Optimality of KLT for High-Rate Transform Coding of Gaussian Vector-Scale Mixtures: Application to Reconstruction, Estimation, and Classification

The Karhunen-Loeacuteve transform (KLT) is known to be optimal for high-rate transform coding of Gaussian vectors for both fixed-rate and variable-rate encoding. The KLT is also known to be suboptimal for some non-Gaussian models. This paper proves high-rate optimality of the KLT for variable-rate encoding of a broad class of non-Gaussian vectors: Gaussian vector-scale mixtures (GVSM), which extend the Gaussian scale mixture (GSM) model of natural signals. A key concavity property of the scalar GSM (same as the scalar GVSM) is derived to complete the proof. Optimality holds under a broad class of quadratic criteria, which include mean-squared error (MSE) as well as generalized f-divergence loss in estimation and binary classification systems. Finally, the theory is illustrated using two applications: signal estimation in multiplicative noise and joint optimization of classification/reconstruction systems