KLT

KLT, short for the Karhunen–Loève transform, is a linear, orthogonal transform used to decorrelate and compress stochastic processes and signals. It identifies a set of orthogonal basis functions (or eigenvectors) of the signal's covariance operator; projecting the signal onto these basis vectors yields uncorrelated components. In practical terms, for finite data, the KLT is equivalent to principal component analysis (PCA).

Mathematically, for a zero-mean random vector x with covariance matrix C, the KLT seeks eigenvectors e_i and corresponding eigenvalues λ_i such that C e_i = λ_i e_i. When the data are centered, the transformed coordinates y_i = e_i^T x have variances λ_i and are uncorrelated. The components are typically ordered by decreasing eigenvalues, so the first few components capture most of the signal’s variance. In the continuous setting, the transform uses eigenfunctions of the covariance operator of the stochastic process.

Procedure and reconstruction: given a data matrix X with samples as columns, compute C = (1/n) X X^T, find its eigenvectors V and eigenvalues Λ, and form y = V^T X. To reconstruct, x̂ ≈ V_k y_k using the top k components. This yields optimal mean-square error reconstruction for a given number of components, under a linear, orthogonal transform.

Applications and limitations: the KLT is widely used for dimensionality reduction, data compression, denoising, and pattern recognition in signal processing and imaging. It is data-dependent, requiring representative training data to compute the transform, and its optimality is specific to the assumed covariance. It can be computationally intensive for high-dimensional data and is sensitive to outliers and non-stationary statistics.