iw.multiresolution

sparsify_matrix

iw.multiresolution.sparsify_matrix.sparsify_matrix()

sparsify_matrix function

• Inputs:

Parameters

• M (or array 1d double values) – M input matrix 1d sparse matrix to sparsified

• row (1d array 1d int_ values) – row array of sparse matrix M

• col (1d int_ values) – column array of sparse matrix M

• shape (int) – shape of M matrix

• threshold (1d array 1d double array) – array of threshold values

• Output:

Returns

tuple of (Msparsedata, Msparserow, Msparsecol, shape) where Msparsedata is M sparsified Matrix Msparserow, Msparsecol, shape corresponding row colunms and shape

Return type

tuple of array and int value

struct_multires_Lbarre

1. Tab_Struct_multires_Lbarre

class iw.multiresolution.struct_multires_Lbarre.Tab_Struct_multires_Lbarre

Bases: object

This Class computes the new subgraph, the matrix of the Laplacian of the new subgraph, the matrix Lambda (which is built through Lambdabreve and Lambdabarre from the approximate solution 1 of Diaconis-Fill equation. This means:

• Lbarre is the Schur complement of L_Rc in L with Rc the complement of the set Xbarre

• Lambda is the matrix whose rows vectors are and and keeps them in two elements Struct_Mana_re (array of Struct_M_ana_recons class) and Struct_Mres_gr (array of Struct_multires_Lbarre class).

• Inputs:

Parameters

• graph (iw.graph_c.Graph_c class.) – graph is the current graph for this step of calculation

• mu (1d double array) – measure of reversibility. In the case the laplacian is symetric it has to be the uniform measure.

• n (int) – size of the set of vertices

• mod (string) –

  define the mod of multiscale calculations:

  • ’step’ determine the number of steps for decomposition

  • ’card’ determine the minimum cardinal of graph

• m (int) – number of maximum nodes to stop the decomposition lowerbound on the size of Xbarre

• steps (int) – number of steps to stop the decomposition

• theta (double) – parameter for sparsification threshold

• Attributes:

Variables

• Struct_Mres_gr – array of iw.multiresolution.struct_multires_Lbarre.Struct_multires_Lbarre

• Struct_Mana_re – array of iw.multiresolution.struct_multires_Lbarre.Struct_M_ana_recons

• steps (int) – number of level of decomposition

Struct_Mana_re

Struct_Mres_gr

steps

2. Struct_multires_Lbarre

class iw.multiresolution.struct_multires_Lbarre.Struct_multires_Lbarre

Bases: object

Class Struct_multires_Lbarre this class saves Inputs of analysis computation in its attribute

• Inputs:

Parameters

• Lbarre (1d double array) – Lbarre matrix 1d sparse matrix Lbarre is Schur complement of [L]_Rc in L iw.diaconis_fill.complementschur() with Rc the complement of the set Xbarre

• row1b (1d int_ array) – row array of sparse matrix Lbarre

• col1b (1d int_ array) – column array of sparse matrix Lbarre

• shapelb (int) – shape of Lbarre matrix

• Lbarres (1d double array) – Lbarres matrix 1d sparse matrix sparcified Schur complement of [L]_Rc in L

• rowlbs (1d int_ array) – row array of sparse matrix Lbarres

• collbs (1d int_ array) – column array of sparse matrix Lbarres

• shapelbs (int) – shape of Lbarres matrix

• alphabar (double) – max(abs(L(x,x))

• mu (1d double array) – measure of reversibility. In the case the laplacian is symetric it has to be the uniform measure.

• Xbarre (1d int_ array) – vector of nR indices corresponding to the part of matrix L

• gamma (double) – value of gamma : numeric. 1/gamma= maximum Hitting time

• beta (double) – value of beta : mean time of return after the first step

• q (double) – parameter to sample the vertices of the new graph

• qprime (double) – parameter to compute the solution of Diaconis-Fill equation

• Attributes:

Variables

• Lbarre – Lbarre matrix 1d sparse matrix Lbarre is Schur complement of [L]_Rc in L iw.diaconis_fill.complementschur() with Rc the complement of the set Xbarre

• rowLbarre – row array of sparse matrix Lbarre

• colLbarre – column array of sparse matrix Lbarre

• shapeLbarre (int) – shape of Lbarre matrix

• Lbarres – Lbarres matrix 1d sparse matrix sparcified Schur complement of [L]_Rc in L

• rowLbarres – row array of sparse matrix Lbarres

• colLbarres – column array of sparse matrix Lbarres

• shapeLbarres (int) – shape of Lbarres matrix

• alphabar (double) – max(abs(L(x,x))

• mubarre – measure of reversibility. In the case the laplacian is symetric it has to be the uniform measure.

• Xbarre – vector of nR indices corresponding to the part of matrix L

• gamma (double) – value of gamma : numeric. 1/gamma= maximum Hitting time

• beta (double) – value of beta : mean time of return after the first step

• q (double) – parameter to sample the vertices of the new graph

• qprime (double) – parameter to compute the solution of Diaconis-Fill equation

Lbarre

Lbarres

Xbarre

alphabar

beta

colLbarre

colLbarres

gamma

mubarre

q

qprime

rowLbarre

rowLbarres

shapeLbarre

shapeLbarres

3. Struct_M_ana_recons

class iw.multiresolution.struct_multires_Lbarre.Struct_M_ana_recons

Bases: object

Class Struct_M_ana_recons this class saves Inputs for reconstruction computation in its attributes

• Inputs:

Parameters

• Lambdabarre (1d double array) – Lambdabarre matrix 1d sparse matrix matrix whose rows are the nu_xbarre

• rowla (1d int_ array) – row array of sparse matrix Lambdabarre

• colla (1d int_ array) – column array of sparse matrix Lambdabarre

• shape0la (int) – shape dimension 0 of Lambdabarre matrix

• shape1la (int) – shape dimension 1 of Lambdabarre matrix

• Lambdabreve (1d double array) – Lambdabreve matrix 1d sparse matrix matrix whose rows are the psi_xbreve

• rowlb (1d int_ array) – row array of sparse matrix Lambdabreve

• collb (1d int_ array) – column array of sparse matrix Lambdabreve

• shape0lb (int) – shape dimension 0 of Lambdabreve matrix

• shape1lb (int) – shape dimension 1 of Lambdabreve matrix

• Reconsbarre (1d double array) – Reconsbarre matrix 1d sparse matrix Reconstruction matrix whose rows are the nu_xbarre

• rowlra (1d int_ array) – row array of sparse matrix Reconsbarre

• collra (1d int_ array) – column array of sparse matrix Reconsbarre

• shape0lra (int) – shape dimension 0 of Reconsbarre matrix

• shape1lra (int) – shape dimension 1 of Reconsbarre matrix

• Reconsbreve (1d double array) – Reconsbreve matrix 1d sparse matrix Reconstruction matrix matrix whose rows are the psi_xbreve

• rowlrb (1d int_ array) – row array of sparse matrix Reconsbreve

• collrb (1d int_ array) – column array of sparse matrix Reconsbreve

• shape0lrb (int) – shape dimension 0 of Reconsbreve matrix

• shape1lrb (int) – shape dimension 1 of Reconsbreve matrix

• Attributes:

Variables

• Lambdabarre – Lambdabarre matrix 1d sparse matrix matrix whose rows are the nu_xbarre

• rowLambdabarre – row array of sparse matrix Lambdabarre

• colLambdabarre – column array of sparse matrix Lambdabarre

• shape0Lambdabarre (int) – shape dimension 0 of Lambdabarre matrix

• shape1Lambdabarre (int) – shape dimension 1 of Lambdabarre matrix

• Lambdabreve – Lambdabreve matrix 1d sparse matrix matrix whose rows are the psi_xbreve

• rowLambdabreve – row array of sparse matrix Lambdabreve

• colLambdabreve – column array of sparse matrix Lambdabreve

• shape0Lambdabreve (int) – shape dimension 0 of Lambdabreve matrix

• shape1Lambdabreve (int) – shape dimension 1 of Lambdabreve matrix

• Reconsbarre – Reconsbarre matrix 1d sparse matrix Reconstruction matrix whose rows are the nu_xbarre

• Recons_row_barre – row array of sparse matrix Reconsbarre

• Recons_col_barre – column array of sparse matrix Reconsbarre

• Recons_shape0_barre (int) – shape dimension 0 of Reconsbarre matrix

• Recons_shape1_barre (int) – shape dimension 1 of Reconsbarre matrix

• Reconsbreve – Reconsbreve matrix 1d sparse matrix Reconstruction matrix matrix whose rows are the psi_xbreve

• Recons_row_breve – row array of sparse matrix Reconsbreve

• Recons_col_breve – column array of sparse matrix Reconsbreve

• Recons_shape0_breve (int) – shape dimension 0 of Reconsbreve matrix

• Recons_shape1_breve (int) – shape dimension 1 of Reconsbreve matrix

Lambdabarre

Lambdabreve

Recons_col_barre

Recons_col_breve

Recons_row_barre

Recons_row_breve

Recons_shape0_barre

Recons_shape0_breve

Recons_shape1_barre

Recons_shape1_breve

Reconsbarre

Reconsbreve

colLambdabarre

colLambdabreve

rowLambdabarre

rowLambdabreve

shape0Lambdabarre

shape0Lambdabreve

shape1Lambdabarre

shape1Lambdabreve

tab_one_step_Lambda

iw.multiresolution.tab_one_step_Lambda.tab_one_step_Lambda()

Intermediate function which returns Lambdabarre, Lambdabreve matrix

• Inputs:

Parameters

• L (1d double array) – L is the laplacien matrix L n x n matrix; Markov generator

• row (1d int_ array) – row array of sparse matrix L

• col (1d int_ array) – column array of sparse matrix L

• shape (int) – shape of Laplacien matrix

• Xbarre (1d int_ array) – vector of nR indices corresponding to the part of matrix L

• Xbreve (1d int_ array) – vector of nR-n indices corresponding to complement of Xbarre the root indices

• n (int) – size of the set of vertices

• Outputs:

Returns

tuple of (Lambdabarre and its row, col and shapes, Lambdabreve ,and its row, col and shapes qprime), where matrix whose rows are the nu_xbarre and Lambdabreve matrix whose rows are the psi_xbreve qprime parameter to compute the solution of Diaconis-Fill equation

Return type

tuple of arrays and int