# 11.9 Nearest Correlation Matrix Problem¶

A correlation matrix is a symmetric positive definite matrix with unit diagonal. This term has origins in statistics, since the matrix whose entries are the correlation coefficients of a sequence of random variables has all these properties.

In this section we study variants of the problem of approximating a given symmetric matrix $$A$$ with correlation matrices:

• find the correlation matrix $$X$$ nearest to $$A$$ in the Frobenius norm,

• find an approximation of the form $$D+X$$ where $$D$$ is a diagonal matrix with positive diagonal and $$X$$ is a positive semidefinite matrix of low rank, using the combination of Frobenius and nuclear norm.

Both problems are related to portfolio optimization, where one can often have a matrix $$A$$ that only approximates the correlations of stocks. For subsequent optimizations one would like to approximate $$A$$ with a correlation matrix or, in the factor model, with $$D+VV^T$$ with $$VV^T$$ of small rank.

## 11.9.1 Nearest correlation with the Frobenius norm¶

The Frobenius norm of a real matrix $$M$$ is defined as

$\|M\|_F = \left(\sum_{i,j}M_{i,j}^2\right)^{1/2}$

and with respect to this norm our optimization problem can be expressed simply as:

(11.36)$\begin{split}\begin{array}{ll} \minimize & \|A-X\|_F\\ \st & \mathbf{diag}(X) = e,\\ & X \succeq 0.\\ \end{array}\end{split}$

We can exploit the symmetry of $$A$$ and $$X$$ to get a compact vector representation. To this end we make use of the following mapping from a symmetric matrix to a flattened vector containing the (scaled) lower triangular part of the matrix:

(11.37)$\begin{split}\begin{array}{ll} \mbox{vec}: & \R^{n\times n} \rightarrow \real^{n(n+1)/2} \\ \mbox{vec}(M) = & (\alpha_{11}M_{11},\alpha_{21}M_{21},\alpha_{22}M_{22},\ldots,\alpha_{n1}M_{n1},\ldots,\alpha_{nn}M_{nn}) \\ \alpha_{ij}=&\begin{cases}1 & j=i\\ \sqrt{2} & j<i\end{cases} \end{array}\end{split}$

Note that $$\|M\|_F=\|\mbox{vec}(M)\|_2$$. The Fusion implementation of $$\mbox{vec}$$ is as follows:

Listing 11.17 Implementation of function $$vec$$ in (11.37). Click here to download.
    public static Expression Vec(Expression e)
{
int N       = e.GetShape();
int[] msubi = new int[N * (N + 1) / 2],
msubj = new int[N * (N + 1) / 2];
double[] mcof = new double[N * (N + 1) / 2];

for (int i = 0, k = 0; i < N; ++i)
for (int j = 0; j < i + 1; ++j, ++k)
{
msubi[k] = k;
msubj[k] = i * N + j;
if (i == j) mcof[k] = 1.0;
else        mcof[k] = Math.Sqrt(2);
}

var S = Matrix.Sparse(N * (N + 1) / 2, N * N, msubi, msubj, mcof);
return Expr.Mul(S, Expr.Flatten(e));
}


That leads to an optimization problem with both conic quadratic and semidefinite constraints:

(11.38)$\begin{split}\begin{array}{ll} \minimize & t\\ \st & (t, \mbox{vec} (A-X)) \in \Q,\\ & \mathbf{diag}(X) = e,\\ & X \succeq 0.\\ \end{array}\end{split}$

Code example

Listing 11.18 Implementation of problem (11.38). Click here to download.
    public static void nearestcorr_frobenius(Matrix A)
{
int N = A.NumRows();
using (var M = new Model("NearestCorrelation"))
{
// Setting up the variables
var X = M.Variable("X", Domain.InPSDCone(N));
var t = M.Variable("t", 1, Domain.Unbounded());

// (t, vec (A-X)) \in Q
M.Constraint( Expr.Vstack(t, Vec(Expr.Sub(A, X))), Domain.InQCone() );

// diag(X) = e
M.Constraint(X.Diag(), Domain.EqualsTo(1.0));

// Objective: Minimize t
M.Objective(ObjectiveSense.Minimize, t);

// Solve the problem
M.Solve();

// Get the solution values
Console.WriteLine("X = \n{0}", mattostr(X.Level(), N));
Console.WriteLine("t = {0}", mattostr(t.Level(), N));
}
}


We use the following input

Listing 11.19 Input for the nearest correlation problem.
      int N = 5;
var A = Matrix.Dense( new double[,]
{ { 0.0,  0.5,  - 0.1,  -0.2,   0.5},
{ 0.5,  1.25, -0.05, -0.1,   0.25},
{ -0.1, -0.05,  0.51,  0.02, -0.05},
{ -0.2, -0.1,   0.02,  0.54, -0.1},
{ 0.5,  0.25, -0.05, -0.1,   1.25}
});


The expected output is the following (small differences may apply):

X =
[[ 1.          0.50001941 -0.09999994 -0.20000084  0.50001941]
[ 0.50001941  1.         -0.04999551 -0.09999154  0.24999101]
[-0.09999994 -0.04999551  1.          0.01999746 -0.04999551]
[-0.20000084 -0.09999154  0.01999746  1.         -0.09999154]
[ 0.50001941  0.24999101 -0.04999551 -0.09999154  1.        ]]


## 11.9.2 Nearest Correlation with Nuclear-norm Penalty¶

Next, we consider the approximation of $$A$$ of the form $$D+X$$ where $$D=\diag(w),\ w\geq 0$$ and $$X\succeq 0$$. We will also aim at minimizing the rank of $$X$$. This can be approximated by a relaxed linear objective penalizing the trace $$\trace(X)$$ (which in this case is the nuclear norm of $$X$$ and happens to be the sum of its eigenvalues).

The combination of these constraints leads to a problem:

$\begin{split}\begin{array}{ll} \minimize & \left\|X+\diag(w)-A\right\|_F + \gamma \trace(X),\\ \st & X \succeq 0, w \geq 0, \end{array}\end{split}$

where the parameter $$\gamma$$ controls the tradeoff between the quality of approximation and the rank of $$X$$.

Exploit the mapping $$\mbox{vec}$$ defined in (11.37) we can express this problem as:

(11.39)$\begin{split}\begin{array}{ll} \minimize & t + \gamma\trace(X) \\ \st & (t, \mbox{vec} (X + \diag(w) - A) ) \in \Q, \\ & X \succeq 0 , w \geq 0. \end{array}\end{split}$

Code example

Listing 11.20 Implementation of problem (11.39). Click here to download.
    public static void nearestcorr_nn(Matrix A, double[] gammas, double[] res, int[] rank)
{
int N = A.NumRows();
using (var M = new Model("NucNorm"))
{
// Setup variables
var t = M.Variable("t", 1, Domain.Unbounded());
var X = M.Variable("X", Domain.InPSDCone(N));
var w = M.Variable("w", N, Domain.GreaterThan(0.0));

// (t, vec (X + diag(w) - A)) in Q
var D = Expr.MulElm( Matrix.Eye(N), Var.Repeat(w, 1, N) );
M.Constraint( Expr.Vstack( t, Vec(Expr.Sub(Expr.Add(X, D), A)) ), Domain.InQCone() );

for (var k = 0; k < gammas.Length; ++k)
{
// Objective: Minimize t + gamma*Tr(X)
var gamm_trX = Expr.Mul( gammas[k], Expr.Sum(X.Diag()) );
M.Solve();

// Find the eigenvalues of X and approximate rank
var d = new double[N];
mosek.LinAlg.syeig(mosek.uplo.lo, N, X.Level(), d);
var rnk = 0; foreach (var v in d) if (v > 1e-6) ++rnk;

res[k] = t.Level();
rank[k] = rnk;
}
}
}


We feed MOSEK with the same input as in Sec. 11.9.1 (Nearest correlation with the Frobenius norm). The problem is solved for a range of values $$\gamma$$ values, to demonstrate how the penalty term helps achieve a low rank solution. To this extent we report both the rank of $$X$$ and the residual norm $$\left\|X+\diag(w)-A\right\|_F$$.

--- Nearest Correlation with Nuclear Norm---
gamma=0.000000, res=3.076163e-01, rank=4
gamma=0.100000, res=4.251692e-01, rank=2
gamma=0.200000, res=5.112082e-01, rank=1
gamma=0.300000, res=5.298432e-01, rank=1
gamma=0.400000, res=5.592686e-01, rank=1
gamma=0.500000, res=6.045702e-01, rank=1
gamma=0.600000, res=6.764402e-01, rank=1
gamma=0.700000, res=8.009913e-01, rank=1
gamma=0.800000, res=1.062385e+00, rank=1
gamma=0.900000, res=1.129513e+00, rank=0
gamma=1.000000, res=1.129513e+00, rank=0