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arXiv · 2409.08970

Fast DCT+: A Family of Fast Transforms Based on Rank-One Updates of the Path Graph

Samuel Fernández-Menduiña·Eduardo Pavez·Antonio Ortega
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Abstract

This paper develops fast graph Fourier transform (GFT) algorithms with O(n log n) runtime complexity for rank-one updates of the path graph. We first show that several commonly-used audio and video coding transforms belong to this class of GFTs, which we denote by DCT+. Next, starting from an arbitrary generalized graph Laplacian and using rank-one perturbation theory, we provide a factorization for the GFT after perturbation. This factorization is our central result and reveals a progressive structure: we first apply the unperturbed Laplacian's GFT and then multiply the result by a Cauchy matrix. By specializing this decomposition to path graphs and exploiting the properties of Cauchy matrices, we show that Fast DCT+ algorithms exist. We also demonstrate that progressivity can speed up computations in applications involving multiple transforms related by rank-one perturbations (e.g., video coding) when combined with pruning strategies. Our results can be extended to other graphs and rank-k perturbations. Runtime analyses show that Fast DCT+ provides computational gains over the naive method for graph sizes larger than 64, with runtime approximately equal to that of 8 DCTs.

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Samuel Fernández-Menduiña, Eduardo Pavez, Antonio Ortega. 2024-09-13. Fast DCT+: A Family of Fast Transforms Based on Rank-One Updates of the Path Graph. https://arxiv.org/abs/2409.08970

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