Observable Clocks for Sequential Recovery in Tensor PCA

Online stochastic gradient descent on multi-spiked tensor PCA recovers the planted spikes one at a time. Long plateaus, during which an overlap grows from its random-initialization scale, are followed by rapid recovery. These stages are defined through overlaps with the unknown spikes, so they cannot be observed directly. We show that they can nevertheless be read off online from a quantity SGD already computes. For each estimator column, the contraction of the fresh sample with the current iterate—a rescaled per-sample loss term—tracks the overlap-defined column signal uniformly over polynomially many steps. A windowed threshold on this signal gives an observable stopping rule. From initializations at the random scale with separated greedy weights, it reports the columns in the latent greedy recovery order, and each report falls within a vanishing fraction of its stage of the corresponding certified recovery time. The proof combines a reconstruction of the known sequential-recovery dynamics with new two-sided bounds on stage durations and threshold margins. We further show that the lower bulk of the transverse Riemannian Hessian carries the same signal as a spectral shift, uniformly along the trajectory under non-Gaussian noise. This spectral readout is consistent, but it is a noisier readout of the same quantity rather than an improvement. Simulations illustrate both monitors at finite size.