Support Recovery in One-bit Compressed Sensing with Near-Optimal Measurements and Sublinear Time
Pith reviewed 2026-05-17 21:42 UTC · model grok-4.3
The pith
One-bit compressed sensing recovers k-sparse supports using near-optimal measurements and sublinear decoding.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For universal exact support recovery there exist measurement matrices and group-testing decoders that succeed with m = O(k² log(n/k) log n) measurements in time O(k m). For ε-approximate universal recovery the same framework uses m = O(k ε^{-1} log(n/k) log n) measurements in time O(ε^{-1} m). For probabilistic exact recovery a separate scheme achieves m = O(k (log k / log log k) log n) measurements in time O(m) with vanishing error probability.
What carries the argument
Group-testing-based decoding procedures that use sign measurements to test groups of coordinates for the presence of nonzero entries.
If this is right
- Exact universal recovery is possible with quadratic-in-k measurements but only O(k m) decoding time.
- Approximate recovery reduces the measurement count to linear in k while preserving sublinear decoding in n.
- Probabilistic recovery reaches nearly linear-in-k measurements with decoding time linear in the number of measurements and error probability tending to zero.
Where Pith is reading between the lines
- The same group-testing bridge could be tested on noisy one-bit models or on other quantized sensing tasks.
- The sublinear decoding opens the possibility of applying one-bit sensing to problems where n reaches billions.
- Constants hidden in the big-O expressions could be tightened by refining the group-testing primitives.
Load-bearing premise
Suitable measurement matrices exist that let the group-testing decoders achieve the stated recovery guarantees for any k-sparse signal under the noiseless sign model.
What would settle it
A k-sparse vector together with a measurement matrix for which the proposed group-testing decoder returns an incorrect support after using the claimed number of measurements would disprove the bounds.
read the original abstract
One-bit compressed sensing (1bCS) addresses the recovery of sparse signals from highly quantized measurements, retaining only the sign of each linear measurement. In the support recovery setting, the goal is to identify $\text{supp}(x)$, the nonzero coordinates of an unknown signal $x \in \mathbb{R}^n$ from $y = \text{sign}(Ax)$, where $A \in \mathbb{R}^{m \times n}$ and $|\text{supp}(x)| \le k \ll n$. Existing methods minimize the number of measurements but often incur $\Omega(n)$ decoding complexity, limiting large-scale applicability. We propose new 1bCS schemes that achieve sublinear decoding complexity while maintaining near-optimal measurement bounds. For universal support recovery, our framework provides: (i) exact recovery with $m = O(k^2 \log(n/k) \log n)$ measurements and decoding complexity $D=O(km)$, and (ii) $\epsilon$-approximate recovery with $m = O(k \epsilon^{-1} \log(n/k) \log n)$ and $D=O(\epsilon^{-1} m)$. For probabilistic exact recovery, we design a scheme with $m = O\big(k \frac{\log k}{\log\log k} \log n\big)$ and $D=O(m)$, achieving vanishing error probability. Our approach leverages ideas from group testing to bridge classical sparse recovery techniques with modern algorithmic efficiency considerations, highlighting a new trade-off between compression efficiency and computational complexity.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes new one-bit compressed sensing schemes for support recovery of k-sparse signals from y = sign(Ax). It adapts group testing ideas to achieve sublinear decoding complexity with near-optimal measurements: universal exact recovery with m = O(k² log(n/k) log n) and D = O(km); ε-approximate recovery with m = O(k ε^{-1} log(n/k) log n) and D = O(ε^{-1} m); and probabilistic exact recovery with m = O(k (log k / log log k) log n) and D = O(m) with vanishing error probability.
Significance. If the derivations hold, the work is significant for bridging group testing with 1bCS to enable efficient large-scale support recovery where linear-time decoding is prohibitive. The explicit trade-offs between measurement count and decoding time, along with the probabilistic scheme's improved bound, represent a useful algorithmic contribution in the field.
major comments (2)
- [Section 2] Section 2 (model and assumptions): The problem is stated for arbitrary k-sparse x ∈ ℝ^n under the noiseless sign model with no explicit lower bound on |x_i| for i in the support. The group-testing reduction for decoding may fail to achieve the claimed bounds in general, since small nonzero magnitudes or opposing signs can produce sign cancellations in Ax that are not present in standard combinatorial group testing. This assumption (or its absence) is load-bearing for all three main results and requires explicit treatment or a robustness proof.
- [Section 4] Section 4 (main theorems): The measurement bounds are stated as O(·) but it is unclear whether the matrix constructions are explicit and efficient or merely existential; if the latter, the sublinear decoding claim may require additional verification that the decoder can be implemented without Ω(n) preprocessing.
minor comments (2)
- [Abstract] The abstract and introduction would benefit from a short explicit comparison of the new bounds against known information-theoretic lower bounds for 1bCS support recovery.
- [Abstract] Notation for the probabilistic bound (log k / log log k) should be parenthesized for clarity in the displayed equation.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for highlighting these important points. We address each major comment below with clarifications and indicate the changes we will make.
read point-by-point responses
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Referee: Section 2 (model and assumptions): The problem is stated for arbitrary k-sparse x ∈ ℝ^n under the noiseless sign model with no explicit lower bound on |x_i| for i in the support. The group-testing reduction for decoding may fail to achieve the claimed bounds in general, since small nonzero magnitudes or opposing signs can produce sign cancellations in Ax that are not present in standard combinatorial group testing. This assumption (or its absence) is load-bearing for all three main results and requires explicit treatment or a robustness proof.
Authors: We agree that the model statement in Section 2 would benefit from an explicit assumption to ensure the group-testing reduction holds without sign cancellations. In the current analysis the nonzero entries are treated as having magnitudes large enough to dominate the linear combinations under the chosen matrix normalization, but this is not stated. We will revise Section 2 to add the standard assumption that min_{i ∈ supp(x)} |x_i| ≥ 1 (after possible rescaling of x) and include a short paragraph explaining why this prevents cancellations for the matrix constructions used. We will also note that the results extend to smaller magnitudes by adjusting the matrix scaling factor, at the cost of a constant factor in the measurement bound. revision: yes
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Referee: Section 4 (main theorems): The measurement bounds are stated as O(·) but it is unclear whether the matrix constructions are explicit and efficient or merely existential; if the latter, the sublinear decoding claim may require additional verification that the decoder can be implemented without Ω(n) preprocessing.
Authors: The matrix constructions underlying all three main results are explicit. For the universal exact-recovery scheme we use explicit (k, d)-disjunct matrices constructed via the standard combinatorial method based on Reed-Solomon codes (or the explicit sparse random constructions with known support), which admit O(1)-time row access and require no Ω(n) preprocessing. The decoder only queries the m rows of A that correspond to the received measurements y and runs the group-testing identification procedure directly on those rows. We will add a clarifying paragraph in Section 4 that specifies the construction, cites the explicit disjunct-matrix reference, and confirms that the claimed sublinear decoding time holds with constant-time access to matrix entries. revision: yes
Circularity Check
Derivation is self-contained via external group-testing reductions
full rationale
The paper adapts known combinatorial group-testing decoders to the sign-measurement model y=sign(Ax) and states explicit measurement and time bounds derived from those external combinatorial results. No equation or claim reduces a stated bound to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the constructions are presented as direct translations of standard group-testing guarantees into the 1bCS setting against independent combinatorial benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The unknown signal x is at most k-sparse
- domain assumption Measurements are exact signs with no noise
Reference graph
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discussion (0)
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