invertiforms (version 0.1.2)

RegularizedLaplacian: Construct and use the Regularized Laplacian

Description

Construct and use the Regularized Laplacian

Usage

RegularizedLaplacian(A, tau_row = NULL, tau_col = NULL)
# S4 method for RegularizedLaplacian,Matrix
transform(iform, A)
# S4 method for RegularizedLaplacian,matrix
transform(iform, A)
# S4 method for RegularizedLaplacian,sparseLRMatrix
transform(iform, A)
# S4 method for RegularizedLaplacian,Matrix
inverse_transform(iform, A)
# S4 method for RegularizedLaplacian,matrix
inverse_transform(iform, A)
# S4 method for RegularizedLaplacian,vsp_fa
inverse_transform(iform, A)

Value

• RegularizedLaplacian() creates a RegularizedLaplacian object.

• transform() returns the transformed matrix, typically as a Matrix::Matrix.

• inverse_transform() returns the inverse transformed matrix, typically as a Matrix::Matrix.

Arguments

A

A matrix to transform.

tau_row

Additive regularizer for row sums of abs(A). Typically this corresponds to inflating the (absolute) out-degree of each node by tau_row. Defaults to NULL, in which case we set tau_row to the mean (absolute) row sum of A.

tau_col

Additive regularizer for column sums of abs(A). Typically this corresponds to inflating the (absolute) in-degree of each node by tau_col. Defaults to NULL, in which case we set tau_col to the mean (absolute) column sum of A.

iform

An Invertiform object describing the transformation.

Details

We define the regularized Laplacian \(L^\tau(A)\) of an \(n \times n\) graph adjacency matrix \(A\) as

$$ L^\tau(A)_{ij} = \frac{A_{ij}}{\sqrt{d^{out}_i + \tau_{row}} \sqrt{d^{in}_j + \tau_{col}}} $$

where

$$ d^{out}_i = \sum_{j=1}^n \| A_{ij} \| $$

and

$$ d^{in}_j = \sum_{i=1}^n \| A_{ij} \|. $$

When \(A_{ij}\) denotes the present of an edge from node \(i\) to node \(j\), which is fairly standard notation, \(d^{out}_i\) denotes the (absolute) out-degree of node \(i\) and \(d^{in}_j\) denotes the (absolute) in-degree of node \(j\). Then \(\tau_{row}\) is an additive out-degree regularizer and \(\tau_{col}\) is an additive in-degree regularizer.

Note that this documentation renders more clearly at https://rohelab.github.io/invertiforms/.

Examples

library(igraph)
library(igraphdata)

data("karate", package = "igraphdata")

A <- get.adjacency(karate)

iform <- RegularizedLaplacian(A)

L <- transform(iform, A)
L

A_recovered <- inverse_transform(iform, L)

all.equal(A, A_recovered)