Nonnegative matrix factorization
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which follows from the fact, at optimality, | which follows from the fact, at optimality, | ||
| - | <math> Z^*=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I | + | <math> Z^*=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I\;\;W^{\rm T}\;H\,\textbf{]} |
\end{array}</math> | \end{array}</math> | ||
Revision as of 17:10, 6 May 2026
Exercise from Convex Optimization & Euclidean Distance Geometry, ch.4:
Given rank-2 nonnegative matrix
find a nonnegative factorization
by solving
which follows from the fact, at optimality,
Use the known closed-form solution for a direction vector to regulate rank (rank constraint is replaced) by Convex Iteration;
set to a nonincreasingly ordered diagonalization and
,
then
In summary, initialize then alternate solution of
with
Global convergence occurs, in this example, in only a few iterations.