球面与球体上的可微且加速的小波变换
Differentiable and accelerated wavelet transforms on the sphere and ball
AI总结：
本文提出在球面和球体上可微且加速的方向性小波变换，较现有软件分别提速300倍和21800倍，并开源S2WAV和S2BALL库以支持数据驱动分析。
摘要：
方向性小波字典是分层表示，能够高效地捕获和分割跨尺度、位置和方向的信息。这类表示对物理信号表现出特别的亲和力，这些信号通常展现出高度各向异性、局部化的多尺度结构。许多物理上重要的信号是在球面域上观测到的，例如宇宙学中的天球。利用计算调和分析的最新进展，我们设计了在二维球面 $\mathbb{S}^2$ 和三维球体 $\mathbb{B}^3 = \mathbb{R}^+ \times \mathbb{S}^2$（即由球面与径向半直线增广形成的空间）上高度可分布且自动可微的方向性小波变换。与现有软件相比，我们观察到球面和球体上的信号分别加速了高达 300 倍和 21800 倍，同时保持 64 位机器精度。这些算法不仅显著加速了现有的球面小波变换，自动微分提供的梯度信息还解锁了许多以前在这些空间上不可能实现的数据驱动分析技术。我们公开发布了 S2WAV 和 S2BALL，这是我们的变换的开源 JAX 库，它们自动可微，并且易于在硬件加速器（如 GPU 和 TPU）集群上部署。
英文摘要：
Directional wavelet dictionaries are hierarchical representations which efficiently capture and segment information across scale, location and orientation. Such representations demonstrate a particular affinity to physical signals, which often exhibit highly anisotropic, localised multiscale structure. Many physically important signals are observed over spherical domains, such as the celestial sky in cosmology. Leveraging recent advances in computational harmonic analysis, we design new highly distributable and automatically differentiable directional wavelet transforms on the $2$-dimensional sphere $\mathbb{S}^2$ and $3$-dimensional ball $\mathbb{B}^3 = \mathbb{R}^+ \times \mathbb{S}^2$ (the space formed by augmenting the sphere with the radial half-line). We observe up to a $300$-fold and $21800$-fold acceleration for signals on the sphere and ball, respectively, compared to existing software, whilst maintaining 64-bit machine precision. Not only do these algorithms dramatically accelerate existing spherical wavelet transforms, the gradient information afforded by automatic differentiation unlocks many data-driven analysis techniques previously not possible for these spaces. We publicly release both S2WAV and S2BALL, open-sourced JAX libraries for our transforms that are automatically differentiable and readily deployable both on and over clusters of hardware accelerators (e.g. GPUs & TPUs).
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