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REVIEW 4 major objections 5 minor 1 cited by

Subset Multivariate Collective And Point Anomaly Detection

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single penalised statistic can locate collective anomalies in multivariate time series with a finite-sample accuracy guarantee.

desk verdict A genuine first for subset multivariate collective-anomaly consistency, but the theorem as stated covers only the known-baseline/specific-penalty version, and two load-bearing lemmas are unproved; still worth serious refereeing. read the letter →

arxiv 1909.01691 v1 pith:OUSTCIVN submitted 2019-09-04 stat.ME math.STstat.COstat.MLstat.TH

Alexander T M Fisch , Idris A Eckley , Paul Fearnhead This is my paper · ORCID
classification stat.MEmath.STstat.COstat.MLstat.TH
keywords multivariatetimeseriescollectiveanomaliesepidemicchangepointspointdynamicprogrammingcopynumbervariationpenalisedlikelihoodhigh-dimensionaldetection
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 argues that locating collective anomalies, intervals during which some components of a multivariate time series behave atypically, can be framed as maximising a penalised saving statistic, and that this maximisation is both computationally tractable and statistically consistent. The proposed method, MVCAPA, is designed to detect anomalies that affect only a few strong components as well as those that affect many weak components, and it can also report point anomalies and imperfectly aligned anomaly windows. The central finite-sample result states that, under a suitable penalty regime, the estimated number of anomalies equals the true number and every estimated window lies within $10C/\Delta_k^2$ of its true start and end, with error probability at most $A n^3 e^{-\psi}$. This matters because earlier multivariate epidemic changepoint methods came with no such consistency guarantee, and because the result allows the number of anomalies to grow with the sample size. A careful reader should note that the guarantee assumes the baseline parameter is known; the paper estimates it robustly in applications but supplies no theorem for that case.

What carries the argument

The central object is the penalised saving statistic $S(s,e)=\max_k\left(\sum_{i=1}^k S_{(i)}(s,e)-\beta_i\right)-\alpha$, where $S_{(i)}(s,e)$ are the ordered per-component log-likelihood savings from fitting a window $(s,e)$ as anomalous. This statistic is maximised over all windows by a pruned dynamic programme, which makes the maximisation exact rather than an approximation like binary segmentation. The argument is carried by the choice of penalties: for sparse anomalies the penalty grows like $C\psi+C|J|\log p$, and for dense anomalies like $p+C\psi+C\sqrt{p\psi}$, with the sparse-dense boundary at $k^*=p^{1/2}\psi/\log p$. The proof then controls a family of events that rule out false-positive windows, over-splitting, over-merging, and boundary errors, and combines them to prove the finite-sample consistency bound.

What would settle it

Simulate data under the null model with an estimated baseline: generate Gaussian series with $p$ components, estimate $\theta_0$ from the full no-anomaly data, run MVCAPA with the penalties from (8), and count the false-positive rate. If this rate exceeds $A n^3 e^{-\psi}$ once $\hat{\theta}_0$ is plugged in, the known-baseline condition is doing essential work. A deterministic check would be to compare the distribution of the saving statistic $S(s,e)$ under the estimated baseline against the $\chi^2$ tail bounds used in Propositions 1-3.

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Extended reading notes

Core claim

The core claim is that the penalised saving statistic, maximised jointly over all possible anomaly windows by a pruned dynamic programme, consistently estimates the number and location of subset multivariate collective anomalies in a Gaussian mean-shift model. Theorem 1 states that, with the penalty regime in (8), the MVCAPA partition $\hat{\tau}=(\hat{s}_1,\hat{e}_1,\hat{J}_1),\dots,(\hat{s}_{\hat{K}},\hat{e}_{\hat{K}},\hat{J}_{\hat{K}})$ satisfies $P(\hat{K}=K,\ |\hat{s}_k-s_k|<10C/\Delta_k^2,\ |\hat{e}_k-e_k|<10C/\Delta_k^2)>1-A n^3 e^{-\psi}$, provided each true window and the gaps around it are at least $40C/\Delta_k^2$ long. In the single-anomaly setting, Proposition 4 shows that the test has asymptotic power for every sparse or dense change that lies within the established detection boundary. The same machinery also detects point anomalies and allows start and end lags up to a maximum $w$, with computational cost that increases only linearly in $w$.

Load-bearing premise

The entire false-positive control and consistency theory assumes the baseline parameter $\theta_0$ (and $\sigma_0$ for point anomalies) is known, and the paper proves no theorem for the case where the baseline is estimated from data.

Editorial extensions

If this is right

  • If Theorem 1 is correct, MVCAPA's estimated partition converges to the true partition with window-boundary error at most $O(\Delta_k^{-2})$, and the error probability decays exponentially in the user-set parameter $\psi$.
  • The consistency result allows the number of anomalies $K$ to grow with $n$, so the method is not restricted to a fixed, small number of anomalous windows.
  • The single-window test reaches the optimal detection boundary for both sparse and dense anomalies, meaning no method can asymptotically do better in the large-$p$ regime described by the paper.
  • Allowing lags of up to $w$ increases power to detect shared anomalies that are not perfectly aligned across components, as illustrated by the copy-number-variation application.
  • With pruning, the exact dynamic-programming maximisation runs in close-to-linear time when the number of anomalies grows linearly with $n$, avoiding the approximations of binary-segmentation-like approaches.

Reading between the lines

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

  • A natural extension the authors leave implicit is a plug-in version where $\theta_0$ is estimated robustly before running MVCAPA; a testable prediction is that the false-positive rate under the estimated baseline deviates from $A n^3 e^{-\psi}$, and the deviation should grow with the estimation error of $\hat{\theta}_0$.
  • The penalty regime in (8) fixes $k^*$ through $\psi/\log p$; one could instead estimate $k^*$ from the data, for instance by running MVCAPA over a grid of $k^*$ values and choosing the partition with best cross-validated predictive fit.
  • The lagged-window extension suggests a connection to event-alignment problems in other multivariate applications, such as multi-sensor detection, where the same pruned dynamic programme could be applied with a component-specific lag penalty instead of a global $w$.
  • The finite-sample bound depends on $n^3$, so one concrete stress test is to check whether the bound is tight by simulating the worst-case configuration of many short anomalies separated by minimal gaps.
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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

4 major / 5 minor

Summary. The paper develops a penalised-saving statistic for detecting subset multivariate collective anomalies, where an anomalous window affects only a subset of components. It derives finite-sample false-positive bounds for three penalty regimes and a composite regime, analyzes power in a sparse/dense high-dimensional asymptotic regime, and introduces MVCAPA, a pruned dynamic programming algorithm that maximizes the penalised saving exactly. The main theoretical result, Theorem 1, claims that under Gaussian noise with known baseline and a specific penalty regime, MVCAPA consistently estimates the number and locations of anomalies up to O(Δ_k^{-2}) error with probability at least 1 - A n^3 e^{-ψ}. The paper also extends the methodology to lagged anomalies and point anomalies, compares MVCAPA with PASS and Inspect in simulations, and applies it to copy number variation data.

Significance. If Theorem 1 and the supporting propositions are fully established, this would be a valuable contribution: it would provide the first finite-sample consistency guarantee for subset multivariate collective anomaly detection with a growing number of anomalies, and the penalty constructions are derived from chi-square tail bounds rather than arbitrary tuning. The exact dynamic programming and pruning scheme is also a useful algorithmic contribution, and the availability of the R package anomaly supports reproducibility. However, the guarantees as stated are conditional on assumptions that do not match the implemented procedure: the baseline parameters are assumed known, the consistency theorem is proved for penalty (8) rather than the composite penalty used in the software, and two lemmas used in the proof of Theorem 1 are omitted. These gaps are load-bearing for the paper's headline claims, so the current version overstates what is formally established.

major comments (4)
  1. [Section 2.1 and Theorem 1] The theoretical framework assumes throughout that the baseline parameter θ0 (and σ0 for point anomalies) is known; Section 2.1 only suggests estimating it robustly 'as in [12]' without any guarantee. Theorem 1 and Propositions 1-5 are proved under this known-baseline assumption. The implemented MVCAPA, including the CNV application and the anomaly package, estimates the baseline from data. No result quantifies the effect of estimation error on the false-positive bound, the power claim, or the consistency bound, so the abstract's statement that MVCAPA is 'shown to consistently estimate' anomalies is not supported for the procedure actually recommended. Please either add a theorem covering estimated baselines or substantially weaken the claims.
  2. [Section 5 and Supplementary Material 10.1.8, 10.2.20, 10.2.21] The proof of Theorem 1 relies on Lemmas 20 and 21, which are used in the proofs of Propositions 14 and 15 and hence in establishing Property I and Proposition 9. The proofs of these two lemmas are omitted, with the text saying only that they are 'very similar' to the proofs of Lemmas 18 and 17. Because these lemmas are load-bearing for the central consistency theorem, the theorem as printed is not fully verifiable. Complete proofs should be provided.
  3. [Sections 2.2, 5, and 7] Theorem 1 is proved for the penalty regime specified in equation (8), which is only described as 'very similar' to the pointwise minimum of penalty regimes 1 and 2. The implemented and recommended procedure, however, uses the composite penalty regime that is the pointwise minimum of regimes 1, 2, and 3, with constants rescaled to achieve a user-specified false-positive rate; Proposition 4 also uses the composite regime. No theorem shows that the false-positive bound, the power result, or the consistency guarantee holds for this composite penalty. This creates a gap between the theoretical claims and the method evaluated in the simulations and application.
  4. [Section 2.3 and Proposition 4] Proposition 4 states power for sparse changes when r_p > ρ_- and for dense changes when r_p < (1/2)ρ_+. The proof of the dense case indeed establishes only r_p < (1/2)(1/2 - ξ). Since the preceding text identifies ρ_+ = 1/2 - ξ as the dense detectability boundary, the proposition does not establish power throughout the dense detection region: anomalies with r_p between (1/2)ρ_+ and ρ_+ are not covered. The claim of power 'at the optimal detection boundary' is therefore not supported as stated; please either extend the proof or revise the claim.
minor comments (5)
  1. [Section 2.2] There is a typo 'Under the the null model' in the paragraph defining M0.
  2. [Section 3.1] The sentence 'The penalised savings S′(·) and S(·), as we assume point anomalies to be sparse.' is incomplete; it should be reworded.
  3. [Section 4] In the subsection on collective and point anomalies, 'The saving in (6) can be minimised exactly' should read 'maximised exactly', since the algorithm maximizes the penalised saving.
  4. [Supplementary Material, Lemma 14] The statement of Lemma 14 uses j″ but the lemma statement does not define j″ before its use; please clarify the indexing.
  5. [Section 10.1.10] In the proof of Proposition 11, 'The third equality holds for large enough values of C' should presumably say 'inequality' rather than 'equality'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: penalty regimes are derived from null-distribution tail bounds, Theorem 1 is proved from explicit events, and self-citations to [12]/[16] are auxiliary rather than load-bearing.

full rationale

The paper's central derivations are self-contained rather than circular. Propositions 1-3 derive the three penalty regimes from tail bounds on chi-squared statistics under the null model M0, and prove false-positive control by Bonferroni correction over windows; the penalties are analytic choices, not fitted to the anomalies later estimated, and Propositions 4 and Theorem 1 do not assume the conclusions they prove. Theorem 1 is established through a self-contained argument: the proof defines the events E1-E11, proves in Lemma 7 that these events hold with probability 1 - An^3 e^{-psi}, and then uses Lemmas 8-22 to show the optimal partition lies in the set B_C; the penalty (8) is an explicit analytic construction and Delta_k^2 is a definition used in the theorem statement, not a fitted parameter. The self-citations to the authors' own work are not load-bearing: [12] is used for the point-anomaly construction and for suggesting robust baseline estimation, but the point-anomaly false-positive control is proved in Proposition 5, and the pruning statements are proved as Propositions 6 and 8 rather than being imported from [16]. The main caveats are matters of scope and verifiability, not circularity: Section 2.1 assumes theta_0 is known, while the application estimates it robustly as in [12], and Lemmas 20 and 21 in Sections 10.2.20-10.2.21 have proofs omitted as 'very similar' to earlier arguments. These limit the formal applicability of the theorem to the implemented procedure and make parts of the proof non-fully-verifiable from the manuscript, but they do not make any derivation equivalent to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The central claim rests on four user-set or unspecified numeric constants (penalty scaling, epsilon, C, psi), on the known-baseline assumption, on Gaussian noise, on the non-overlap and constant-shift model, and on a subadditivity condition for costs. These are stated, but the first four are not fully pinned down, which limits exact reproducibility and makes the theory partially non-constructive.

free parameters (4)
  • Penalty scaling constant = not reported (tuned to target false positive rate)
    Section 2.2 final paragraph: user-specified false positive rate achieved by scaling alpha and beta_i by a single constant tuned on anomaly-free simulations. This constant affects the operating point and is not provided.
  • epsilon in Penalty Regime 2 and 2' = unspecified small positive constant
    Section 2.2 Penalty Regime 2 and Section 6.2 Regime 2': epsilon > 0 is a constant in the penalty formulas and in the false positive bounds; its value is never stated, and it changes the penalty magnitude.
  • Universal constant C in Theorem 1 and penalty (8) = unspecified, existence asserted
    Section 5, Theorem 1 and Eq. (8): the theorem requires C to exceed a global constant but does not provide a value; the penalty regime (8) depends on C, so the result is non-constructive regarding this constant.
  • Penalty exponent psi = chosen as functions of n and p (e.g., 2 log n + 2 log log p in Proposition 4; log p in Theorem 1 context)
    Section 2.2 and 5: psi controls the false positive probability bound and appears in all penalty regimes; it is set by hand depending on the asymptotic setting.
assumptions (5)
  • domain assumption Baseline parameters theta_0 (and sigma_0) are known.
    Section 2.1 states theta_0 is known and, if not, an approximation can be obtained by robust estimation as in [12]; all theoretical false positive and consistency guarantees are derived under known theta_0, so estimated theta_0 is not covered.
  • domain assumption Observations are i.i.d. Gaussian white noise under the null, with known mean 0 and variance 1.
    Section 2.2 assumes M0: x_t,c ~ N(0,1) i.i.d.; Theorem 1 uses eta_t ~ N(0, I_p). The authors say results extend to sub-Gaussian noise for Theorem 1 (Section 5) but do not prove it there.
  • domain assumption Anomalous windows do not overlap, and each affected component within a window has a constant mean shift.
    Model (1) in Section 2.1 imposes e_k <= s_{k+1}; Theorem 1 model (7) assumes the same mean mu_k for all affected components within an anomaly. Real anomalies may not follow this.
  • domain assumption Cost functions satisfy subadditivity min_theta(sum_{a+1}^c Ci) >= min_theta(sum_{a+1}^b Ci) + min_theta(sum_{b+1}^c Ci) for the pruning propositions.
    Proposition 6 and 8 require this condition; it holds for negative log-likelihood and sum of squares, but the algorithm's pruning guarantees are conditional on it.
  • standard math Tail bounds for chi-squared and normal distributions from [17] and [4] are valid.
    Used in proofs of Propositions 1-3 and Lemma 1.

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Cite this review

Pith. "Pith review of Subset Multivariate Collective And Point Anomaly Detection." pith.science (2026). https://pith.science/paper/OUSTCIVN

@misc{pith2026190901691,
  author       = {Pith},
  title        = {Pith review of: Subset Multivariate Collective And Point Anomaly Detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUSTCIVN}},
  note         = {Machine review of arXiv:1909.01691}
}
read the original abstract

In recent years, there has been a growing interest in identifying anomalous structure within multivariate data streams. We consider the problem of detecting collective anomalies, corresponding to intervals where one or more of the data streams behaves anomalously. We first develop a test for a single collective anomaly that has power to simultaneously detect anomalies that are either rare, that is affecting few data streams, or common. We then show how to detect multiple anomalies in a way that is computationally efficient but avoids the approximations inherent in binary segmentation-like approaches. This approach, which we call MVCAPA, is shown to consistently estimate the number and location of the collective anomalies, a property that has not previously been shown for competing methods. MVCAPA can be made robust to point anomalies and can allow for the anomalies to be imperfectly aligned. We show the practical usefulness of allowing for imperfect alignments through a resulting increase in power to detect regions of copy number variation.

Figures

Figures reproduced from arXiv: 1909.01691 by the authors.

Figure 1
Figure 1. An example time series with K = 2 collective anomalies, highlighted in dark blue. Using the notation from Section 2.1 these collective anomalies occur between times s1 = 150 to e1 = 200 and s2 = 350 to e2 = 400; and the affected components are J1 = {1, 2, 3} and J2 = {3, 4}. Markov chain, determines whether the time series’ behaviour is typical or anomalous. Examples of state space models for anomaly detection can b… view at source ↗
Figure 2
Figure 2. A comparison of the 3 penalty regimes for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Example series and ROC curves for setting 1. MVCAPA is in red, PASS in green, and Inspect in blue. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Example series and ROC curves for setting 3. MVCAPA is in red, PASS in green, and Inspect in blue. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Example series and ROC curves for setting 4. MVCAPA is in red, PASS in green, and Inspect in blue. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Precision of true positives detected by all methods measured in mean absolute distance for MVCAPA, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: A comparison of run-times for MVCAPA (red squared), PASS (green circles), and Inspect (blue [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: A comparison between PASS, MVCAPA without lags, and MVCAPA with a lag of up to 40 for two [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Example series and ROC curves for setting 2. MVCAPA is in red, PASS in green, and Inspect in blue. [PITH_FULL_IMAGE:figures/full_fig_p051_9.png]

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  28. [37]

    if |Jk|≤ k∗ C (xi:j, Jk)≥C ( x(sk+1):j, Jk ) + 6C(ψ + log(p)) if the kth anomalous window is dense; i.e

    if the kth anomalous window is sparse; i.e. if |Jk|≤ k∗ C (xi:j, Jk)≥C ( x(sk+1):j, Jk ) + 6C(ψ + log(p)) if the kth anomalous window is dense; i.e. if |Jk|>k∗ C (xi:j, 1)≥C ( x(sk+1):j, 1 ) + 6C(ψ + √ pψ) providedC exceeds some global constant and the event E holds. Lemma 13....

  29. [38]

    sk <i ≤j≤sk+1 and min(ek−i + 1,j−ek)≥ 10C ∆2 k

  30. [39]

    Then the corresponding holds for all J given E 1

    ek−1 <i ≤j≤ek and min(sk−i + 1,j−sk)≥ 10C ∆2 k . Then the corresponding holds for all J given E 1. C (xi:j, J)≥C (xi:ek, 1) + 4C(ψ + √ pψ) 2. C (xi:j, J)≥C ( x(sk+1):j, 1 ) + 4C(ψ + √ pψ) providedC exceeds some global constant and the event E holds. For Part II, we will requir...

  31. [40]

    We need to show that the optimality of τ implies that each true anomalous segment overlaps with at least one fitted segment in τ

  32. [41]

    We prove both statements by contradiction: First assume that τ is optimal but that there exists a k such that [sk + 1,ek] is not covered at all by any fitted segment inτ

    We need to show that the optimality of τ implies that each true anomalous segment overlaps with at most one fitted segment in τ. We prove both statements by contradiction: First assume that τ is optimal but that there exists a k such that [sk + 1,ek] is not covered at all by an...

  33. [42]

    Lemma 14, implies that it is sufficient to show that C(xa′:b, 1)−C (xa:b, J1)≤ 5 40C ( ψ + √ pψ ) and C(xc:d′, 1)−C (xc:d, J2)≤ 5 40C ( ψ + √ pψ )

    J k is dense : We will show that replacing ( a,b, J1), (c,d, J2) with (a′,d′, 1) reduces the penalised cost. Lemma 14, implies that it is sufficient to show that C(xa′:b, 1)−C (xa:b, J1)≤ 5 40C ( ψ + √ pψ ) and C(xc:d′, 1)−C (xc:d, J2)≤ 5 40C ( ψ + √ pψ ) . We limit ourselves to...

  34. [43]

    Lemma 14, implies that it is sufficient to show that C(xa′:b, Jk)−C (xa:b, J1)≤ 5 40C (ψ +|Jk| log(p)) and C(xc:d′, Jk)−C (xc:d, J2)≤ 5 40C (ψ +|Jk| log(p))

    J k is sparse : We will show that replacing ( a,b, J1), (c,d, J2) with (a′,d′, Jk) reduces the penalised cost. Lemma 14, implies that it is sufficient to show that C(xa′:b, Jk)−C (xa:b, J1)≤ 5 40C (ψ +|Jk| log(p)) and C(xc:d′, Jk)−C (xc:d, J2)≤ 5 40C (ψ +|Jk| log(p)). These proo...

  35. [44]

    No further transformation is required in this case, i.e

    |τ′ k∩(A1∪A 2)| = 0, which happens whenτ′ does not contain a short fitted segment at either the beginning or the end of the kth anomalous region. No further transformation is required in this case, i.e. τ′′ k =τ′ k

  36. [45]

    |τ′ k∩ (A1∪A 2)| = 1

  37. [46]

    We will only explicitly describe the transformation for the second case, as applying it twice yields a transfor- mation for the third case

    |τ′ k∩ (A1∪A 2)| = 2. We will only explicitly describe the transformation for the second case, as applying it twice yields a transfor- mation for the third case. Without loss of generality we further assume that τ′ k∩ (A1∪A 2) = (s,ek, J), i.e. that the short fitted segment lie...

  38. [47]

    No such segment can overlap a true fitted dense change, the k′th say, by more than 10C ∆′2 k as this would contradict the optimality of τ

  39. [48]

    All other fitted segments, overlapping with at least two anomalous regions, including, potentially, a certain number of sparse changes by more that10C ∆ can be replaced by fitted segments each overlapping with exactly one true anomalous segment in a way which strictly bounds the...

  40. [49]

    Otherwise, the interval [ s + 1,e ] would also contain at least 10C ∆′2 k observations belonging to the typical distribution

    First of all we can show that the optimality of τ implies that no partition (s,e, J)∈τ\T 2 can overlap a dense change (the k′th change say) by more than 10C ∆′2 k . Otherwise, the interval [ s + 1,e ] would also contain at least 10C ∆′2 k observations belonging to the typical ...

  41. [50]

    For this segment define the set of true anomalous segments it overlaps by more than 10C ∆2 k to be De,s := { k :|[s + 1,e ]∩ [sk + 1,ek + 1]|≥ 10C ∆2 k }

    Consider now, a segment (s,e, J)∈τ\T 2 not overlapping with any dense changes by more than 10C ∆2 k . For this segment define the set of true anomalous segments it overlaps by more than 10C ∆2 k to be De,s := { k :|[s + 1,e ]∩ [sk + 1,ek + 1]|≥ 10C ∆2 k } . and note that|Jk| is...

  42. [51]

    The beginning of the fitted segment ( s,e, J)∈ τ\T 2 overlaps with a true anomalous region [ sk′ + 1,ek′], but does so by less than 10C ∆2 k′ . i.e. ∃k′ :ek′− 10C ∆2 k′ ≤s + 1≤ek′

  43. [52]

    The end of the fitted segment ( s,e, J)∈ τ\T 2 overlaps with a true anomalous region [ sk′′ + 1,ek′′], but does so by less than 10C ∆2 k′′ . i.e. ∃k′′ :sk′′ + 1 + 10C ∆2 k′′ ≥e≥sk′′ + 1

  44. [53]

    None of 1 and 2 apply. Note that this allows for the beginning and or the end of ( s,e, J)∈τ\T 2 to lie in a truly anomalous region provided the overlap with that region exceeds the critical threshold of 10C ∆2 . We then replace (s,e, J) in τ to obtain a new partition ˜τ. depe...

  45. [55]

    11 20C (ψ +|J| log(p)) if J is sparse or 11 20C ( ψ +√pψ ) if J is dense

  46. [56]

    22 20C (ψ +|J| log(p)) if J is sparse or 22 20C ( ψ +√pψ ) if J is dense

  47. [57]

    The fourth case follows directly from the following Lemma: Lemma 22

    0 depending on the case in order to prove the proposition. The fourth case follows directly from the following Lemma: Lemma 22. Let the event E hold and C exceed some global constant. Let s and e be such the fourth scenario applies, i.e

  48. [58]

    ∄k′ :ek′− 10C ∆2 k′ ≤s + 1≤ek′

  49. [59]

    This Lemma can also be used to bound the increase in penalised cost obtained for the other three cases

    ∄k′′ :sk′′ + 1 + 10C ∆2 k′′ ≥e≥sk′′ + 1 Then, the following holds true for all sparse J C (xs,e, J)≥ 19 20C (ψ +|J| log(p)) + ∑ k∈Ds,e ( C ( x(sk+1):ek, Jk )) Moreover, the following statement is also true: C (xs,e, 1)≥ 19 20C ( ψ + √ pψ ) + ∑ k∈Ds,e ( C ( x(sk+1):ek, Jk )) . ...

  50. [60]

    We consider two cases separately: Case 1: µ−(1−a)−2λaP ( χ2 1 <a ) ≥ 0. The lower bound in 17 shows that d dλ ( log ( E ( eλZ))) −µ is bounded by 1 1− 2λ [ µ− (1−a)− 2λaP ( χ2 1 <a ) P (χ2 1 <a ) + 1√1−2λ P (χ2 1 >a ) + (1−a−µ) + 2(µ +a)λ ] ≤ 1 1− 2λ [ µ− (1−a)− 2λaP ( χ2 1 <a...

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Reviewed August 14, 2026 · model on record in the stance chip above.