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A note on the sample complexity of multi-target detection

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For multi-target detection in high noise, the sample-complexity cost is of order σ^6.

desk verdict A clean MTD-to-MRA reduction and a plausible Θ(σ^6) rate for circular translations; the SO(2) lower bound is asserted rather than proved. read the letter →

arxiv 2501.11980 v1 pith:UJDJZUWO submitted 2025-01-21 eess.SP cs.ITmath.IT

Amnon Balanov , Shay Kreymer , Tamir Bendory This is my paper · ORCID
classification eess.SPcs.ITmath.IT MSC 94A1262F12
keywords multi-targetdetectionsamplecomplexityautocorrelationanalysismulti-referencealignmentcryo-EMhigh-noiseregimecirculartranslationsFourier-Besselbasis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how many signal occurrences are needed to recover the shape of a target signal from one long, very noisy recording in which the target appears many times at unknown locations and in unknown orientations. Its central result is that in the high-noise regime the answer is of order $\sigma^6$, where $\sigma$ is the noise level, for the one-dimensional model with uniform circular translations: the authors prove both that autocorrelations up to third order give a recovery method needing only on the order of $\sigma^6$ occurrences and that no estimator can succeed with substantially fewer. The same third-order mechanism gives a lower bound of order $\sigma^6$ for two-dimensional rotations and an upper bound of order $\sigma^6$ when there is no group action, with matching lower bound conjectured. The broader message is that the sample complexity of these models is governed by the lowest order of autocorrelation that uniquely determines the orbit of the signal under the group, and that a reduction to multi-reference alignment turns known sample-complexity results into lower bounds for the harder unknown-location problem.

What carries the argument

The load-bearing objects are the empirical autocorrelations of the recorded trace, $a_z^{(d)}[\ell_1,\dots,\ell_{d-1}]$, which average products of entries at fixed lags and are invariant to unknown target locations. Under the well-separated assumption, Proposition III.3 shows that these empirical autocorrelations converge almost surely to the ensemble mean of a single randomized target, which turns location uncertainty into a clean statistical limit. The two other load-bearing tools are a reduction from MTD to multi-reference alignment (any estimator for MTD would solve the easier known-location MRA problem, so MTD's sample complexity is at least MRA's) and a compactness-plus-continuity argument that promotes convergence of autocorrelation tensors to convergence of the estimated orbit.

What would settle it

For a concrete check, pick a signal in $\mathbb{R}^L$ with non-vanishing DFT, simulate the 1-D circular-translation MTD model at noise level $\sigma$ with $N \sim \sigma^4$ well-separated occurrences, and see whether an estimator built from the empirical covariance (second-order autocorrelation) drives the orbit MSE to zero as $\sigma \to \infty$. The paper's lower bound predicts that no such second-order estimator can succeed at this rate; if one does, the claimed $\Theta(\sigma^6)$ sample complexity is wrong.

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Extended reading notes

Core claim

For the multi-target detection (MTD) model, in the high-noise regime where $\sigma, N, M\to\infty$ with fixed density $\gamma=N/M$, the paper establishes that sample complexity is controlled by the lowest autocorrelation order that identifies the signal orbit. For one-dimensional MTD with uniformly distributed circular translations, Propositions V.1 and III.4 together give an upper bound of order $\sigma^6$ for well-separated targets and a lower bound of order $\sigma^6$ for merely non-overlapping targets, so the high-noise sample complexity is exactly $\Theta(\sigma^6)$. For two-dimensional band-limited images rotated by uniform SO(2), the paper proves the lower bound of order $\sigma^6$ and conjectures the upper bound; for the no-group-action case it proves the $\sigma^6$ upper bound and conjectures the lower. The mechanism behind the upper bounds is Proposition III.4: whenever autocorrelation ensemble means up to order $\bar d$ uniquely determine the orbit, recovery from the empirical autocorrelations is achievable with sample complexity at most order $\sigma^{2\bar d}$. The mechanism behind the lower bounds is Proposition IV.1: unknown locations cannot make estimation any easier than the corresponding multi-reference alignment problem, so any sample-complexity lower bound for MRA transfers directly to MTD.

Load-bearing premise

The matching $\sigma^6$ story depends on the claim that the third autocorrelation moment is the first one that can uniquely identify the signal's orbit: the upper bounds require third-order autocorrelations to be sufficient, and the lower bound for rotation-invariant images imports from a cited theorem that the power spectrum and bispectrum together determine an image up to rotation, but does not independently prove that no lower-order statistic could determine the orbit at finite sample size.

Editorial extensions

If this is right

  • For the 1-D uniform circular-translation model, the sample complexity is $\Theta(\sigma^6)$ in high noise; any estimator, autocorrelation-based or not, must use on the order of $\sigma^6$ signal occurrences to reach a fixed small MSE, and the autocorrelation method achieves this rate.
  • For non-uniform translation distributions, the lower bound falls to order $\sigma^4$, so the distribution over the group, not just the group itself, determines the exponent in the sample complexity.
  • For 2-D band-limited images under uniform SO(2) rotations, the sample complexity is at least order $\sigma^6$; if the conjectured upper bound holds, the exponent matches the 1-D circular-translation case.
  • When no group acts on the signal, the third-order autocorrelation still gives an upper bound of order $\sigma^6$, so unknown locations alone already force this rate.
  • General principle: whenever the autocorrelation ensemble mean up to order $\bar d$ uniquely determines the orbit, recovery is possible with at most order $\sigma^{2\bar d}$ occurrences; this is the engine behind all the upper bounds.

Reading between the lines

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

  • The reduction in Proposition IV.1 suggests a testable invariance: the asymptotic sample-complexity exponent for MTD should equal that of the corresponding MRA problem whenever the target spacing satisfies the non-overlapping condition, so empirical comparisons of MRA and MTD estimators at matched densities would directly check the tightness.
  • The general $\sigma^{2\bar d}$ upper bound implies that searching for group actions and measurement geometries with uniquely identifying low-order moments is a practical design principle: lower-order identification means cheaper recovery from noisy micrographs.
  • If the conjectured 2-D SO(2) upper bound is confirmed by a provable algorithm, the same machinery would likely extend to cryo-EM projection models, where the relevant moments are projection-rotation invariants; the paper's outlook already points in that direction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the sample complexity of multi-target detection (MTD), where a signal x appears N times at unknown locations in a long noisy observation, each occurrence acted on by a random group element. The authors propose a general framework: an upper bound via autocorrelation analysis (if the d-th order ensemble autocorrelations uniquely determine the orbit, then N = ω(σ^{2d}) observations suffice), and a lower bound via a reduction to multi-reference alignment (MRA). They apply the framework to three models: 1D circular translations (matching upper/lower bounds of ω(σ^6)), 2D images with uniform SO(2) rotations (a claimed lower bound of ω(σ^6), with the upper bound left as a conjecture), and 1D signals without group action (an upper bound of ω(σ^6)). The proofs are in the appendix, including an almost-sure convergence result for empirical autocorrelations in the well-separated case and a measurable-mapping argument for the MRA reduction.

Significance. If correct, the 1D result would provide a clean Θ(σ^6) characterization and the reduction to MRA is a useful conceptual contribution. The autocorrelation convergence theorem under the well-separated assumption is a genuine extension of previous work, and the paper is careful to identify which upper bounds are conjectural. However, the only 2D result (Proposition V.2) rests on an invalid proof: moment-order identifiability does not imply a sample-complexity lower bound, and an upper-bound theorem is invoked to prove a lower bound. Consequently the 2D claim is currently unsupported. The paper is perhaps best viewed as a working note consolidating the 1D bounds and identifying open problems.

major comments (2)
  1. [Appendix E (proof of Proposition V.2)] The proof of the lower bound N*_{MTD_SO(2)} ≥ ω(σ^6) is invalid. The statement 'as the minimal moment required to determine uniquely the signal x is the third moment' is an unsupported inference from identifiability to sample complexity: [20, Theorem II.1] shows only that the combined power spectrum and bispectrum determine the image up to rotation, which is an identifiability result, not a minimax or Fisher-information lower bound. Furthermore, the final step 'applying Proposition III.4 and Theorem A.9 proves the lower bound' is a category error, since Proposition III.4 is an upper-bound statement (it gives N = ω(σ^{2\bar d}) sufficient for consistency when autocorrelations identify the orbit). An upper-bound theorem cannot yield N* ≥ ω(σ^6); a lower bound would require a dedicated argument (e.g., a minimax bound for the uniform-SO(2) MRA model, as in [4] for specific regimes) or a citation to such a result. As written, Proposition V.2 is unsupported, and because Section V.B leaves the upper bound as a conjecture, no 2D MTD rate is established.
  2. [Appendix B (Lemma A.7, Proposition III.4)] The proof of the upper-bound framework is incomplete. Lemma A.7 applies [22, Theorem 2.1] to the extremum estimator defined in (A.39), but it verifies only pointwise convergence of the empirical autocorrelations (Corollary A.6), not the uniform convergence in probability over the compact parameter space that the theorem requires. Without uniform convergence (or an explicit stochastic equicontinuity argument for the polynomial objective), consistency of the estimated orbit does not follow. In addition, the variance bound E[(a_z^{(d)} - \bar\mu^{(d)})^2] = O(σ^{2d}/N) in Corollary A.5 is asserted on the basis of a nonexistent 'Lemma A.3' (likely Corollary A.3) and with no derivation; this bound is the quantitative engine behind the claimed N = ω(σ^{2\bar d}) rate, so it needs a proof or citation. These gaps affect the upper-bound claims in Propositions V.1(1) and V.3.
minor comments (5)
  1. [Throughout] The notation ω(σ^6) is used in a nonstandard way: phrases such as 'bounded from above by ω(σ^6)' are not meaningful in standard asymptotic notation, since ω(f) denotes a class of functions growing faster than f. Please define the intended meaning (e.g., 'N = ω(σ^6) is sufficient' vs. 'N = O(σ^6)') or replace with Θ/O notation.
  2. [Appendix A, Eq. (A.16)] The set C_M^{(i)} is defined by |m - r_i| < L, but the sum in (A.19) ranges over k = -L,...,L-1, including the index r_i - L. The definition should be r_i - L ≤ m ≤ r_i + L - 1 to match the decomposition.
  3. [Appendix B, proof of Corollary A.5] The text refers to 'Lemma A.3' when the variance bound is established earlier in Corollary A.3; please correct the citation.
  4. [Appendix C] The last sentence of the proof of Proposition IV.1 says 'which completes the proof of the proportion'; this should be 'proposition'.
  5. [Section V.B] Please clarify the status of Proposition V.2 in the main text: the proof in Appendix E is not valid, and the upper bound is explicitly conjectural, so the proposition should be labeled as a conjecture or re-proven.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the claimed rates follow from independent autocorrelation uniqueness results and a reduction to MRA, not from fitted parameters or self-referential definitions.

full rationale

The derivation chain is not circular. For the 1D translation model, the upper bound is obtained by invoking an independent uniqueness result ([11, Thm 3.2]) for the first three autocorrelation orders and feeding it into Proposition III.4, whose proof derives the sigma^{2d} rate from variance and concentration estimates on the empirical autocorrelations (Corollaries A.5-A.6 and Lemma A.7); no parameter is fitted to the claimed sigma^6 rate. The lower bound is a reduction (Proposition IV.1) to the MRA model, whose sample complexity is imported from the published MRA result [1]; Lemma A.8 is a standard data-processing inequality, so the reduction does not assume the conclusion. The well-separated assumption is a model condition, not a fitted input. The only concern is Appendix E: Proposition V.2's proof moves from third-moment identifiability ([20, Thm II.1], a theorem from overlapping authors but with an independent proof) to a minimax lower bound without supplying the required Fisher-information or minimax argument, and it invokes the upper-bound Proposition III.4 in the same sentence. That is a missing-support/rigor gap in the 2D lower bound, not a circular reduction: the cited uniqueness theorem is not defined in terms of the target sample-complexity rate, and the 1D Theta(sigma^6) claim rests on independent MRA bounds. Hence no step is equivalent to its inputs by construction; score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. It relies on standard statistical asymptotics plus domain assumptions about placement, compactness, and uniqueness of autocorrelations. One ad hoc step appears in Appendix E, where the SO(2) lower bound is inferred from moment order without proof.

assumptions (6)
  • domain assumption High-noise asymptotic regime with fixed density γ = N/M < 1 and known noise variance σ^2.
    Section II.A states this asymptotic regime; all bounds are stated as σ, N, M → ∞ with γ fixed.
  • domain assumption Non-overlapping or well-separated placement of signal occurrences (separation ≥ L or ≥ 2L).
    Definition II.1; needed for the i.i.d. slice decomposition in Lemma A.1 and the empirical autocorrelation convergence in Proposition III.3.
  • domain assumption Compact parameter space Θ for the signal.
    Assumed in Proposition III.4 and Lemma A.7 to apply the likelihood estimation theorem [22, Thm 2.1].
  • domain assumption The autocorrelation ensemble mean up to order ā uniquely determines the orbit.
    Assumed in Proposition III.4; for the concrete cases it is imported from cited uniqueness theorems ([11, Thm 3.2], [7, Prop 4.1], [20, Thm II.1]).
  • ad hoc to paper For SO(2) MRA, the minimal moment order (third) being required for uniqueness implies a ω(σ^6) sample complexity lower bound.
    Used in Appendix E to prove Proposition V.2; no derivation is given in this paper.
  • domain assumption The group elements and noise are i.i.d. and independent of each other.
    Part of the MTD model (I.1) and used throughout the autocorrelation analysis.

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Cite this review

Pith. "Pith review of A note on the sample complexity of multi-target detection." pith.science (2026). https://pith.science/paper/UJDJZUWO

@misc{pith2026250111980,
  author       = {Pith},
  title        = {Pith review of: A note on the sample complexity of multi-target detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJDJZUWO}},
  note         = {Machine review of arXiv:2501.11980}
}
read the original abstract

This work studies the sample complexity of the multi-target detection (MTD) problem, which involves recovering a signal from a noisy measurement containing multiple instances of a target signal in unknown locations, each transformed by a random group element. This problem is primarily motivated by single-particle cryo-electron microscopy (cryo-EM), a groundbreaking technology for determining the structures of biological molecules. We establish upper and lower bounds for various MTD models in the high-noise regime as a function of the group, the distribution over the group, and the arrangement of signal occurrences within the measurement. The lower bounds are established through a reduction to the related multi-reference alignment problem, while the upper bounds are derived from explicit recovery algorithms utilizing autocorrelation analysis. These findings provide fundamental insights into estimation limits in noisy environments and lay the groundwork for extending this analysis to more complex applications, such as cryo-EM.

Figures

Figures reproduced from arXiv: 2501.11980 by the authors.

Figure 1
Figure 1. (a) Single-particle electron microscopy reconstructs 3D structures from 2D projections [6]. Particles in vitrified ice form a micrograph, modeled using the MTD framework (I.1), where xi = Π(gi · x), with gi a 3D rotation and Π as a tomographic projection. In high noise levels, direct particle detection is infeasible, but 3D reconstruction might be feasible by directly processing the micrographs [8]. (b) The estimati… view at source ↗

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Reference graph

Works this paper leans on

36 extracted references · 34 canonical work pages

  1. [4]

    Estimation under group actions: recovering orbits from invariants

    Afonso S Bandeira, Ben Blum-Smith, Joe Kileel, Jonathan Niles- Weed, Amelia Perry, and Alexander S Wein. Estimation under group actions: recovering orbits from invariants. Applied and Computational Harmonic Analysis , 66:236–319, 2023

    work page 2023

  2. [1]

    Multireference alignment is easier with an aperiodic translation distribution

    Emmanuel Abbe, Tamir Bendory, William Leeb, Jo ˜ao M Pereira, Nir Sharon, and Amit Singer. Multireference alignment is easier with an aperiodic translation distribution. IEEE Transactions on Information Theory, 65(6):3565–3584, 2018

    work page 2018

  3. [2]

    Fundamental limits in multi-image align- ment

    Cecilia Aguerrebere, Mauricio Delbracio, Alberto Bartesaghi, and Guillermo Sapiro. Fundamental limits in multi-image align- ment. IEEE Transactions on Signal Processing , 64(21):5707– 5722, 2016

    work page 2016

  4. [3]

    Einstein from noise: Statistical analysis

    Amnon Balanov, Wasim Huleihel, and Tamir Bendory. Einstein from noise: Statistical analysis. bioRxiv, pages 2024–07, 2024

    work page 2024

  5. [5]

    Multireference alignment using semidefinite programming

    Afonso S Bandeira, Moses Charikar, Amit Singer, and Andy Zhu. Multireference alignment using semidefinite programming. In Proceedings of the 5th conference on Innovations in theoretical computer science, pages 459–470, 2014

    work page 2014

  6. [6]

    Single- particle cryo-electron microscopy: Mathematical theory, compu- tational challenges, and opportunities

    Tamir Bendory, Alberto Bartesaghi, and Amit Singer. Single- particle cryo-electron microscopy: Mathematical theory, compu- tational challenges, and opportunities. IEEE signal processing magazine, 37(2):58–76, 2020

    work page 2020

  7. [7]

    Multi-target detection with application to cryo- electron microscopy

    Tamir Bendory, Nicolas Boumal, William Leeb, Eitan Levin, and Amit Singer. Multi-target detection with application to cryo- electron microscopy. Inverse Problems, 35(10):104003, 2019

    work page 2019

  8. [8]

    Toward single particle reconstruction without particle picking: Breaking the detection limit

    Tamir Bendory, Nicolas Boumal, William Leeb, Eitan Levin, and Amit Singer. Toward single particle reconstruction without particle picking: Breaking the detection limit. SIAM Journal on Imaging Sciences, 16(2):886–910, 2023

    work page 2023

Show all 36 references
  1. [9]

    A transversality theorem for semi-algebraic sets with application to signal recovery from the second moment and cryo-EM

    Tamir Bendory, Nadav Dym, Dan Edidin, and Arun Suresh. A transversality theorem for semi-algebraic sets with application to signal recovery from the second moment and cryo-EM. arXiv preprint arXiv:2405.04354, 2024

    2024 arXiv

  2. [10]

    Dihedral multi-reference alignment

    Tamir Bendory, Dan Edidin, William Leeb, and Nir Sharon. Dihedral multi-reference alignment. IEEE Transactions on In- formation Theory, 68(5):3489–3499, 2022

    2022

  3. [11]

    Multi-target detection with rotations

    Tamir Bendory, Ti-Yen Lan, Nicholas F Marshall, Iris Rukshin, and Amit Singer. Multi-target detection with rotations. Inverse problems and imaging (Springfield, Mo.) , 17(2):362, 2023

    2023

  4. [12]

    Detection and recovery of hidden submatrices

    Marom Dadon, Wasim Huleihel, and Tamir Bendory. Detection and recovery of hidden submatrices. IEEE Transactions on Signal and Information Processing over Networks , 2024

    2024

  5. [13]

    Generic orbit recovery from invariants of very low degree

    Dan Edidin and Josh Katz. Generic orbit recovery from invariants of very low degree. arXiv preprint arXiv:2408.09599 , 2024

    2024

  6. [14]

    The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of un- stained biological molecules

    Richard Henderson. The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of un- stained biological molecules. Quarterly reviews of biophysics , 28(2):171–193, 1995

    1995

  7. [15]

    Avoiding the pitfalls of single particle cryo- electron microscopy: Einstein from noise

    Richard Henderson. Avoiding the pitfalls of single particle cryo- electron microscopy: Einstein from noise. Proceedings of the National Academy of Sciences , 110(45):18037–18041, 2013

    2013

  8. [16]

    Two-dimensional multi- target detection: An autocorrelation analysis approach

    Shay Kreymer and Tamir Bendory. Two-dimensional multi- target detection: An autocorrelation analysis approach. IEEE Transactions on Signal Processing , 70:835–849, 2022

    2022

  9. [17]

    An approxi- mate expectation-maximization for two-dimensional multi-target detection

    Shay Kreymer, Amit Singer, and Tamir Bendory. An approxi- mate expectation-maximization for two-dimensional multi-target detection. IEEE signal processing letters , 29:1087–1091, 2022

    2022

  10. [18]

    A stochas- tic approximate expectation-maximization for structure deter- mination directly from cryo-EM micrographs

    Shay Kreymer, Amit Singer, and Tamir Bendory. A stochas- tic approximate expectation-maximization for structure deter- mination directly from cryo-EM micrographs. arXiv preprint arXiv:2303.02157, 2023

    2023

  11. [19]

    Multi-target detection with an arbitrary spacing distribution

    Ti-Yen Lan, Tamir Bendory, Nicolas Boumal, and Amit Singer. Multi-target detection with an arbitrary spacing distribution. IEEE Transactions on Signal Processing , 68:1589–1601, 2020

    2020

  12. [20]

    Heterogeneous multireference alignment for images with application to 2D classification in single particle reconstruc- tion

    Chao Ma, Tamir Bendory, Nicolas Boumal, Fred Sigworth, and Amit Singer. Heterogeneous multireference alignment for images with application to 2D classification in single particle reconstruc- tion. IEEE Transactions on Image Processing , 29:1699–1710, 2019

    2019

  13. [21]

    Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions

    Nicholas F Marshall, Oscar Mickelin, and Amit Singer. Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions. SIAM Journal on Scientific Computing , 45(5):A2431–A2457, 2023

    2023

  14. [22]

    Chapter 36 large sample estimation and hypothesis testing

    Whitney K Newey and Daniel McFadden. Chapter 36 large sample estimation and hypothesis testing. volume 4 of handbook of econometrics. Elsevier, 12:2111–2245, 1994

    1994

  15. [23]

    The development of cryo-EM into a mainstream structural biology technique

    Eva Nogales. The development of cryo-EM into a mainstream structural biology technique. Nature methods, 13(1):24–27, 2016

    2016

  16. [24]

    Multi- reference alignment in high dimensions: sample complexity and phase transition

    Elad Romanov, Tamir Bendory, and Or Ordentlich. Multi- reference alignment in high dimensions: sample complexity and phase transition. SIAM Journal on Mathematics of Data Science , 3(2):494–523, 2021

    2021

  17. [25]

    Theory of statistics

    Mark J Schervish. Theory of statistics . Springer Science & Business Media, 2012

    2012

  18. [26]

    Computational methods for single-particle electron cryomicroscopy

    Amit Singer and Fred J Sigworth. Computational methods for single-particle electron cryomicroscopy. Annual review of biomedical data science , 3:163–190, 2020. APPENDIX A. Proof of Proposition III.3 Let z ∈ RLM represent observations sampled from the well-separated MTD model (...

    2020

  19. [27]

    For every 0 ≤ k ≤ L − 1, E [F [r0 + k]] = E [A [k] A [k + ℓ1] ...A [k + ℓd−1]] . (A.26)

  20. [28]

    (A.27) Combining (A.26) and (A.27) with (A.5) and (A.6), and summing over −L ≤ k < Lcompletes the proof

    For every −L ≤ k <0, E [F [r0 + k]] = E [B [k] B [k + ℓ1] ...B [k + ℓd−1]] . (A.27) Combining (A.26) and (A.27) with (A.5) and (A.6), and summing over −L ≤ k < Lcompletes the proof. B. Proof of Proposition III.4 To establish this proposition, our objective is to show that the ...

  21. [29]

    Now, we must examine the convergence of the entire empirical autocorrelation tensor (in the Frobenius norm) to the ensemble mean

    In proposition III.3, we addressed the convergence of each entry in the empirical autocorrelation indi- vidually. Now, we must examine the convergence of the entire empirical autocorrelation tensor (in the Frobenius norm) to the ensemble mean

  22. [30]

    To achieve this, it is essential that some additional mild conditions would hold, as proved in Lemma A.7

    Even if the empirical autocorrelation converges to the ensemble mean with a sample complexity of ω σ2d , this does not guarantee that the orbit estimator converges at the same sample complexity to the orbit of the signal x. To achieve this, it is essential that some additional...

  23. [31]

    Identification - By the assumption that the orbit is uniquely defined by all the autocorrelations up to order ¯d, it means that ˆQM (z; G · x) is identifiable, that is, for every x ̸= x0, ˆQM (z; G · x) ̸= ˆQM (z; G · x0)

  24. [32]

    − − →Q0 (G · x) , (A.40) where Q0 (G · x) indicates the expected value of the right-hand-side of (A.38), given by ¯µ( ¯d) G·x,

    Convergence of ˆQM - By corollary A.6, for N = ω σ2 ¯d , as N → ∞, ˆQM (z; G · x) a.s. − − →Q0 (G · x) , (A.40) where Q0 (G · x) indicates the expected value of the right-hand-side of (A.38), given by ¯µ( ¯d) G·x, . . .¯µ(1) G·x , where ¯µ(d) G·x is defined by (A.29)

  25. [33]

    Continuity of ˆQM (z; G · x) in x - Since ˆQM is polynomial in x, it is also continuous in x

  26. [34]

    Then, by applying [22, Theorem 2.1], where all the condition of the theorem are satisfied, we obtain the result: [G · x P − →G · x

    Compactness - By assumption, the parameter space Θ is compact. Then, by applying [22, Theorem 2.1], where all the condition of the theorem are satisfied, we obtain the result: [G · x P − →G · x. (A.41) Since (A.40) holds for N = ω σ2 ¯d , (A.41) also holds under this condition...

  27. [35]

    Place the MRA instances {Zi}N −1 i=0 within the vec- tor Y in any valid non-overlapping configuration (Definition II.1)

  28. [36]

    Clearly, the map defined above is a measurable map and defines the non-overlapping MTD model in (I.1), according to any unknown locations {si}N −1 i=0

    In all the remaining indices of Y , which do not contain the instances {Zi}N −1 i=0 , add additive i.i.d random noise with variance σ2. Clearly, the map defined above is a measurable map and defines the non-overlapping MTD model in (I.1), according to any unknown locations {si...

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