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\author{Adam Rahman}
\title{Toeplitz Approximation}


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\begin{document}
\SweaveOpts{concordance=FALSE}

\maketitle

Given a symmetric matrix $\m{F}$, the Toeplitz approximation problem seeks to find the nearest symmetric positive definite Toeplitz matrix. In general, a Toeplitz matrix is one with constant descending diagonals, i.e.

\[
\m{T} = \left[
\begin{array}{ccccc}
a & b & c & d & e \\
f & a & b & c & d \\
g & f & a & b & c \\
h & g & f & a & b \\
i & h & g & f & a
\end{array}
\right]
\]

is a general Toeplitz matrix. For our specific problem, we seek a \emph{symmetric} Toeplitz matrix, i.e.,

\[
\m{T}^{\ast} = \left[
\begin{array}{ccccc}
a & b & c & d & e \\
b & a & b & c & d \\
c & b & a & b & c \\
d & c & b & a & b \\
e & d & f & b & a
\end{array}
\right]
\]

The problem is formulated as the following optimization problem 

\[
\begin{array}{ll}
\underset{\m{X}}{\text{maximize}} & -y_{n+1} \\
\text{subject to} & \\
&
\begin{array}{rl}
\left[\begin{array}{cc}
\m{I} & \m{0} \\
\m{0} & -\beta
\end{array}\right] ~~+~~ \sum_{k=1}^{n}y_{k}
\left[\begin{array}{cc}
\m{0} & \gamma_{k}\ve{e}_{k} \\
\gamma_{k}\ve{e}_{k}^{T} & -2q_{k}
\end{array}\right] ~~+~~y_{n+1}\m{B} &\geq~~ \m{0} \\
\tr{[y_{1},...,y_{n}]} + y_{n+1}\m{B} & \geq~~ \ve{0}
\end{array}
\end{array}
\]

\noindent where $\m{B}$ is an $(n+1)\times(n+1)$ matrix of zeros, and $\m{B}_{(n+1)(n+1)} = 1$, $q_{1} = -tr(\m{F})$, $q_{k} =$ sum of $k^{th}$ diagonal upper and lower triangular matrix, $\gamma_{1} = \sqrt{n}$, $\gamma_{k} = \sqrt{2*(n-k+1)},~k=2,...,n$, and $\beta = ||\m{F}||_{F}^{2}$. 

The function \verb!toep! takes as input a symmetric matrix \verb!F! for which we would like to find the nearest Toeplitz matrix, and returns the optimal solution using \verb!sqlp!.

\begin{verbatim}
R> out <- toep(F)
\end{verbatim}

\section*{Numerical Example}

Consider the following symmetric matrix for which we would like to find the nearest Toeplitz matrix

\begin{verbatim}
R> data(Ftoep)

          V1     V2     V3     V4     V5     V6     V7     V8     V9    V10
 [1,]  0.170  0.127  0.652 -0.490  0.963  0.372 -0.707 -0.250 -0.022  1.087
 [2,]  0.127 -1.637  0.031  1.276 -1.475 -1.842 -0.529  1.534 -2.810  0.923
 [3,]  0.652  0.031  3.339 -0.246  0.249 -2.367  4.327  0.876 -1.832  0.507
 [4,] -0.490  1.276 -0.246 -1.556 -1.415 -0.022 -0.052  1.564 -1.140 -0.982
 [5,]  0.963 -1.475  0.249 -1.415 -0.656 -0.059 -3.101  0.337 -1.526 -0.737
 [6,]  0.372 -1.842 -2.367 -0.022 -0.059  2.617 -0.919  0.869  2.574  0.669
 [7,] -0.707 -0.529  4.327 -0.052 -3.101 -0.919  0.936  1.458 -0.622  1.632
 [8,] -0.250  1.534  0.876  1.564  0.337  0.869  1.458  0.013  1.348  1.736
 [9,] -0.022 -2.810 -1.832 -1.140 -1.526  2.574 -0.622  1.348 -3.817  0.925
[10,]  1.087  0.923  0.507 -0.982 -0.737  0.669  1.632  1.736  0.925  0.527
\end{verbatim}

Using \verb!sqlp!, we are interested in the output \verb!Z!, the optimal solution to the dual problem, which will be the nearest symmetric Toeplitz matrix. Note that the final row/column should be removed.

\begin{verbatim}
R> out <- toep(Ftoep)

R> F <- out$Z[[1]]
R> F <- F[-nrow(F),]
R> F <- F[,-ncol(F)]

       [,1]   [,2]   [,3]   [,4]   [,5]   [,6]   [,7]   [,8]   [,9]  [,10]
 [1,]  0.563  0.098 -0.038 -0.113  0.343 -0.054 -0.237 -0.369  0.228  0.077
 [2,]  0.098  0.563  0.098 -0.038 -0.113  0.343 -0.054 -0.237 -0.369  0.228
 [3,] -0.038  0.098  0.563  0.098 -0.038 -0.113  0.343 -0.054 -0.237 -0.369
 [4,] -0.113 -0.038  0.098  0.563  0.098 -0.038 -0.113  0.343 -0.054 -0.237
 [5,]  0.343 -0.113 -0.038  0.098  0.563  0.098 -0.038 -0.113  0.343 -0.054
 [6,] -0.054  0.343 -0.113 -0.038  0.098  0.563  0.098 -0.038 -0.113  0.343
 [7,] -0.237 -0.054  0.343 -0.113 -0.038  0.098  0.563  0.098 -0.038 -0.113
 [8,] -0.369 -0.237 -0.054  0.343 -0.113 -0.038  0.098  0.563  0.098 -0.038
 [9,]  0.228 -0.369 -0.237 -0.054  0.343 -0.113 -0.038  0.098  0.563  0.098
[10,]  0.077  0.228 -0.369 -0.237 -0.054  0.343 -0.113 -0.038  0.098  0.563
\end{verbatim}
\end{document}