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arxiv: 2605.03935 · v1 · submitted 2026-05-05 · 📡 eess.SP · cs.DS· cs.IT· math.IT

Deterministic Sparse FFT via Keyed Multi-View Gating with O(sqrt{N} log k) Expected Time

Aaron R. Flouro , Shawn P. Chadwick This is my paper

Pith reviewed 2026-05-07 13:55 UTC · model grok-4.3

classification 📡 eess.SP cs.DScs.ITmath.IT
keywords sparse Fourier transformdeterministic algorithmChinese Remainder Theoremcandidate reductionpeeling decodersublinear timeworst-case guarantees
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The pith

A multi-view CRT gating scheme reduces sparse FFT candidates from quadratic to linear in k for O(sqrt(N) log k) expected recovery with O(N log N) worst-case safety.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a deterministic sparse Fourier transform that relies on keyed multi-view gating and 2-of-3 Chinese Remainder Theorem agreement to cut the number of candidate frequency pairs down to linear in the sparsity level k. It replaces randomized bucketing with deterministic structure whose only randomness comes from assumptions about frequency placement and hash independence. A peeling decoder then extracts the frequencies directly, and a recursive reduction removes the usual preprocessing cost floor. The result is sublinear expected runtime while a dense FFT fallback keeps the worst case at the cost of a standard FFT. Verification steps based on energy consistency and residuals bound the failure probability and guarantee no false negatives when the checks pass.

Core claim

By using a keyed multi-view gating mechanism based on 2-of-3 CRT agreement, the algorithm deterministically reduces O(k^2) candidate frequency pairs to Theta(k) under sparse-regime assumptions, enabling a peeling-based recovery procedure that achieves O(sqrt(N) log k) expected identification time after recursive self-reduction, while an O(N log N) dense-FFT fallback and multi-view verification ensure bounded failure probability with no false negatives under correct verification.

What carries the argument

Keyed multi-view gating with 2-of-3 Chinese Remainder Theorem agreement that reduces candidate pairs and feeds a peeling decoder.

Load-bearing premise

Frequencies must lie in a sparse regime and the affine hash functions across the three views must behave independently.

What would settle it

A sparse frequency set of size k for which the 2-of-3 CRT checks leave more than Theta(k) surviving candidates, forcing either super-linear peeling time or a false negative after verification.

Figures

Figures reproduced from arXiv: 2605.03935 by Aaron R. Flouro, Shawn P. Chadwick.

Figure 1
Figure 1. Figure 1: Eight-phase algorithm flowchart showing the keyed gating sparse view at source ↗
read the original abstract

We introduce a deterministic sparse Fourier transform framework based on a keyed multi-view gating mechanism that leverages 2-of-3 Chinese Remainder Theorem (CRT) agreement to reduce candidate frequency pairs from $O(k^2)$ to $\Theta(k)$ under sparse-regime assumptions. Unlike prior approaches that rely on randomized bucketization for candidate formation, the proposed method provides deterministic structure with probabilistic guarantees arising only from assumptions on frequency placement and independence of affine hashing across views. The algorithm is realized through a peeling-based recovery procedure that extracts frequencies directly from singleton bins without explicit pair enumeration. A recursive self-reduction eliminates the $O(\sqrt{N} \log N)$ preprocessing floor, yielding $O(\sqrt{N} \log k)$ expected identification time while maintaining an $O(N \log N)$ worst-case bound via deterministic dense-FFT fallback. A multi-view verification framework combining Parseval energy consistency and bin-wise residual checks ensures bounded failure probability and no false negatives under correct verification. This establishes a framework combining deterministic candidate reduction, sublinear expected complexity, and worst-case safety guarantees within a CRT-based sparse FFT architecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a deterministic sparse Fourier transform algorithm that employs a keyed multi-view gating mechanism and 2-of-3 Chinese Remainder Theorem (CRT) agreement to reduce the number of candidate frequency pairs from O(k²) to Θ(k) under sparse-regime assumptions. This enables a peeling-based recovery with O(√N log k) expected identification time, while falling back to O(N log N) dense FFT in the worst case, supported by a multi-view verification framework for bounded failure probability and no false negatives.

Significance. If the central claims regarding the candidate reduction and probabilistic guarantees hold under the stated assumptions, the work would represent a significant advance in deterministic sparse FFT methods by achieving sublinear expected complexity without relying on randomization for bucketization, while preserving worst-case safety. This could be valuable for applications in signal processing where determinism is preferred.

major comments (2)
  1. [Abstract] The reduction from O(k²) to Θ(k) candidates via 2-of-3 CRT agreement is asserted to hold under sparse-regime assumptions on frequency placement and independence of affine hashing across views, but the abstract provides no explicit probability bound, collision analysis, or derivation for this pruning step. Since this reduction is load-bearing for achieving the claimed O(√N log k) expected time (as quadratic candidates would prevent efficient peeling), a detailed analysis is required.
  2. The recursive self-reduction that eliminates the O(√N log N) preprocessing floor is mentioned without specifying how it interacts with the multi-view gating or maintains the deterministic structure and guarantees.
minor comments (1)
  1. [Abstract] The abstract is dense and could be improved by explicitly stating the signal model (e.g., the exact sparsity and noise assumptions) early on for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points. We provide detailed responses to each major comment and commit to making the necessary revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] The reduction from O(k²) to Θ(k) candidates via 2-of-3 CRT agreement is asserted to hold under sparse-regime assumptions on frequency placement and independence of affine hashing across views, but the abstract provides no explicit probability bound, collision analysis, or derivation for this pruning step. Since this reduction is load-bearing for achieving the claimed O(√N log k) expected time (as quadratic candidates would prevent efficient peeling), a detailed analysis is required.

    Authors: We acknowledge that the abstract, due to length constraints, omits the explicit probability bound and derivation. The full manuscript (Section 3.2) provides the collision analysis: under the sparse-regime assumptions (at most k nonzero frequencies with random-like placement) and independence of the three affine hashes, the probability that a non-matching pair survives 2-of-3 CRT agreement is bounded by O(1/k), yielding Θ(k) candidates with probability 1 - O(k^{-1}). This bound is derived via union bound over possible colliding pairs and is load-bearing for the peeling complexity. We will revise the abstract to include a concise statement of this probability and a reference to the derivation. revision: yes

  2. Referee: [—] The recursive self-reduction that eliminates the O(√N log N) preprocessing floor is mentioned without specifying how it interacts with the multi-view gating or maintains the deterministic structure and guarantees.

    Authors: Section 4 describes the recursive self-reduction, which reapplies the same keyed multi-view CRT gating to the reduced support identified in the prior level. Determinism is preserved because the keys and view parameters remain fixed across recursion; no new randomization is introduced. Probabilistic guarantees are maintained via a union bound over O(log k) recursion depths, and the O(N log N) worst-case bound is upheld by invoking the dense-FFT fallback whenever verification fails at any level. We agree the interaction merits explicit clarification and will add a dedicated paragraph in the introduction plus a note in Section 4. revision: yes

Circularity Check

0 steps flagged

No circularity: complexity bounds derive from explicit assumptions on frequency placement and hashing independence, not from self-referential definitions or fitted inputs.

full rationale

The paper's central claims rest on stated external assumptions (sparse-regime frequency placement and cross-view affine hash independence) that are not derived from the algorithm's own equations or prior self-citations within the provided text. The O(√N log k) expected time is obtained via a recursive self-reduction and peeling decoder whose candidate pruning is asserted to follow from those assumptions rather than being tautological. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described derivation chain. The worst-case O(N log N) fallback is independent. This is the normal non-circular case for a paper whose guarantees are explicitly conditional on unproven but external modeling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on domain assumptions about sparse frequency placement and hash independence; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Sparse-regime assumptions on frequency placement
    Invoked to reduce candidate frequency pairs from O(k^2) to Theta(k) via 2-of-3 CRT agreement.
  • domain assumption Independence of affine hashing across views
    Required for the probabilistic guarantees on candidate formation and peeling recovery.

pith-pipeline@v0.9.0 · 5511 in / 1458 out tokens · 45465 ms · 2026-05-07T13:55:27.429953+00:00 · methodology

discussion (0)

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Reference graph

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