# Source code for graspy.match.skp

```
# Title: sinkhorn_knopp Source Code
# Author: Tabanpour, B
# Date: 2018
# Code version: 0.2
# Availability: https://pypi.org/project/sinkhorn_knopp/
#
# The MIT License
#
# Copyright (c) 2016 Baruch Tabanpour
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
import warnings
import numpy as np
[docs]class SinkhornKnopp:
"""
Sinkhorn Knopp Algorithm
Takes a non-negative square matrix P, where P =/= 0
and iterates through Sinkhorn Knopp's algorithm
to convert P to a doubly stochastic matrix.
Guaranteed convergence if P has total support [1]:
Parameters
----------
max_iter : int, default=1000
The maximum number of iterations.
epsilon : float, default=1e-3
Metric used to compute the stopping condition,
which occurs if all the row and column sums are
within epsilon of 1. This should be a very small value.
Epsilon must be between 0 and 1.
Attributes
----------
_max_iter : int, default=1000
User defined parameter. See above.
_epsilon : float, default=1e-3
User defined paramter. See above.
_stopping_condition: string
Either "max_iter", "epsilon", or None, which is a
description of why the algorithm stopped iterating.
_iterations : int
The number of iterations elapsed during the algorithm's
run-time.
_D1 : 2d-array
Diagonal matrix obtained after a stopping condition was met
so that _D1.dot(P).dot(_D2) is close to doubly stochastic.
_D2 : 2d-array
Diagonal matrix obtained after a stopping condition was met
so that _D1.dot(P).dot(_D2) is close to doubly stochastic.
References
----------
.. [1] Sinkhorn, Richard & Knopp, Paul. (1967). "Concerning nonnegative
matrices and doubly stochastic matrices," Pacific Journal of
Mathematics. 21. 10.2140/pjm.1967.21.343.
"""
def __init__(self, max_iter=1000, epsilon=1e-3):
if type(max_iter) is int or type(max_iter) is float:
if max_iter > 0:
self._max_iter = int(max_iter)
else:
msg = "max_iter must be greater than 0"
raise ValueError(msg)
else:
msg = "max_iter is not of type int or float"
raise TypeError(msg)
if type(epsilon) is int or type(epsilon) is float:
if epsilon > 0 and epsilon < 1:
self._epsilon = int(epsilon)
else:
msg = "epsilon must be between 0 and 1 exclusively"
raise ValueError(msg)
else:
msg = "epsilon is not of type int or float"
raise TypeError(msg)
self._stopping_condition = None
self._iterations = 0
self._D1 = np.ones(1)
self._D2 = np.ones(1)
[docs] def fit(self, P):
"""
Fit the diagonal matrices in Sinkhorn Knopp's algorithm
Parameters
----------
P : 2d array-like
Must be a square non-negative 2d array-like object, that
is convertible to a numpy array. The matrix must not be
equal to 0 and it must have total support for the algorithm
to converge.
Returns
-------
P_eps : A double stochastic matrix.
"""
P = np.asarray(P)
assert np.all(P >= 0)
assert P.ndim == 2
assert P.shape[0] == P.shape[1]
N = P.shape[0]
max_thresh = 1 + self._epsilon
min_thresh = 1 - self._epsilon
# Initialize r and c, the diagonals of D1 and D2
# and warn if the matrix does not have support.
r = np.ones((N, 1))
pdotr = P.T.dot(r)
total_support_warning_str = (
"Matrix P must have total support. " "See documentation"
)
if not np.all(pdotr != 0):
warnings.warn(total_support_warning_str, UserWarning)
c = 1 / pdotr
pdotc = P.dot(c)
if not np.all(pdotc != 0):
warnings.warn(total_support_warning_str, UserWarning)
r = 1 / pdotc
del pdotr, pdotc
P_eps = np.copy(P)
while (
np.any(np.sum(P_eps, axis=1) < min_thresh)
or np.any(np.sum(P_eps, axis=1) > max_thresh)
or np.any(np.sum(P_eps, axis=0) < min_thresh)
or np.any(np.sum(P_eps, axis=0) > max_thresh)
):
c = 1 / P.T.dot(r)
r = 1 / P.dot(c)
self._D1 = np.diag(np.squeeze(r))
self._D2 = np.diag(np.squeeze(c))
P_eps = self._D1.dot(P).dot(self._D2)
self._iterations += 1
if self._iterations >= self._max_iter:
self._stopping_condition = "max_iter"
break
if not self._stopping_condition:
self._stopping_condition = "epsilon"
self._D1 = np.diag(np.squeeze(r))
self._D2 = np.diag(np.squeeze(c))
P_eps = self._D1.dot(P).dot(self._D2)
return P_eps
```