# Sprecher Networks: A Parameter-Efficient Kolmogorov-Arnold Architecture

arXiv: https://arxiv.org/abs/2512.19367 · HTML: https://mlsanity.com/paper/2512.19367/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [Convolutional Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.13155/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [Kolmogorov-Arnold Transformer](https://mlsanity.com/paper/2409.10594/) (2024)
- [A Temporal Kolmogorov-Arnold Transformer for Time Series Forecasting](https://mlsanity.com/paper/2406.02486/) (2024)
- [Learning Activation Functions: A new paradigm for understanding Neural Networks](https://mlsanity.com/paper/1906.09529/) (2019)
- [Neural Network Architecture Beyond Width and Depth](https://mlsanity.com/paper/2205.09459/) (2022)
- [GS-KAN: Parameter-Efficient Kolmogorov-Arnold Networks via Sprecher-Type Shared Basis Functions](https://mlsanity.com/paper/2512.09084/) (2025)

## Abstract (arXiv)

We introduce Sprecher Networks (SNs), a family of trainable architectures derived from David Sprecher's 1965 constructive form of the Kolmogorov-Arnold representation. Each SN block implements a "sum of shifted univariate functions" using only two shared learnable splines per block, a monotone inner spline φ and a general outer spline Phi, together with a learnable shift parameter η and a mixing vector λ shared across all output dimensions. Stacking these blocks yields deep, compositional models; for vector-valued outputs we append an additional non-summed output block. We also propose an optional lateral mixing operator enabling intra-block communication between output channels with only O(dout) additional parameters. Owing to the vector (not matrix) mixing weights and spline sharing, SNs scale linearly in width, approximately O(sumℓ(dℓ-1+dℓ+G)) parameters for G spline knots, versus O(sumℓ dℓ-1dℓ) for dense MLPs and O(Gsumℓ dℓ-1dℓ) for edge-spline KANs. This linear width-scaling is particularly attractive for extremely wide, shallow models, where low depth can translate into low inference latency. Finally, we describe a sequential forward implementation that avoids materializing the din× dout shifted-input tensor, reducing peak forward-intermediate memory from quadratic to linear in layer width, relevant for memory-constrained settings such as on-device/edge inference; we demonstrate deployability via fixed-point real-time digit classification on resource-constrained embedded device with only 4 MB RAM. We provide empirical demonstrations on supervised regression, Fashion-MNIST classification (including stable training at 25 hidden layers with residual connections and normalization), and a Poisson PINN, with controlled comparisons to MLP and KAN baselines.