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Greedy Algorithms for Hybrid Compressed Sensing

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proposes two greedy algorithms for hybrid compressed sensing—one-bit measurements for support detection, analog measurements for residue updates—and proves success-probability bounds for both.

desk verdict The hybrid-CS algorithms and Theorem 4 are worth a look, but the main theory is vacuous: an m_r×n matrix with m_r<n cannot satisfy RIP of order n, so Theorems 5–6 do not apply to the paper's own experiments. read the letter →

arxiv 1908.06359 v1 pith:XABULZA2 submitted 2019-08-18 eess.SP cs.ITmath.IT

Ching-Lun Tai , Sung-Hsien Hsieh , Chun-Shien Lu This is my paper · ORCID
classification eess.SPcs.ITmath.IT MSC 94A1294A20
keywords hybridcompressedsensingone-bitgreedyalgorithmssupportdetectionrandomuniformtessellationsrestrictedisometrypropertysparserecoverybitbudget
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that hybrid compressed sensing—collecting a small number of ordinary linear measurements alongside a larger number of one-bit sign measurements—can be decoded by greedy algorithms. Its first algorithm detects the support of a sparse signal by repeatedly testing candidate supports against the one-bit inequalities, then updating the residue with a least-squares projection from the linear measurements. Its second algorithm takes an initial support guess and greedily swaps one index at a time until the guess stops changing. For both algorithms the paper proves lower bounds on the probability of success, and for one-bit measurements it proves an error bound based on random uniform tessellations. The practical point is that one-bit measurements are cheap in storage and robust to noise, while linear measurements carry the scale information that one-bit measurements discard; if the claims hold, hybrid CS gives a bit-efficient recovery route when signal energy is unknown.

What carries the argument

The load-bearing object is the hybrid measurement pair $y_r = A_r\tilde{x}$ and $y_o = \operatorname{sign}(A_o\tilde{x})$ with $\tilde{x}=x+u$, together with a greedy selection rule. For a candidate support set $S$, the algorithms form the least-squares estimate $\tau_n(A_{rS}^{\dagger} y_r, S)$ and count how many of the one-bit inequalities $[y_o]_i \langle a_{o,i}, \hat{x}\rangle \ge 0$ it satisfies; the candidate with the largest count wins. Algorithm 1 then removes the contribution of the selected support from the linear measurements via the residue update $r_j = y_r - A_{r\Omega_j} A_{r\Omega_j}^{\dagger} y_r$, while Algorithm 2 repeatedly adds the best new index and prunes the worst current one. The theoretical bridge is the random-uniform-tessellation bound, restated for sparse signals in Theorem 4, which links normalized recovery error to the fraction of separating hyperplanes; the probability theorems then convert the sign-change probability for a single Gaussian hyperplane into binomial success counts.

What would settle it

Take any $m_r \times n$ measurement matrix $A_r$ with $m_r < n$, for instance the paper's simulation choice $m_r = 6$, $n = 256$. Pick a nonzero vector $v$ with $A_r v = 0$; then $\lVert A_r v \rVert_2 = 0$ while $\lVert v \rVert_2 > 0$, so the restricted-isometry inequality $(1-\delta_n)\lVert v \rVert_2^2 \le \lVert A_r v \rVert_2^2$ forces $\delta_n \ge 1$. Since the theorems assume $\delta_n \in (0, 0.5]$, their probability bounds cannot apply to any matrix of the intended size.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that support information can be extracted from binary measurements by a greedy counting rule, provided the candidate estimates are formed from the traditional linear measurements, and that a second greedy pass can repair a wrong initial support. Theorem 4 says that for sparse signals the normalized recovery error is bounded by $\delta + d_{A_o}(\tilde{x}, \hat{x})$, where $d_{A_o}$ is the fraction of one-bit hyperplanes separating the noisy signal from the estimate. Theorem 5 gives a lower bound on the probability that Algorithm 1 detects the true support after $s$ iterations, and Theorem 6 gives a lower bound on the probability that Algorithm 2 turns the initial support into the true support. The simulation section reports that both algorithms outperform three classic greedy algorithms for traditional compressed sensing under the same bit budget in noisy experiments.

Load-bearing premise

The main theorems assume the small linear measurement matrix preserves distances for every vector in the full $n$-dimensional space, but a matrix with more columns than rows always sends some nonzero vector to zero, so the assumption cannot hold in the regime the algorithms are designed for.

Editorial extensions

If this is right

  • At a fixed storage budget, moving some budget from high-precision linear measurements to one-bit measurements can preserve or improve recovery accuracy; the simulations show the hybrid algorithms at 64s bits at least matching the classic greedy methods.
  • Support detection can proceed without prior knowledge of signal energy: one-bit measurements decide direction, while linear measurements provide scale and the residue for the next iteration.
  • The support-modification pass of Algorithm 2 means hybrid CS can be initialized from any rough support guess and then refined, so it can be chained after another detector.
  • The random-tessellation error bound implies that the one-bit reconstruction error shrinks as the estimate satisfies more binary inequalities, which is exactly the score each greedy step maximizes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural repair, not explored in the paper, would replace the order-$n$ restricted isometry assumption with an order-$2s$ or coherence condition; whether the same proof skeleton then yields non-vacuous probability bounds is left open.
  • The success bounds in Theorems 5 and 6 depend on per-iteration thresholds and reference values that the paper does not specify, so choosing those parameters is an empirical degree of freedom that could change which algorithm wins.
  • The same greedy binary-inequality scoring could be applied to multi-bit or dithered quantized measurements and to block-sparse signals, since only the sign agreement between measurement and estimate is used in the selection rule.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript studies hybrid compressed sensing, in which measurements consist of noisy linear measurements y_r = A_r x + e_r and noisy binary measurements y_o = sign(A_o x + e_o). The authors first prove a theorem on random uniform tessellations for sparse signals (Theorem 4), and then propose two greedy algorithms that use the one-bit measurements for support detection and the linear measurements for residue updates and signal estimates. The main advertised theoretical results are Theorem 5, a lower bound on the probability that Algorithm 1 detects the true support after s iterations, and Theorem 6, a lower bound on the probability that Algorithm 2 turns an initial support into the true support. Simulations compare the two algorithms with OMP, SP, and CoSaMP under equal bit budgets for n=256 and several sparsity levels.

Significance. Hybrid compressed sensing is a relevant and under-studied problem, and the proposed algorithms are clearly specified and intuitively motivated. The simulation study covers a reasonable range of parameters and shows consistent gains over the selected classical greedy algorithms, which is an encouraging empirical result. The paper is also clearly written and does not rely on circular reasoning or fitted parameters in its theoretical statements. However, the central theoretical contribution is invalid: both algorithm theorems are conditioned on an impossible restricted-isometry assumption, and the proofs contain additional algebraic steps that fail within the range of the stated parameters. If the theorems were corrected, the paper could make a useful contribution; as it stands, the claimed guarantees are vacuous in the paper's own measurement regime.

major comments (2)
  1. [Section IV-A, Theorem 5; Section IV-B, Theorem 6] The assumption that A_r ∈ R^{m_r × n} satisfies the restricted isometry property of order n with δ_n ∈ (0, 0.5] is impossible whenever m_r < n, which is exactly the regime considered in this paper. For example, Section V-A sets n = 256 and m_r = ⌈1.5s⌉, so m_r = 6 for s = 4. By the rank-nullity theorem, every such A_r has a nonzero vector z in its nullspace; the RIP lower bound of order n would require (1 - δ_n)||z||_2^2 ≤ ||A_r z||_2^2 = 0, forcing δ_n ≥ 1. This contradicts δ_n < 1 and therefore the hypotheses of Theorems 5 and 6 are empty. Consequently, the probability lower bounds (25) and (26) do not apply to the algorithms as implemented in the simulations, and the principal theoretical claims of the paper are vacuous.
  2. [Appendix B, around Eq. (51); Appendix C, Eq. (60)] Even if one disregarded the impossible RIP order-n assumption, the proof of Theorem 5 contains a false inequality. The step (1 + sqrt((1 + δ_n)/(1 - δ_j))) ≤ 1 + sqrt(2) is asserted in the derivation of Eq. (51). Since Lemma 1 gives δ_j ≤ δ_n, the ratio (1 + δ_n)/(1 - δ_j) can be as large as (1 + δ_n)/(1 - δ_n), which equals 3 when δ_n = 0.5; hence sqrt(3) > sqrt(2) and the displayed bound fails for parameters allowed by the theorem. Similarly, the inference from the bound on ||e_r||_2 to the bound on ||u||_2 in Eq. (50) requires δ_n ≤ 1 - (1 + sqrt(2))/(2 + sqrt(2)) ≈ 0.293, which is not implied by δ_n ∈ (0, 0.5]. Theorem 6 uses the same argument through Eq. (60), so both probability bounds are demonstrably incorrect even under a hypothetical valid RIP assumption.
minor comments (4)
  1. [Theorem 6 statement] The initial detected support set is written as \tilde{Ω}_0 = {i_{s'+1}, i_{s'+2}, ..., i_{s'}}; the last index should be i_s, not i_{s'}. The proof in Appendix C correctly uses the index set {s'+1, ..., s}.
  2. [Figures 1 and 2] The legends in Figures 1 and 2 contain the typographical errors 'Algotithm 1' and 'Algotithm 2'; these should read 'Algorithm 1' and 'Algorithm 2'.
  3. [Section V-A and V-B] The simulation section does not report how the free threshold parameters c_j, n_j for Algorithm 1 and \hat{n}_j, \tilde{n}_j for Algorithm 2 are chosen; without this information, the reader cannot tell whether the simulation configuration is compatible with the conditions in Theorems 5 and 6, even setting aside the vacuous RIP assumption.
  4. [Definition 2] Definition 2 contains a typo: 'F or' should be 'For' at the beginning of the definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the probability theorems derive from external RIP and tessellation results; the unsatisfiable RIP-of-order-n assumption is a validity flaw, not a circular reduction.

full rationale

This paper's derivation chain is self-contained with respect to circularity concerns. Theorems 5 and 6 are proved from standard RIP lemmas (Lemma 1-3), normal tail bounds, binomial CDFs, and Lemma 4 (Goemans-Williamson sign-change probability); Theorem 4 uses Plan and Vershynin's tessellation theorem (Theorem 3) and the Baraniuk et al. quotient property (Corollary 1). These are external, parameter-free results reused as hypotheses; the paper does not fit a parameter and then rename it a prediction, and it does not invoke a self-citation as load-bearing. The only notable issue is internal: Theorems 5 and 6 assume Ar satisfies RIP of order n with delta_n in (0,0.5], but the intended hybrid-CS regime has m_r < n (e.g., Section V-A uses n=256, m_r = ceil(1.5s), so m_r <= 48); by rank-nullity any m_r x n matrix with m_r < n has a nonzero null vector z, forcing (1 - delta_n)||z||_2^2 <= 0 and hence delta_n >= 1, contradicting delta_n <= 0.5. That makes the theorem hypotheses empty in the simulation regime and invalidates the stated probability guarantees, but it is a correctness/validity defect, not a circularity: the conclusion is not assumed in the premises, nor is the success probability a fitted input. No circular step can be exhibited, so the score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central theorems rest on standard random matrix theory plus two ad hoc assumptions: an impossible n-th order RIP condition on A_r, and a noise bound tailored to the proof. The unspecified thresholds c_j, n_j, \hat n_j, \tilde n_j further weaken the theoretical statements.

free parameters (2)
  • Thresholds c_j, n_j (Algorithm 1 theorem)
    Theorem 5 introduces positive constants c_j and integer thresholds n_j without specifying how to choose them. The probability bound depends on these values, so the stated guarantee is not explicit or directly usable.
  • Thresholds \hat n_j, \tilde n_j (Algorithm 2 theorem)
    Theorem 6 similarly introduces auxiliary thresholds \hat n_j and \tilde n_j for support augmentation and pruning, again without a selection rule, making the bound dependent on unspecified choices.
assumptions (4)
  • domain assumption The measurement matrices A_r and A_o have independent standard normal entries divided by sqrt(m_r) and sqrt(m_o).
    This Gaussian measurement model is stated in Section IV and used in the proofs of Theorems 4, 5, and 6.
  • ad hoc to paper A_r satisfies the restricted isometry property of order n with δ_n in (0,0.5].
    This assumption appears in Theorems 5 and 6, but it is impossible when m_r < n because a nonzero nullspace vector forces δ_n ≥ 1. The theorems are vacuous under this condition.
  • ad hoc to paper The linear measurement noise satisfies ||e_r||2 ≤ (||x||2 - sqrt(2)||[x]_{Ω \ i1}||2) / (2 + sqrt(2)) with ||x||2 / ||[x]_{Ω \ i1}||2 ≥ sqrt(2).
    This technical noise bound is used in the proofs of Theorems 5 and 6 to control the distance between the true signal and the estimate built from selected support.
  • standard math Standard RIP lemmas (Lemmas 1, 2, 3) and Gaussian mean width bounds hold as cited from prior work.
    The proofs import these results from references [48], [49], [36], and [52] without new derivation.

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Pith. "Pith review of Greedy Algorithms for Hybrid Compressed Sensing." pith.science (2026). https://pith.science/paper/XABULZA2

@misc{pith2026190806359,
  author       = {Pith},
  title        = {Pith review of: Greedy Algorithms for Hybrid Compressed Sensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XABULZA2}},
  note         = {Machine review of arXiv:1908.06359}
}
read the original abstract

Compressed sensing (CS) is a technique which uses fewer measurements than dictated by the Nyquist sampling theorem. The traditional CS with linear measurements achieves efficient recovery performances, but it suffers from the large bit consumption due to the huge storage occupied by those measurements. Then, the one-bit CS with binary measurements is proposed and saves the bit budget, but it is infeasible when the energy information of signals is not available as a prior knowledge. Subsequently, the hybrid CS which combines the traditional CS and one-bit CS appears, striking a balance between the pros and cons of both types of CS. Considering the fact that the one-bit CS is optimal for the direction estimation of signals under noise with a fixed bit budget and that the traditional CS is able to provide residue information and estimated signals, we focus on the design of greedy algorithms, which consist of the main steps of support detection and recovered signal update, for the hybrid CS in this paper. We first propose a theorem on the random uniform tessellations for sparse signals to further investigate the properties of one-bit CS. Afterwards, we propose two greedy algorithms for the hybrid CS, with the one-bit CS responsible for support detection and traditional CS offering updated residues and signal estimates. For each of the proposed algorithms, we provide the corresponding theorem with proof to analyze their capabilities theoretically. Simulation results have demonstrated the efficacy of the proposed greedy algorithms under a limited bit budget in noisy environments.

Figures

Figures reproduced from arXiv: 1908.06359 by the authors.

Figure 2
Figure 2. Recovery performances of Experiment 2 the feasibility under a small bit budget of the proposed algorithms. This is because the proposed algorithms strike a balance between the pros and cons of both the traditional CS and one-bit CS. Particularly, the traditional CS can be regarded as a special case of the hybrid CS. Note that the recovery performances of Algorithm 2 is better than those of Algorithm 1, which demonst… view at source ↗
Figure 1
Figure 1. Recovery performances of Experiment 1 5 10 15 20 25 30 Sparsity s -8 -6 -4 -2 0 2 4 6 Recovery SNR r (dB) Algotithm 1 Algorithm 2 OMP [18] CoSaMP [21] SP [22] (a) ξs = 0 dB 5 10 15 20 25 30 Sparsity s -5 0 5 10 15 20 Recovery SNR r (dB) Algotithm 1 Algorithm 2 OMP [18] CoSaMP [21] SP [22] (b) ξs = 10 dB [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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