Consistency and convergence rates in Lagrangian empirical design of variable-rate vector quantizers

The Lagrangian formulation of variable-rate vector quantization is known to yield useful necessary conditions for quantizer optimality and generalized Lloyd algorithms for quantizer design. In this work we show the consistency of empirical design based on minimizing the Lagrangian performance over a stationary and ergodic training sequence. We also study the finite sample performance for independent training data drawn from a source distribution with bounded support.