Macroscale analogs [1,2,3] of microscopic spin systems offer direct insights into fundamental physical principles, thereby advancing our understanding of synchronization phenomena [4] and informing the design of novel classes of chiral metamaterials [5,6,7]. Here, we introduce hydrodynamic spin lattices (HSLs) of walking droplets as a new class of active spin systems with particle-wave coupling. HSLs reveal a variety of non-equilibrium symmetry-breaking phenomena, including transitions from anti-ferromagnetic to ferromagnetic order that can be controlled by varying lattice geometry and system rotation [8]. Theoretical predictions based on a generalized Kuramoto model [4] derived from first principles rationalize our experimental observations, establishing HSLs as a versatile platform for exploring active phase oscillator dynamics. The tunability of HSLs suggests exciting directions for future research, from active spin-wave dynamics to hydrodynamic analog computation and droplet-based topological insulators.

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Rectified-linear-unit (ReLU) neural networks, which play a prominent role in deep learning, generate continuous and piecewise-linear (CPWL) functions. While they provide a powerful parametric representation, the mapping between the parameter and function spaces lacks stability. In this paper, we investigate an alternative representation of CPWL functions that relies on local hat basis functions. It is predicated on the fact that any CPWL function can be specified by a triangulation and its values at the grid points. We give the necessary and sufficient condition on the triangulation (in any number of dimensions) for the hat functions to form a Riesz basis, which ensures that the link between the parameters and the corresponding CPWL function is stable and unique. In addition, we provide an estimate of the 2 → L2 condition number of this local representation. Finally, as a special case of our framework, we focus on a systematic parametrization of R d with control points placed on a uniform grid. In particular, we choose hat basis functions that are shifted replicas of a single linear box spline. In this setting, we prove that our general estimate of the condition number is optimal. We also relate our local representation to a nonlocal one based on shifts of a causal ReLU-like function.

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