Interesting Open Problem Related to Complexity of Computing the Fourier Transform and Group Theory
Pith reviewed 2026-05-24 19:55 UTC · model grok-4.3
The pith
A 2013 lower bound on the normalized Fourier transform under unitary pair operations raises a natural open problem in group representation theory.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The author describes a natural open problem arising from the 2013 Omega(n log n) lower bound for computing the normalized Fourier transform assuming only unitary operations on pairs of coordinates is allowed, and this problem is related to group theory, and in particular to representation theory.
What carries the argument
The model of computation restricted to unitary operations on pairs of coordinates, which the paper uses to connect Fourier transform complexity to representation theory of groups.
If this is right
- The lower bound on the Fourier transform can be studied by examining how group representations decompose into pairwise unitary operations.
- Insights from representation theory may yield either matching upper bounds or stronger lower bounds in the same computational model.
- The connection could extend the analysis to other linear transformations whose matrices arise from group actions.
Where Pith is reading between the lines
- Solving the open problem might allow importing techniques from representation theory to prove lower bounds for related transforms such as the Hadamard or discrete cosine transforms.
- If the group-theoretic view succeeds, it could suggest that the minimal complexity is governed by the dimensions of irreducible representations rather than by information-theoretic arguments alone.
- The same modeling approach may apply to quantum circuit complexity, where unitary gates on pairs of qubits already play a central role.
Load-bearing premise
The 2013 model restricting computation to unitary operations on pairs of coordinates provides a meaningful entry point for group-theoretic analysis of Fourier transform complexity.
What would settle it
An explicit group representation or sequence of unitary pair operations that computes the normalized Fourier transform in o(n log n) steps would settle the open problem by showing the 2013 bound does not capture the full algebraic structure.
read the original abstract
The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$. From a lower bound perspective, relatively little is known. Ailon shows in 2013 an $\Omega(n\log n)$ bound for computing the normalized Fourier Transform assuming only unitary operations on pairs of coordinates is allowed. The goal of this document is to describe a natural open problem that arises from this work, which is related to group theory, and in particular to representation theory.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript describes an open problem in the complexity of computing the Fourier transform. It recalls Ailon's 2013 Omega(n log n) lower bound under the model of unitary operations on pairs of coordinates and posits that this gives rise to a natural open problem related to group theory and representation theory.
Significance. If the connection to representation theory can be made precise, the open problem could encourage new approaches to lower bounds for the FFT using algebraic methods. However, the manuscript offers no formal statement of the problem, no partial results, and no specific group-theoretic formulation, so its potential impact is unclear and likely limited as a standalone publication.
major comments (1)
- [Abstract] Abstract: the central claim that Ailon's unitary-pair model 'gives rise to a natural open problem' related to representation theory is asserted without any construction, mapping, or reference showing how the model interacts with group representations or characters.
Simulated Author's Rebuttal
We thank the referee for their review. Our manuscript is a short note intended to pose an open problem rather than resolve it. We address the major comment below.
read point-by-point responses
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Referee: [Abstract] Abstract: the central claim that Ailon's unitary-pair model 'gives rise to a natural open problem' related to representation theory is asserted without any construction, mapping, or reference showing how the model interacts with group representations or characters.
Authors: The manuscript deliberately poses the connection as an open question rather than claiming to construct it. The naturalness of the link follows from the fact that the normalized DFT is the Fourier transform on the cyclic group (whose irreps are well-known) and that unitary pair operations preserve the L2 norm in a manner compatible with unitary representations. We acknowledge that no explicit mapping or reference is supplied because developing one is part of the open problem. We will revise the abstract and add a short section that formally states the open problem in representation-theoretic language to the extent currently possible. revision: yes
Circularity Check
No derivation or quantitative claim; open-problem statement only
full rationale
The document poses an open problem arising from the author's own 2013 lower bound but contains no derivation chain, theorem, prediction, or quantitative claim. No equation or step reduces to a self-citation, fitted input, or self-definition. The reference to group theory is motivational framing rather than a constructed result, so no circularity is present.
discussion (0)
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