{y} - \mathbf{V}\|^2
Parameters
----------
V: 1-dim or 2-dim ndarray
z: float or array
If array, len(z) must be compatible with :math:`\mathbf{V}`
axis: None or int
- axis=None: project :math:`\mathbf{V}` by :math:`P(\mathbf{V}.\mathrm{ravel}(), max_nz, z)`
- axis=1: project each :math:`\mathbf{V}_i` by :math:`P(\mathbf{V}_i, max_nz, z_i)`
- axis=0: project each :math:`\mathbf{V}_{:, j}` by :math:`P(\mathbf{V}_{:, j}, max_nz, z_j)`
Returns
-------
projection: ndarray, shape :math:`\mathbf{V}`.shape
References:
Sparse projections onto the simplex
Anastasios Kyrillidis, Stephen Becker, Volkan Cevher and, Christoph Koch
ICML 2013
https://arxiv.org/abs/1206.1529
"""
if nx is None:
nx = get_backend(V)
if V.ndim == 1:
return projection_sparse_simplex(
# V[nx.newaxis, :], max_nz, z, axis=1).ravel()
V[None, :],
max_nz,
z,
axis=1,
).ravel()
if V.ndim > 2:
raise ValueError("V.ndim must be <= 2")
if axis == 1:
# For each row of V, find top max_nz values; arrange the
# corresponding column indices such that their values are
# in a descending order.
max_nz_indices = nx.argsort(V, axis=1)[:, -max_nz:]
max_nz_indices = nx.flip(max_nz_indices, axis=1)
row_indices = nx.arange(V.shape[0])
row_indices = row_indices.reshape(-1, 1)
print(row_indices.shape)
# Extract the top max_nz values for each row
# and then project to simplex.
U = V[row_indices, max_nz_indices]
z = nx.ones(len(U)) * z
cssv = nx.cumsum(U, axis=1) - z[:, None]
ind = nx.arange(max_nz) + 1
cond = U - cssv / ind > 0
# rho = nx.count_nonzero(cond, axis=1)
rho = nx.sum(cond, axis=1)
theta = cssv[nx.arange(len(U)), rho - 1] / rho
nz_projection = nx.maximum(U - theta[:, None], 0)
# Put the projection of max_nz_values to their original column indices
# while keeping other values zero.
sparse_projection = nx.zeros(V.shape, type_as=nz_projection)
if isinstance(nx, JaxBackend):
# in Jax, we need to use the `at` property of `jax.numpy.ndarray`
# to do in-place array modifications.
sparse_projection = sparse_projection.at[row_indices, max_nz_indices].set(
nz_projection
)
else:
sparse_projection[row_indices, max_nz_indices] = nz_projection
return sparse_projection
elif axis == 0:
return projection_sparse_simplex(V.T, max_nz, z, axis=1).T
else:
V = V.ravel().reshape(1, -1)
return projection_sparse_simplex(V, max_nz, z, axis=1).ravel()
[docs]
def unif(n, type_as=None):
r"""
Return a uniform histogram of length `n` (simplex).
Parameters
----------
n : int
number of bins in the histogram
type_as : array_like
array of the same type of the expected output (numpy/pytorch/jax)
Returns
-------
h : array_like (`n`,)
histogram of length `n` such that :math:`\forall i, \mathbf{h}_i = \frac{1}{n}`
"""
if type_as is None:
return np.ones((n,)) / n
else:
nx = get_backend(type_as)
return nx.ones((n,), type_as=type_as) / n
[docs]
def clean_zeros(a, b, M):
r"""Remove all components with zeros weights in :math:`\mathbf{a}` and :math:`\mathbf{b}`"""
M2 = M[a > 0, :][:, b > 0].copy() # copy force c style matrix (froemd)
a2 = a[a > 0]
b2 = b[b > 0]
return a2, b2, M2
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def euclidean_distances(X, Y, squared=False):
r"""
Considering the rows of :math:`\mathbf{X}` (and :math:`\mathbf{Y} = \mathbf{X}`) as vectors, compute the
distance matrix between each pair of vectors.
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends.
Parameters
----------
X : array-like, shape (n_samples_1, n_features)
Y : array-like, shape (n_samples_2, n_features)
squared : boolean, optional
Return squared Euclidean distances.
Returns
-------
distances : array-like, shape (`n_samples_1`, `n_samples_2`)
"""
nx = get_backend(X, Y)
a2 = nx.einsum("ij,ij->i", X, X)
b2 = nx.einsum("ij,ij->i", Y, Y)
c = -2 * nx.dot(X, Y.T)
c += a2[:, None]
c += b2[None, :]
c = nx.maximum(c, 0)
if not squared:
c = nx.sqrt(c)
if X is Y:
c = c * (1 - nx.eye(X.shape[0], type_as=c))
return c
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def dist(x1, x2=None, metric="sqeuclidean", p=2, w=None):
r"""Compute distance between samples in :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends.
Parameters
----------
x1 : array-like, shape (n1,d)
matrix with `n1` samples of size `d`
x2 : array-like, shape (n2,d), optional
matrix with `n2` samples of size `d` (if None then :math:`\mathbf{x_2} = \mathbf{x_1}`)
metric : str | callable, optional
'sqeuclidean' or 'euclidean' on all backends. On numpy the function also
accepts from the scipy.spatial.distance.cdist function : 'braycurtis',
'canberra', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice',
'euclidean', 'hamming', 'jaccard', 'kulczynski1', 'mahalanobis',
'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean',
'sokalmichener', 'sokalsneath', 'sqeuclidean', 'wminkowski', 'yule'.
p : float, optional
p-norm for the Minkowski and the Weighted Minkowski metrics. Default value is 2.
w : array-like, rank 1
Weights for the weighted metrics.
Returns
-------
M : array-like, shape (`n1`, `n2`)
distance matrix computed with given metric
"""
if x2 is None:
x2 = x1
if metric == "sqeuclidean":
return euclidean_distances(x1, x2, squared=True)
elif metric == "euclidean":
return euclidean_distances(x1, x2, squared=False)
else:
if not get_backend(x1, x2).__name__ == "numpy":
raise NotImplementedError()
else:
if isinstance(metric, str) and metric.endswith("minkowski"):
return cdist(x1, x2, metric=metric, p=p, w=w)
if w is not None:
return cdist(x1, x2, metric=metric, w=w)
return cdist(x1, x2, metric=metric)
[docs]
def dist0(n, method="lin_square"):
r"""Compute standard cost matrices of size (`n`, `n`) for OT problems
Parameters
----------
n : int
Size of the cost matrix.
method : str, optional
Type of loss matrix chosen from:
* 'lin_square' : linear sampling between 0 and `n-1`, quadratic loss
Returns
-------
M : ndarray, shape (`n1`, `n2`)
Distance matrix computed with given metric.
"""
res = 0
if method == "lin_square":
x = np.arange(n, dtype=np.float64).reshape((n, 1))
res = dist(x, x)
return res
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def cost_normalization(C, norm=None, return_value=False, value=None):
r"""Apply normalization to the loss matrix
Parameters
----------
C : ndarray, shape (n1, n2)
The cost matrix to normalize.
norm : str
Type of normalization from 'median', 'max', 'log', 'loglog'. Any
other value do not normalize.
Returns
-------
C : ndarray, shape (`n1`, `n2`)
The input cost matrix normalized according to given norm.
"""
nx = get_backend(C)
if norm is None:
pass
elif norm == "median":
if value is None:
value = nx.median(C)
C /= value
elif norm == "max":
if value is None:
value = nx.max(C)
C /= float(value)
elif norm == "log":
C = nx.log(1 + C)
elif norm == "loglog":
C = nx.log(1 + nx.log(1 + C))
else:
raise ValueError(
"Norm %s is not a valid option.\n"
"Valid options are:\n"
"median, max, log, loglog" % norm
)
if return_value:
return C, value
else:
return C
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def dots(*args):
r"""dots function for multiple matrix multiply"""
nx = get_backend(*args)
return reduce(nx.dot, args)
[docs]
def is_all_finite(*args):
r"""Tests element-wise for finiteness in all arguments."""
nx = get_backend(*args)
return all(not nx.any(~nx.isfinite(arg)) for arg in args)
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def label_normalization(y, start=0, nx=None):
r"""Transform labels to start at a given value
Parameters
----------
y : array-like, shape (n, )
The vector of labels to be normalized.
start : int
Desired value for the smallest label in :math:`\mathbf{y}` (default=0)
nx : Backend, optional
Backend to perform computations on. If omitted, the backend defaults to that of `y`.
Returns
-------
y : array-like, shape (`n1`, )
The input vector of labels normalized according to given start value.
"""
if nx is None:
nx = get_backend(y)
diff = nx.min(y) - start
return y if diff == 0 else (y - diff)
[docs]
def labels_to_masks(y, type_as=None, nx=None):
r"""Transforms (n_samples,) vector of labels into a (n_samples, n_labels) matrix of masks.
Parameters
----------
y : array-like, shape (n_samples, )
The vector of labels.
type_as : array_like
Array of the same type of the expected output.
nx : Backend, optional
Backend to perform computations on. If omitted, the backend defaults to that of `y`.
Returns
-------
masks : array-like, shape (n_samples, n_labels)
The (n_samples, n_labels) matrix of label masks.
"""
if nx is None:
nx = get_backend(y)
if type_as is None:
type_as = y
labels_u, labels_idx = nx.unique(y, return_inverse=True)
n_labels = labels_u.shape[0]
masks = nx.eye(n_labels, type_as=type_as)[labels_idx]
return masks
[docs]
def parmap(f, X, nprocs="default"):
r"""parallel map for multiprocessing.
The function has been deprecated and only performs a regular map.
"""
return list(map(f, X))
[docs]
def check_params(**kwargs):
r"""check_params: check whether some parameters are missing"""
missing_params = []
check = True
for param in kwargs:
if kwargs[param] is None:
missing_params.append(param)
if len(missing_params) > 0:
print("POT - Warning: following necessary parameters are missing")
for p in missing_params:
print("\n", p)
check = False
return check
[docs]
def check_random_state(seed):
r"""Turn `seed` into a np.random.RandomState instance
Parameters
----------
seed : None | int | instance of RandomState
If `seed` is None, return the RandomState singleton used by np.random.
If `seed` is an int, return a new RandomState instance seeded with `seed`.
If `seed` is already a RandomState instance, return it.
Otherwise raise ValueError.
"""
if seed is None or seed is np.random:
return np.random.mtrand._rand
if isinstance(seed, (int, np.integer)):
return np.random.RandomState(seed)
if isinstance(seed, np.random.RandomState):
return seed
raise ValueError(
"{} cannot be used to seed a numpy.random.RandomState" " instance".format(seed)
)
[docs]
def get_coordinate_circle(x):
r"""For :math:`x\in S^1 \subset \mathbb{R}^2`, returns the coordinates in
turn (in [0,1[).
.. math::
u = \frac{\pi + \mathrm{atan2}(-x_2,-x_1)}{2\pi}
Parameters
----------
x: ndarray, shape (n, 2)
Samples on the circle with ambient coordinates
Returns
-------
x_t: ndarray, shape (n,)
Coordinates on [0,1[
Examples
--------
>>> u = np.array([[0.2,0.5,0.8]]) * (2 * np.pi)
>>> x1, y1 = np.cos(u), np.sin(u)
>>> x = np.concatenate([x1, y1]).T
>>> get_coordinate_circle(x)
array([0.2, 0.5, 0.8])
"""
nx = get_backend(x)
x_t = (nx.atan2(-x[:, 1], -x[:, 0]) + np.pi) / (2 * np.pi)
return x_t
[docs]
def reduce_lazytensor(a, func, axis=None, nx=None, batch_size=100):
"""Reduce a LazyTensor along an axis with function fun using batches.
When axis=None, reduce the LazyTensor to a scalar as a sum of fun over
batches taken along dim.
.. warning::
This function works for tensor of any order but the reduction can be done
only along the first two axis (or global). Also, in order to work, it requires that the slice of size `batch_size` along the axis to reduce (or axis 0 if `axis=None`) is can be computed and fits in memory.
Parameters
----------
a : LazyTensor
LazyTensor to reduce
func : callable
Function to apply to the LazyTensor
axis : int, optional
Axis along which to reduce the LazyTensor. If None, reduce the
LazyTensor to a scalar as a sum of fun over batches taken along axis 0.
If 0 or 1 reduce the LazyTensor to a vector/matrix as a sum of fun over
batches taken along axis.
nx : Backend, optional
Backend to use for the reduction
batch_size : int, optional
Size of the batches to use for the reduction (default=100)
Returns
-------
res : array-like
Result of the reduction
"""
if nx is None:
nx = get_backend(a[0:1])
if axis is None:
res = 0.0
for i in range(0, a.shape[0], batch_size):
res += func(a[i : i + batch_size])
return res
elif axis == 0:
res = nx.zeros(a.shape[1:], type_as=a[0])
if nx.__name__ in ["jax", "tf"]:
lst = []
for j in range(0, a.shape[1], batch_size):
lst.append(func(a[:, j : j + batch_size], 0))
return nx.concatenate(lst, axis=0)
else:
for j in range(0, a.shape[1], batch_size):
res[j : j + batch_size] = func(a[:, j : j + batch_size], axis=0)
return res
elif axis == 1:
if len(a.shape) == 2:
shape = a.shape[0]
else:
shape = (a.shape[0], *a.shape[2:])
res = nx.zeros(shape, type_as=a[0])
if nx.__name__ in ["jax", "tf"]:
lst = []
for i in range(0, a.shape[0], batch_size):
lst.append(func(a[i : i + batch_size], 1))
return nx.concatenate(lst, axis=0)
else:
for i in range(0, a.shape[0], batch_size):
res[i : i + batch_size] = func(a[i : i + batch_size], axis=1)
return res
else:
raise (NotImplementedError("Only axis=None, 0 or 1 is implemented for now."))
[docs]
def get_lowrank_lazytensor(Q, R, d=None, nx=None):
"""Get a low rank LazyTensor T=Q@R^T or T=Q@diag(d)@R^T
Parameters
----------
Q : ndarray, shape (n, r)
First factor of the lowrank tensor
R : ndarray, shape (m, r)
Second factor of the lowrank tensor
d : ndarray, shape (r,), optional
Diagonal of the lowrank tensor
nx : Backend, optional
Backend to use for the reduction
Returns
-------
T : LazyTensor
Lowrank tensor T=Q@R^T or T=Q@diag(d)@R^T
"""
if nx is None:
nx = get_backend(Q, R, d)
shape = (Q.shape[0], R.shape[0])
if d is None:
def func(i, j, Q, R):
return nx.dot(Q[i], R[j].T)
T = LazyTensor(shape, func, Q=Q, R=R)
else:
def func(i, j, Q, R, d):
return nx.dot(Q[i] * d[None, :], R[j].T)
T = LazyTensor(shape, func, Q=Q, R=R, d=d)
return T
[docs]
def get_parameter_pair(parameter):
r"""Extract a pair of parameters from a given parameter
Used in unbalanced OT and COOT solvers
to handle marginal regularization and entropic regularization.
Parameters
----------
parameter : float or indexable object
nx : backend object
Returns
-------
param_1 : float
param_2 : float
"""
if isinstance(parameter, float) or isinstance(parameter, int):
param_1, param_2 = parameter, parameter
elif len(parameter) == 1:
param_1, param_2 = parameter[0], parameter[0]
else:
if len(parameter) > 2:
raise ValueError(
"Parameter must be either a scalar, \
or an indexable object of length 1 or 2."
)
else:
param_1, param_2 = parameter[0], parameter[1]
return param_1, param_2
[docs]
class deprecated(object):
r"""Decorator to mark a function or class as deprecated.
deprecated class from scikit-learn package
https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/utils/deprecation.py
Issue a warning when the function is called/the class is instantiated and
adds a warning to the docstring.
The optional extra argument will be appended to the deprecation message
and the docstring.
.. note::
To use this with the default value for extra, use empty parentheses:
>>> from ot.deprecation import deprecated # doctest: +SKIP
>>> @deprecated() # doctest: +SKIP
... def some_function(): pass # doctest: +SKIP
Parameters
----------
extra : str
To be added to the deprecation messages.
"""
# Adapted from http://wiki.python.org/moin/PythonDecoratorLibrary,
# but with many changes.
def __init__(self, extra=""):
self.extra = extra
def __call__(self, obj):
r"""Call method
Parameters
----------
obj : object
"""
if isinstance(obj, type):
return self._decorate_class(obj)
else:
return self._decorate_fun(obj)
def _decorate_class(self, cls):
msg = "Class %s is deprecated" % cls.__name__
if self.extra:
msg += "; %s" % self.extra
# FIXME: we should probably reset __new__ for full generality
init = cls.__init__
def wrapped(*args, **kwargs):
warnings.warn(msg, category=DeprecationWarning)
return init(*args, **kwargs)
cls.__init__ = wrapped
wrapped.__name__ = "__init__"
wrapped.__doc__ = self._update_doc(init.__doc__)
wrapped.deprecated_original = init
return cls
def _decorate_fun(self, fun):
r"""Decorate function fun"""
msg = "Function %s is deprecated" % fun.__name__
if self.extra:
msg += "; %s" % self.extra
def wrapped(*args, **kwargs):
warnings.warn(msg, category=DeprecationWarning)
return fun(*args, **kwargs)
wrapped.__name__ = fun.__name__
wrapped.__dict__ = fun.__dict__
wrapped.__doc__ = self._update_doc(fun.__doc__)
return wrapped
def _update_doc(self, olddoc):
newdoc = "DEPRECATED"
if self.extra:
newdoc = "%s: %s" % (newdoc, self.extra)
if olddoc:
newdoc = "%s\n\n%s" % (newdoc, olddoc)
return newdoc
def _is_deprecated(func):
r"""Helper to check if func is wrapped by our deprecated decorator"""
if sys.version_info < (3, 5):
raise NotImplementedError("This is only available for python3.5 " "or above")
closures = getattr(func, "__closure__", [])
if closures is None:
closures = []
is_deprecated = "deprecated" in "".join(
[c.cell_contents for c in closures if isinstance(c.cell_contents, str)]
)
return is_deprecated
[docs]
class BaseEstimator(object):
r"""Base class for most objects in POT
Code adapted from sklearn BaseEstimator class
Notes
-----
All estimators should specify all the parameters that can be set
at the class level in their ``__init__`` as explicit keyword
arguments (no ``*args`` or ``**kwargs``).
"""
nx: Backend = None
def _get_backend(self, *arrays):
nx = get_backend(*[input_ for input_ in arrays if input_ is not None])
if nx.__name__ in ("tf",):
raise TypeError("Domain adaptation does not support TF backend.")
self.nx = nx
return nx
@classmethod
def _get_param_names(cls):
r"""Get parameter names for the estimator"""
# fetch the constructor or the original constructor before
# deprecation wrapping if any
init = getattr(cls.__init__, "deprecated_original", cls.__init__)
if init is object.__init__:
# No explicit constructor to introspect
return []
# introspect the constructor arguments to find the model parameters
# to represent
init_signature = signature(init)
# Consider the constructor parameters excluding 'self'
parameters = [
p
for p in init_signature.parameters.values()
if p.name != "self" and p.kind != p.VAR_KEYWORD
]
for p in parameters:
if p.kind == p.VAR_POSITIONAL:
raise RuntimeError(
"POT estimators should always "
"specify their parameters in the signature"
" of their __init__ (no varargs)."
" %s with constructor %s doesn't "
" follow this convention." % (cls, init_signature)
)
# Extract and sort argument names excluding 'self'
return sorted([p.name for p in parameters])
[docs]
def get_params(self, deep=True):
r"""Get parameters for this estimator.
Parameters
----------
deep : bool, optional
If True, will return the parameters for this estimator and
contained subobjects that are estimators.
Returns
-------
params : mapping of string to any
Parameter names mapped to their values.
"""
out = dict()
for key in self._get_param_names():
# We need deprecation warnings to always be on in order to
# catch deprecated param values.
# This is set in utils/__init__.py but it gets overwritten
# when running under python3 somehow.
warnings.simplefilter("always", DeprecationWarning)
try:
with warnings.catch_warnings(record=True) as w:
value = getattr(self, key, None)
if len(w) and isinstance(w[0].category, DeprecationWarning):
# if the parameter is deprecated, don't show it
continue
finally:
warnings.filters.pop(0)
# XXX: should we rather test if instance of estimator?
if deep and hasattr(value, "get_params"):
deep_items = value.get_params().items()
out.update((key + "__" + k, val) for k, val in deep_items)
out[key] = value
return out
[docs]
def set_params(self, **params):
r"""Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects
(such as pipelines). The latter have parameters of the form
``<component>__<parameter>`` so that it's possible to update each
component of a nested object.
Returns
-------
self
"""
if not params:
# Simple optimisation to gain speed (inspect is slow)
return self
valid_params = self.get_params(deep=True)
# for key, value in iteritems(params):
for key, value in params.items():
split = key.split("__", 1)
if len(split) > 1:
# nested objects case
name, sub_name = split
if name not in valid_params:
raise ValueError(
"Invalid parameter %s for estimator %s. "
"Check the list of available parameters "
"with `estimator.get_params().keys()`." % (name, self)
)
sub_object = valid_params[name]
sub_object.set_params(**{sub_name: value})
else:
# simple objects case
if key not in valid_params:
raise ValueError(
"Invalid parameter %s for estimator %s. "
"Check the list of available parameters "
"with `estimator.get_params().keys()`."
% (key, self.__class__.__name__)
)
setattr(self, key, value)
return self
[docs]
class UndefinedParameter(Exception):
r"""
Aim at raising an Exception when a undefined parameter is called
"""
pass
[docs]
class OTResult:
"""Base class for OT results.
Parameters
----------
potentials : tuple of array-like, shape (`n1`, `n2`)
Dual potentials, i.e. Lagrange multipliers for the marginal constraints.
This pair of arrays has the same shape, numerical type
and properties as the input weights "a" and "b".
value : float, array-like
Full transport cost, including possible regularization terms and
quadratic term for Gromov Wasserstein solutions.
value_linear : float, array-like
The linear part of the transport cost, i.e. the product between the
transport plan and the cost.
value_quad : float, array-like
The quadratic part of the transport cost for Gromov-Wasserstein
solutions.
plan : array-like, shape (`n1`, `n2`)
Transport plan, encoded as a dense array.
log : dict
Dictionary containing potential information about the solver.
backend : Backend
Backend used to compute the results.
sparse_plan : array-like, shape (`n1`, `n2`)
Transport plan, encoded as a sparse array.
lazy_plan : LazyTensor
Transport plan, encoded as a symbolic POT or KeOps LazyTensor.
status : int or str
Status of the solver.
batch_size : int
Batch size used to compute the results/marginals for LazyTensor.
Attributes
----------
potentials : tuple of array-like, shape (`n1`, `n2`)
Dual potentials, i.e. Lagrange multipliers for the marginal constraints.
This pair of arrays has the same shape, numerical type
and properties as the input weights "a" and "b".
potential_a : array-like, shape (`n1`,)
First dual potential, associated to the "source" measure "a".
potential_b : array-like, shape (`n2`,)
Second dual potential, associated to the "target" measure "b".
value : float, array-like
Full transport cost, including possible regularization terms and
quadratic term for Gromov Wasserstein solutions.
value_linear : float, array-like
The linear part of the transport cost, i.e. the product between the
transport plan and the cost.
value_quad : float, array-like
The quadratic part of the transport cost for Gromov-Wasserstein
solutions.
plan : array-like, shape (`n1`, `n2`)
Transport plan, encoded as a dense array.
sparse_plan : array-like, shape (`n1`, `n2`)
Transport plan, encoded as a sparse array.
lazy_plan : LazyTensor
Transport plan, encoded as a symbolic POT or KeOps LazyTensor.
marginals : tuple of array-like, shape (`n1`,), (`n2`,)
Marginals of the transport plan: should be very close to "a" and "b"
for balanced OT.
marginal_a : array-like, shape (`n1`,)
Marginal of the transport plan for the "source" measure "a".
marginal_b : array-like, shape (`n2`,)
Marginal of the transport plan for the "target" measure "b".
"""
def __init__(
self,
potentials=None,
value=None,
value_linear=None,
value_quad=None,
plan=None,
log=None,
backend=None,
sparse_plan=None,
lazy_plan=None,
status=None,
batch_size=100,
):
self._potentials = potentials
self._value = value
self._value_linear = value_linear
self._value_quad = value_quad
self._plan = plan
self._log = log
self._sparse_plan = sparse_plan
self._lazy_plan = lazy_plan
self._backend = backend if backend is not None else NumpyBackend()
self._status = status
self._batch_size = batch_size
# I assume that other solvers may return directly
# some primal objects?
# In the code below, let's define the main quantities
# that may be of interest to users.
# An OT solver returns an object that inherits from OTResult
# (e.g. SinkhornOTResult) and implements the relevant
# methods (e.g. "plan" and "lazy_plan" but not "sparse_plan", etc.).
# log is a dictionary containing potential information about the solver
# Dual potentials --------------------------------------------
def __repr__(self):
s = "OTResult("
if self._value is not None:
s += "value={},".format(self._value)
if self._value_linear is not None:
s += "value_linear={},".format(self._value_linear)
if self._plan is not None:
s += "plan={}(shape={}),".format(
self._plan.__class__.__name__, self._plan.shape
)
if self._lazy_plan is not None:
s += "lazy_plan={}(shape={}),".format(
self._lazy_plan.__class__.__name__, self._lazy_plan.shape
)
if s[-1] != "(":
s = s[:-1] + ")"
else:
s = s + ")"
return s
@property
def potentials(self):
"""Dual potentials, i.e. Lagrange multipliers for the marginal constraints.
This pair of arrays has the same shape, numerical type
and properties as the input weights "a" and "b".
"""
return self._potentials
@property
def potential_a(self):
"""First dual potential, associated to the "source" measure "a"."""
if self._potentials is not None:
return self._potentials[0]
else:
return None
@property
def potential_b(self):
"""Second dual potential, associated to the "target" measure "b"."""
if self._potentials is not None:
return self._potentials[1]
else:
return None
# Transport plan -------------------------------------------
@property
def plan(self):
"""Transport plan, encoded as a dense array."""
# N.B.: We may catch out-of-memory errors and suggest
# the use of lazy_plan or sparse_plan when appropriate.
return self._plan
@property
def sparse_plan(self):
"""Transport plan, encoded as a sparse array."""
if self._sparse_plan is not None:
return self._sparse_plan
elif self._plan is not None:
return self._backend.tocsr(self._plan)
else:
return None
@property
def lazy_plan(self):
"""Transport plan, encoded as a symbolic KeOps LazyTensor."""
return self._lazy_plan
# Loss values --------------------------------
@property
def value(self):
"""Full transport cost, including possible regularization terms and
quadratic term for Gromov Wasserstein solutions."""
return self._value
@property
def value_linear(self):
"""The "minimal" transport cost, i.e. the product between the transport plan and the cost."""
return self._value_linear
@property
def value_quad(self):
"""The quadratic part of the transport cost for Gromov-Wasserstein solutions."""
return self._value_quad
# Marginal constraints -------------------------
@property
def marginals(self):
"""Marginals of the transport plan: should be very close to "a" and "b"
for balanced OT."""
if self._plan is not None:
return self.marginal_a, self.marginal_b
else:
return None
@property
def marginal_a(self):
"""First marginal of the transport plan, with the same shape as "a"."""
if self._plan is not None:
return self._backend.sum(self._plan, 1)
elif self._lazy_plan is not None:
lp = self._lazy_plan
bs = self._batch_size
nx = self._backend
return reduce_lazytensor(lp, nx.sum, axis=1, nx=nx, batch_size=bs)
else:
return None
@property
def marginal_b(self):
"""Second marginal of the transport plan, with the same shape as "b"."""
if self._plan is not None:
return self._backend.sum(self._plan, 0)
elif self._lazy_plan is not None:
lp = self._lazy_plan
bs = self._batch_size
nx = self._backend
return reduce_lazytensor(lp, nx.sum, axis=0, nx=nx, batch_size=bs)
else:
return None
@property
def status(self):
"""Optimization status of the solver."""
return self._status
@property
def log(self):
"""Dictionary containing potential information about the solver."""
return self._log
# Barycentric mappings -------------------------
# Return the displacement vectors as an array
# that has the same shape as "xa"/"xb" (for samples)
# or "a"/"b" * D (for images)?
@property
def a_to_b(self):
"""Displacement vectors from the first to the second measure."""
raise NotImplementedError()
@property
def b_to_a(self):
"""Displacement vectors from the second to the first measure."""
raise NotImplementedError()
# # Wasserstein barycenters ----------------------
# @property
# def masses(self):
# """Masses for the Wasserstein barycenter."""
# raise NotImplementedError()
# @property
# def samples(self):
# """Sample locations for the Wasserstein barycenter."""
# raise NotImplementedError()
# Miscellaneous --------------------------------
@property
def citation(self):
"""Appropriate citation(s) for this result, in plain text and BibTex formats."""
# The string below refers to the POT library:
# successor methods may concatenate the relevant references
# to the original definitions, solvers and underlying numerical backends.
return """POT library:
POT Python Optimal Transport library, Journal of Machine Learning Research, 22(78):1−8, 2021.
Website: https://pythonot.github.io/
Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z. Alaya, Aurélie Boisbunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, Léo Gautheron, Nathalie T.H. Gayraud, Hicham Janati, Alain Rakotomamonjy, Ievgen Redko, Antoine Rolet, Antony Schutz, Vivien Seguy, Danica J. Sutherland, Romain Tavenard, Alexander Tong, Titouan Vayer;
@article{flamary2021pot,
author = {R{\'e}mi Flamary and Nicolas Courty and Alexandre Gramfort and Mokhtar Z. Alaya and Aur{\'e}lie Boisbunon and Stanislas Chambon and Laetitia Chapel and Adrien Corenflos and Kilian Fatras and Nemo Fournier and L{\'e}o Gautheron and Nathalie T.H. Gayraud and Hicham Janati and Alain Rakotomamonjy and Ievgen Redko and Antoine Rolet and Antony Schutz and Vivien Seguy and Danica J. Sutherland and Romain Tavenard and Alexander Tong and Titouan Vayer},
title = {{POT}: {Python} {Optimal} {Transport}},
journal = {Journal of Machine Learning Research},
year = {2021},
volume = {22},
number = {78},
pages = {1-8},
url = {http://jmlr.org/papers/v22/20-451.html}
}
"""
[docs]
class LazyTensor(object):
"""A lazy tensor is a tensor that is not stored in memory. Instead, it is
defined by a function that computes its values on the fly from slices.
Parameters
----------
shape : tuple
shape of the tensor
getitem : callable
function that computes the values of the indices/slices and tensors
as arguments
kwargs : dict
named arguments for the function, those names will be used as attributed
of the LazyTensor object
Examples
--------
>>> import numpy as np
>>> v = np.arange(5)
>>> def getitem(i,j, v):
... return v[i,None]+v[None,j]
>>> T = LazyTensor((5,5),getitem, v=v)
>>> T[1,2]
array([3])
>>> T[1,:]
array([[1, 2, 3, 4, 5]])
>>> T[:]
array([[0, 1, 2, 3, 4],
[1, 2, 3, 4, 5],
[2, 3, 4, 5, 6],
[3, 4, 5, 6, 7],
[4, 5, 6, 7, 8]])
"""
def __init__(self, shape, getitem, **kwargs):
self._getitem = getitem
self.shape = shape
self.ndim = len(shape)
self.kwargs = kwargs
# set attributes for named arguments/arrays
for key, value in kwargs.items():
setattr(self, key, value)
def __getitem__(self, key):
k = []
if isinstance(key, int) or isinstance(key, slice):
k.append(key)
for i in range(self.ndim - 1):
k.append(slice(None))
elif isinstance(key, tuple):
k = list(key)
for i in range(self.ndim - len(key)):
k.append(slice(None))
else:
raise NotImplementedError(
"Only integer, slice, and tuple indexing is supported"
)
return self._getitem(*k, **self.kwargs)
def __repr__(self):
return "LazyTensor(shape={},attributes=({}))".format(
self.shape, ",".join(self.kwargs.keys())
)
[docs]
def proj_SDP(S, nx=None, vmin=0.0):
"""
Project a symmetric matrix onto the space of symmetric matrices with
eigenvalues larger or equal to `vmin`.
Parameters
----------
S : array_like (n, d, d) or (d, d)
The input symmetric matrix or matrices.
nx : module, optional
The numerical backend module to use. If not provided, the backend will
be fetched from the input matrix `S`.
vmin : float, optional
The minimum value for the eigenvalues. Eigenvalues below this value will
be clipped to vmin.
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends.
Returns
-------
P : ndarray (n, d, d) or (d, d)
The projected symmetric positive definite matrix.
"""
if nx is None:
nx = get_backend(S)
w, P = nx.eigh(S)
w = nx.clip(w, vmin, None)
if len(S.shape) == 2: # input was (d, d)
return P @ nx.diag(w) @ P.T
else: # input was (n, d, d): broadcasting
Q = nx.einsum("ijk,ik->ijk", P, w) # Q[i] = P[i] @ diag(w[i])
# R[i] = Q[i] @ P[i].T
return nx.einsum("ijk,ikl->ijl", Q, nx.transpose(P, (0, 2, 1)))
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