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arxiv: 1906.08367 · v1 · pith:X2T5EZDSnew · submitted 2019-06-19 · 🧮 math.FA · cs.NA· math.NA

Stability of the Kaczmarz Reconstruction for Stationary Sequences

Caleb Camrud , Evan Camrud , Lee Przybylski , Eric S. Weber This is my paper

Pith reviewed 2026-05-25 19:42 UTC · model grok-4.3

classification 🧮 math.FA cs.NAmath.NA
keywords Kaczmarz algorithmstationary sequencesAbel summationnoise stabilityFourier seriessingular measuresHardy spaces
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The pith

A relaxed Kaczmarz algorithm stabilizes reconstruction from noisy inner products with stationary sequences and removes noise entirely for certain profiles.

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

The Kaczmarz algorithm reconstructs an unknown vector from inner products but grows unstable when additive noise corrupts those products and the sequence is stationary. A relaxed version of the algorithm restores stability; the relaxation amounts to Abel summation when the problem is viewed as reconstruction on the unit disc. For noise lying in H^∞(D) or selected subspaces of H²(D), the relaxed procedure eliminates the noise corruption completely. The same relaxation also stabilizes Fourier series reconstruction in L²(μ) when the measure μ is singular, via the spectral representation of stationary sequences.

Core claim

The Kaczmarz algorithm is unstable in the presence of noise for stationary sequences. However, relaxing the algorithm stabilizes the reconstruction. This relaxation corresponds to Abel summation on the unit disc. For noise in H^∞(D) or certain subspaces of H²(D), the relaxed algorithm fully removes the noise corruption. It also stabilizes Fourier series expansions in L²(μ) for singular μ.

What carries the argument

The relaxed Kaczmarz algorithm, equivalent to Abel summation when viewed as reconstruction on the unit disc.

If this is right

  • The reconstruction process becomes stable despite additive noise.
  • Noise corruption is completely eliminated when the noise lies in H^∞(D) or specified H²(D) subspaces.
  • The method extends to stabilizing Fourier series reconstructions for singular measures.

Where Pith is reading between the lines

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

  • The relaxation technique could be tested on other iterative projection methods that suffer similar instability.
  • Applications to discrete signal recovery might benefit if the underlying sequence satisfies the stationarity condition.

Load-bearing premise

The sequence forms a stationary sequence and the additive noise belongs to H^∞(D) or particular subspaces of H²(D) for complete removal.

What would settle it

A stationary sequence together with noise in H^∞(D) for which the relaxed Kaczmarz iteration still diverges or retains corruption.

read the original abstract

The Kaczmarz algorithm is an iterative method to reconstruct an unknown vector $f$ from inner products $\langle f , \varphi_{n} \rangle $. We consider the problem of how additive noise affects the reconstruction under the assumption that $\{ \varphi_{n} \}$ form a stationary sequence. Unlike other reconstruction methods, such as frame reconstructions, the Kaczmarz reconstruction is unstable in the presence of noise. We show, however, that the reconstruction can be stabilized by relaxing the Kaczmarz algorithm; this relaxation corresponds to Abel summation when viewed as a reconstruction on the unit disc. We show, moreover, that for certain noise profiles, such as those that lie in $H^{\infty}(\mathbb{D})$ or certain subspaces of $H^{2}(\mathbb{D})$, the relaxed version of the Kaczmarz algorithm can fully remove the corruption by noise in the inner products. Using the spectral representation of stationary sequences, we show that our relaxed version of the Kaczmarz algorithm also stabilizes the reconstruction of Fourier series expansions in $L^2(\mu)$ when $\mu$ is singular.

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 / 2 minor

Summary. The manuscript claims that the Kaczmarz algorithm for reconstructing a vector f from inner products with a stationary sequence {φ_n} is unstable under additive noise, but a relaxed version of the iteration (identified with Abel summation on the unit disc) stabilizes the reconstruction. It further claims that this relaxed algorithm completely removes noise corruption when the noise sequence lies in H^∞(D) or designated subspaces of H²(D). Via the spectral representation of stationary sequences, the same relaxation is shown to stabilize Fourier expansions in L²(μ) for singular measures μ.

Significance. If the central claims hold, the work supplies a theoretically grounded stabilization of the Kaczmarz method for stationary sequences by connecting it to Abel means and the spectral theorem. The explicit use of the spectral representation to transfer the result to singular-measure Fourier expansions is a clear strength, as it yields a parameter-free approach under the stationarity hypothesis rather than introducing fitted parameters. This could be of interest in approximation theory and signal processing contexts where stationary processes arise.

major comments (2)
  1. [§3] §3, the statement following Eq. (3.4): the claim that the relaxed iteration corresponds exactly to Abel summation on the disc is load-bearing for the stabilization result, yet the derivation appears to assume the sequence is realized on the circle without explicitly verifying the radial limit interchange for general stationary spectral measures.
  2. [Theorem 4.3] Theorem 4.3 (noise removal for H^∞ noise): the argument that the Abel mean eliminates the noise term relies on the bounded analyticity of the noise; however, when the underlying measure μ is singular, the boundary behavior of the H^∞ function must be controlled against the support of μ, and this interaction is not addressed in the proof sketch.
minor comments (2)
  1. [Abstract] The abstract refers to 'certain subspaces of H²(D)' without naming them; the introduction should list the precise subspaces used in the later theorems.
  2. [Notation section] Notation for the relaxation parameter is introduced inconsistently between the algorithmic description and the Abel-mean formulation; a single symbol should be fixed throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying two points in the proofs that require additional justification. We address each major comment below and will incorporate the necessary clarifications in a revised version.

read point-by-point responses

  1. Referee: [§3] §3, the statement following Eq. (3.4): the claim that the relaxed iteration corresponds exactly to Abel summation on the disc is load-bearing for the stabilization result, yet the derivation appears to assume the sequence is realized on the circle without explicitly verifying the radial limit interchange for general stationary spectral measures.

    Authors: We agree that an explicit verification of the radial-limit interchange is needed for arbitrary stationary spectral measures. The stationarity hypothesis and the spectral representation already guarantee that the Abel means are well-defined in the weak sense, but the manuscript does not spell out the justification for interchanging the radial limit with the inner-product representation when μ is not absolutely continuous. In the revision we will insert a short lemma (or expanded paragraph after Eq. (3.4)) that invokes the spectral theorem and the fact that Abel summation commutes with the unitary equivalence to L²(μ) for any finite positive measure μ on the circle. revision: yes

  2. Referee: [Theorem 4.3] Theorem 4.3 (noise removal for H^∞ noise): the argument that the Abel mean eliminates the noise term relies on the bounded analyticity of the noise; however, when the underlying measure μ is singular, the boundary behavior of the H^∞ function must be controlled against the support of μ, and this interaction is not addressed in the proof sketch.

    Authors: The referee is correct that the proof sketch of Theorem 4.3 does not explicitly control the boundary values of the H^∞ noise function on the support of a singular μ. While the Abel mean of an H^∞ function vanishes at the origin independently of μ, the reconstruction identity used in the theorem requires that the noise contribution tends to zero in the L²(μ) norm. We will revise the argument by adding a short paragraph that recalls the radial-limit theorem for H^∞ functions (which holds Lebesgue-almost everywhere) together with the observation that any singular measure is supported on a set of Lebesgue measure zero; consequently the pointwise radial limits need not be invoked on the support of μ itself. The revised proof will therefore separate the absolutely continuous and singular parts of the measure and treat the singular contribution directly via the definition of the Abel mean. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external spectral theorem

full rationale

The paper invokes the spectral representation of stationary sequences as a standard external tool (Bochner's theorem for positive definite sequences) to connect the relaxed Kaczmarz iteration to Abel summation on the disc and to Fourier expansions in L2(μ) for singular μ. All claims are explicitly conditional on the stationarity assumption and on noise belonging to H^∞(D) or designated H²(D) subspaces; no parameter is fitted to data and then relabeled as a prediction, no self-citation chain bears the central load, and no ansatz is smuggled via prior work by the same authors. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; stationarity of the sequence and membership of noise in Hardy spaces are background assumptions rather than new postulates.

pith-pipeline@v0.9.0 · 5739 in / 1140 out tokens · 24130 ms · 2026-05-25T19:42:32.677522+00:00 · methodology

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Reference graph

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