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Computing the SVD efficiently with photonic chips

Johannes Maly, Korbinian Neuner, Samarth Vadia

TL;DR

This work investigates the potential of linear photonic chips for accelerating the computation of the singular value decomposition (SVD) of a matrix and finds hybrid systems of digital controller and photonic chip asymptotically perform on par with large-scale CPU/GPU systems in terms of runtime.

Abstract

In light of today's massive data processing, digital computers are reaching fundamental performance limits due to physical limitations and energy consumption. For specific applications, tailored analog systems offer promising alternatives to digital processors. In this work, we investigate the potential of linear photonic chips for accelerating the computation of the singular value decomposition (SVD) of a matrix. The SVD is a key primitive in linear algebra and forms a crucial component of various modern data processing algorithms. Our main insights are twofold: first, hybrid systems of digital controller and photonic chip asymptotically perform on par with large-scale CPU/GPU systems in terms of runtime. Second, such hybrid systems clearly outperform digital systems in terms of energy consumption.

Computing the SVD efficiently with photonic chips

TL;DR

This work investigates the potential of linear photonic chips for accelerating the computation of the singular value decomposition (SVD) of a matrix and finds hybrid systems of digital controller and photonic chip asymptotically perform on par with large-scale CPU/GPU systems in terms of runtime.

Abstract

In light of today's massive data processing, digital computers are reaching fundamental performance limits due to physical limitations and energy consumption. For specific applications, tailored analog systems offer promising alternatives to digital processors. In this work, we investigate the potential of linear photonic chips for accelerating the computation of the singular value decomposition (SVD) of a matrix. The SVD is a key primitive in linear algebra and forms a crucial component of various modern data processing algorithms. Our main insights are twofold: first, hybrid systems of digital controller and photonic chip asymptotically perform on par with large-scale CPU/GPU systems in terms of runtime. Second, such hybrid systems clearly outperform digital systems in terms of energy consumption.
Paper Structure (11 sections, 1 theorem, 6 equations, 9 figures, 10 tables, 7 algorithms)

This paper contains 11 sections, 1 theorem, 6 equations, 9 figures, 10 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Synopsis of our results
  3. Related work
  4. Notation
  5. Computing the SVD on the specified hybrid system
  6. SVD via alternating QR decomposition
  7. Optimizing GRK-SVD for our hybrid system
  8. Experiments
  9. Discussion
  10. Operation counts of algorithms
  11. Pseudocode for all presented algorithms

Key Result

Lemma 2.1

Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ be an arbitrary matrix, and let $\boldsymbol{\Sigma}^{(1)},\boldsymbol{\Sigma}^{(\frac{3}{2})},\boldsymbol{\Sigma}^{(2)},\dots$ be the sequence generated by Algorithm alg:SVDviaQR. If $\boldsymbol{\Sigma}^{(k/2)}$ converges, then there exists an SVD repre for all $k \ge k_\varepsilon$.

Figures (9)

  • Figure 1: Schematic representation of the hybrid system comprising a digital controller (DC) and an optical chip (OC). Artwork generated using ChatGPT.
  • Figure 2: MZI configurations of the optical chip.
  • Figure 3: MZI configurations of the optical chip for accelerating (i) the QR-decomposition in QR-SVD and the bidiagonalization step of GRK-SVD, and (ii) the composition of the singular vector matrices in the chasing phase of GRK-SVD. Note that flipping the chip can be done without changing the hardware by reallocating which optical channel corresponds to which coordinate.
  • Figure 4: Number of iterations to reach target precision of $10^{-6}$ for QR-SVD and GRK-SVD. Linear best fit for medians is $y=mx+b$, with $m \approx 13.88$ and $b \approx -78.61$ (QR-SVD) and $m \approx 1.47$ and $b\approx 0.83$ (GRK-SVD).
  • Figure 5: Error on off-diagonal entries of $\boldsymbol{\Sigma}^{(k)}$ when applying QR-SVD and GRK-SVD to $200$ randomly drawn matrices in $\mathbb{R}^{15\times 15}$.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma 2.1
  • proof