Rank Stabilization for Sparse Coordinate Completion of Unit-Norm Tight Frames

Source abstract

We determine generic completion-fiber dimensions for unit-norm tight frames when each partially observed column has exactly two missing coordinates. We work on the complex algebraic variety defined by the real frame equations. The missing coordinate pairs form a labelled multigraph. For every $d\ge4$ and every frame length $R\ge N_d:=\binom{d+1}{2}-1$, we prove that the missing-pair Jacobian matroid is the Rado matroid of an explicit subspace arrangement. Its rank is therefore given by a graph-theoretic minimum formula and is independent of $R$; the complementary rank defect is the generic fiber dimension. The main step is a realization theorem at length $N_d$. It combines a transversal--graphic matroid partition with a zero-coordinate moment submersion and a compatible decomposition of a generic residual matrix. Basis extension and stability under fully observed columns then yield the rank formula at every larger length. We also give an explicit defect formula in dimension four and a local real counterpart on the smooth real frame locus. A common-rotation obstruction bounds the smallest uniform stabilization threshold from below by $2d-1$.