Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this collection, which define algebraic maps from parameter space to tensor space.

Designing expressive yet computationally efficient layers for high-dimensional tensor data (e.g., images) remains a significant challenge. While sequence modeling has seen a shift toward linear-time architectures, extending these benefits to higher-order tensors is non-trivial. In this work, we introduce the Fast Iterated Sums (FIS) layer, a novel tensor-to-tensor primitive with linear time and space complexity relative to the input size. Theoretically, our framework bridges deep learning and algorithmic combinatorics: it leverages “corner tree” structures from permutation pattern counting to efficiently compute 2D iterated sums. This formulation admits dual interpretations as both a higher-order state-space model (SSM) and a multiparameter extension of the Signature Transform. Practically, the FIS layer serves as a drop-in replacement for standard layers in vision backbones. We evaluate its performance on image classification and anomaly detection. When replacing layers in a smaller ResNet, the FIS-based model achieves accuracy of a larger ResNet baseline while reducing both trainable parameters and multiply-add operations. When replacing layers in ConvNeXt tiny, the FIS-based model saves around 2% of parameters, has around 8% shorter time per epoch and improves accuracy by around 0.6% on CIFAR-10 and around 2% on CIFAR-100. Furthermore, on the texture subset of MVTec AD, it attains an average AUROC of 97.3%. The code is available at https://github.com/diehlj/fast-iterated-sums .

Non-commutative Gröbner bases of two-sided ideals are not necessarily finite. Motivated by this, we provide a closed-form description of a finite and reduced Gröbner bases for the two-sided ideal used in the construction of Wang's quantum symmetric group. In particular, this constructively proves that the word problem for quantum symmetric groups is decidable.