A lattice vector quantizer is a NNVQ $Q_{\Lambda}$ with codebook $\Lambda$. - Encoding rule: $$Q_{\Lambda}(\mathbf{x})=\underset{\mathbf{y}\in\Lambda}{\arg\min}\ \lVert \mathbf{x}-\mathbf{y} \rVert$$ - Quantizer Cell for $\mathbf{y}_{i}$: $$R_{i}=\{ \mathbf{x}\in\mathbb{R}^{k}:\lVert \mathbf{x}-\mathbf{y}_{i} \rVert \le \lVert \mathbf{x}-\mathbf{y}_{j} \rVert , \ \mathbf{y}_{j}\in\Lambda \}$$

Assume MSE and $\mathbf{X}\sim f$. Then the distortion of a LVQ $Q_{\Lambda}$ can be expressed as $$D(Q_{\Lambda})=\sum_{i}\int\limits _{R_{i}}\lVert \mathbf{x}-\mathbf{y}_{i} \rVert ^{2}f(\mathbf{x}) \, d\mathbf{x}$$