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arxiv: 2509.12889 · v4 · pith:GR7H5OT3new · submitted 2025-09-16 · 🧮 math.ST · stat.ML· stat.TH

Gaussian Mixture Model with unknown diagonal covariances via continuous sparse regularization

Romane Giard (ECL , ICJ , PSPM) , Yohann de Castro (ICJ , ECL , PSPM , IUF) , Cl\'ement Marteau (PSPM
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UCBL)
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Pith reviewed 2026-05-18 16:50 UTC · model grok-4.3

classification 🧮 math.ST stat.MLstat.TH
keywords Gaussian mixture modelBeurling-LASSOsparse regularizationdiagonal covariancenon-asymptotic recoverydual certificatesFisher-Rao geometryseparation condition
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The pith

The Beurling-LASSO framework recovers Gaussian mixture model parameters with unknown per-component diagonal covariances at nearly parametric rates when components are sufficiently separated.

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

This paper sets out to establish that convex sparse regularization via the Beurling-LASSO can jointly recover the unknown number of components, their means, individual diagonal covariance matrices, and weights from i.i.d. samples of a multivariate Gaussian mixture. The setting is more general than earlier work because each component is allowed its own unknown diagonal covariance instead of requiring them to be known or identical. If the claims hold, the method delivers non-asymptotic statistical guarantees with convergence rates close to the parametric optimum for both parameter estimation and density prediction. The proofs rest on constructing non-degenerate dual certificates under an explicit separation condition between components, using the Fisher-Rao geometry of the model together with a new semi-distance.

Core claim

Extending the Beurling-LASSO to multivariate Gaussian mixtures whose components have distinct unknown diagonal covariance matrices yields non-asymptotic recovery guarantees that achieve nearly parametric convergence rates for the component means, the diagonal covariance entries, the mixture weights, and the density estimator, once the components satisfy an explicit separation condition that permits the construction of non-degenerate dual certificates; the analysis is carried out in the Fisher-Rao geometry and introduces a novel semi-distance adapted to the setting.

What carries the argument

The Beurling-LASSO convex program that promotes sparsity over the space of measures, extended to component-specific diagonal covariances by means of a separation condition that enables non-degenerate dual certificates.

If this is right

  • The number of mixture components is recovered simultaneously with their parameters without being supplied in advance.
  • Nearly parametric rates hold for estimating means, diagonal covariance entries, and weights.
  • Density prediction inherits the same non-asymptotic guarantees.
  • The framework applies to multivariate data with flexible per-component covariance structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation-based analysis may extend to other exponential-family mixture models if analogous dual-certificate constructions can be found.
  • Practical performance could be checked by applying the estimator to moderately separated components in real or simulated high-dimensional data.
  • The novel semi-distance may prove useful for analyzing recovery in related sparse-measure estimation problems beyond Gaussian mixtures.

Load-bearing premise

The mixture components must satisfy an explicit separation condition that permits construction of non-degenerate dual certificates.

What would settle it

Generate samples from a two-component mixture whose means or covariances are closer than the paper's separation threshold and verify whether the BLASSO estimator recovers the correct number of components or the claimed rates.

Figures

Figures reproduced from arXiv: 2509.12889 by Cl\'ement Marteau (PSPM, ECL, ICJ, IUF), PSPM, PSPM), Romane Giard (ECL, UCBL), Yohann de Castro (ICJ.

Figure 1
Figure 1. Figure 1: Schematic representation for Gaussian mixture models in d [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

This paper addresses the statistical estimation of Gaussian Mixture Models (GMMs) with unknown diagonal covariances from independent and identically distributed samples. We employ the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures, to simultaneously estimate the number of components and their parameters. Our main contribution extends the BLASSO methodology to multivariate GMMs with component-specific unknown diagonal covariance matrices. This setting is significantly more flexible than previous approaches, which required known and identical covariances. We establish non-asymptotic recovery guarantees with nearly parametric convergence rates for component means, diagonal covariances, and weights, as well as for density prediction. A key theoretical contribution is the identification of an explicit separation condition on mixture components that enables the construction of non-degenerate dual certificates-essential tools for establishing statistical guarantees for the BLASSO. Our analysis leverages the Fisher-Rao geometry of the statistical model and introduces a novel semi-distance adapted to our framework, providing new insights into the interplay between component separation, parameter space geometry, and achievable statistical recovery.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a Beurling-LASSO (BLASSO) convex optimization framework for estimating the parameters of a multivariate Gaussian mixture model with unknown, component-specific diagonal covariance matrices from i.i.d. samples. It simultaneously recovers the number of components and their means, diagonal covariances, and weights. The central theoretical contribution is a set of non-asymptotic recovery guarantees with nearly parametric convergence rates for the means, covariances, weights, and the estimated density, obtained under an explicit separation condition on the mixture components that permits construction of non-degenerate dual certificates. The analysis relies on the Fisher-Rao geometry of the model and introduces a novel semi-distance adapted to the setting.

Significance. If the guarantees can be established rigorously, the work meaningfully extends prior BLASSO analyses (which required known or identical covariances) to a substantially more flexible GMM setting. The nearly parametric rates and the explicit separation condition for dual-certificate construction would constitute a solid theoretical advance. The introduction of a novel semi-distance in Fisher-Rao geometry is a genuine strength that could inform future geometric analyses of mixture estimation.

major comments (2)
  1. [§3] §3 (Main Results) and the separation condition stated in the abstract: the condition is formulated primarily in terms of a mean-based semi-distance. Because the local Fisher information metric scales with each component’s diagonal covariance entries, it is possible to satisfy the stated separation while still admitting a small perturbation that trades a mean shift against a covariance adjustment in one coordinate. Such a perturbation can keep the BLASSO objective nearly flat, causing the dual-certificate norm to reach or exceed 1. This directly threatens the exact-recovery and rate arguments for the covariance parameters in particular. An additional covariance-separation requirement or a covariance-aware redefinition of the semi-distance appears necessary to guarantee non-degenerate certificates.
  2. [§4] Proof outline for the dual certificate (presumably §4 or the appendix): the construction is described at a high level but lacks an explicit verification that the certificate norm remains strictly below 1 uniformly over the range of admissible diagonal covariances. Without this quantitative bound, the non-asymptotic nearly-parametric rates claimed for the covariance estimators cannot be verified from the given argument.
minor comments (2)
  1. [Introduction] The notation for the novel semi-distance is introduced without a compact symbol; adopting a single symbol (e.g., d_FR) would improve readability when it is used repeatedly in the rate statements.
  2. [§2] A few typographical inconsistencies appear in the display of the BLASSO objective (missing parentheses around the integral term in one displayed equation).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We are pleased that the significance of extending BLASSO to GMMs with unknown diagonal covariances is recognized. Below, we address the major comments point by point.

read point-by-point responses

  1. Referee: [§3] §3 (Main Results) and the separation condition stated in the abstract: the condition is formulated primarily in terms of a mean-based semi-distance. Because the local Fisher information metric scales with each component’s diagonal covariance entries, it is possible to satisfy the stated separation while still admitting a small perturbation that trades a mean shift against a covariance adjustment in one coordinate. Such a perturbation can keep the BLASSO objective nearly flat, causing the dual-certificate norm to reach or exceed 1. This directly threatens the exact-recovery and rate arguments for the covariance parameters in particular. An additional covariance-separation requirement or a covariance-aware redefinition of the semi-distance appears necessary to guarantee non-degenerate certificates.

    Authors: We appreciate this careful analysis of the separation condition. The semi-distance introduced in our work is explicitly constructed using the Fisher-Rao geometry of the Gaussian family with diagonal covariances. This geometry naturally incorporates the scaling of the local information metric by the covariance parameters. Consequently, the separation condition is not purely mean-based but accounts for the covariance structure through the metric. We will revise the manuscript to clarify this point in §3 and the abstract, providing a more explicit statement that the semi-distance is covariance-aware and explaining why mean-covariance trade-offs are controlled by the condition. If the referee's concern persists after clarification, we are open to adding an explicit covariance separation term. revision: partial

  2. Referee: [§4] Proof outline for the dual certificate (presumably §4 or the appendix): the construction is described at a high level but lacks an explicit verification that the certificate norm remains strictly below 1 uniformly over the range of admissible diagonal covariances. Without this quantitative bound, the non-asymptotic nearly-parametric rates claimed for the covariance estimators cannot be verified from the given argument.

    Authors: We acknowledge that the current presentation of the dual certificate construction in the proof outline is at a high level. In the revised version, we will expand the proof to include an explicit quantitative bound showing that the certificate norm is strictly less than 1, uniformly over the admissible range of diagonal covariance matrices. This will be detailed in the appendix or §4, ensuring the non-asymptotic rates for the covariance estimators are fully verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under stated separation condition

full rationale

The paper derives non-asymptotic recovery guarantees for GMM parameters and density via BLASSO, relying on an explicit separation condition to construct non-degenerate dual certificates in Fisher-Rao geometry with a novel semi-distance. This is a standard conditional theorem structure: the separation is an assumption, not defined in terms of the recovery rates or dual norms it enables. No equations reduce predictions to fitted inputs by construction, no load-bearing self-citations are invoked for uniqueness or ansatz, and the central claims remain independent of the target results. The analysis is self-contained against external benchmarks once the separation holds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard convex optimization theory for measures, the Fisher-Rao information geometry of the Gaussian family, and the existence of a separation condition that produces non-degenerate dual certificates. No free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • standard math Standard results from convex analysis and sparse regularization in the space of measures (Beurling-LASSO framework)
    Invoked to promote sparsity and obtain recovery guarantees
  • domain assumption Fisher-Rao geometry on the statistical manifold of Gaussians with diagonal covariances
    Used to define the novel semi-distance between mixture components

pith-pipeline@v0.9.0 · 5748 in / 1326 out tokens · 46825 ms · 2026-05-18T16:50:28.265660+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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    Statistical gua rantees for the EM algorithm: From population to sample-based analysis

    doi: 10.1214/aop/1176989128. Balakrishnan S., Wainwright M. J., and Yu B. (2017). “Statistical gua rantees for the EM algorithm: From population to sample-based analysis”. The Annals of Statistics 45.1, pp. 77–120. doi: 10.1214/16-AOS1435. Beardon A. F. (1983). The geometry of discrete groups . Springer. doi: 10.1007/978-1-4612-1146-4 . Bouveyron C., Cele...

    work page doi:10.1214/aop/1176989128 2017

  2. [2]

    So using Cauchy-Schwarz inequality, (

    As η = Ψ ∗p, recalling ( 10) we have ∫ X η d(ˆµ n,ω − µ 0 ω ) = ⟨ p,L ◦ Φ(ˆµ n − µ 0) ⟩ L . So using Cauchy-Schwarz inequality, (

    work page

  3. [3]

    leads to   L ˆfn − L ◦ Φˆµ n    2 L + 2κD η ( ˆµ n,ω ,µ 0 ω ) − 2κ ∥p∥L  L ◦ Φµ 0 − L ◦ Φˆµ n   L ≤ ∥ Γn∥2 L . The triangle inequality  L ◦ Φµ 0 − L ◦ Φˆµ n   L ≤ ∥ Γn∥L +   L ˆfn − L ◦ Φˆµ n    L then leads to (  L ◦ ˆfn − L ◦ Φˆµ n    L − κ ∥p∥L )2 + 2κD η ( ˆµ n,ω ,µ 0 ω ) ≤ (∥Γn∥L +κ ∥p∥L)2 . As the Bregman divergence is posi...

    work page

  4. [4]

    To control the mass of the estimator on the jth near region, we make use of the local no n-degenerate certificate ηj (Definition 3.3)

    and ( 31), we deduce the bound for the far region (item 1 of Theorem 3.1). To control the mass of the estimator on the jth near region, we make use of the local no n-degenerate certificate ηj (Definition 3.3). We have, for all j = 1,...,s , |ω 0 j − ˆµ n,ω (X near j (r))|= ⏐ ⏐ ⏐ ⏐ ⏐ω 0 j − ∫ ηj dˆµ n,ω + ∫ ηj dˆµ n,ω − ∫ X near j (r) dˆµ n,ω ⏐ ⏐ ⏐ ⏐ ⏐ , ≤ ⏐...

    work page

  5. [5]

    Using again ( 31), we deduce that |ω 0 j − ˆµ n,ω (X near j (re))| ≤ ∥pj∥L (2 ∥Γn∥L + 2κ ∥p∥L) + max { 1 − ˜ε0 ε0 , 1 ε2 ( ˜ε3 r2e + ˜ε2 )} Dη (ˆµ n,ω ,µ 0 ω )

    we get 1 ≤ ˜ε3 r2e dg(x0 j,x )2 for all x ∈ X near j (r) \ Xnear j (re), so 1 + ˜ε2dg(x0 j,x )2 ≤ ( ˜ε3 r2e + ˜ε2 ) dg(x0 j,x )2. Using again ( 31), we deduce that |ω 0 j − ˆµ n,ω (X near j (re))| ≤ ∥pj∥L (2 ∥Γn∥L + 2κ ∥p∥L) + max { 1 − ˜ε0 ε0 , 1 ε2 ( ˜ε3 r2e + ˜ε2 )} Dη (ˆµ n,ω ,µ 0 ω ). (37) We can conclude the proof using the controls on E [∥Γn∥L], ∥p...

    work page

  6. [6]

    F.2 Proof of Corollary 3.1 Let 0 <r e ≤ r

    we get E [ |ω 0 j − ˆµ n,ω (X near j (re))| ] ≤ 4√ cpρn + max { 1 − ˜ε0 ε0 , 1 ε2 ( ˜ε3 r2e + ˜ε2 )} 2ρn √ cps, ≲ √ s τd/ 2√ nr2e (38) keeping only the dependence on s,τ,r e,n . F.2 Proof of Corollary 3.1 Let 0 <r e ≤ r. We first prove that ⏐ ⏐ ⏐ ⏐ ˆµ n,ω W (X near j (re)) − a0 j ⏐ ⏐ ⏐ ⏐ ≤ (1 +H(r)re)W (x0 j )− 1 ⏐ ⏐ω 0 j − ˆµ n,ω (X near j (re)) ⏐ ⏐ +a0 j...

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  7. [7]

    we get ⏐ ⏐ ⏐ ⏐ ˆµ n,ω W (X near j (re)) − a0 j ⏐ ⏐ ⏐ ⏐ ≤ (er2 e + √ e2r2e − 1)d/ 2W (x0 j )− 1|ω 0 j − ˆµ n,ω (X near j (re))|+a0 j ( (er2 e + √ e2r2e − 1)d/ 2 − 1 ) . We conclude the proof of ( 39) by noticing that h :re ∈ R+ ↦→ (er2 e + √ e2r2e − 1)d/ 2 is convex (for all d ∈ N∗ ), hence for re ≤ r we have (er2 e + √ e2r2 e − 1)d/ 2 ≤ h(0) + h(r) − h(0)...

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  8. [8]

    with Lemmas G.1 and 3.1. We get E [ Φˆµ n − Φµ 0 2 L2(Rd) ] ≤ eE [ L ◦ Φ(ˆµ n − µ 0)  2 L ] + 2 (2π )d ( E [ ∥ˆµ n∥2 TV ] +  µ 0 2 TV ) τddd/ 2 2d/ 2u2d min e− 2 u2 min τ 2 , ≤ 4e(ρn +κ√ cps)2 + ((2π )d/ 2(2u2 max +τ2)d/ 2 ˜CΓρ4 n κ 2 + 2 ( 2 (umax umin )d + 1 )  µ 0 2 TV )(2π )− dτddd/ 2 2d/ 2u2d min e− 2 u2 min τ 2 , ≤ 4e(ρn +κ√ cps)2 + ...

    work page 2025

  9. [9]

    So as tanh is 1-Lipschitz, dg(x,x 0) = l ≥ |c1||t − t0|, ≥ re √ w√ 1 + r2 2 + 1− w w e2r2 − 1 2r2

    and ( 52), 1 |c1||t − t0| = √ 1 2 ( − 1 + u2 0 − u2 (t − t0)2 )2 + 2u2 0 +τ2 (t − t0)2 , = √ 1 2 + 1 2 (u2 0 − u2)2 (t − t0)4 + u2 0 +u2 +τ2 (t − t0)2 , (54) ≤ √ 1 2 + e2(1− w)r2e − 1 2w2r4e + 1 wr2e , ≤ 1√ wre √ r2 2 + 1 − w w e2r2 − 1 2r2 + 1 where we used that y ↦→ e2y − 1 is convex along with (1 − w)r2 e ≤ r2, and wr2 e ≤ r2. So as tanh is 1-Lipschitz...

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  10. [10]

    that cosh (c2 2 ) cosh (c2 2 + √ 2l )≥ e(1− w)r2 e + √ e2(1− w)r2 e − 1. As cosh (c2 2 ) cosh (c2 2 + √ 2l )= cosh( √ 2l) − sinh (c2 2 ) sinh( √ 2l) cosh (c2 2 ) ≤ cosh( √ 2l) + |sinh( √ 2l)|=e| √ 2l|, it comes that e √ 2l ≥ e(1− w)r2 e + √ e2(1− w)r2e − 1 leading again to l ≥ 1√ 2 ln ( e(1− w)r2 e + √ e2(1− w)r2e − 1 ) . As 1√ 2 ln ( ey2 + √ e2y2 − 1 ) ≥...

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  11. [11]

    and recalling that u0 ≤ u, we have 1 ≤ cosh (c2 2 ) cosh (c2 2 + √ 2l )= √ 2u2 +τ2 2u2 0 +τ2 ≤ er2 e + √ e2r2 e − 1. (59) Using ( 59) along with the equality cosh(A+B) cosh(A) = cosh(B) + tanh(A) sinh(B), we deduce that cosh (√ 2l ) + tanh (c2 2 ) sinh (√ 2l ) ≤ 1 and that cosh (√ 2l ) − tanh (c2 2 + √ 2l ) sinh (√ 2l ) ≤ er2 e + √ e2r2e − 1. It comes 2 c...

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  12. [12]

    So if d (x,x 0)< √ 2, dg(x,x 0)2 ≤ F (d (x,x 0)) with F :y ∈ R+ ↦→ 1 2 arcosh   1 2 ( ey2 + √ e2y2 − 1 + 1 ) 1 − 1√ 2y   2

    As 1 − 1√ 2re ≤ 1 and √ e2r2e − 1 ≥ er2 e − 1, the bound found for the first case (see ( 60)) is larger than the one found for the second case (see ( 61)). So if d (x,x 0)< √ 2, dg(x,x 0)2 ≤ F (d (x,x 0)) with F :y ∈ R+ ↦→ 1 2 arcosh   1 2 ( ey2 + √ e2y2 − 1 + 1 ) 1 − 1√ 2y   2 . (62) Lemma H.7 (Local control of dg with the semi-distance, upper bound, ...

    work page

  13. [13]

    H.4 Proof of Lemma 5.4 We use Lemma H.5

    As for all k = 1,...,d , d (xk,x 0,k ) ≤ d (x,x 0), by Lemma H.6 it comes that dg(x,x 0)2 = d∑ k=1 dg(xk,x 0,k )2 ≤ d∑ k=1 F (d (xk,x 0,k )), ≤ dF (d (x,x 0)) where we used that F is non-decreasing. H.4 Proof of Lemma 5.4 We use Lemma H.5. It comes that we can choose ˜ε3 = 1 + 1 R (

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  14. [14]

    We aim to find a more interpretable parameter

    3025√ d ) withR defined by ( 57) in the appendix. We aim to find a more interpretable parameter. Remark that r ∈ R+ ↦→ e2r2 − 1 r2 is increasing, along with e2r2 − 1 r2 ≥ 2. We deduce that for all 0 < r ≤ 0. 3025, R(r) ≥ − ( 1+ 0. 30252 2 ) + √ 5 e2× 0. 30252 − 1

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  15. [15]

    So 1 + 1 R (

    30252 . So 1 + 1 R (

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  16. [16]

    84 (see [Giard, 2025, Section III])

    3025√ d ) ≤ 2. 84 (see [Giard, 2025, Section III]). I Proof of Theorem 5.2 Construction of the certificates by solving a linear system We give explicit formulas for η, ηj of Theorem 5.2. These certificates are of the form ( 18). In the following, we write η = ηα,β and ηj = ηα j ,β j . Recalling Definitions 3.2 and 3.3, we want that for all j = 1,...,s ,ηα,β ...

    work page 2025

  17. [17]

    g − 1 2 x0 s         ∈ Rs(d+1)× s(d+1), ˜Υ := DgΥDg, using the results of [Poon et al., 2023, p

    and defining Dg :=         Ids g − 1 2 x0 1 . . . g − 1 2 x0 s         ∈ Rs(d+1)× s(d+1), ˜Υ := DgΥDg, using the results of [Poon et al., 2023, p. 268] we have ∥pα,β ∥2 L =   α β   T Υ   α β   = uT s ˜Υ− 1us. We can apply [Poon et al., 2023, Lemma 3] that gives    ˜Υ− 1    2 ≤ 2. So ∥pα,β ∥2 L ≤    ˜Υ− 1    2 ∥us∥2 2 ≤ 2...

    work page 2023

  18. [18]

    48 Proof

    below). 48 Proof. Obtaining ¯ε0: By definition of the semi-distance (see ( 21)), d (x,x ′) ≥ r implies thatKnorm(x,x ′) ≤ e− r2/ 2. Obtaining ¯ε2, general overview: We use that −vTH g 2Knorm(x,x ′)v ≥ ¯ε2 ∥v∥2 x′ ∀v ∈ R2 ⇐ ⇒ − vT g− 1/ 2 x′ H g 2Knorm(x,x ′)g− 1/ 2 x′ v ≥ ¯ε2 ∥v∥2 2 ∀v ∈ R2. Defining ˜H 02(x,x ′) := Knorm(x,x ′)− 1g− 1/ 2 x′ H g 2Knorm(x,x ...

    work page 2025

  19. [19]

    (78) Bounds for ˜H 02 b′ kb′ k (x,x ′): We use ( 78)

    we have ˜H 02 t′ kt′ k (x,x ′) ≤ ˜H 02 t′ kt′ k (xk,x ′ k) = − 1, ˜H 02 u′ ku′ k (x,x ′) ≤ ˜H 02 u′ ku′ k (xk,x ′ k), |˜H 02 t′ ku′ k (x,x ′)| ≤ | ˜H 02 t′ ku′ k (xk,x ′ k)|. (78) Bounds for ˜H 02 b′ kb′ k (x,x ′): We use ( 78). From ( 74) and ( 75), we have ˜H 02 t′ kt′ k (x,x ′) ∨ ˜H 02 u′ ku′ k (x,x ′) ≤ 1 2r4(er2 + √ e2r2 − 1)2 + 3r2 √ e2r2 − 1(er2 + ...

    work page 2025

  20. [20]

    and ( 79) along with √ Knorm(x,x ′) ≤ 1, it comes |H 10 t (x,x ′)| ≤ 2√ e and |H 10 u (x,x ′)| ≤ ( 4 √ 2 e + 1√ 2 ) , (81) hence  H 10(x,x ′)   2 ≤ √ 4 e + ( 4 √ 2 e + 1√ 2 )2 and   K (10) norm(x,x ′)    x = √ Knorm(x,x ′)  H 10(x,x ′)   2 ≤ e− ∆2/ 4    √ 4 e + ( 4 √ 2 e + 1√ 2 )2 .   K (11) norm(x,x ′)    x,x ′ : We have H 11(x,x ′...

    work page 2025

  21. [21]

    We have |g− 1/ 2 bkbk ∂bkKnorm(x,x ′)| ≤    √ d∏ l=1 Knorm(xl,x ′ l) ⏐ ⏐ ⏐ ⏐ √ Knorm(xk,x ′ k) − 1 g− 1/ 2 bkbk ∂bkKnorm(xk,x ′ k) ⏐ ⏐ ⏐ ⏐

    for Knorm(xk,x ′ k), we get that ∀q ≥ 1, |tk − t′ k|q (u′2 k +u2 k +τ2) q 2 √ Knorm(xk,x ′ k) ≤ (2q e )q/ 2 . We have |g− 1/ 2 bkbk ∂bkKnorm(x,x ′)| ≤    √ d∏ l=1 Knorm(xl,x ′ l) ⏐ ⏐ ⏐ ⏐ √ Knorm(xk,x ′ k) − 1 g− 1/ 2 bkbk ∂bkKnorm(xk,x ′ k) ⏐ ⏐ ⏐ ⏐ . The term in the right-hand side has already been dealt with in Lemma J.5: ( 81) gives a bound for ⏐ ⏐ ⏐...

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  22. [22]

    3025, ¯ε2(0

    88 + ln(s − 1), r = 0. 3025, ¯ε2(0. 3025) = 0. 13139, ¯ε0(0. 3025) = 0. 04472. Proof. To establish that Knorm satisfies the LPC, we first determine the size of the near regions r, giving the constraint on the minimal separation ∆ (see Definition 5.2). We want to pick r such that ¯ε0(r), ¯ε2(r) exist and 1 64 min { ¯ε0(r) B0 , ¯ε2(r) B2 } is maximal, using Le...

    work page 2025

  23. [23]

    Proposition J.2 (Knorm satisfies the LPC, d ≥ 1)

    05 cr ) (see [Giard, 2025, Section VII.1]). Proposition J.2 (Knorm satisfies the LPC, d ≥ 1). Let s ≥ 2. Assume that X ⊂ Rd × [umin,u max]d and that τ ≤ umin. Then Knorm satisfies the LPC with parameters s, r = 0. 3025√ d , ¯ε2(r) = 0. 13139, ¯ε0(r) = 0. 0894 2d , ∆(s) = 2 √

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  24. [24]

    62) + ln(s − 1)

    9 + 3 ln(d + 6. 62) + ln(s − 1). Proof. We take Proposition J.1 as a starting point. We make use of Lemmas J.2, J.4 and J.6. Setting r = 0. 3025√ d , we can take ¯ε2(r) = 0. 13139 (as in dimension d = 1) because d ∈ N∗ ↦→ e− r2 0 2d |G(r0)| with r0 = 0. 3025 is non- decreasing. 56 Furthermore, for r ≤ 0. 3025 we have 1 − e− r2/ 2 ≥ 0. 977 r2 2 . In fact, ...

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  25. [25]

    The minimal separation should verify ( s − 1) √ 2d(170

    0894 2d(2 + √ 2d) =:cd,r . The minimal separation should verify ( s − 1) √ 2d(170. 5 + 25. 78d)e− ∆(s)2/ 4 ≤ cd,r , i.e. ∆(s) ≥ 2 √ ln ( 64

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  26. [26]

    0894 ) + ln ( (170. 5 + 25. 78d)2d(2 + √ 2d) √ 2d ) + ln(s − 1). It suffices that ∆( s) ≥ 2 √

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  27. [27]

    62) + ln(s − 1) (see [Giard, 2025, Section VII.2])

    9 + 3 ln(d + 6. 62) + ln(s − 1) (see [Giard, 2025, Section VII.2]). K Proofs of Section 6 K.1 Proof of Lemma 6.1 We construct ηNDSC in the same way as in Theorem 5.2, but we do not track the constants and dependence on d. We first determine rNDSC such that ηNDSC can satisfy ( NDSC). Then we prove that Knorm verifies the LPC (Definition 5.2) with r = rNDSC an...

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  28. [28]

    We remark that F is continuous on [0, √ 2[ and that F (0) = 0

    in the appendix. We remark that F is continuous on [0, √ 2[ and that F (0) = 0. Furthermore, if r ≤ 0. 32√ d we can take ¯ε2(r) = e− 0. 322 2 |G(0. 32)| (see Lemma J.4). This quantity does not depend on r. So there exists 0<r NDSC ≤ 0. 32√ d (that depends only on d) such that ( 83) is satisfied. LPC and existence of ηNDSC: It remains to show the existence ...

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  29. [29]

    Then Knorm satisfies the LPC (Definition 5.2) with these parameters

    It depends only on d (through ¯ε0(rNDSC) and ¯ε2(rNDSC)) and s. Then Knorm satisfies the LPC (Definition 5.2) with these parameters. Finally, Theorem 5.2 applies. The minimal separation ∆ NDSC follows from Lemma 5.3. It depends on d, umin,umax,τ,rNDSC, ∆(s). Under this separation, we can construct a non-degenerate ce rtificateηNDSC verifying ηNDSC > − 1. We ...

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