Package 'admmDensestSubmatrix'

densub

Description

Iteratively solves the convex optimization problem using ADMM.

Usage

densub(G, m, n, tau = 0.35, gamma = 6/(sqrt(m * n) * (q - p)),
  opt_tol = 1e-04, maxiter, quiet = TRUE)

Arguments

G	sampled binary matrix
m	number of rows in dense submatrix
n	number of columns in dense submatrix
tau	penalty parameter for equality constraint violation
gamma	$l_1$ regularization parameter
opt_tol	stopping tolerance in algorithm
maxiter	maximum number of iterations of the algorithm to run
quiet	toggles between displaying intermediate statistics

Details

$min |X|_* + gamma* |Y|_1 + 1_Omega_W (W) + 1_Omega_Q (Q) + 1_Omega_Z (Z)$

s.t $X - Y = 0$, $X = W$, $X = Z$,

where $Omega_W (W)$, $Omega_Q (Q)$, $Omega_Z (Z)$ are the sets: $Omega_W = {W in R^MxN | e^TWe = mn}$

$Omega_Q = {Q in R^MxN | Projection of Q on not N = 0}$

$Omega_Z = {Z in R^MxN | Z_ij <= 1 for all (i,j) in M x N}$

$Omega_Q = {Q in R^MxN | Projection of Q on not N = 0}$

$Omega_Z = {Z in R^MxN | Z_ij <= 1 for all (i,j) in M x N}$

$1_S$ is the indicator function of the set $S$ in $R^MxN$ such that $1_S(X) = 0$ if $X$ in $S$ and +infinity otherwise

Value

Rank one matrix with $mn$ nonzero entries, matrix $Y$ that is used to count the number of disagreements between $G$ and $X$

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Soft threshholding operator.

Description

Applies the shrinkage operator for singular value tresholding.

Usage

mat_shrink(K, tau)

Arguments

K	matrix
tau	regularization parameter

Value

Matrix

Examples

mat_shrink(matrix(c(1,0,0,0,1,1,1,1,1), nrow=3, ncol=3, byrow=TRUE),0.35)

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Sample matrix

Description

Generates binary $(M,N)$ - matrix sampled from dense $(m,n)$ - submatrix.

Usage

plantedsubmatrix(M, N, m, n, p, q)

Arguments

M	number of rows in sampled matrix
N	number of rows in sampled matrix
m	number of rows in dense submatrix
n	natural number used to calculate number of rows in dense submatrix
p	density outside planted submatrix
q	density inside planted submatrix

Details

Let $U*$ and $V*$ be $m$ and $n$ index sets. For each i in U*, j in V* we let $a_ij = 1$ with probability $q$ and $0$ otherwise. For each remaining $ij$ we set $a_ij = 1$ with probability $p < q$ and take $a_ij = 0$ otherwise.

Value

Matrix $G$ sampled from the planted dense $(mn)$-submatrix model, dense sumbatrix $X0$, matrix $Y0$ used to count the number of disagreements between $G$ and $X0$

Examples

plantedsubmatrix(10,10,1,2,0.25,0.75)

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