# Structure of the Circular-Dyadic Convolution Error

arXiv: https://arxiv.org/abs/2607.15293 · HTML: https://mlsanity.com/paper/2607.15293/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 27 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)
- [Fast Walsh-Hadamard Transform and Smooth-Thresholding Based Binary Layers in Deep Neural Networks](https://mlsanity.com/paper/2104.07085/) (2021)
- [Block Walsh-Hadamard Transform Based Binary Layers in Deep Neural Networks](https://mlsanity.com/paper/2201.02711/) (2022)
- [A Hybrid Quantum-Classical Approach based on the Hadamard Transform for the Convolutional Layer](https://mlsanity.com/paper/2305.17510/) (2023)
- [Uniform Approximations for Randomized Hadamard Transforms with Applications](https://mlsanity.com/paper/2203.01599/) (2022)

## Abstract (arXiv)

Dyadic and circular convolution can both be computed in O(Nlog N) time using the Hadamard transform and the FFT-computed discrete Fourier transform (DFT), respectively. The Hadamard transform is preferable for its real-valued sign flips, yet its substitution for the DFT introduces algebraic error. We present three complementary results that characterize this error. First, we identify exact error cancellation: two input and two output positions are universally error-free, and no reordering of the output can eliminate this error. Second, the error operator is nearly full rank, while its null space has only logarithmic dimension. Third, the expected error is governed by a single alignment scalar, with a closed-form expression obtained by averaging over random filters. In general, the substitution error asymptotically doubles the output energy, except for filters in the universal zero-error subspace, which incur no error. Collectively, these results show that the substitution error is structured, predictable, and governed by alignment.