Package {symTensor}

Generate all permutations of 1:n

Description

Generate all permutations of 1:n

Usage

.allPermutations(n)

Arguments

Value

List of integer vectors

Compute Beta-divergence D_beta(X || Y)

Description

Compute Beta-divergence D_beta(X || Y)

Usage

.betaDiv(X, Y, beta)

Arguments

Value

Scalar divergence value

Contract an array along a specific mode with a vector

Description

Contract an array along a specific mode with a vector

Usage

.contractMode(arr, v, mode)

Arguments

Value

Array with one fewer dimension

MU exponent for Beta-divergence

Description

Returns the exponent gamma for the multiplicative update rule. gamma = 1 for Beta in [1,2], otherwise adjusted for convergence guarantee.

Usage

.muExponent(beta)

Arguments

Value

Exponent gamma

References

Fevotte, C. & Idier, J. (2011). Neural Computation, 23(9).

Compute repeated outer product of a vector

Description

Compute repeated outer product of a vector

Usage

.outerRepeat(v, d)

Arguments

Value

d-dimensional array

Contract a tensor with a vector along all modes except the first

Description

Contract a tensor with a vector along all modes except the first

Usage

.tensorVectorContract(arr, v, d)

Arguments

Value

Numeric vector of length n

Full contraction of tensor with vector along all modes

Description

Full contraction of tensor with vector along all modes

Usage

.tensorVectorContractAll(arr, v, d)

Higher-order Power Method for Symmetric Tensor Decomposition

Description

Decomposes a symmetric tensor into a sum of rank-1 symmetric terms using the higher-order power method (S-HOPM) with deflation.

Usage

HigherOrderPower(
  x,
  R = 1L,
  max_iter = 1000L,
  tol = 1e-08,
  init = c("random", "svd")
)

Arguments

Details

For a d-th order symmetric tensor T \in R^{n \times n \times \cdots \times n}, finds T \approx \sum_{r=1}^{R} \lambda_r v_r \otimes v_r \otimes \cdots \otimes v_r.

Value

A list with components:

lambdas

Numeric vector of length R with eigenvalues

vectors

Matrix (n x R) where each column is a unit eigenvector

residual

Residual tensor after deflation

converged

Logical vector of length R

iters

Integer vector of iterations per component

References

Kolda, T. G., & Mayo, J. R. (2011). Shifted Power Method for Computing Tensor Eigenpairs. SIAM Journal on Matrix Analysis and Applications.

De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000). On the Best Rank-1 and Rank-(R1, R2, ..., RN) Approximation of Higher-Order Tensors. SIAM Journal on Matrix Analysis and Applications.

Examples

library(rTensor)
# Create a symmetric 3-mode tensor
n <- 5
v1 <- rnorm(n); v1 <- v1 / sqrt(sum(v1^2))
# T = 3.0 * v1 o v1 o v1
arr <- 3.0 * outer(outer(v1, v1), v1)
tnsr <- as.tensor(arr)
result <- HigherOrderPower(tnsr, R = 1)
result$lambdas  # should be close to 3.0

Label Propagation via PageRank

Description

Semi-supervised label propagation on a graph. Uses Personalized PageRank internally to propagate labels from seed nodes to the rest of the graph.

Usage

LabelPropagation(A, seeds, damping = 0.85, max_iter = 1000L, tol = 1e-08)

Arguments

Value

A list with components:

labels

Integer vector of predicted labels for all nodes

scores

Matrix (n x K) of PageRank-based scores per class

classes

Unique class labels

Examples

# 6-node graph with 2 communities
A <- matrix(0, 6, 6)
A[1,2] <- A[2,1] <- 1; A[1,3] <- A[3,1] <- 1; A[2,3] <- A[3,2] <- 1
A[4,5] <- A[5,4] <- 1; A[4,6] <- A[6,4] <- 1; A[5,6] <- A[6,5] <- 1
A[3,4] <- A[4,3] <- 0.5  # weak bridge
seeds <- c("1" = 1, "6" = 2)  # node 1 -> class 1, node 6 -> class 2
result <- LabelPropagation(A, seeds)
result$labels

(Personalized) PageRank

Description

Computes the PageRank vector for a given adjacency or transition matrix using power iteration.

Usage

PageRank(
  A,
  damping = 0.85,
  personalization = NULL,
  max_iter = 1000L,
  tol = 1e-08
)

Arguments

Value

A list with components:

vector

PageRank vector (sums to 1)

iter

Number of iterations until convergence

converged

Logical, whether the algorithm converged

Examples

# Simple web graph
A <- matrix(c(0,1,1,0, 1,0,0,1, 0,1,0,1, 1,0,1,0), 4, 4)
pr <- PageRank(A)
pr$vector

Symmetrize a matrix or tensor

Description

For a matrix, computes (A + A^T) / 2. For a tensor, averages over all permutations of indices.

Usage

Symmetrize(x)

Arguments

Value

Symmetrized matrix or Tensor

Examples

A <- matrix(c(1, 2, 3, 4), 2, 2)
Symmetrize(A)

library(rTensor)
arr <- array(rnorm(27), dim = c(3, 3, 3))
S <- Symmetrize(as.tensor(arr))
isSymmetricTensor(S)  # TRUE

TOPHITS: Tensor HITS Algorithm

Description

Computes hub and authority scores for higher-order link analysis using Tucker decomposition of a tensor. Generalization of HITS (Hyperlink-Induced Topic Search) to tensors.

Usage

TOPHITS(x, R = 2L, max_iter = 100L, tol = 1e-08)

Arguments

Details

For a 3-mode tensor T \in R^{I \times J \times K}, TOPHITS computes Tucker decomposition T \approx C \times_1 U_1 \times_2 U_2 \times_3 U_3 and derives hub/authority scores from the factor matrices.

For a symmetric tensor (all modes equal), a single set of scores is returned.

Value

A list with components:

scores

List of score matrices, one per mode (each n_i x R_i). For symmetric tensors, a single matrix.

core

Core tensor from Tucker decomposition

rankings

List of integer vectors giving node rankings per mode (ordered by dominant score)

decomp

Full Tucker decomposition result from rTensor

References

Kolda, T. G., & Bader, B. W. (2006). The TOPHITS Model for Higher-Order Web Link Analysis. Workshop on Link Analysis, Counterterrorism and Security.

Examples

library(rTensor)
# Create a 3-mode tensor
arr <- array(abs(rnorm(125)), dim = c(5, 5, 5))
tnsr <- as.tensor(arr)
result <- TOPHITS(tnsr, R = 2)
result$rankings

Check if a matrix or tensor is symmetric

Description

For a matrix, checks if A[i,j] == A[j,i] for all i,j. For a tensor, checks if the tensor is invariant under all permutations of its indices (i.e., fully symmetric).

Usage

isSymmetricTensor(x, tol = 1e-10)

Arguments

Value

Logical indicating whether x is symmetric

Examples

# Matrix
A <- matrix(c(1,2,2,3), 2, 2)
isSymmetricTensor(A)  # TRUE

# 3-mode symmetric tensor
library(rTensor)
arr <- array(0, dim = c(3, 3, 3))
for (i in 1:3) for (j in 1:3) for (k in 1:3) {
  arr[i, j, k] <- i + j + k
}
isSymmetricTensor(as.tensor(arr))  # TRUE

Symmetric Non-negative Matrix Factorization (symNMF)

Description

Decomposes a symmetric non-negative matrix \Omega \approx M S M^\top using multiplicative update (MU) rules with Beta-divergence.

Usage

symNMF(
  Omega,
  K,
  model = c("MSM", "MM"),
  Beta = 2,
  initM = NULL,
  initS = NULL,
  lambda = 0,
  num.iter = 100L,
  thr = 1e-10,
  verbose = FALSE
)

Arguments

Details

Two models are supported:

• MSM: \Omega \approx M S M^\top where M is N x K and S is K x K

• MM: \Omega \approx M M^\top where M is N x K (special case with S = I)

The loss function is the Beta-divergence D_\beta(\Omega \| \hat{\Omega}):

• \beta = 2: Frobenius (Euclidean) distance

• \beta = 1: Kullback-Leibler divergence

• \beta = 0: Itakura-Saito divergence

• Other values of \beta interpolate/extrapolate these cases

Value

A list with components:

M

Factor matrix (N x K)

S

Coefficient matrix (K x K). Identity matrix when model = "MM".

RecError

Vector of reconstruction errors at each iteration

RelChange

Vector of relative changes at each iteration

converged

Logical

iter

Number of iterations performed

References

Kuang, D., Park, H., & Ding, C. H. Q. (2012). Symmetric Nonnegative Matrix Factorization for Graph Clustering. SDM 2012.

Fevotte, C. & Idier, J. (2011). Algorithms for Nonnegative Matrix Factorization with the Beta-Divergence. Neural Computation, 23(9).

Examples

set.seed(42)
M_true <- matrix(c(rep(c(1,0), 5), rep(c(0,1), 5)), 10, 2)
Omega <- M_true %*% t(M_true) + matrix(runif(100, 0, 0.1), 10, 10)
Omega <- (Omega + t(Omega)) / 2

# Frobenius (Beta=2, default)
res_fro <- symNMF(Omega, K = 2, Beta = 2)

# KL divergence (Beta=1)
res_kl <- symNMF(Omega, K = 2, Beta = 1)

# Itakura-Saito (Beta=0)
res_is <- symNMF(Omega, K = 2, Beta = 0)