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arxiv: 1907.03692 · v1 · pith:NT7SBYPZnew · submitted 2019-07-08 · 🧮 math.NA · cs.NA

Multiscale High-Dimensional Sparse Fourier Algorithms for Noisy Data

Bosu Choi , Andrew Christlieb , Yang Wang This is my paper

Pith reviewed 2026-05-25 01:00 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords sparse Fourier transformhigh-dimensional signalsnoisy samplesmultiscale algorithmnearly sparsebandlimitedrobust recoverynumerical analysis
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The pith

A multiscale sparse Fourier algorithm recovers high-dimensional signals from noisy samples while remaining efficient.

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

The paper extends a 2016 sparse Fourier algorithm, which worked for exactly s-sparse bandlimited signals in d dimensions, to the case of noisy and nearly sparse samples. It introduces a multiscale procedure that isolates the energetic frequencies despite noise perturbations. This matters because practical measurements almost always contain noise, so the extension aims to make the recovery technique usable beyond idealized settings. The method preserves the original efficiency targets under the same bandlimited assumptions. If the claim holds, it enables accurate Fourier mode identification with sampling that scales linearly with dimension and sparsity even when the input is imperfect.

Core claim

We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier work handled exactly s-sparse bandlimited signals in d dimensions with Theta(ds log s) runtime and Theta(ds) sampling. The new multiscale algorithm proves robust against noise and efficient when signals remain nearly s-sparse, meaning they are well approximated by the best s Fourier modes, under the bandlimited assumption that energetic frequencies lie in [-N/2, N/2)^d intersect Z^d.

What carries the argument

The multiscale sparse Fourier algorithm, a hierarchical procedure that isolates energetic frequencies in the presence of noise.

If this is right

  • The algorithm handles signals well approximated by s Fourier modes rather than requiring exact sparsity.
  • It achieves the same Theta(ds log s) average-case runtime as the noiseless version.
  • Sampling complexity remains Theta(ds) under the bandlimited assumption.
  • Recovery stays possible when measurements contain additive noise that preserves near-sparsity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • One could apply the same multiscale isolation idea to other high-dimensional sparse recovery tasks such as sparse polynomial approximation.
  • Testing on real sensor arrays with measured noise statistics would show where the near-sparsity assumption breaks in practice.
  • The method suggests a general pattern: hierarchical refinement can trade extra scales for noise tolerance without changing the leading complexity term.

Load-bearing premise

The input signals remain nearly s-sparse in the d-dimensional Fourier domain even after noise is added, so the multiscale steps can isolate the main frequencies.

What would settle it

Generate synthetic d-dimensional signals with controlled noise variance that pushes the Fourier coefficients outside any s-term approximation, then measure whether the algorithm's recovered frequencies deviate beyond the claimed error bound.

Figures

Figures reproduced from arXiv: 1907.03692 by Andrew Christlieb, Bosu Choi, Yang Wang.

Figure 1
Figure 1. Figure 1: (a)Average ℓ1 error vs. noise level σ in logarithm. (b)Average samples vs. noise level in logarithm. (c)Average runtime vs. noise level in logarithm. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a)Average ℓ1 error vs. sparsity s in logarithm. (b)Average samples vs. sparsity s in logarithm. (c)Average runtime vs. sparsity s in logarithm. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
read the original abstract

We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $\Theta(ds \log s)$ average-case runtime and $\Theta(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best $s$ Fourier modes. In this paper, we propose a multiscale sparse Fourier algorithm for noisy samples that proves to be both robust against noise and efficient.

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

0 major / 2 minor

Summary. The manuscript develops a multiscale high-dimensional sparse Fourier algorithm for noisy samples. It extends the 2016 noiseless algorithm (with claimed Θ(ds log s) average-case runtime and Θ(ds) sampling complexity for exactly s-sparse bandlimited signals in d dimensions) to the noisy, nearly s-sparse setting, asserting that the new procedure remains both robust to noise and efficient under the same bandlimited model.

Significance. If the robustness and efficiency claims are substantiated by explicit error bounds, runtime analysis, and sampling guarantees that hold for noisy nearly-sparse inputs, the work would meaningfully extend sparse FFT methodology to practical high-dimensional settings where noise is unavoidable. The multiscale construction could provide a concrete route to isolating energetic frequencies without inflating the sampling or computational cost beyond the noiseless baseline.

minor comments (2)
  1. The abstract states that the algorithm 'proves to be both robust against noise and efficient,' but the provided text supplies no theorem statements, error bounds, or runtime derivations; these must be supplied explicitly in the main body with numbered equations and assumptions clearly listed.
  2. Notation for the frequency support set [-N/2, N/2)^d ∩ Z^d and the nearly-sparse model should be defined once at the outset rather than introduced piecemeal.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the summary of our contributions. We address the key point raised regarding substantiation of the robustness and efficiency claims.

read point-by-point responses

  1. Referee: If the robustness and efficiency claims are substantiated by explicit error bounds, runtime analysis, and sampling guarantees that hold for noisy nearly-sparse inputs, the work would meaningfully extend sparse FFT methodology to practical high-dimensional settings where noise is unavoidable. The multiscale construction could provide a concrete route to isolating energetic frequencies without inflating the sampling or computational cost beyond the noiseless baseline.

    Authors: The manuscript provides explicit error bounds, runtime analysis, and sampling guarantees for the noisy nearly s-sparse case under the bandlimited model. These are derived in the theoretical sections extending the 2016 noiseless algorithm, showing that the multiscale procedure maintains the Θ(ds log s) average-case runtime and Θ(ds) sampling complexity while remaining robust to noise. The analysis isolates energetic frequencies without increasing costs beyond the noiseless baseline. revision: no

Circularity Check

0 steps flagged

Minor self-citation to 2016 base algorithm; new multiscale procedure for noise is independent

full rationale

The paper cites its own 2016 noiseless sparse Fourier work only to import the bandlimited s-sparse model and baseline runtime; the central contribution is a distinct multiscale procedure whose robustness and efficiency claims are asserted under that model but are not shown to reduce to a redefinition, fit, or self-referential collapse of the 2016 inputs. No equation or step in the provided material equates a new prediction to a fitted parameter or prior result by construction. This is the normal, non-circular pattern of algorithmic extension.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract supplies no explicit free parameters or invented entities; it rests on domain assumptions about sparsity and bandlimitedness carried over from the referenced 2016 paper.

axioms (2)
  • domain assumption Signals are at most s-sparse and bandlimited in [-N/2, N/2)^d cap Z^d
    Stated as the setting inherited from the 2016 paper.
  • domain assumption Measurements contain additive noise yet the signal remains nearly sparse
    Explicitly identified as the practical case the new algorithm targets.

pith-pipeline@v0.9.0 · 5683 in / 1154 out tokens · 40182 ms · 2026-05-25T01:00:29.465010+00:00 · methodology

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Reference graph

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