Perron-Frobenius matrix completion
==================================
The DGP atom library has several functions of positive matrices,
including the trace, (matrix) product, sum, Perron-Frobenius eigenvalue,
and :math:`(I - X)^{-1}` (eye-minus-inverse). In this notebook, we use
some of these atoms to formulate and solve an interesting matrix
completion problem.
In this problem, we are given some entries of an elementwise positive
matrix :math:`A`, and the goal is to choose the missing entries so as to
minimize the Perron-Frobenius eigenvalue or spectral radius. Letting
:math:`\Omega` denote the set of indices :math:`(i, j)` for which
:math:`A_{ij}` is known, the optimization problem is
.. math::
\begin{equation}
\begin{array}{ll}
\mbox{minimize} & \lambda_{\text{pf}}(X) \\
\mbox{subject to} & \prod_{(i, j) \not\in \Omega} X_{ij} = 1 \\
& X_{ij} = A_{ij}, \, (i, j) \in \Omega,
\end{array}
\end{equation}
which is a log-log convex program. Below is an implementation of this
problem, with specific problem data
.. math::
A = \begin{bmatrix}
1.0 & ? & 1.9 \\
? & 0.8 & ? \\
3.2 & 5.9& ?
\end{bmatrix},
where the question marks denote the missing entries.
.. code:: python
import cvxpy as cp
n = 3
known_value_indices = tuple(zip(*[[0, 0], [0, 2], [1, 1], [2, 0], [2, 1]]))
known_values = [1.0, 1.9, 0.8, 3.2, 5.9]
X = cp.Variable((n, n), pos=True)
objective_fn = cp.pf_eigenvalue(X)
constraints = [
X[known_value_indices] == known_values,
X[0, 1] * X[1, 0] * X[1, 2] * X[2, 2] == 1.0,
]
problem = cp.Problem(cp.Minimize(objective_fn), constraints)
problem.solve(gp=True)
print("Optimal value: ", problem.value)
print("X:\n", X.value)
.. parsed-literal::
Optimal value: 4.702374203221372
X:
[[1. 4.63616907 1.9 ]
[0.49991744 0.8 0.37774148]
[3.2 5.9 1.14221476]]