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

Samuel Fernández-Menduiña, Eduardo Pavez, Antonio Ortega

TL;DR

This paper develops fast graph Fourier transform (GFT) algorithms with O(nlogn) runtime complexity for rank-one updates of the path graph by providing a factorization for the GFT after perturbation and exploiting the properties of Cauchy matrices.

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.

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

TL;DR

This paper develops fast graph Fourier transform (GFT) algorithms with O(nlogn) runtime complexity for rank-one updates of the path graph by providing a factorization for the GFT after perturbation and exploiting the properties of Cauchy matrices.

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.
Paper Structure (13 sections, 10 theorems, 7 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 10 theorems, 7 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rank-one relationships
  4. Main results
  5. Proofs and extensions
  6. Proofs of Theorem \ref{['thm:main_eq']} and Theorem \ref{['th:fast_general']}
  7. Proof of Theorem \ref{['th:fast_line']}
  8. Pre-processing
  9. Reduction to trigonometric polynomials
  10. Non-uniform Fast Fourier transform
  11. Extension to other graphs and rank-$k$ updates
  12. Numerical Experiments
  13. Conclusion

Key Result

Proposition 2.4

Let $\hbox{$\bf e$}_j$ be the $j$th canonical vector. Then, the Laplacian $\hbox{$\bf L$}$ of an undirected graph can be written as where $w_{ij}$ denotes the edge weights, for $i, j = 1, \hdots, \vert \mathcal{V} \vert$.

Figures (4)

  • Figure 1: Path graph (a) and examples of DCT+ graphs, related to the path graph by a rank-one update of the Laplacian (b-d).
  • Figure 2: (a) Classical video coding approach to compute transforms for RDO. GBT is a graph-based tranform. In the best case, this approach has complexity $O(kn\log n)$. (b) Pruning method exploiting progressivity, with complexity $O(n\log n + (k-1)c_p^2)$.
  • Figure 3: Top row: average runtime for transforming $10000$ AR($0.99$) signals of size $\lbrace 8, 16, 32, 64, 96, 128, 160, 192, 224, 256\rbrace$ using the Fast DCT+ (DCT+), NMVP for DCT+ (Naive), and as a baseline, the time to compute the DCT. Bottom row: results normalized by the DCT time. We consider the three graphs in Fig. \ref{['fig:path_graph']}(b-d), letting $\alpha = 1.5$. For edge updates, we update the edge $(2, 3)$. For edge addition, we add $(3, 5)$.
  • Figure 4: Aggregated runtime and reconstruction quality versus the number of components after pruning. We consider DCT, ADST, and the transform of a path graph with a self-loop of $1.5$. The direct method computes each transform with its fastest implementation. Signals are rows from blocks of size $32\times 32$, obtained from $20$ images of the CLIC dataset CLIC2022. Reconstruction quality is measured against the output of the direct method; we compute the mean-squared error and take mean and maximum among all signals.

Theorems & Definitions (15)

  • Definition 2.1: Cauchy matrix gastinel1960inversion
  • Definition 2.2: GFT ortega2018graph
  • Definition 2.3: Rank-one updates
  • Proposition 2.4: batson2014twice
  • Definition 2.5: DCT+
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 4.1: thompson1976behavior
  • ...and 5 more