{"id":"b032ac0c-b857-4150-95b2-4cab6ce08799","arxiv_id":"1909.01966","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"When Poisson-count data fall outside the image cone, likelihood maximizers and ML-EM cluster points concentrate on sparse point masses, while inside the cone they retain full support.","lead":"This paper explains why maximum-likelihood reconstructions in PET imaging look spiky. It proves that when measured data cannot be produced by any nonnegative image, the optimal image must concentrate on a few points, and it quantifies how likely that situation is.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sparsity claim is vacuous without linear independence: an m=2 example with a1=a2=1/2 and y=(0.6,0.4) makes every probability measure optimal, including absolutely continuous ones.","rationale":"The reader's weakest assumption flags linear independence of the a_i as important, but focuses on the interior-cone and boundary theorems (Lemma 3.12, Prop 3.13, Theorem 4.10). The deeper issue is that Corollary 3.8, the paper's central sparsity theorem, is vacuous without linear independence: λ*≠0 does not prevent A^*λ* from vanishing identically. The m=2 example with a1=a2=1/2 satisfies all standing assumptions (nonnegative continuous a_i, sum normalized to 1, y outside the cone) and yields a continuum of absolutely continuous minimizers, directly contradicting the abstract's 'must be sparse'. This is not a mere technical gap: the main advertised phenomenon fails in a simple admissible instance. The paper can be repaired by explicitly assuming linear independence (and the generic regularity of Remark 3.9) in the statement of the sparsity theorem, and by adjusting the abstract to say 'generically' rather than unconditionally. Other flagged issues—the incorrect definition of K~ in Prop 3.13 and the reversed inequality in Theorem 5.2—are substantial but localized and fixable. Because the central idea remains plausible under the added hypotheses, a conditional accept with mandatory revisions is appropriate rather than outright rejection.","tokens_in":20481,"tokens_out":16526,"duration_ms":171066,"concrete_test":"Check the explicit m=2 example: K=[0,1], a1=a2=1/2, y=(0.6,0.4). Verify that A(M+) is the diagonal ray, y is outside it, and 𝓁(µ)=µ(K)-log(µ(K)/2) is minimized by every measure of mass 1, e.g., normalized Lebesgue measure, contradicting the unconditional sparsity claim. Then, to test the repaired statement, add assumption (28) and verify whether the generic conditions in Remark 3.9 actually imply that the zero set of A^*λ* has empty interior; otherwise the 'sum of Diracs' conclusion still does not follow from the stated theorems.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Corollary 3.8 defines 'sparse' as supp(µ*) ⊂ argmin(A^*λ*), but this is only meaningful when A^*λ* is not identically zero. If the detector functions a_i are linearly dependent, λ*≠0 can still give A^*λ* ≡ 0, so argmin = K and the condition is vacuous. Concretely, take K=[0,1], m=2, a1=a2=1/2, y=(0.6,0.4). The cone A(M+) is the ray {(c/2,c/2)}, so y∉A(M+). The negative log-likelihood is 𝓁(µ)=µ(K)-log(µ(K)/2), minimized by every measure of mass 1, including normalized Lebesgue measure. The dual maximizer is λ*=(-0.2,0.2), with A^*λ*=0. Thus the paper's central advertised claim, that likelihood maximisers must be sparse when y is outside the cone, is false as stated. Remark 3.9 adds linear independence plus generic Hessian conditions to obtain sums of Diracs, but those assumptions are not part of Corollary 3.8 and are not proven for the PET setting. This is the most load-bearing gap: the headline result depends on an unstated, nontrivial hypothesis, and when that hypothesis fails the conclusion collapses completely.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Poisson maximum-likelihood problem for a linear operator A acting on nonnegative Radon measures on a compact set K, with finite-dimensional observations y. It derives optimality conditions for minimizers of the negative log-likelihood, proves a sparsity characterization when y lies outside the image cone A(M+), invokes a theorem of Georgiou to obtain absolutely continuous solutions when y is in the interior, analyzes ML-EM iterates (monotonicity, fixed-point cluster points, support properties), and gives concentration bounds for the probability that Poisson data fall outside the cone. Numerical experiments with a PET operator illustrate the predicted sparsity. The paper is a theoretical contribution with a clear applied motivation.","tokens_in":20756,"tokens_out":20980,"duration_ms":205178,"significance":"If the main claims hold, this would be a substantial step toward explaining the spiky artifacts of ML-EM in PET, since the dichotomy between sparse solutions outside the cone and absolutely continuous optimal limits inside the cone is governed by a parameter-free, checkable condition (membership of y in A(M+)). The convex-analysis core is clean, the dual-certificate interpretation is useful, and the numerical experiments are reproducible. The main theorems have no fitted parameters and the statistical bounds are explicit. However, the correctness of the headline claims currently depends on hypotheses that are either unstated or mis-stated: linear independence of the a_i is needed for the sparsity conclusion, the reduced compact K-tilde in Section 3.4 is defined with the wrong set, Proposition 3.14 is false as stated, and the concentration-bound proof in Section 5 contains a reversed inequality. The significance can only be assessed after those fixes.","major_comments":[{"comment":"The sparsity conclusion of Corollary 3.8 is vacuous unless A*λ* is not identically zero. The proof only shows λ* ≠ 0, but when the functions a_i are linearly dependent it can still happen that A*λ* = 0, in which case arg min(A*λ*) = K and condition (27) holds for every probability measure. Concretely, on K=[0,1] take m=2, a1=a2=1/2 and y=(0.6,0.4); then y ∉ A(M+), the negative log-likelihood is minimized by every measure of mass 1, and the dual maximizer is λ*=(-0.2,0.2) with A*λ*=0. Thus the conclusion \"must be sparse, i.e., typically a sum of point masses\" is false as stated. The paper should add the linear-independence assumption (28) to Corollary 3.8 (and to Corollary 4.4 and Remark 3.9), or otherwise prove that a nontrivial zero set is obtained. Moreover, even under (28), condition (27) only places the support in the zero set of a nonnegative continuous function; the additional Hessian/analyticity hypotheses of Remark 3.9 are needed to conclude a sum of Dirac masses, and they are not verified for PET detector responses.","section":"§3.2, Corollary 3.8 and Remark 3.9"},{"comment":"The reduced set K-tilde is defined as K \\ ∪_{i∉supp(y)} a_i^{-1}({0}); this is the set where at least one zero-count detector response is positive, which is the opposite of what the proof requires. To have ⟨µ,a_i⟩=0 for every i∉supp(y), the support of the extended measure must be contained in ∩_{i∉supp(y)} a_i^{-1}({0}) = K \\ ∪_{i∉supp(y)} {a_i>0}. As written, the extension argument in Proposition 3.13 can produce a measure with positive integrals against zero-count detectors. For example, if a_3 is positive everywhere on K and y_3=0, the printed K-tilde equals K and the proof would construct a measure with ⟨µ,a_3⟩>0, contradicting Aµ=y. Since Proposition 3.13 is used by Corollary 4.9 and Theorem 4.10, those results are invalid as printed. Please correct the definition of K-tilde, re-state assumptions (31)-(32) on the corrected set, and check that the corrected K-tilde satisfies the hypotheses of Theorem 3.11 (for instance, having nonempty interior if \"absolutely continuous positive density\" is meant with respect to Lebesgue measure).","section":"§3.4, definition before Eq. (29) and Proposition 3.13"},{"comment":"Proposition 3.14 is false as stated. An extreme point y of the convex set A(M+)∩S can have multiple preimages x with a(x)=y, and then any probability measure supported on that preimage set satisfies Aµ=y, not only Dirac masses. For instance, if a_1 has a flat plateau at its maximum and a_2=1-a_1, the corresponding y is an extreme point of conv(a(K)) but every measure supported on the plateau is a solution. The proof sketch \"the only extremal points among probability measures are the Dirac masses\" confuses extremality in the image convex set with extremality in the domain simplex; the map a need not be injective. The proposition needs an additional assumption, such as the level set {x: a(x)=y} being a singleton, before its conclusion holds. This affects the boundary-extremal case mentioned at the start of Section 4.2.","section":"§3.4, Proposition 3.14"},{"comment":"The derivation of the concentration bounds in Theorem 5.2 uses the inequality 1-e^{-u} ≥ u for u>0, which is false; in fact 1-e^{-u} < u. In the second bound, the text asserts e^{-γt(1-exp(-ε/m))} ≤ e^{-γt ε/m}, but since 1-exp(-ε/m) < ε/m the inequality is reversed. The same problem occurs in the final step of the first bound, where e^{-γt(1-exp(-ε))} is replaced by e^{-γtε}. Thus the displayed concentration bounds in Theorem 5.2 are not justified by the given proof; corrected exponents or constants are needed. Since these bounds are one of the paper's stated contributions in Section 5, this is a load-bearing error.","section":"§5, proof of Theorem 5.2"}],"minor_comments":[{"comment":"There are several typographical errors, including \"in the sense of of the divergenced\" in Section 1 and \"the the sequence\" in the proof of Proposition 4.3; a careful proofread is needed.","section":"General"},{"comment":"The assertion that the iterates (14) are the EM algorithm for the continuous model is explicitly left unproved; a reference or a short proof would strengthen the paper, especially because the EM interpretation is invoked in later discussions.","section":"§2.2.4"},{"comment":"The closedness of A(M+) is asserted without proof; a one-sentence justification, for example writing A(M+) as the cone over the compact convex set conv{a(x): x∈K}, would be useful.","section":"§2.4"},{"comment":"The statement \"Under (28), A(M+) has non-empty interior\" is used without proof; this is a short exercise but should be spelled out because Theorem 3.11 explicitly assumes nonempty interiors of both the cone and its dual cone.","section":"§3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper has a valuable core and the main issues are repairable, but the headline sparsity theorem currently lacks a necessary hypothesis, Section 3.4 contains a wrong set definition that invalidates Proposition 3.13 and Theorem 4.10, Proposition 3.14 is false as stated, and the Section 5 concentration-bound proof has a reversed inequality. I recommend major revision rather than rejection because the corrections are local and the convex-analysis framework is sound. The authors should also re-check all statements that rely on the corrected K-tilde and on the linear-independence assumption."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper deserves a serious referee, but not as-is. The core idea—that likelihood maximizers concentrate when the data fall outside the image cone—is the right explanation for PET speckle, and the proof via duality is mostly sound. However, the advertised sparsity claim needs linear independence of the detector functions; without it, Corollary 3.8 is vacuous. The boundary-case section has a wrong definition of the reduced set, and the concentration bounds in Theorem 5.2 use a reversed inequality.\n\nWhat is genuinely new and good: the paper sets the continuum problem as a true Poisson point process likelihood, gives clean optimality conditions, proves uniqueness of the dual maximizer, and shows an explicit ML-EM limit with Dirac masses in a one-detector case. The numerical dual certificates are a nice practical touch, and the authors are honest about unresolved convergence questions. This is real mathematical work, not a repackaging of the discrete results.\n\nThe soft spots are real, and one is load-bearing. The stress-test example is correct: take two identical constant detector functions and y outside the cone; every probability measure is optimal and A*lambda* is identically zero, so the support condition in Corollary 3.8 holds trivially for absolutely continuous measures. Linear independence is added only in Remark 3.9, along with regularity assumptions, but it is not in the theorem statement or abstract. Since the paper's central promise is that sparsity is forced, omitting this hypothesis is a significant gap. The abstract overclaims; the theorem should say \"under generic nondegeneracy.\"\n\nThe K-tilde definition is also wrong as printed: to force zero integrals against off-support detectors, you need the intersection of their zero sets, not the complement of the union. As written, Proposition 3.13 and Theorem 4.10 do not follow. This is a typo-level fix in spirit, but it invalidates the printed proofs. In Theorem 5.2, the inequality e^{-gamma t(1-e^{-epsilon/m})} is larger than e^{-gamma t epsilon/m}, so the stated concentration bound is reversed. Again, fixable by keeping the exponent as 1-e^{-epsilon/m} or adjusting constants.\n\nNone of these flaws kill the main idea. The duality framework is sound, and the sparsity phenomenon is real under generic conditions. Who gets value: anyone working on EM algorithms, Poisson inverse problems, or measure-valued optimization. A careful referee can help the authors correct the statements and make the paper trustworthy.\n\nRecommendation: send to peer review and request revision. The corrected version should state the nondegeneracy hypothesis prominently and repair K-tilde and the concentration inequalities.","headline":"Worth engaging: a mostly sound core duality argument, but the headline sparsity claim needs linear independence and two later results contain fixable errors.","tokens_in":21305,"tokens_out":4485,"would_cite":true,"duration_ms":47102,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","49J27","62F10","44A60","60F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"When Poisson data fall outside the cone of achievable measurements, every maximum-likelihood reconstruction is sparse—generically a sum of Dirac masses—and inside the cone ML-EM cluster points are optimal with full support.","keywords":["Poisson inverse problem","ML-EM algorithm","sparse measures","Radon measures","Kullback-Leibler divergence","dual cone","concentration bounds","positron emission tomography"],"falsifier":"Take $K=[0,1]$, detectors $a_1(x)=1$ and $a_2(x)=x$, so the cone is $\\{(\\alpha,\\beta):\\alpha\\ge\\beta\\ge 0\\}$; choose normalized Poisson data $y=(0.4,0.6)$, which lies outside the cone. Solving $\\min_{\\mu\\ge 0} d(y\\,\\|\\,A\\mu)$ numerically and checking whether every minimizer's support lies in $\\arg\\min(\\lambda^*_1+\\lambda^*_2 x)$ for the unique dual maximizer $\\lambda^*$ would test the sparsity characterization directly. Independently, for the boundary case, construct data on $\\partial A(\\mathcal{M}_+)$ satisfying assumptions (31) and (32) and check whether the asserted absolutely continuous solution exists, since the printed proof's reduced set is defined incorrectly.","tokens_in":20230,"feed_emoji":"🩻","tokens_out":15040,"duration_ms":135995,"temperature":0.7,"pith_summary":"This paper treats image reconstruction for Poisson inverse problems—most concretely PET—with the unknown image modelled as a non-negative Radon measure on a compact set rather than as a discretized grid. It proves a sharp cone dichotomy for the maximum-likelihood problem $\\min_{\\mu\\ge 0} d(y\\,\\|\\,A\\mu)$. If the measured data $y$ falls outside the cone $A(\\mathcal{M}_+)=\\{A\\mu:\\mu\\ge 0\\}$, which is typical at low dose or short exposure, then every minimizer is sparse: its support is contained in the minimum set of an explicit dual function $A^*\\lambda^*$, and for generic smooth detectors it is a sum of Dirac masses. If $y$ lies in the interior of the cone and the detector response functions are linearly independent, absolutely continuous minimizers exist and ML-EM cluster points are optimal with full support. The paper also derives exponential concentration bounds for the probability that Poisson noise pushes the data outside the cone, explaining the long-observed spiky artefacts of ML-EM.","feed_headline":"Poisson data outside the cone force sparse reconstructions","feed_subtitle":"Low-dose Poisson data falls outside the feasible cone, so optimal reconstructions collapse to sums of point masses.","key_machinery":"The load-bearing object is the image cone $A(\\mathcal{M}_+)\\subset\\mathbb{R}^m$ and its dual cone $(A(\\mathcal{M}_+))^*=\\{\\lambda\\in\\mathbb{R}^m:A^*\\lambda=\\sum_i\\lambda_i a_i\\ge 0\\text{ on }K\\}$. The argument runs through the identity $\\nabla\\ell(\\mu)=A^*\\lambda(A\\mu)$ with $\\lambda_i(A\\mu)=1-y_i/\\langle\\mu,a_i\\rangle$, which turns the KKT conditions into support localization: any minimizer's support is contained in the zero set of the nonnegative dual function $A^*\\lambda^*$, and $\\lambda^*$ is unique. For the interior case, the second piece of machinery is the imported moment-problem theorem: $y\\in\\operatorname{int}A(\\mathcal{M}_+)$ implies the existence of a positive continuous density solution $A\\mu=y$, and once such a reference solution exists, a relative-entropy monotonicity argument ($D(\\mu^*\\,\\|\\,\\mu_{k+1})\\le D(\\mu^*\\,\\|\\,\\mu_k)$) forces ML-EM cluster points to have full support and, by linear independence of the detectors, to be optimal.","core_discovery":"The central discovery is that the feasible set for Poisson measurements has a cone geometry that dictates solution structure. After normalizing $\\sum_i a_i=1$, the negative log-likelihood is $\\ell(\\mu)=\\langle\\mu,1\\rangle-\\sum_i y_i\\log\\langle\\mu,a_i\\rangle$, and its gradient is $A^*\\lambda(A\\mu)$ with $\\lambda_i=1-y_i/\\langle\\mu,a_i\\rangle$. Optimality (KKT) gives $A^*\\lambda^*\\ge 0$ and $A^*\\lambda^*=0$ on $\\operatorname{supp}\\mu^*$, where $\\lambda^*$ is the unique maximizer of the dual problem $g(\\lambda)=\\sum_i y_i\\log(1-\\lambda_i)$ over the dual cone. When $y\\notin A(\\mathcal{M}_+)$, $\\lambda^*\\neq 0$, so the zero set of the nonnegative function $A^*\\lambda^*$ is a proper closed set and every minimizer's support lies in it; with $C^2$ detectors and nondegenerate Hessians, the interior part of the support is a sum of Dirac masses. When $y\\in\\operatorname{int}A(\\mathcal{M}_+)$, the moment-problem theorem quoted from reference [14] produces an absolutely continuous solution with positive continuous density, and the ML-EM iteration started from an absolutely continuous initial measure has the property that every cluster point is optimal and has full support.","pith_inferences":["The paper does not state a crossover dose, but its bounds imply a threshold roughly $m/(\\gamma\\varepsilon)$ below which the sparse regime is typical; this could be tested by sweeping dose in the numerical experiments.","The continuum mechanism suggests that the spike positions seen at finite ML-EM iterations are selected by the dynamics approaching the sharp peaks of $A^*\\lambda^*$; comparing reconstructed spike locations across noise realizations with the argmin set of the dual function would test this.","For motion-corrected PET, where deformations act on the continuum measure, the same cone dichotomy should transfer, meaning low-dose motion-corrected reconstructions should also be atomic unless the deformed data cone is entered; regularisation could then be designed to favour absolutely continuous components at high dose."],"forward_implications":["In low-dose or short-exposure PET, the spiky appearance of ML-EM images is a structural feature of the maximum-likelihood problem itself, not merely an artefact of early stopping: every exact minimizer concentrates on the zero set of the dual function.","In the long-exposure regime, if the data enter the interior of the cone and the detectors are linearly independent, ML-EM started from a smooth positive image cannot converge to Dirac masses: its cluster points are optimal and have full support.","The probability of landing in the sparse regime is at most $2^m e^{-n\\varepsilon/m}$ conditional on $n$ counts and at most $C(m)(1+(\\gamma t)^m)e^{-\\gamma t\\varepsilon}$ for a dose $t$, with $\\varepsilon$ the Kullback--Leibler distance from the true distribution to the complement of the cone; dose therefore controls sparsity exponentially.","Boundary data at an extremal point of the cone force any solution to be a Dirac mass, so sparse solutions can persist even on the boundary of the feasible set.","In the sparse regime, the limiting locations of the point masses depend on the initial measure $\\mu_0$, shown explicitly when only one detector receives counts."],"supporting_citations":[{"why":"Supplies the moment-problem theorem (Theorem 3.11) guaranteeing an absolutely continuous solution with positive continuous density for data in the interior of the cone.","marker":"[14]"},{"why":"Provides the uniqueness of the dual maximizer (their Theorem 4.1) used in Corollary 3.8 and the fixed-point argument for ML-EM cluster points in Proposition 4.3.","marker":"[21]"},{"why":"Gives the convex-optimization KKT and Slater framework used to derive the optimality conditions and the dual formulation.","marker":"[6]"},{"why":"Supplies the classical large-deviations bound for empirical distributions used as the first concentration inequality.","marker":"[36]"},{"why":"Provides the tighter multinomial concentration inequality used as the second bound in Proposition 5.1.","marker":"[22]"},{"why":"Bounds the combinatorial constant $C(m)$ in the dose-dependent concentration bound.","marker":"[4]"},{"why":"Provides the asymptotic expansion of Laplace-type integrals used in Proposition 4.6 to exhibit an explicit ML-EM limit that is a sum of Dirac masses, confirming the sparsity mechanism.","marker":"[40]"}],"fun_headline_variants":["Outside the feasible cone, ML-EM solutions become sparse sums","Sparse spikes appear when Poisson data misses the cone","Low-dose Poisson data forces point-mass reconstructions","Feasible cone boundary decides sparse vs smooth ML-EM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dichotomy rests on the detector response functions being linearly independent, so the cone of achievable data has a genuine interior, together with the imported moment-problem theorem that interior data admit an absolutely continuous solution; the boundary-case theorem as printed also defines its reduced set with the wrong object.","fun_headline_variants_meta":{"raw":{"variants":["Outside the feasible cone, ML-EM solutions become sparse sums","Sparse spikes appear when Poisson data misses the cone","Low-dose Poisson data forces point-mass reconstructions","Feasible cone boundary decides sparse vs smooth ML-EM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000622,"raw_usage":{"total_tokens":2916,"prompt_tokens":1011,"completion_tokens":1905,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":1839}},"tokens_in":627,"tokens_out":1905,"duration_ms":14248,"temperature":1.0,"reasoning_tokens":1839,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:04:37.384443+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $K=[0,1]$, detectors $a_1(x)=1$ and $a_2(x)=x$, so the cone is $\\{(\\alpha,\\beta):\\alpha\\ge\\beta\\ge 0\\}$; choose normalized Poisson data $y=(0.4,0.6)$, which lies outside the cone. Solving $\\min_{\\mu\\ge 0} d(y\\,\\|\\,A\\mu)$ numerically and checking whether every minimizer's support lies in $\\arg\\min(\\lambda^*_1+\\lambda^*_2 x)$ for the unique dual maximizer $\\lambda^*$ would test the sparsity characterization directly. Independently, for the boundary case, construct data on $\\partial A(\\mathcal{M}_+)$ satisfying assumptions (31) and (32) and check whether the asserted absolutely continuous solution exists, since the printed proof's reduced set is defined incorrectly.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the moment-problem theorem (Theorem 3.11) guaranteeing an absolutely continuous solution with positive continuous density for data in the interior of the cone."},{"cited_title":"Positron emission tomography, Borel measures and weak convergence.Inverse Problems 12 , 6 (1996), 965","cited_arxiv_id":null,"evidence_quote":"Provides the uniqueness of the dual maximizer (their Theorem 4.1) used in Corollary 3.8 and the fixed-point argument for ML-EM cluster points in Proposition 4.3."},{"cited_title":"Convex optimization","cited_arxiv_id":null,"evidence_quote":"Gives the convex-optimization KKT and Slater framework used to derive the optimality conditions and the dual formulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical large-deviations bound for empirical distributions used as the first concentration inequality."},{"cited_title":"Improved bounds on Bell numbers and on moments of sums of random variables","cited_arxiv_id":null,"evidence_quote":"Bounds the combinatorial constant $C(m)$ in the dose-dependent concentration bound."},{"cited_title":"Asymptotic approximations of integrals , vol","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic expansion of Laplace-type integrals used in Proposition 4.6 to exhibit an explicit ML-EM limit that is a sum of Dirac masses, confirming the sparsity mechanism."}],"review_version":1}