Maintaining the Right Abstraction: A Finite-Sample Theory of Adaptive Low-Rank Representations under Drift

When the dimension and orientation of a representation are learned simultaneously under distribution drift, their errors form a feedback loop. Off-equilibrium subspace error perturbs the statistic used to select rank, while each rank change moves the learner to a new Grassmann manifold and creates new orientation error. We develop a finite-sample theory that closes this loop. For a linear Gaussian channel, a single spectral benefit function determines the globally optimal rank and orientation, and its Riemannian Hessian determines the local recovery rate. Our main theorem for blockwise empirical updates gives an explicit invariant tracking radius, margins that prevent erroneous rank changes, re-entry guarantees after every accepted birth or death, and a finite recovery time while orientation remains out of equilibrium. At spectral crossings, we bound detection delay. Near a selection boundary, a two-point lower bound identifies an unavoidable $n^{-1/2}$ ambiguity in dimension. Experiments follow the structure of the theory by testing the spectral selection boundary, curvature-controlled recovery, hybrid reorganization, and low-rank scaling; scientific snapshots and a gated MNIST relaxation place the same mechanisms in broader representation settings. The resulting picture is a temporarily stable projection between excessive complexity and an evolving, noisy world.