{"id":"33c5399b-de3f-4422-8cb3-29f82a34a881","arxiv_id":"1908.05423","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If a signal generator satisfies a new zero-set condition, causal signals in complex-generated shift-invariant spaces can be recovered from three random magnitude samples per unit interval, with probability one.","lead":"A mathematical team proves that phaseless sampling, recovering a signal from magnitude-only measurements, is possible in complex-valued shift-invariant spaces using just a few random samples per unit interval, provided the signal generator meets a new generalized Haar condition. The result gives a concrete reconstruction algorithm and shows why a previously standard algebraic condition fails for complex signals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The complex GHC-generator class rests on an uncertified numerical determinant check; Theorem 2.5's guarantee has no rigorously verified nonempty domain in the complex case.","rationale":"The reader's weakest_assumption correctly identifies GHC and its numerical verification as the load-bearing premise. I agree: the main theorem is a conditional, and the proof chain from GHC through Lemmas 2.10-2.12 appears internally consistent. I also checked the secondary link, Proposition 2.1: for Lambda_{phi,1}, the zero set of a real combination sum a_k phi(x+k) is contained in the zero set of phi(x) times its conjugate, whose real and imaginary parts are real-linear combinations of Xi_phi; hence the inheritance is justifiable, so the omitted proof is a presentation gap rather than a fatal flaw. Theorem 1.1's equation (1.3) is indeed wrong as written, but the claimed impossibility can be obtained by a conjugation construction with coefficients d_k = e^{2 i alpha k} \\bar{c}_k, so the motivation survives. The remaining substantive issue is that the only complex GHC examples are supported by an uncertified determinant evaluation. This is fixable by a rigorous computation, and it does not disprove the conditional theorem, so the reader's CONDITIONAL verdict stands unchanged.","tokens_in":26484,"tokens_out":37574,"duration_ms":391500,"concrete_test":"Certify the determinant in (1.11) for the two stated chirp parameter sets (a,b,p) = (4, 4/5, 1) and (50, 4/5, 1) at a rational tuple (x1,...,x7) in (0,1)^7 using interval arithmetic or exact symbolic computation with tight error bounds. A certified nonzero determinant proves the seven components of Xi_phi are linearly independent; together with their analyticity on (0,1), this proves GHC and eliminates the numerical gap. If the certified determinant is zero at every tested rational tuple, the claimed complex GHC-generator examples are unsubstantiated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central guarantee of the paper, Theorem 2.5, is conditional on the generalized Haar condition (GHC) in (1.8). In the complex-valued case, where GHC is new, the only evidence that any GHC-generator exists is the random numerical determinant check in (1.11) for the chirp generator (1.10). The reasoning 'analytic components plus a nonzero determinant at one tuple implies linear independence, and analyticity then gives the zero-measure property' is sound in principle. But the paper reports only that uniform random search 'found' a nonzero determinant; it does not certify that value with rigorous error bounds. A floating-point nonzero determinant is not a proof. Thus the applicability of the sampling-density-3 theorem to the complex chirp examples rests on an unverified numerical computation. This does not expose an internal inconsistency in the conditional theorem, but it is the least secure load-bearing premise for the paper's claimed complex-domain contribution. A secondary gap is Proposition 2.1, whose one-line proof 'can be easily concluded' is the bridge by which Lemmas 2.10-2.12 obtain probability-one zero-set statements for Lambda_{phi,1} and Lambda_{phi,2}; it is fillable, but the paper does not fill it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies phaseless sampling (PLS) of causal signals in shift-invariant spaces (SISs) generated by complex-valued generators. It first claims an impossibility theorem (Theorem 1.1) for PLS in SISs generated by a real-valued full-spark function multiplied by a linear phase, then introduces a generalized Haar condition (GHC) on a system of quadratic products of the generator's real and imaginary parts. Under GHC, the paper proves that with probability 1, random sampling density 3 suffices to determine any nonseparable causal signal in a complex-generated SIS up to a unimodular scalar (Theorem 2.5), with density 2 in the real-valued case (Theorem 3.1). The proofs are constructive and lead to recursive phase-decoding/coefficient-recovery algorithms (Approaches II-B and III-B), including local reconstruction guarantees (Propositions 2.6 and 3.2). Numerical simulations for chirp-modulated generators are reported, along with noise-robustness tables.","tokens_in":26685,"tokens_out":9047,"duration_ms":86057,"significance":"The central conditional theorems are potentially significant. The zero-distribution/GHC perspective is a genuinely new tool for phaseless sampling, and the paper gives the first sampling-density guarantees of this type for complex-generated SISs. The constructive nature of the reconstruction algorithms is also a strength, as is the explicit formulation of the GHC hypothesis rather than a fitted assumption. However, the significance is tempered by three gaps: the proof of Theorem 1.1 is invalid as written; the existence of complex GHC-generators is supported only by a floating-point numerical check; and Proposition 2.1, which is load-bearing for the probability-one zero-set lemmas, is asserted without proof. If these gaps are fixed, the paper would be a solid contribution; in its current form the complex-domain claims rest on an unverified premise.","major_comments":[{"comment":"The identity (1.3) is false in general. For φ(x)=e^{iαx}ϕ(x), the two sums are e^{iαx}[c0ϕ(x)+c1e^{-iα}ϕ(x-1)] and e^{iαx}[c0ϕ(x)+c1e^{iα}ϕ(x-1)]. Their moduli are |c0ϕ(x)+c1e^{-iα}ϕ(x-1)| and |c0ϕ(x)+c1e^{iα}ϕ(x-1)|, which are not equal for arbitrary c0,c1. For instance, with s=2, N=1, c0=1, c1=e^{iβ}, α=π/4, β=π/3 and ϕ(x)=ϕ(x-1)=1, the two moduli are |1+e^{i(β-α)}| and |1+e^{i(β+α)}|, which differ. Thus the proof of Theorem 1.1 does not establish the claimed impossibility. Since this theorem is a principal motivation for the complex-valued part of the paper, a corrected proof (or a corrected statement) is needed.","section":"§I-B, Theorem 1.1, Eq. (1.3)"},{"comment":"The assertion that φ in (1.10) is a complex-valued GHC-generator is not proved. The paper states that a uniform random search 'found' a nonzero determinant in (1.11), and concludes that the analytic components are linearly independent and hence GHC holds. A finite-precision floating-point determinant value is not a rigorous certificate of linear independence, and the phrase 'with probability 1' is an empirical observation, not a mathematical proof. Because Theorems 2.5 and the related propositions apply only to GHC-generators, the complex-domain contribution currently has no rigorously verified nonempty example class. Please provide either an exact/interval-arithmetic verification for a concrete parameter tuple, or a rigorous proof for a family of generators, or explicitly state that the complex examples are conditional on a numerical conjecture.","section":"§I-C1, Eq. (1.11) and the chirp example (1.10)"},{"comment":"Proposition 2.1 is load-bearing but its proof is omitted with the phrase 'can be easily concluded.' The implication is not immediate: GHC is defined for the real span of the quadratic system Ξ_φ, whereas Λ_{φ,1} and Λ_{φ,2} are systems of complex-valued functions, and the zero-measure property must be transferred to their complex spans. This proposition is used in Lemmas 2.10 and 2.11 and in the proof of Theorem 2.5, so a complete proof should be supplied. A natural route is to show that for h in the span of Λ_{φ,1} (or Λ_{φ,2}), the function |h|^2, restricted to (0,1), is a real linear combination of elements of Ξ_φ, so that GHC on Ξ_φ controls the zero set of h.","section":"§II-A, Proposition 2.1"}],"minor_comments":[{"comment":"The comparison paragraph says 'section III-C (complex-valued case)' when it appears to refer to the complex-valued simulation in Section II-H; please correct the cross-reference.","section":"§III-C, text after Eq. (3.48)"},{"comment":"The affiliation string contains a typo: 'Naning' should be 'Nanning'.","section":"Affiliation, first page"},{"comment":"In the displayed formula for step 2 of Approach II-B, the notation zn,k and zn,l is introduced but the argmin expression is a little hard to parse; consider rewriting the tie-breaking/selection rule with explicit definitions of the four candidate roots.","section":"§II-B, Eq. (2.31)"},{"comment":"The proof argues separately that the real and imaginary parts of an,f+ibn,f are nonzero with probability 1, then concludes the phase is not jπ/2. This is correct once Proposition 2.1 is available, but the dependence on Proposition 2.1 should be stated explicitly in this lemma, since it is not immediately visible.","section":"§V-B, proof of Lemma 2.11"}],"recommendation":"major_revision","confidential_remarks":"The paper's conditional theorems seem defensible, and the GHC framework is interesting. The main obstacles are (i) the invalid proof of Theorem 1.1, (ii) the purely numerical verification of GHC for complex generators, and (iii) the omitted proof of Proposition 2.1. None of these appears impossible to fix within the paper's scope, but they are load-bearing enough that the current version should not be accepted as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the conditional results are new and likely valuable, but the paper needs a proper fix before I'd trust the advertised guarantees.\n\nWhat's genuinely good: the zero-distribution/GHC perspective is new, and the paper uses it to get random sampling density 3 for complex-generated shift-invariant spaces and density 2 for the real case. That is a real step beyond the deterministic results in Sun and in Chen–Cheng–Sun–Wang. The reconstruction algorithms (PD-CR) are constructive, and the numerical experiments on chirp-modulated spaces are consistent with the theory. The local-reconstruction point—few random samples versus very many deterministic samples for highly oscillatory signals—is well illustrated.\n\nNow the soft spots, in proportion. First, Theorem 1.1, which motivates the whole paper, has a flawed proof. Equation (1.3) is simply false as written: for φ(x)=e^{iαx}ϕ(x), the two coefficient sequences {c_k} and {e^{i2αk}c_k} do not give the same magnitude unless extra conditions hold. The reader's example with α=π/4, β=π/3 shows the claimed identity fails. The theorem may be true via a different construction, but the paper doesn't provide one. This isn't fatal for the main theorems, but it's a bad look in the introduction.\n\nSecond, the complex GHC-generator class rests on a numerical determinant check, not a proof. The reasoning that analytic components plus a nonzero determinant at one tuple imply independence, and hence the zero-measure property, is sound in principle. But the paper only reports that uniform random search found a nonzero determinant; it doesn't certify the value rigorously. So Theorem 2.5's hypothesis is not known to be nonempty in the complex case. That is a real gap, and the stress-test note gets it right.\n\nThird, Proposition 2.1 is load-bearing for the probability-one lemmas but its proof is one line: \"can be easily concluded.\" It is probably fillable, but the paper should fill it. The real-valued case is in better shape, because B-splines and refinable functions give known GHC examples.\n\nThe citation pattern looks fine; the relevant prior work is acknowledged. The math in the appendix is long but structurally coherent.\n\nWho is this for? People working on phase retrieval, sampling theory, and shift-invariant spaces. It deserves a serious referee, not a desk reject, but the referee should demand a corrected Theorem 1.1, a rigorous GHC existence proof or a clearer honest statement that the complex case is conditional, and a full proof of Proposition 2.1. My recommendation: send to review, but expect major revision.","headline":"The random-sampling theorems are plausible and worth a serious referee, but the paper ships with a demonstrably wrong proof in its motivating Theorem 1.1 and a key hypothesis (GHC) that has no rigorously verified nonempty domain in the complex case.","tokens_in":27291,"tokens_out":3722,"would_cite":false,"duration_ms":36957,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A12","42C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that random phaseless sampling at density 3 succeeds for complex-generated shift-invariant spaces when the generator satisfies a generalized Haar condition, and gives a constructive recursive reconstruction algorithm.","keywords":["phaseless sampling","phase retrieval","shift-invariant space","generalized Haar condition","random sampling density","causal signals","chirp signals","zero distribution"],"falsifier":"Evaluate the determinant in Eq. (1.11) symbolically for the chirp generator family used in the paper: if there are parameter values for which the determinant is identically zero as a function of the sampled points, GHC fails and the density-3 guarantee does not apply to that family. Alternatively, exhibit a nonzero real-linear combination of the functions in $\\Xi_\\varphi$ whose zero set on $(0,1)$ has positive Lebesgue measure.","tokens_in":26224,"feed_emoji":"📶","tokens_out":11669,"duration_ms":101369,"temperature":0.7,"pith_summary":"This paper proves that random phaseless sampling—recovering a signal from magnitude-only measurements—is possible for causal signals in shift-invariant spaces generated by complex-valued functions, provided the generator satisfies a generalized Haar condition. It first shows that the full spark property, which permits phaseless sampling for real-valued generators, is not enough when the generator is multiplied by a linear phase: such signals are never uniquely recoverable from magnitudes. Replacing the cardinality view of zero sets with a measure-zero condition, the paper proves that three independent uniform samples per unit interval determine any nonseparable causal signal in a complex-generated space up to a global unimodular constant; the real-valued analogue needs only two samples per unit interval. The proof is constructive, giving an alternating phase-decoding and coefficient-recovery algorithm, and numerical experiments recover highly oscillatory chirp signals from a few random samples where deterministic methods would need hundreds.","feed_headline":"Three random magnitude samples per interval recover chirp signals","feed_subtitle":"A generalized Haar condition guarantees recovery from just three random intensity samples per unit interval.","key_machinery":"The generalized Haar condition (GHC) is the load-bearing object. For $\\varphi=\\varphi_{\\Re}+i\\varphi_{\\Im}$, the $2s-1$ functions in $\\Xi_\\varphi=\\{\\varphi_{\\Re}\\varphi_{\\Re}(\\cdot+k)+\\varphi_{\\Im}\\varphi_{\\Im}(\\cdot+k),\\ \\varphi_{\\Re}\\varphi_{\\Im}(\\cdot+k)-\\varphi_{\\Im}\\varphi_{\\Re}(\\cdot+k)\\}_{k=1}^{s-1}\\cup\\{\\varphi_{\\Re}^2+\\varphi_{\\Im}^2\\}$ must be linearly independent, and every nonzero real-linear combination must have a zero set of Lebesgue measure zero on $(0,1)$. This measure-zero property ensures that an auxiliary quantity $A_{n,f}(x,y)+iB_{n,f}(x,y)$ is nonzero with probability 1 and that the two quadratic equations in (2.21) share exactly one solution. The reconstruction then alternates between phase decoding, where the roots of a quadratic provide candidate phases for $f(n+t_{n1})$, and coefficient recovery, where the decoded phase gives $c_n$; the recursion over $n$ advances one unit interval at a time.","core_discovery":"The central claim is Theorem 2.5: if $\\varphi$ is a complex-valued GHC-generator with support in $(0,s)$, then every nonseparable causal signal $f=\\sum_{k=0}^\\infty c_k\\varphi(\\cdot-k)$ in $V_{\\mathrm{ca}}(\\varphi)$ is determined up to a unimodular scalar with probability 1 by the random magnitude samples $\\{|f(t_0)|\\}\\cup\\{|f(n+t_{n1})|,|f(n+t_{n2})|,|f(n+t_{n3})|: n=1,\\dots,\\infty\\}$, where all offsets are i.i.d. uniform on $(0,1)$. A signal is nonseparable when it cannot be written as a sum of two nonzero elements of the space with disjoint supports; separability is the only obstruction, since separated parts can be rephased independently. The same argument yields Theorem 3.1 for real-valued generators and real-valued signals, where density 2 suffices and the reconstruction is up to a sign. Because the phase decoding is recursive, the restriction of a signal to $[0,L]$ is recovered with probability 1 from finitely many samples, and the number of samples does not grow with the oscillation rate.","pith_inferences":["The GHC condition is likely to hold generically among analytic generator families: for generators whose quadratic products are real-analytic on $(0,1)$, GHC reduces to linear independence of those products, since a nonzero analytic function cannot vanish on a positive-measure set. The paper checks this numerically for its chirp generators; a symbolic proof for a parameter family would be a natural","A plausible extension is multi-generator shift-invariant spaces: if the analogue of $\\Xi_\\varphi$ for several generators has the same zero-measure property, the phase-decoding recursion may carry over with a higher sampling density.","The noise experiments suggest a qualitative trade-off: binary-phase (real-valued) signals tolerate much lower SNR because the phase decision is a sign decision, while rapidly rotating phases accumulate errors; one testable design rule is to oversample randomly in intervals where the phase function changes quickly.","The paper does not address whether density 3 is minimal for complex generators; its impossibility result for linear-phase modulations hints that fewer samples may fail, but the minimum is open."],"forward_implications":["In any complex-generated shift-invariant space whose generator satisfies GHC, three random samples per unit interval suffice to recover a nonseparable causal signal up to one global constant of modulus 1.","Local reconstruction is cheap: the restriction of such a signal to $[0,L]$ is recovered with probability 1 from $3L-2$ random samples, a count independent of the signal's oscillation rate.","For real-valued generators and real-valued signals, the density drops to two random samples per unit interval, with recovery up to a sign.","The recursive nature of the algorithm means no global search over phases or coefficients is needed; each step solves a fixed small quadratic system.","The result excludes the conjugation ambiguity that plagues conjugate phase retrieval: a complex signal and its conjugate cannot both be valid reconstructions unless the signal is real up to a global phase."],"supporting_citations":[{"why":"Establishes deterministic phaseless sampling for nonseparable real-valued signals in compactly generated SISs via full spark, the baseline this paper extends to random sampling for complex generators.","marker":"[26]"},{"why":"Gives the deterministic spline-space phaseless sampling result whose sample count is contrasted with the random local reconstruction.","marker":"[24]"},{"why":"Defines conjugate phase retrieval in Paley-Wiener space, used to show Theorem 2.5 has no conjugation ambiguity.","marker":"[29]"},{"why":"Supplies the chirp-modulated shift-invariant spaces whose generators the paper verifies as GHC-generators and uses in numerical simulations.","marker":"[34]"},{"why":"Motivates highly oscillatory chirp signals as the natural signal class for the local reconstruction guarantee.","marker":"[40]"},{"why":"Supplies the fact that zeros of nonzero analytic functions have measure zero, the key ingredient for checking GHC on analytic generators.","marker":"[45]"}],"fun_headline_variants":["Three random magnitude samples per interval recover causal signals","Random phaseless sampling achieves density three for chirps","Causal signals pinned down by three random magnitude samples","Generalized Haar condition enables sparse random sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the generalized Haar condition: every nonzero real-linear combination of the generator's quadratic products must vanish only on a set of measure zero on $(0,1)$, and for the chirp generators this is verified numerically rather than by a closed-form proof.","fun_headline_variants_meta":{"raw":{"variants":["Three random magnitude samples per interval recover causal signals","Random phaseless sampling achieves density three for chirps","Causal signals pinned down by three random magnitude samples","Generalized Haar condition enables sparse random sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00028,"raw_usage":{"total_tokens":1740,"prompt_tokens":1103,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":576}},"tokens_in":719,"tokens_out":637,"duration_ms":6828,"temperature":1.0,"reasoning_tokens":576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:16:54.822614+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the determinant in Eq. (1.11) symbolically for the chirp generator family used in the paper: if there are parameter values for which the determinant is identically zero as a function of the sampled points, GHC fails and the density-3 guarantee does not apply to that family. Alternatively, exhibit a nonzero real-linear combination of the functions in $\\Xi_\\varphi$ whose zero set on $(0,1)$ has positive Lebesgue measure.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes deterministic phaseless sampling for nonseparable real-valued signals in compactly generated SISs via full spark, the baseline this paper extends to random sampling for complex generators."},{"cited_title":"Conjugate Phase Retrieval in Paley-Wiener Space","cited_arxiv_id":"1910.12975","evidence_quote":"Defines conjugate phase retrieval in Paley-Wiener space, used to show Theorem 2.5 has no conjugation ambiguity."},{"cited_title":"Bhandari and A","cited_arxiv_id":null,"evidence_quote":"Supplies the chirp-modulated shift-invariant spaces whose generators the paper verifies as GHC-generators and uses in numerical simulations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates highly oscillatory chirp signals as the natural signal class for the local reconstruction guarantee."},{"cited_title":"Garnett, Bounded analytic functions, Graduate Texts in Mathe- matics, 236, Springer, 2007","cited_arxiv_id":null,"evidence_quote":"Supplies the fact that zeros of nonzero analytic functions have measure zero, the key ingredient for checking GHC on analytic generators."}],"review_version":1}