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

Soumya Jana and Pierre Moulin

The  Karhunen-Lo\`{e}ve 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.