New paper reveals structure of substitution error when using Hadamard for convolution

In a new paper uploaded to arXiv (2607.15293), researchers Ben Fauber and Alireza Moradzadeh analyze what happens when the Hadamard transform is used in place of the FFT for computing convolution. The Hadamard transform is attractive for its real-valued sign flips and O(N log N) runtime, but it introduces an algebraic error compared to the exact FFT-based circular convolution. The authors present three complementary results that characterize this error in detail.

First, they identify exact error cancellation at two input and two output positions that are universally error-free, and show that no output reordering can eliminate the error entirely. Second, the error operator is nearly full rank, with a null space of only logarithmic dimension. Third, the expected error is governed by a single alignment scalar—a closed-form expression obtained by averaging over random filters. In general, the substitution error asymptotically doubles the output energy, except for filters that lie in the universal zero-error subspace. These findings demonstrate that the error is not random but structured, predictable, and governed by alignment.