Matching

Fast Approximate QAP

class graspy.match.FastApproximateQAP(n_init=1, init_method='barycenter', max_iter=30, shuffle_input=True, eps=0.1, gmp=False)[source]

Fast Approximate QAP Algorithm (FAQ) The FAQ algorithm solves the Quadratic Assignment Problem (QAP) finding an alignment of the vertices of two graphs which minimizes the number of induced edge disagreements [1].

Parameters:
n_init : int, positive (default = 1)

Number of random initializations of the starting permutation matrix that the FAQ algorithm will undergo. n_init automatically set to 1 if init_method = 'barycenter'

init_method : string (default = 'barycenter')

The initial position chosen

"barycenter" : the non-informative “flat doubly stochastic matrix,” \(J=1*1^T /n\) , i.e the barycenter of the feasible region

"rand" : some random point near \(J, (J+K)/2\), where K is some random doubly stochastic matrix

max_iter : int, positive (default = 30)

Integer specifying the max number of FW iterations. FAQ typically converges with modest number of iterations

shuffle_input : bool (default = True)

Gives users the option to shuffle the nodes of A matrix to avoid results from inputs that were already matched

eps : float (default = 0.1)

A positive, threshold stopping criteria such that FW continues to iterate while Frobenius norm of \((P_{i}-P_{i+1}) > eps\)

gmp : bool (default = False)

Gives users the option to the Graph Matching Problem (GMP) rather than the Quadratic Assignment (QAP). This is accomplished through trivial negation of the objective function.

Attributes:
perm_inds_ : array, size (n,) where n is the number of vertices in the graphs fitted.

The indices of the optimal permutation on the nodes of B, found via FAQ, to best minimize the objective function \(f(P) = trace(A^T PBP^T )\).

score_ : float

The objective function value of for the optimal permutation found.

References

[R4d04029f4490-1]J. T. Vogelstein, J. M. Conroy, V. Lyzinski, L. J. Podrazik, S. G. Kratzer, E. T. Harley, D. E. Fishkind, R. J. Vogelstein, and C. E. Priebe, “Fast approximate quadratic programming for graph matching,” PLOS one, vol. 10, no. 4, p. e0121002, 2015.
fit(self, A, B)[source]

Fits the model with two assigned adjacency matrices

A : 2d-array, square, positive
A square, positive adjacency matrix
B : 2d-array, square, positive
A square, positive adjacency matrix
Returns:
self : returns an instance of self
fit_predict(self, A, B)[source]

Fits the model with two assigned adjacency matrices, returning optimal permutation indices

A : 2d-array, square, positive
A square, positive adjacency matrix
B : 2d-array, square, positive
A square, positive adjacency matrix
Returns:
perm_inds_ : 1-d array, some shuffling of [0, n_vert)

The optimal permutation indices to minimize the objective function

Sinkhorn-Knopp Algorithm

class graspy.match.SinkhornKnopp(max_iter=1000, epsilon=0.001)[source]

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

[Rf58a442f29b7-1]Sinkhorn, Richard & Knopp, Paul. (1967). "Concerning nonnegative matrices and doubly stochastic matrices," Pacific Journal of Mathematics. 21. 10.2140/pjm.1967.21.343.
fit(self, P)[source]

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.