Inference from Quantized Data via Normal Variance-Mean Mixtures

Inference from quantized data has received significant attention in recent years due to its broad applications in machine learning and signal processing. Existing likelihood-based approaches are often restricted to Gaussian assumptions or low-bit quantization settings, limiting modeling flexibility and robustness under complex data distributions. In this work, we study inference from quantized observations under the general normal variance-mean mixture (NVMM) framework, which encompasses distributions including Gaussian, $t$, generalized hyperbolic skew-$t$, and generalized hyperbolic distributions. Optimization under the NVMM framework is challenging because the underlying likelihood function involves multidimensional integrals that are difficult to evaluate due to multidimensional quantization and latent mixture variables. To address this difficulty, we propose an expectation conditional maximization (ECM) algorithm with latent-variable augmentations for both quantization and mixture modeling. By leveraging the conditional Gaussian structure of the NVMM family, the proposed method admits closed-form updates for all model parameters at each iteration, leading to an efficient and tractable optimization procedure. We further establish global linear convergence guarantees for the proposed ECM algorithm. Beyond basic parameter estimation, the proposed framework naturally extends to several structured learning and recovery tasks under the NVMM framework, including quantized regression, matrix completion, compressive sensing, and covariance estimation. Numerical experiments demonstrate the effectiveness and robustness of the proposed framework across a variety of quantized inference problems.