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Fisher Information and the Minimum Mean-Square Error for Locating Closely Spaced Point Sources

Chuling Wen, Jian Lu, Qingtang Jiang, Charles K. Chui

arXiv:2610.32018v1cs.ITmath.IT

TL;DR

The paper asks how accurately closely spaced sources can be located from noisy observations when unknown phases can make nearby configurations difficult to distinguish. It derives Bayes-risk bounds jointly in separation and noise and gives conditions linking this risk to capped Fisher information. The bounds depend on source-count parity, while priors concentrated near poorly identified configurations can change the risk exponents.

  • Problem

    When information degenerates near phase-cancellation configurations, local Fisher information alone does not determine the minimum average location error under a prior.

    Source: §I · pp. 1–2, §VIII · p. 26

  • Method

    The paper proves joint separation-and-noise Bayes-risk bounds and gives an information criterion based on a global inverse bound and prior-controlled paths.

    Source: §I · pp. 2–3, A. A Uniform Gram Condition on a Window · p. 24

  • Results

    For fixed K with independent uniform phases, matching bounds hold for sufficiently small separations at all noise levels; odd and even K have different risk regimes.

    Source: §I · p. 2

  • Takeaways & Limitations

    The Bayes risk depends on prior mass near poorly identified configurations, and under stated conditions it has the order of the capped Cramér–Rao bound.

    Source: §I · p. 3, §VIII · p. 26

    The capped-information result is not asserted for arbitrary priors; the unknown-center extension retains one fine-scale true cluster, and unknown-count recovery needs a stronger noise threshold.

    Source: §VIII · p. 26

Abstract

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We determine the optimal average error of locating closely spaced sources from noisy superpositions of a known smooth response with unknown complex coefficients. Relative phases control how changes in one location can be offset by the other unknowns, so local Fisher information alone does not determine the attainable error. For every fixed number $K\ge2$ of sources with independent uniform phases, we prove matching Bayes risk bounds jointly in separation and noise, for all sufficiently small separations and all noise levels. Odd source counts have two risk regimes; even counts have an intermediate regime in which mean squared error is linear in the noise standard deviation. A projected least-squares estimator attains these orders without knowing the phases or the prior density. Bounded nonuniform phase laws preserve the orders under stated support conditions, whereas priors concentrated near poorly identified configurations can change the exponents. A common finite comparison yields both a global inverse bound and paths controlled by the original prior. These give sufficient conditions under which the risk has the order of the pointwise capped Cram\'er--Rao bound, and a lower-tail calculation explains the power and logarithmic regimes. We also give conditional guarantees when the cluster center, separation, and source count are unknown, with an additional low-noise requirement for count selection. A calibrated covariance factor model illustrates the same criterion beyond source localization.

I. INTRODUCTION

The paper studies how accurately closely spaced sources can be located from noisy observations when unknown phases and coefficients affect how distinguishable nearby configurations are. It develops Bayes-risk bounds jointly in separation and noise, alongside conditions connecting that risk to Fisher information.

I. INTRODUCTION

  • Phase cancellation can make efficient information degenerate, so pointwise or phase-averaged Fisher information alone need not determine minimum average error.The relevant risk depends on how much prior mass lies near poorly identified configurations.Source: §I · p. 2
  • For fixed K with independent uniform phases, matching Bayes-risk bounds hold at all noise levels for sufficiently small separations, with two regimes for odd K and an additional linear-noise regime for even K.For K = 2, the risk order is min{ε^2, ζ_n, ζ_n^2 ε^-3}; projected least squares attains the stated orders without knowing the phases or prior density.Source: §I · p. 2
  • A sufficient criterion for capped Cramér–Rao risk combines a global inverse bound with a matching lower bound from paths controlled by the prior.The criterion also applies to a calibrated covariance factor model, whose risk orders vary with the number of reference variables.Source: §I · p. 3

A. Observation Model

  • The model observes noisy samples of a known response to K sources with complex coefficients, and studies location error as cluster separation and effective noise vary.The sources lie at τ_j = τ_0 + εs_j, with normalized offsets s_j and a prior on offsets, moduli, and phases.Source: §I · pp. 1–2, A. Observation Model · p. 4

B. Cluster Class, Prior, and Bayes Risk

  • The Bayes localization risk is the minimum prior-and-noise mean-square error for estimating an ordered source location from the full data.The baseline prior uses independent uniform phases and a node–modulus density bounded below near an interior point.Source: B. Cluster Class, Prior, and Bayes Risk · p. 4

C. Response Condition

  • The response condition requires bounded derivatives through order 2K and a uniformly nondegenerate Gram matrix for the response derivatives at the cluster center.The stated examples include line spectra, sampled sinusoids, uniform linear arrays, and Gaussian responses under the specified design conditions.Source: C. Response Condition · p. 5

D. Efficient Fisher Information

The paper derives nuisance-adjusted Fisher information for source locations, then establishes Bayes-risk orders and identifies when capped Cramér–Rao bounds describe them. The results cover uniform and qualifying nonuniform phase laws, with parity-dependent regimes and a prior-independent estimator.

D. Efficient Fisher Information

  • Efficient information for a location removes the effects of nuisance parameters through a Schur complement, yielding a noise-scaled distance from its tangent to the nuisance tangent space.The efficient information is expressed using J_j, with the coefficient magnitude included when the argument is the full parameter x.Source: D. Efficient Fisher Information · pp. 5–6

A. Uniform Phases

  • Theorem 1 gives matching Bayes-risk bounds for every fixed K, all sufficiently small separations, and every noise level; odd and even K have different orders.For even K, the intermediate regime is linear in the noise standard deviation.Source: A. Uniform Phases · p. 6
  • The projected global least-squares estimator attains these orders without knowing the phases or prior density.The stated upper bound applies to every node–modulus law with independent uniform phases.Source: A. Uniform Phases · p. 6
  • Figure 1 plots the risk orders on logarithmic axes, with segment slopes showing noise exponents and the plateau marking the ε² cap.The parity-dependent intermediate regime appears for even source counts.Source: Figure 1 · p. 7, 1) Implications · p. 7
  • When center, separation, or source count is unknown, attaining the below-cap orders requires additional conditions; count selection also needs a further noise threshold and incurs a count-error penalty.The source count is treated as unknown but bounded by a known K_max.Source: A. Uniform Phases · p. 7

1) Implications:

  • For odd K, averaging inverse efficient information before capping gives the risk order; for even K, that average exceeds the risk in the intermediate regime by at most ε^-1/2.The stated maximum factor occurs at ζ_n = ε^(3K−1)/2.Source: 1) Implications · p. 7
  • For two sources, phase cancellation creates the intermediate order: when ε^3 ≪ ζ_n ≪ ε^2, a near-cancellation arc contributes risk proportional to ζ_n.Capping the phase-averaged local bound afterward misses this regime.Source: 1) Implications · p. 7

2) Dependent and Nonuniform Phase Laws:

  • The same risk orders hold for dependent, nonuniform phase laws under stated density conditions, including bounded-concentration correlated von Mises laws when the node–modulus density is bounded below on the relevant projection.The result does not require phase independence or a global lower bound on the phase density.Source: 2) Dependent and Nonuniform Phase Laws · p. 8

3) Comparisons With Prior Works:

Earlier phase-dependent Cramér–Rao results largely concern fixed configurations, while this section compares prior-averaged risk and shows how concentration near poorly identified phases can change its rate. The weighted and singular priors yield different intermediate-risk behavior, attained by the same projected least-squares estimator without prior knowledge.

3) Comparisons With Prior Works:

  • Earlier work studied phase effects mainly at fixed configurations; the present question concerns phase contributions under a prior, including shrinking neighborhoods of degenerate phases.Swingler [8] found an inverse-cube separation law for two complex sinusoids, matching the low-noise separation dependence for K = 2, but without a proof or Bayes risk.Source: 3) Comparisons With Prior Works · p. 8

1) Power Weights Near an Even-Source Degeneracy:

  • For even K, power-weighting phases near the information-degenerate set D = 0 changes the intermediate risk law according to the prior’s weight.The specified local information satisfies J_j ≍ ε^(3K−4)(D^2 + ε^2) for j in A; negative β places more prior mass near D = 0.Source: 1) Power Weights Near an Even-Source… · p. 9

3) Risk Under the Weighted and Singular Priors:

  • The projected global least-squares estimator attains both prior-specific risk orders without knowing the prior, uniformly over the stated density classes.The uniformity conditions differ: part (a) fixes a lower density bound, while part (b) fixes the upper-to-lower density ratio.Source: 3) Risk Under the Weighted and… · p. 10

4) Implications:

  • For −1 < β < 0, risk decays more slowly in noise than under uniform phases; β = 1 adds a logarithmic factor, while β > 1 removes the intermediate range.For 0 < β < 1, risk decays faster; mixing a negative-β prior with a uniform-phase baseline preserves the slower law.Source: 4) Implications · p. 10

2) A Singular Joint Prior With Three Sources:

IV. RELATED WORKS

Prior work studies local Cramér–Rao accuracy, worst-case recovery, and related Bayes lower bounds using different error criteria. The analysis here combines a uniform finite comparison with prior-weighted statistical bounds to control source-location risk.

IV. RELATED WORKS:

  • Prior work spans local Cramér–Rao bounds, worst-case super-resolution, nonstandard rates in singular models, and Bayes lower-bound techniques, often using different error criteria.For example, two-source separation criteria differ from individual-location error, and the cited singular-information analyses address local behavior or drifting sequences rather than joint Bayes localization risk.Source: §IV · p. 11

A. Key Lemmas

  • A uniform finite–tangent comparison controls both source-signal differences and derivatives, yielding an inverse bound for upper risk and fixed-moment paths for lower risk.Hermite–Newton interpolation transfers moment norms to response norms uniformly even when node lists have repetitions.Source: A. Key Lemmas · pp. 11–12

1) One Finite Comparison for Signals and Tangents:

  • Hermite interpolation bounds response norms through divided differences with constants independent of node gaps and combination coefficients.The comparison applies for sufficiently small separation and remains uniform at repeated nodes.Source: 1) One Finite Comparison for Signals… · p. 12

2) Prony Coordinates and Finite Differences:

  • Prony coordinates give a common identity for finite differences and tangents, with a smooth local inverse and uniformly bounded Jacobians on compact admissible subsets.The moment comparison holds uniformly over phases, while fixed-moment paths additionally require control of the original prior along those paths.Source: 2) Prony Coordinates and Finite… · pp. 12–13

3) One Statistical Conversion:

  • A minimum-distance fit gives an upper risk bound from finite comparisons, while prior-weighted pairs with small signal differences give a lower bound through two-point testing.The lower-bound argument requires both endpoints to carry prior mass; the two resulting integrals need not match for an arbitrary prior.Source: 3) One Statistical Conversion · pp. 13–14

4) Truncated Power Integrals:

The analysis averages phase-dependent information and evaluates the resulting risk bounds through truncated power integrals. Uniform-in-phase estimates capture configurations that approach degeneracy as separation shrinks.

4) Truncated Power Integrals:

1) Information and Its Phase Average:

  • The uniform moment estimate holds for all phases, including phases that depend on separation, so fixed-configuration exponents alone do not control the shrinking-separation regime.Its constants are uniform over the stated compact node set and separated nodes.Source: 1) Information and Its Phase Average · p. 15
  • For even K, the nuisance-phase average of λ_max(R)^−1/2 is bounded by Cε^(1−K/2), controlling the phase-averaged information ellipse.The ellipse determinant satisfies det R = ε^(K−1)|d_j|{1 + O(ε)}.Source: 1) Information and Its Phase Average · p. 15
  • For every fixed K ≥ 2, phase-averaged inverse information has order ε^−(3K−3)|d_j|^−3, uniformly over admissible nodes and coefficient moduli.The result assumes independent uniform phases and holds for sufficiently small separation.Source: 1) Information and Its Phase Average · p. 15

2) The Inverse Bound:

A global inverse bound and fixed-moment paths provide the upper and lower sides of the source-location risk analysis. Evaluating the resulting bounds gives matching risk orders for the stated source models and priors.

2) The Inverse Bound:

  • The inverse bound compares projected signal differences with changes in a source location, uniformly over arbitrary phases, including phases that vary with separation.The bound applies to all pairs in X_K when separation is sufficiently small.Source: 2) The Inverse Bound · p. 16
  • For every prior on X_K, the estimator’s Bayes risk is at most C E_π min{ε^2, ζ_n^2/J_min}.The bound combines the separation-limited error with the inverse-information term.Source: 2) The Inverse Bound · p. 16

3) Fixed-Moment Paths:

  • Fixed-moment paths move a target node at speed of order ε while fixing the first few complex moments; the mean changes only through the first moment that changes, and prior control uses density ratios, not derivatives.Source: 3) Fixed-Moment Paths · p. 16
  • For even K with the weighted prior, paths translate u at fixed other chart coordinates and yield an integral weighted by |b|^β.The path construction also covers uniform phases when β = 0.Source: 3) Fixed-Moment Paths · p. 17

4) Closing the Two Risk Bounds:

  • Closing the bounds reduces weighted-prior risks to truncated power integrals; for the three-source singular prior, the dyadic sum has the order of ε^2 I_(3−2γ)(ε, ζ_n/ε^4).The integral evaluations include logarithmic cases.Source: 4) Closing the Two Risk Bounds · p. 17

VI. FISHER INFORMATION AND THE BAYES RISK: A CRITERION

The section gives a criterion for when Bayes risk matches a prior-averaged, pointwise capped Cramér–Rao bound. It applies the criterion to source models and shows why local information alone can misstate risk.

VI. FISHER INFORMATION AND THE BAYES RISK: A CRITERION

  • The criterion uses an information surrogate and requires a global inverse bound plus a matching lower bound from paths controlled by the prior.The section applies these conditions to source models and a one-factor model with information loss through cancellation.Source: §VI · pp. 17–18, B. The Criterion · p. 19

A. Setting

  • Gaussian observations with a projected mean map provide a setting for comparing scalar-target risk with capped information bounds.The comparison depends on the projection rank, and the target’s local modulus describes its change relative to the mean map.Source: A. Setting · p. 18

B. The Criterion

  • Inverse bounds and prior-dominated paths together make Bayes risk comparable to the prior average of the pointwise capped Cramér–Rao bound.Under the stated uniformity conditions, a minimum-distance estimator attains this order without using the prior.Source: B. The Criterion · p. 19
  • The same efficient information, I(x) = 4x², can accompany different Bayes risks, while signal symmetry on [−1, 1] makes the inverse bound fail.Thus capped inverse information can either overestimate or underestimate Bayes risk.Source: B. The Criterion · p. 20

C. When the Averaged Cram´er–Rao Bound Suffices

The section relates the lower tail of an information surrogate to whether averaging inverse information preserves the capped risk order. It applies this relationship to source priors and a calibrated covariance factor model.

C. When the Averaged Cramér–Rao Bound Suffices

  • κ > 1 is necessary and sufficient for the averaged capped bound to uniformly match the pointwise capped bound as the smallest information scale decreases.For κ < 1 the intermediate noise exponent is 2κ, with a logarithmic correction at κ = 1.Source: C. When the Averaged Cram´er–Rao Bound… · p. 21

D. Application to Theorems 1 and 2

  • For the source priors and targets covered by Theorems 1 and 2, Bayes risk has the order of the prior average of the capped inverse information.The surrogate Jmin has lower-tail exponent κ = (1 + β)/2 under πβ and κ = (3 − 2γ)/2 under πγ.Source: D. Application to Theorems 1 and 2 · p. 22

E. A One-Factor Model

  • For S and I, κ = m/4, and Proposition 5 implies the averaged Cramér–Rao bound has the order of the risk if and only if m > 4.Source: E. A One-Factor Model · p. 23

F. Comparisons With Prior Works

The paper distinguishes its fixed-prior Bayes-risk criterion from local minimax and fixed-model Bayesian bounds. It then extends localization guarantees to unknown cluster parameters and source count, with count selection restricted to a low-noise range.

F. Comparisons With Prior Works

  • Theorem 3 compares fixed-prior Bayes risk with capped efficient information when an all-candidate inverse bound and prior-dominated paths satisfy its conditions.The cited prior benchmarks instead address local two-point minimax risk or fixed-model asymptotic Bayesian bounds.Source: F. Comparisons With Prior Works · p. 23

A. A Uniform Gram Condition on a Window

B. Estimators and Risk Guarantees

  • With unknown center and separation, the risk has order ΨK under the stated phase-density, design, and prior-mass conditions.The guarantee applies for ζn ≤ ζ∗,K(ε), with constants independent of the center law.Source: B. Estimators and Risk Guarantees · pp. 24–25

C. Selecting the Source Count

  • Source-count selection attains order ΨK only when conditions (45) and (48) hold and ζn ≤ ζ∗,K(ε).The discussion states that unknown-count recovery requires a stronger noise threshold; this is not a full-noise guarantee.Source: C. Selecting the Source Count · p. 25

VIII. DISCUSSION

  • The paper’s criterion depends on information-tail behavior and prior-controlled paths, not on source geometry.The discussion identifies a calibrated covariance factor model as another instance.Source: §VIII · p. 26
  • Unknown-center results retain a single fine-scale true cluster, while unknown-count recovery needs a stronger noise threshold.The paper also leaves full-complexity guarantees for algorithms and leading joint-risk constants open.Source: §VIII · p. 26

APPENDIX A PROOFS OF LEMMA 6 AND PROPOSITION 3

These appendices develop finite-dimensional comparisons and prior-controlled paths used to establish localization-risk bounds. They also show how the risk changes for power-weighted phase configurations and mixtures of priors.

APPENDIX A PROOFS OF LEMMA 6 AND PROPOSITION 3

  • A finite-dimensional elimination identity relates a target-direction minimum to determinant ratios, with each ε-weighted row contributing its squared weight to each minor.The appendix uses this identity in comparisons involving projected signal distances and location changes.Source: Appendix A · p. 26

APPENDIX B PATHS AND PHASE INTEGRALS FOR UNIFORM PHASES

  • For uniform phases, prior-dominated paths are constructed by varying configuration coordinates while controlling occupation under the prior.For even K, phase integration bounds the inverse square-root information uniformly over separated node configurations.Source: Appendix B · pp. 27–28

APPENDIX C PROOF OF THEOREM 2 FOR POWER-WEIGHTED PRIORS

  • For even K, the power-weighted prior proof uses a chart where D = k(x)b and matches risk bounds through an integral weighted by b^β.The integral yields different orders for β < 1, β = 1, and β > 1; adding a fixed uniform-phase component gives the larger component risk order.Source: Appendix C · pp. 28, 30

APPENDIX D PROOF OF THEOREM 2 FOR THE SINGULAR THREE-SOURCE PRIOR

The appendix proves the risk result for a singular prior on three-source configurations by controlling information across the prior support and constructing paths within that support. The argument also covers specified priors on nodes and weights.

APPENDIX D PROOF OF THEOREM 2 FOR THE SINGULAR THREE-SOURCE PRIOR

  • On the stated support, the information calculation reduces the relevant numerator to a quantity comparable to ε²S, with S = u² + (v² + ε²)².The resulting target information comparison uses the separated-node relation in the denominator.Source: Appendix D · p. 31
  • The lower-bound construction shears fixed-moment coordinates while keeping paths inside the original admissible support and controlling their prior occupation.The appendix evaluates the resulting dyadic sum, including its critical logarithmic case, to complete Theorem 2(b).Source: Appendix D · p. 32
  • The same orders apply to priors specified on nodes and weights when their density is comparable to u^-γ on the stated separated, positive-weight box.The moment map and its Jacobian are bounded above and below on that box.Source: Appendix D · p. 32

APPENDIX E PROOF OF THEOREM 3

The proof of Theorem 3 bounds Bayes risk by integrating capped information along paths whose occupation is controlled by the prior. It closes the comparison when the global inverse bound and matching condition hold.

APPENDIX E PROOF OF THEOREM 3

  • Along each path, the target’s squared displacement is bounded by its length cap and the noise-scaled inverse information.Integrating this pathwise inequality under the occupation bound gives the prior-averaged risk bound.Source: Appendix E · p. 33
  • When the global inverse bound compares the surrogate information with efficient information, the two bounds close the Bayes-risk comparison.The proof uses the inequality between capped inverse-information terms to transfer the path bound.Source: Appendix E · p. 33
  • The local modulus envelope is measurable and admissible for the path-integral comparison.The proof obtains this using upper semicontinuity and monotonicity properties of the modulus.Source: Appendix E · p. 33

APPENDIX F PROOFS OF LEMMAS 11 AND 12 AND PROPOSITIONS 4 AND 5

Appendix F establishes comparisons between global signal differences, target error, and Fisher information, then uses prior-controlled paths and lower-tail bounds to assess when risk matches the capped Cramér–Rao bound. Appendix G applies this framework to a calibrated covariance factor model.

APPENDIX F PROOFS OF LEMMAS 11 AND 12 AND PROPOSITIONS 4 AND 5

  • The inverse bound links target changes to signal changes through the local information scale, while capping controls errors beyond that scale.The proof splits at a threshold set by signal distance and the information-related quantity S.Source: Appendix F · p. 33
  • Prior-controlled paths with sufficient prior occupation yield a lower comparison for risk, connecting global configuration changes to Bayes lower bounds.The proof integrates along paths and uses their occupation measure and a tail comparison.Source: Appendix F · p. 33
  • When the lower tail of information scales as (u/s0)^κ, averaging inverse information before capping preserves the capped-bound order uniformly in its lower cutoff exactly for κ > 1.For κ ≤ 1, the proof constructs laws for which no uniform comparison constant works as the lower cutoff tends to zero.Source: Appendix F · pp. 33–34

APPENDIX G PROOF OF COROLLARY 2

  • The factor-model comparison uses paths that vary v^2 while fixing the other chart coordinates, with target derivative bounded below and covariance speed bounded above.The path construction supplies the prior occupation control used in the lower comparison.Source: Appendix G · p. 35
  • For every C, Theorem 3(b) shows Rπ,d0(n), Aπ(n−1/2; S), and Aπ(n−1/2; I) are of one order, attained by (G.1) at bΣn; with all references calibrated, Lemma 12 applies to S = ∥a∥4.Source: Appendix G · p. 35
  • With calibrated references, exact information tends to 1 as v tends to 0 at fixed nonzero a, while S = ∥a∥^4; the two are comparable where |v| is bounded below.Thus the comparison between information and S is restricted to that stated region.Source: Appendix G · p. 35

APPENDIX H PROOF OF PROPOSITION 6

Appendix H establishes a finite signal comparison over a fixed search window and uses it to control location errors and construct matching risk bounds. It also gives projection and design conditions that support those guarantees across knot collisions.

APPENDIX H PROOF OF PROPOSITION 6

  • For sufficiently small separation, sufficiently close projected signals have nodes that match in order, with location errors bounded by projected signal distance divided by J.The inverse comparison holds uniformly over truths and candidates in the fixed window.Source: Appendix H · p. 36
  • Under bounded coefficient density, the even-K polynomial condition has probability at most Cα(b + α) for a small residual and a near-degenerate leading coefficient.For K = 2 with fixed moduli and uniform phases, the stated small-residual probability is (2/π) arcsin(α/2), showing fixed moduli do not suffice for that lemma.Source: Appendix H · p. 36
  • Projected least squares has a uniform upper risk bound, while paths within the prior support provide the corresponding lower comparison.The lower-bound paths are mixed over center support, with occupation controlled by the original prior.Source: Appendix H · p. 37
  • For odd K, the conditional location-risk order includes the minimum of ε^2, ζ^2 ε^-(3K−3), and εζε^(-(3K−3)/2).The displayed bound follows from the conditional expectation of the inverse information and the stated uniform phase domination.Source: Appendix H · p. 37
  • A fixed-window Gram condition gives a moment-norming projection, and the Taylor projection at the center is admissible when the window is sufficiently narrow.The Taylor projection has rank 2K under the local condition; the broader window construction has rank at most qW, which affects projected-noise and count-selection bounds.Source: Appendix H · pp. 36–38
  • A moment-norming projection makes projected signal distance comparable to moment differences over the whole window, including configurations with colliding knots.Its rank and comparison constants are uniform over the design family and independent of sample size, separation, and noise.Source: Appendix H · p. 38
  • For line-spectrum responses, an equispaced block design with hDW < 2 satisfies the window condition; the window restriction matters because a grid can alias locations separated by 2/h.A single N-point block does not suffice for the stated equispaced design condition.Source: Appendix H · pp. 38–39

APPENDIX I PROOF OF PROPOSITION 7

Appendix I proves a uniform separation between a K-source truth and signals with fewer distinct nodes, then uses projected noise control to select the correct source count. The guarantee is conditional on a low-noise threshold.

APPENDIX I PROOF OF PROPOSITION 7

  • A K-source truth is separated from every signal with fewer than K distinct nodes by a uniform projected-distance lower bound of order ε^(2K−2).The bound holds for arbitrary complex coefficients and candidates in the fixed window.Source: Appendix I · p. 39
  • When projected noise is below a threshold smaller than the count-separation bound, the selector returns the K-class minimum-distance fit, ruling out count reduction by deletion or merging.The proof controls the event using a tail bound for the norm of projected Gaussian noise.Source: Appendix I · p. 39
  • The count-selection guarantee combines the correct-count event with the known-count location-risk bound; its stated risk bound includes an additive term proportional to the failure probability.The lower bound follows by converting any count-and-location estimator into a known-K estimator.Source: Appendix I · p. 39

Supplementary Material for “Fisher Information and the Minimum Mean-Square Error for Locating Closely Spaced Point Sources”

The supplement reports numerical checks of the source-risk regimes, prior effects, and factor-model transition, alongside simulation settings and solver diagnostics. The results support several predicted scalings, while the three-source and singular-prior experiments have explicit finite-scale caveats.

S1. NUMERICAL ILLUSTRATIONS

  • The source simulations use a line-spectrum response sampled at 31 equally spaced points, with each trial’s parameters fixed across its run.Projected least squares is approximated by grid search and local refinement; posterior means use the generating prior.Source: S1. NUMERICAL ILLUSTRATIONS · p. 43, Supplementary Material for “Fisher… · p. 43
  • For three sources, normalized-risk ratios across neighboring separations are near 1 in the transition windows, supporting the predicted odd-source scaling rather than the even-source intermediate expression.The reported ratios are 1.03 [0.75, 1.39] and 1.02 [0.77, 1.41].Source: S1. NUMERICAL ILLUSTRATIONS · p. 43, Figure S1 · p. 44
  • The three-source results do not establish a limiting plateau, and posterior means were not computed for K = 3.At λ ≥ 1, most least-squares solutions lie on the estimation-class boundary; at λ ≥ 10 some risks exceed prior-mean risk.Source: S1. NUMERICAL ILLUSTRATIONS · p. 43
  • For two sources, the intermediate-window slopes are near 1 for both posterior mean and projected least squares, and least-squares MSE is 4.8 to 6.4 times the posterior-mean MSE.The visual compares normalized MSE across noise levels and separations; the intermediate window is marked for ε = 0.01.Source: S1. NUMERICAL ILLUSTRATIONS · p. 43, Figure S1 · p. 44
  • For the singular-prior risk ratio, intermediate slopes contain the predicted −1/2 while low-noise slopes contain 0, but the intervals are wide and the finite-scale checks do not establish separated asymptotic ranges.The calculated intermediate-path points still have λ < ε.Source: S1. NUMERICAL ILLUSTRATIONS · p. 44, Figure S1 · p. 44
  • In the factor model, n times MSE grows linearly in log n for m = 4 and is flat for m = 5, 6, consistent with the predicted logarithmic and n^-1 regimes.The fitted slopes of MSE in n for m = 5, 6 are −1.001 and −1.002.Source: Figure S1 · p. 44, S1. NUMERICAL ILLUSTRATIONS · p. 45
  • The reported MSE includes unsuccessful solves, but the optimality check only tests a necessary inequality and does not certify global optimality.Some marked noise levels have violations contributing at least 10% of MSE.Source: Figure S1 · p. 44, S1. NUMERICAL ILLUSTRATIONS · p. 45

S2. SETTINGS OF THE NUMERICAL SUMMARIES IN SECTION S1

The numerical summaries compare projected least squares and posterior means across source configurations, noise levels, and priors, and also test fixed-window localization and a calibrated factor model. The reported comparisons are qualified by solver checks, subset sensitivity, and the sampling and design assumptions of the guarantees.

A. Source Model and Estimator

B. Two-Source Posterior Mean

  • In intermediate windows, posterior-mean MSE and average posterior variance differ by at most 1.9 Monte Carlo standard errors; the ε=0.003 low window is omitted.The omission reflects only three noise levels and least-squares solves violating the optimality check carrying most of the MSE there; Fig. S2 plots first-node MSE divided by ε² for posterior means and projected least squares under the wide two-source prior.Source: B. Two-Source Posterior Mean · p. 46, Figure S2 · p. 47
  • Across five separations, joint-scale fitted slopes are 3.884, 2.505, and 1.904 along noise paths ζn=ε^(7/2), ε^(5/2), and ε^(3/2), respectively.The restriction to the three smallest separations was selected after inspecting the data.Source: B. Two-Source Posterior Mean · p. 46

C. Three Sources

  • For three-source runs, plateau points use a common prefix of 240 trials per separation, and all retained trials, including unsuccessful solves, enter the MSE.The prespecified optimality-check threshold is passed in every window, but passing the check does not certify a global minimum.Source: C. Three Sources · pp. 46–47
  • The four low-noise slopes on the common prefix are 1.91 [1.81, 2.02], 2.00 [1.96, 2.03], 1.98 [1.86, 2.13], and 2.00 [1.98, 2.02]; the authors do not claim an asymptotic plateau.Using all completed plateau trials changes the pooled exponent on P={1, 10^(1/2)} from 0.11 [−0.05, 0.26] to 0.15 [0.006, 0.29].Source: C. Three Sources · p. 47

D. Factor Model

  • For m=4, fitting n MSE = α + β log n over 2^14≤n≤2^22 gives β=0.168 [0.166, 0.171].The local fitting range was chosen after inspecting the data.Source: D. Factor Model · p. 48

E. Fixed-Window Least Squares Above the Cap Scale

  • Within 15%, fixed-window and known-window risks agree for K=2 when ζn≤ζ*,2/3; above 3ζ*,2, fixed-window risk is 0.4–0.8 ζn.For K=3, fixed-window risk stays below the stated bound but is loose by a factor of order 1/ε over part of the tested range; reported risks are minima found by grid search and local refinement, not necessarily global minima.Source: E. Fixed-Window Least Squares Above… · p. 48

S3. SCOPE OF THE STATISTICAL GUARANTEES

  • The scope table covers Gaussian noise and sufficiently small ε; fixed-window guarantees require a uniform Gram condition, which retained designs must satisfy anew.The stated retained-observation guarantee applies when the retained set and sampling offsets are fixed in advance or independent of sources and noise.Source: S3. SCOPE OF THE STATISTICAL GUARANTEES · p. 49, Table S1 · p. 49
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