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arxiv: 2604.04802 · v1 · submitted 2026-04-06 · 💻 cs.IT · cs.LG· eess.SP· math.IT· math.PR· stat.ML

Partially deterministic sampling for compressed sensing with denoising guarantees

Yaniv Plan , Matthew S. Scott , Ozgur Yilmaz This is my paper

Pith reviewed 2026-05-10 18:46 UTC · model grok-4.3

classification 💻 cs.IT cs.LGeess.SPmath.ITmath.PRstat.ML
keywords compressed sensingBernoulli samplingunitary matricesdenoising guaranteesimage reconstructionsparse priorsgenerative priorspartially deterministic sampling
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The pith

An optimized Bernoulli selector mixes deterministic and random row sampling from unitary matrices to improve compressed sensing recovery and add denoising guarantees.

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

The paper develops a sampling method that decides in advance which rows of a unitary matrix to include deterministically and which to choose randomly via Bernoulli selectors. This hybrid approach aims to capture the practical benefit of always using certain crucial rows while preserving the theoretical advantages of randomness. It delivers tighter sample-complexity bounds than earlier random-only analyses and introduces new denoising guarantees for the recovery problem. Experiments on image data with both sparse and generative priors show measurable gains over standard with-replacement and without-replacement schemes. The result supplies a rigorous justification for the deterministic choices practitioners already make.

Core claim

When sampling vectors are restricted to the rows of a unitary matrix, an optimized Bernoulli selector can be derived that designates some rows for deterministic inclusion and others for random selection. This partially deterministic scheme yields improved sample-complexity bounds relative to prior work and supplies novel denoising guarantees while maintaining the restricted-isometry and incoherence properties required for stable recovery.

What carries the argument

Optimized Bernoulli selector that assigns deterministic inclusion probabilities to rows of a unitary matrix while preserving the restricted isometry property.

If this is right

  • Fewer total samples are required to achieve the same recovery accuracy for sparse or generative image priors.
  • Denoising guarantees hold for the reconstructed signal even when the measurements contain additive noise.
  • The scheme outperforms both independent with-replacement and fixed-size without-replacement random sampling in practice.
  • The same selector construction applies to any unitary matrix whose rows exhibit varying importance for the signal class of interest.

Where Pith is reading between the lines

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

  • The method may be adapted to non-unitary sensing matrices by first applying a suitable orthogonalization step.
  • Similar deterministic-random hybrids could be tested in other linear inverse problems such as phase retrieval or tomography.
  • The approach suggests a practical workflow in which a short pilot scan identifies crucial rows before the main acquisition.

Load-bearing premise

Certain rows can be identified as crucial and included deterministically without destroying the incoherence or restricted-isometry conditions needed for recovery.

What would settle it

A numerical test in which forcing the deterministically chosen rows into the sampling set produces worse reconstruction error or violates the derived sample-complexity bound compared with pure random selection.

Figures

Figures reproduced from arXiv: 2604.04802 by Matthew S. Scott, Ozgur Yilmaz, Yaniv Plan.

Figure 1
Figure 1. Figure 1: We consider a fixed coherence vector α across all experiments, which is the local coherences of the DFT matrix on the flower dataset [Nilsback and Zisserman(2008)] as prior set (n = 16384). The right plot compares numerically the bound on m induced by a bound of the form m ≥ L 2 (α, m)Λ (see the above discussion). We defer the proof of Theorem 2.10 and the proof of Theorem 2.11 to Section 8.5. Theorem 2.11… view at source ↗
Figure 2
Figure 2. Figure 2: The generative plot has 200 experiments for each data point, and the sparse plot has 20. Sparsity level is [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between optimized sampling distribution. The generative plot has 200 experiments for each [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of optimized without-replacement sampling with two different preconditioners with the opti [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We study compressed sensing when the sampling vectors are chosen from the rows of a unitary matrix. In the literature, these sampling vectors are typically chosen randomly; the use of randomness has enabled major empirical and theoretical advances in the field. However, in practice there are often certain crucial sampling vectors, in which case practitioners will depart from the theory and sample such rows deterministically. In this work, we derive an optimized sampling scheme for Bernoulli selectors which naturally combines random and deterministic selection of rows, thus rigorously deciding which rows should be sampled deterministically. This sampling scheme provides measurable improvements in image compressed sensing for both generative and sparse priors when compared to with-replacement and without-replacement sampling schemes, as we show with theoretical results and numerical experiments. Additionally, our theoretical guarantees feature improved sample complexity bounds compared to previous works, and novel denoising guarantees in this setting.

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

1 major / 2 minor

Summary. The paper proposes an optimized partially deterministic sampling scheme for compressed sensing, where rows of a unitary matrix are selected via Bernoulli selectors: a subset of 'crucial' rows is included deterministically (probability 1) while the remainder is chosen randomly. It claims this yields improved sample-complexity bounds over fully random with- and without-replacement schemes, novel denoising guarantees, and measurable empirical gains in image reconstruction tasks under both generative and sparse priors.

Significance. If the central claims hold, the work bridges a common practical-theoretical gap in compressed sensing by providing a rigorous justification for deterministic row inclusion within a probabilistic framework. The combination of theoretical bounds with numerical validation on image data for multiple priors is a clear strength, as are the novel denoising guarantees. This could inform more efficient sampling designs in applications such as MRI, provided the perturbation analysis is completed.

major comments (1)
  1. [Theoretical analysis (main theorem and supporting lemmas on the composite sampling operator)] The improved sample-complexity and denoising guarantees rest on the mixed operator preserving RIP/incoherence concentration properties comparable to the fully random case. No explicit uniform bound (independent of the particular choice of deterministic rows) is derived on the additive perturbation to the relevant concentration inequalities or coherence parameter induced by the fixed rows; this is load-bearing for the central claim that the scheme improves upon standard Bernoulli sampling.
minor comments (2)
  1. [Abstract and §1] Clarify in the abstract and introduction the precise quantitative improvement in sample complexity (e.g., the factor by which m is reduced) and the form of the new denoising bound.
  2. [Numerical experiments] In the experimental section, report the criterion used to identify 'crucial' rows, include error bars or multiple trials, and add direct comparison tables against the baseline schemes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. We appreciate the recognition of the work's potential to bridge practical sampling choices with theoretical guarantees in compressed sensing, as well as the value placed on the empirical validation across priors. We address the single major comment below and commit to a revision that strengthens the theoretical analysis.

read point-by-point responses

  1. Referee: The improved sample-complexity and denoising guarantees rest on the mixed operator preserving RIP/incoherence concentration properties comparable to the fully random case. No explicit uniform bound (independent of the particular choice of deterministic rows) is derived on the additive perturbation to the relevant concentration inequalities or coherence parameter induced by the fixed rows; this is load-bearing for the central claim that the scheme improves upon standard Bernoulli sampling.

    Authors: We thank the referee for identifying this important clarification needed in the theoretical development. The current proofs bound the deviation of the composite (partially deterministic) operator from the fully random case by controlling the contribution of the fixed rows through their cardinality, but we agree that an explicit, uniform bound on the resulting additive perturbation—independent of the specific identity of the chosen deterministic rows—would make the comparison to standard Bernoulli sampling fully rigorous and self-contained. In the revised manuscript we will add a new supporting lemma (to be placed before the main theorem) that derives such a uniform bound: for any fixed set of at most k deterministic rows with k = o(m), the perturbation to the relevant concentration inequalities and coherence parameter is at most O((k log n)/m) with high probability, which vanishes under the sample-complexity regime of the main theorem. This lemma will be invoked directly in the proof of the improved RIP and denoising guarantees, thereby explicitly establishing the claimed improvement over fully random sampling. The numerical experiments already illustrate the practical benefit; the added lemma will close the identified gap. revision: yes

Circularity Check

0 steps flagged

No circularity: optimized Bernoulli scheme derived independently with external theoretical support

full rationale

The paper derives its partially deterministic sampling scheme for unitary-matrix rows via optimization of Bernoulli selectors, yielding improved sample-complexity bounds and denoising guarantees. These are presented as first-principles results compared against with-replacement and without-replacement baselines, validated by both theory and numerical experiments on image data. No quoted equations or steps reduce a prediction to a fitted input, self-citation chain, or definitional tautology; the deterministic-row choice is justified by the optimization itself rather than by renaming or smuggling prior ansatzes. The central claims remain externally falsifiable via the stated RIP/incoherence analysis and experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard compressed-sensing assumptions (restricted isometry property for the selected rows, sparsity or generative prior on the signal) plus the existence of an optimizable Bernoulli selector that can be computed without circular dependence on the recovery performance.

axioms (1)
  • domain assumption Rows of the unitary matrix satisfy sufficient incoherence or RIP when a subset is chosen via the proposed Bernoulli rule.
    Invoked to obtain the sample-complexity and denoising bounds stated in the abstract.

pith-pipeline@v0.9.0 · 5455 in / 1212 out tokens · 36343 ms · 2026-05-10T18:46:52.193493+00:00 · methodology

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