Proving orthogonality properties of residuals for $L^p$ optimization

by aghostinthefigures   Last Updated June 23, 2018 02:20 AM

Consider an $L^2$ (also called least-squares) optimization problem over a vector space $V$, in which one is trying to minimize the $L^2$-norm of the difference between some "target" vector $y$ and vectors in some subspace $W$ of $V$. (This "difference vector" is often called the residual vector.)

The Hilbert projection theorem guarantees that there is a unique vector $x$ in $W$ that minimizes the norm of this residual vector, and consequently also guarantees that the residual vector $x - y$ is orthogonal to the subspace $W$.

If I consider the same problem, but with the key difference that I now want to minimize a $L^{p\neq2}$ norm of the residual, I can provably lose uniqueness of the optimizing vector $x$ (certainly the case in $L^{1}$).

However, I don't know if you also lose orthogonality of the residual vector w.r.t. the subspace $W$, and for which $L^p$ spaces this occurs. Is there a way to show that the residual for such an optimization problem is/isn't orthogonal to the subspace $W$ for the cases where $L^{p\neq2}$?



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