Simmetry constraints in multivariate regression.

Question

Mario 2025-1-24

关闭： John D'Errico 2025-1-25

Hi everyone! I wanted to know if there is the possibility of doing a multivariate regression, forcing the resulting matrix to be symmetrical.

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Answer 1

John D'Errico 2025-1-24

编辑：John D'Errico 2025-1-24

Sure. In fact, I can think of at least 3 ways to do so, almost trivially.

Just estimate the upper (or lower half) of the matrix. Then copy the upper part into the lower part. It is now symmetrical by force.
If your question really is that you want your matri to be SPD (Symmetric & Positive Definite) then estimate what amounts to a Cholesky factor of the matrix, then use that factor to recover the complete matrix, which will now be SPD, again by force.
Finally, symmetriy as a constraint can easily be enforced in terms of linear equality constraints. And almost any of the optimizers in the optimiization or global optimization tools offer linear equality constraints. You do not want to try to force a matrix to be SPD using constraints however. Do that by use of the Cholesky factor trick I mention in #2