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Many source separation algorithms fail to deliver robust performance when applied to signals recorded using high-density sensor arrays where the distance between sensor elements is much less than the wavelength of the signals. This can be attributed to limited dynamic range (determined by analog-to-digital conversion) of the sensor which is insufficient to overcome the artifacts due to large cross-channel redundancy, nonhomogeneous mixing, and high-dimensionality of the signal space. This paper proposes a novel framework that overcomes these limitations by integrating statistical learning directly with the signal measurement (analog-to-digital) process which enables high fidelity separation of linear instantaneous mixtures. At the core of the proposed approach is a min-max optimization of a regularized objective function that yields a sequence of quantized parameters which asymptotically tracks the statistics of the input signal. Experiments with synthetic and real recordings demonstrate significant and consistent performance improvements when the proposed approach is used as the analog-to-digital front-end to conventional source separation algorithms.