Stable Retrieval for Unlimited Sampling via Adaptive Local Representations

We examine theoretical guarantees for our recently proposed local adaptive iterative algorithm with least-squares approach for noisy modulo sampling recovery. We demonstrate that under modest property of the input signal variation, the least-squares formulation remains valid in noisy settings when using a sufficient amount of sub-intervals during preprocessing. We quantify first-iteration retrieval error through noise amplification analysis based on the least singular value of the system matrix. Those findings reveal a fundamental trade-off: while an increasing number of interval suffices for preprocessing effectiveness, it reduces the stability of the retrieval. This characterization provides both theoretical understanding and practical implementation guidance for unlimited sampling for our proposed algorithm. Finally, we show that the retrieval error can be controlled in terms of the noise level with high probability.