Automatic Differentiation

Square-root Kalman filters propagate state covariances in Cholesky-factor form for numerical stability, and are a natural target for gradient-based parameter learning in state-space models. Their core operation, triangularization of a matrix $M \in \mathbb{R}^{n \times m}$, is computed via a QR decomposition in practice, but naively differentiating through it causes two problems: the semi-orthogonal factor is non-unique when $m > n$, yielding undefined gradients; and the standard Jacobian formula involves inverses, which diverges when $M$ is rank-deficient. Both are resolved by the observation that all filter outputs relevant to learning depend on the input matrix only through the Gramian $MM^\top$, so the composite loss is smooth in $M$ even where the triangularization is not. We derive a closed-form chain-rule directly from the differential of this Gramian identity, prove it exact for the Kalman log-marginal likelihood and filtered moments, and extend it to rank-deficient inputs via a two-component decomposition: a column-space term based on the Moore--Penrose pseudoinverse, and a null-space correction for perturbations outside the column space of $M$.

Automatic differentiation (AD) frameworks such as JAX and PyTorch have enabled gradient-based optimization for a wide range of scientific fields. Yet, many "hard" primitives in these libraries such as thresholding, Boolean logic, discrete indexing, and sorting operations yield zero or undefined gradients that are not useful for optimization. While numerous "soft" relaxations have been proposed that provide informative gradients, the respective implementations are fragmented across projects, making them difficult to combine and compare. This work introduces SoftJAX and SoftTorch, open-source, feature-complete libraries for soft differentiable programming. These libraries provide a variety of soft functions as drop-in replacements for their hard JAX and PyTorch counterparts. This includes (i) elementwise operators such as clip or abs, (ii) utility methods for manipulating Booleans and indices via fuzzy logic, (iii) axiswise operators such as sort or rank -- based on optimal transport or permutahedron projections, and (iv) offer full support for straight-through gradient estimation. Overall, SoftJAX and SoftTorch make the toolbox of soft relaxations easily accessible to differentiable programming, as demonstrated through benchmarking and a practical case study. Code is available at github.com/a-paulus/softjax and github.com/a-paulus/softtorch.

Spiking neural networks (SNNs), central to computational neuroscience and neuromorphic machine learning (ML), require efficient simulation and gradient-based training. While AI accelerators offer promising speedups, gradient-based SNNs typically implement sparse spike events using dense, memory-heavy data-structures. Existing exact gradient methods lack generality, and current simulators often omit or inefficiently handle delayed spikes. We address this by deriving gradient computation through spike event queues, including delays, and implementing this into memory-efficient, autodifferentiable spike event queues (ADSEQ). These are benchmarked across CPU, GPU, TPU, and LPU platforms. We find that queue design strongly shapes performance. CPUs, as expected, perform well with traditional tree-based or FIFO implementations, while GPUs excel with ring buffers for smaller simulations, yet under higher memory pressure prefer more sparse data-structures. TPUs seem to favor an implementation based on sorting intrinsics. Selective spike dropping provides a simple performance-accuracy trade-off, which could be enhanced by future autograd frameworks adapting diverging primal/tangent data-structures.