Mesoscopic Rectangular Spikes under Subspace Local Laws: Outlier Values and Singular Subspaces

Abstract

A rectangular data matrix is often modelled as noise plus a signal of low rank. When that rank is fixed the picture is classical: a signal direction must exceed a critical strength before it produces a singular value outside the noise bulk, and above the threshold the singular vectors of the observation retain a definite, computable fraction of the planted direction. We ask what survives when the number of signal directions grows with the dimension. Everything here rests on one hypothesis, which we call a subspace local law: seen from inside the signal subspace, the resolvent of the noise should look like a scalar. We show that this hypothesis alone locates the outliers and counts them, and, in a holomorphic form, determines the outlier singular subspaces as well. The conclusions are stated through spectral projectors rather than individual singular vectors, so they remain meaningful when spike strengths collide or come closer together than the error of the approximation, which at growing rank they must. We then verify the hypothesis in four genuinely different settings, from deterministic noise viewed through randomly oriented signal directions to independent entries with fixed deterministic ones, and for Marchenko--Pastur noise we compute every constant explicitly.