On the rate-distortion performance and computational efficiency of the Karhunen-Loeve transform for lossy data compression

We examine the rate-distortion performance and computational complexity of linear transforms for lossy data compression. The goal is to better understand the performance/complexity tradeoffs associated with using the Karhunen-Loeve transform (KLT) and its fast approximations. Since the optimal transform for transform coding is unknown in general, we investigate the performance penalties associated with using the KLT by examining cases where the KLT fails, developing a new transform that corrects the KLT's failures in those examples, and then empirically testing the performance difference between this new transform and the KLT. Experiments demonstrate that while the worst KLT can yield transform coding performance at least 3 dB worse than that of alternative block transforms, the performance penalty associated with using the KLT on real data sets seems to be significantly smaller, giving at most 0.5 dB difference in our experiments. The KLT and its fast variations studied here range in complexity requirements from O(n^2) to O(n log n) in coding vectors of dimension n. We empirically investigate the rate-distortion performance tradeoffs associated with traversing this range of options. For example, an algorithm with complexity O(n^3/2) and memory O(n) gives 0.4 dB performance loss relative to the full KLT in our image compression experiment

Suboptimality of the Karhunen-Loève transform for transform coding

We examine the performance of the Karhunen-Loeve transform (KLT) for transform coding applications. The KLT has long been viewed as the best available block transform for a system that orthogonally transforms a vector source, scalar quantizes the components of the transformed vector using optimal bit allocation, and then inverse transforms the vector. This paper treats fixed-rate and variable-rate transform codes of non-Gaussian sources. The fixed-rate approach uses an optimal fixed-rate scalar quantizer to describe the transform coefficients; the variable-rate approach uses a uniform scalar quantizer followed by an optimal entropy code, and each quantized component is encoded separately. Earlier work shows that for the variable-rate case there exist sources on which the KLT is not unique and the optimal quantization and coding stage matched to a "worst" KLT yields performance as much as 1.5 dB worse than the optimal quantization and coding stage matched to a "best" KLT. In this paper, we strengthen that result to show that in both the fixed-rate and the variable-rate coding frameworks there exist sources for which the performance penalty for using a "worst" KLT can be made arbitrarily large. Further, we demonstrate in both frameworks that there exist sources for which even a best KLT gives suboptimal performance. Finally, we show that even for vector sources where the KLT yields independent coefficients, the KLT can be suboptimal for fixed-rate coding

Note on New KLT relations

In this short note, we present two results about KLT relations discussed in recent several papers. Our first result is the re-derivation of Mason-Skinner MHV amplitude by applying the S_{n-3} permutation symmetric KLT relations directly to MHV amplitude. Our second result is the equivalence proof of the newly discovered S_{n-2} permutation symmetric KLT relations and the well-known S_{n-3} permutation symmetric KLT relations. Although both formulas have been shown to be correct by BCFW recursion relations, our result is the first direct check using the regularized definition of the new formula.Comment: 15 Pages; v2: minor correction

Note on Soft Graviton theorem by KLT Relation

Recently, new soft graviton theorem proposed by Cachazo and Strominger has inspired a lot of works. In this note, we use the KLT-formula to investigate the theorem. We have shown how the soft behavior of color ordered Yang-Mills amplitudes can be combined with KLT relation to give the soft behavior of gravity amplitudes. As a byproduct, we find two nontrivial identities of the KLT momentum kernel must hold.Comment: 25 page