Filtered γ₂ factorization norm for query mismatch masks

For each i, encode two finite vector families u_{x,i},v_{y,i} in a common finite-dimensional complex inner-product space. Their filtered factorization represents A exactly when A_{xy}=Σ_i(Δ_i){xy}⟨u{x,i}|v_{y,i}⟩_D for every ordered pair (x,y). The Dirac convention [F3] makes this expression conjugate-linear in each u factor and linear in each v factor.

The cost of a representation is max{max_xΣ_i||u_{x,i}||²,max_yΣ_i||v_{y,i}||²}; take the infimum over all finite-dimensional representations. If A_{xx}≠0 for some x, all Δ_i(x,x)=0 and no representation exists. If A has zero diagonal, each off-diagonal pair differs in some coordinate; assigning its entry to the first differing mask and writing $A^{(i)}$ for each resulting matrix, take $u_{x,i}=(\overline{A_{xy}^{(i)}}{)}_{y\in D}$ and $v_{y,i}=e_y$ in ${\mathbb{C}}^{\mathrm{\mid }D\mathrm{\mid }}$. Then $⟨u_{x,i}\mathrm{\mid }{v}_{y,i}{⟩}_D=A_{xy}^{(i)}$, so this is a feasible representation.

Thus the value is +∞ exactly when the diagonal support obstruction occurs, and otherwise the infimum ranges over a nonempty finite-dimensional class with a finite nonnegative cost. For any feasible family, restricting the ambient space for each i to the span of its 2|D| vectors preserves all inner products and norms and gives dimension at most 2|D|. Nonemptiness of D ensures that both maxima in the cost are defined.