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 $D+V{V}^T$ with $V{V}^T$ of small rank.

11.9.1 Nearest correlation with the Frobenius norm

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

$$\Vert M{\Vert }_F={\left(\sum _{i,j}{M}_{i,j}^2\right)}^{1/2}$$

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

(11.36)$$\begin{array}{r}\begin{array}{ll}\text{minimize}& \Vert A-X{\Vert }_F\\ \text{subject to}& \mathbf{diag}(X)=e,\\ & X⪰0.\end{array}\end{array}$$

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)$$\begin{array}{r}\begin{array}{ll}\text{vec}:& {\mathbb{R}}^{n\times n}\to {\mathbb{R}}^{n(n+1)/2}\\ \text{vec}(M)=& ({\alpha }_{11}{M}_{11},{\alpha }_{21}{M}_{21},{\alpha }_{22}{M}_{22},\dots ,{\alpha }_{n1}{M}_{n1},\dots ,{\alpha }_{nn}{M}_{nn})\\ {\alpha }_{ij}=& \{\begin{array}{ll}1& j=i\\ \sqrt{2}& j<i\end{array}\end{array}\end{array}$$

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

Listing 11.18 Implementation of function $vec$ in (11.37). Click here to download.

    public static Expression Vec(Expression e)
    {
      int N       = e.GetShape()[0];
      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)$$\begin{array}{r}\begin{array}{ll}\text{minimize}& t\\ \text{subject to}& (t,\text{vec}(A-X))\in \mathcal{Q},\\ & \mathbf{diag}(X)=e,\\ & X⪰0.\end{array}\end{array}$$

Code example

Listing 11.19 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 11.20 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 $D=\mathbf{diag}(w),w\ge 0$ and $X⪰0$. We will also aim at minimizing the rank of $X$. This can be approximated by a relaxed linear objective penalizing the trace $\mathrm{Tr}(X)$ (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:

$$\begin{array}{r}\begin{array}{ll}\text{minimize}& {\Vert X+\mathbf{diag}(w)-A\Vert }_F+\gamma \mathrm{Tr}(X),\\ \text{subject to}& X⪰0,w\ge 0,\end{array}\end{array}$$

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

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

(11.39)$$\begin{array}{r}\begin{array}{ll}\text{minimize}& t+\gamma \mathrm{Tr}(X)\\ \text{subject to}& (t,\text{vec}(X+\mathbf{diag}(w)-A))\in \mathcal{Q},\\ & X⪰0,w\ge 0.\end{array}\end{array}$$

Code example

Listing 11.21 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, N, 1) );
        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.Objective(ObjectiveSense.Minimize, Expr.Add(t, gamm_trX));
          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()[0];
          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 ${\Vert X+\mathbf{diag}(w)-A\Vert }_F$.

--- 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