When is an inverse problem actually solvable, despite tiny singular values? The (discrete) Picard condition gives the precise test. The reconstruction \hat m = \sum_i \frac{u_i^{\mathsf T} d}{\sigma_i} v_i stays bounded only if the data coefficients |u_i^{\mathsf T} d| decay faster than the singular values \sigma_i.

Then each ratio |u_i^{\mathsf T} d|/\sigma_i stays small and the sum converges to something sensible. If instead the data coefficients level off — as noise always forces them to — the ratios eventually blow up, and the reconstruction is meaningless.

The signature of noise

Clean data satisfies Picard: real signals are smooth, so their high-index coefficients fall away quickly. Noise does not — it spreads roughly equally across all singular directions, so |u_i^{\mathsf T} e| flattens out at the noise floor while \sigma_i keeps shrinking. Past the index where the noise floor overtakes the singular values, the ratio |u_i^{\mathsf T} d|/\sigma_i turns upward and diverges.

That crossover index is exactly where to stop trusting the data — it tells you how many singular components are worth keeping, and so sets the truncation level and the strength of regularization.

The Picard plot

Three quantities versus singular index: the singular values \sigma_i (decaying), the data coefficients |u_i^{\mathsf T} d| (decaying until they hit the noise floor), and their ratio. With no noise the ratio stays flat — Picard satisfied. Add noise and the coefficients flatten; past the crossover the ratio shoots up, marking the useful components from the hopeless ones.