  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).
