Holographic algorithms: From art to science

Leslie Gabriel Valiant

Leslie Gabriel Valiant

Zhiguo Fu, Jin‐Yi Cai

Ali Al-Bashabsheh, Yongyi Mao, Abbas Yongaçoğlu

Holographic algorithms are a recent breakthrough in computer science and has found applications in information theory. This paper provides a proof to the central component of holographic algorithms, namely, the Holant theorem. The proof is simpler and more direct compared with previous works. Along the proof, we also develop a mathematical tool, which we call c-tensor. We expect the notion of c-tensor may be applicable over a wide range of analysis.

Andrii Torchylo, Hunter Swan, Lucas Tellez +1 more

Optimal transport methods have recently established state of the art accuracy and efficiency for holographic laser beam shaping. However, use of such methods is hindered by severe $\mathcal{O}(N^2)$ memory and $\mathcal{O}(N^2)$ time requirements for large scale input or output images with $N$ total pixels. Here we leverage the dual formulation of the optimal transport problem and the separable structure of the cost to implement algorithms with greatly reduced $\mathcal{O}(N)$ memory and $\mathcal{O}(N\log N)$ to $\mathcal{O}(N^{3/2})$ time complexity. These algorithms are parallelizable and can solve megapixel-scale beam shaping problems in tens of seconds on a CPU or seconds on a GPU.

Jin‐Yi Cai, Pinyan Lu