Uniform Sobolev inequalities on geometric graphs

Samuel Mercer; Yves van Gennip

REVIEW 2 major objections 5 minor 38 references

Uniform Sobolev inequalities on geometric graphs

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Pith's one-line read Geometric graphs admit uniform L^q Sobolev inequalities precisely when the concentrating scale satisfies ε_n|V_n|^{1/p−1/q} ≤ C, and the analogous L^∞ bound precisely when ε_n|V_n|^{1/p} ≤ C.

desk verdict Solid, genuinely new uniform Sobolev thresholds; the iff is only proved under quasi-uniform weights, so the abstract claims a bit more than the theorems deliver. read the letter →

arxiv 2607.16948 v1 pith:HR6FZIT2 submitted 2026-07-18 math.AP math.CO

Samuel Mercer , Yves van Gennip This is my paper

classification math.APmath.CO MSC 46E3535R02

keywords geometricgraphsSobolevinequalitiesdiscretegradientsregularisationconnectivitythresholdoptimaltransportgraphLaplacianLqestimates

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 when a sequence of geometric graphs built from points in a Euclidean domain admits a Sobolev inequality with a constant independent of the graph size: controlling the L^q norm of a discrete function by its L^p norm plus the p-th root of its discrete variation. The answer, under quasi-uniform vertex weights, is a single scale condition: the concentrating parameter ε_n times |V_n|^{1/p−1/q} must stay bounded, which at the critical Sobolev exponent forces ε_n to be of order |V_n|^{-1/d}. The same dichotomy is proved for L^∞ control when p>d, with condition ε_n|V_n|^{1/p} ≤ C. Because these inequalities hold for ε_n near the connectivity threshold, they cover scales far below what Γ-convergence analyses require, and they quantify the L^q-regularisation effect of discrete gradients in data-driven problems. Without quasi-uniform weights, only a sufficient condition is proven; the paper leaves the necessity question open.

What carries the argument

The argument is carried by three devices: the k-friendly property (any two adjacent vertices share at least k common neighbours, which upgrades an L^p edge-sum bound to L^q), the envelop property (one graph contains the square of another, used to control oscillations), and a pair of extension lemmas that extend functions and graphs from the domain to a slightly larger domain without losing control of nonlocal or discrete variations. These are combined with optimal-transport lifts of discrete functions to the continuum and mollification, so that classical Sobolev embedding applies only after the discrete regularity has been transferred.

What would settle it

Build a sequence of quasi-uniform geometric graphs with ε_n|V_n|^{1/p−1/q} → ∞ and evaluate the spike function u_n = 1 at one vertex and 0 elsewhere: the ratio of L^q to L^p+variation grows without bound, so the uniform Sobolev inequality fails — this is the paper's own sharpness test. To challenge the sufficiency direction, construct a sequence of non-quasi-uniform weights where ε_n(Λ^+)^{1/q+1/p}/(Λ^-)^{2/p} is unbounded but the Sobolev constant stays finite; if found, it would settle the open problem in Section 6.5.1 in the negative.

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Extended reading notes

Core claim

Theorem 5.10 proves that for 1≤p<d and p≤q≤dp/(d−p), a uniform graph Sobolev inequality holds if ε_n(Λ^+_n)^{1/q+1/p}/(Λ^-_n)^{2/p} is bounded, and—when Assumption 4.3(iv) holds—holds if and only if ε_n|V_n|^{1/p−1/q} is bounded. Theorem 5.15 gives the L^∞ analogue for p>d: uniform control holds if ε_n^p Λ^+_n/(Λ^-_n)^2 is bounded, and under the same quasi-uniformity assumption, if and only if ε_n|V_n|^{1/p} is bounded. The necessity direction is sharp: a single spike at one vertex forces the scale condition; the sufficiency direction constructs a uniform estimate by lifting discrete functions to the continuum, mollifying, applying classical Sobolev embedding, and then comparing back through

Load-bearing premise

The if-and-only-if statement depends on the vertex weights being quasi-uniform, i.e. each weight between C/|V_n| and C/|V_n|; if the weights are arbitrarily uneven, the paper proves only a sufficient condition and leaves necessity open.

Editorial extensions

If this is right

For quasi-uniform geometric graphs, the uniform Sobolev inequality and the critical scale ε_n ~ |V_n|^{-1/d} are equivalent: at the largest admissible exponent q=dp/(d−p), the bound ε_n|V_n|^{1/d} ≤ C is necessary and sufficient.
The inequalities survive at concentrating scales comparable to the connectivity threshold, which is exactly the regime used in numerical data-labelling experiments and excluded by standard Γ-convergence assumptions.
Under the scale condition, the compactness results in T L^r spaces improve: bounded L^p-variation sequences become relatively compact in L^r for r<q, or in L^∞ when p>d and ε_n|V_n|^{1/p} ≤ C.
A uniform Poincaré inequality holds for every p≥1 under the weaker approximation assumption (i) instead of the stronger (i*), so mean-subtraction estimates do not require the fast scale separation.
For random point clouds in dimension d≥3, the optimal Wasserstein rate (log n)^{1/d}/n^{1/d} is compatible with ε_n|V_n|^{1/d} bounded, so Sobolev estimates hold up to, but not including, the critical exponent q=dp/(d−p); in d=1,2 the logarithmic corrections block the critical exponent.

Reading between the lines

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

The scale condition ε_n ≍ |V_n|^{-1/d} looks like a general critical threshold for regularisation by graph gradients: any family of graphs whose connectivity scale decays faster than this should fail to regularise, and any family at or above this scale should satisfy quantitative L^q control; that dichotomy is likely to transfer to manifold settings with a metric-doubling structure.
A testable extension is the Hölder-regularity conjecture stated in the paper: for p>d, uniform C^{0,γ} control should hold iff ε_n^{1−γ}|V_n|^{1/p} is bounded; the spike-sequence computation in the paper already shows necessity.
The proof of the iff direction uses only the worst-case spike function, so the optimal Sobolev constant on any quasi-uniform geometric graph family is controlled by a single test function; this suggests a practical numerical check of whether a given point-cloud sequence lies in the regularising regime.
The open necessity question for non-quasi-uniform weights means the true threshold for general graphs may be expressed through weighted volume rather than cardinality: the sufficient condition ε_n(Λ^+)^{1/q+1/p}/(Λ^-)^{2/p} ≤ C would be necessary exactly if the extremal function is again a spike at a maximum-weight vertex.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Solid paper, worth a serious referee. The main theorem is new and the proofs look honest. The gap between abstract and theorems is real but confined to scope: the if-and-only-if in Theorem 5.10 holds under Assumption 4.3(iv), quasi-uniform vertex weights. Without (iv), the theorem gives a sufficient condition in terms of Λ±, and the authors themselves note in §6.5.1 that necessity is open. The abstract's unqualified 'necessary and sufficient conditions' overstates what is proved for general geometric graphs. That is a framing problem, not a mathematical one.

What is actually new: the sharp scaling threshold ε_n|V_n|^{1/p−1/q} for uniform L^p→L^q Sobolev inequalities, and the fact that inequalities survive when ε_n is at or near the connectivity threshold rather than much larger. The p>d case extends [32] to the full domain and to the same scaling. New tools—the k-friendly graph lemma, the discrete extension operator built through optimal transport, and the boundary handling—are real contributions. The necessity direction via spike functions is valid, constants are explicit, and I did not find a hidden fitted parameter or circular step. The self-citations to [28] are for auxiliary TL^p facts and not load-bearing in a suspicious way.

Soft spots, in order. (1) The iff claim: besides (iv), the second part of Theorem 5.10 also requires a moment condition t>q and the smallness K≤S(Ω,η); these are standard technical assumptions, but they should be in the abstract's summary. More importantly, without (iv) the sufficient condition ε_n(Λ+)^{1/q+1/p}/(Λ−)^{2/p} bounded is much stronger than the necessary condition one gets from spikes, so the exact threshold for non-uniform weights is genuinely open. (2) The p>d theorem has the same structure: iff only under (iv). (3) Some auxiliary results (Proposition 5.14, 5.19) are sketches, but they are compactness corollaries and not central.

Who should read it: anyone working on graph-based regularisation or discrete-to-continuum limits. I would cite it, and I would take it to reading group. Send it to referees; the claim-scope issue is fixable by rewriting the abstract and highlighting Section 6.5.1.

Referee Report

2 major / 5 minor

Summary. The paper proves uniform (in n) Sobolev inequalities for functions on geometric graphs V_n ⊂ Ω with vertex measures μ_n and concentrating parameters ε_n, controlling L^q norms by the L^p norm plus the discrete p-variation GE_p^n. For 1 ≤ p < d and p ≤ q ≤ dp/(d−p), Theorem 5.10 gives a sufficient condition in terms of Λ_n^±, and, under Assumption 4.3(iv), an if-and-only-if condition in terms of ε_n |V_n|^{1/p−1/q}. Theorem 5.15 gives an analogous L^∞ statement for p > d, with an iff under the same quasi-uniformity assumption. The proofs combine optimal transport, mollification, new extension lemmas, k-friendly and envelop combinatorial estimates, and classical Sobolev embeddings, and the paper also derives Poincaré inequalities and improved compactness results.

Significance. If the results hold as stated, this is a substantial contribution: it provides quantitative discrete Sobolev estimates at connectivity-scale length parameters, well beyond the usual Γ-convergence regime, and introduces reusable combinatorial tools (k-friendly graphs, enveloping graphs) plus boundary-extension machinery on Lipschitz domains. The proofs are detailed and internally consistent, with explicit constants and no fitted parameters. The main advertised achievement, however, is an exact necessary-and-sufficient scaling, and this is only proved under the quasi-uniform weight assumption Assumption 4.3(iv); without it the paper itself leaves the converse open. That claim-scope issue is central and needs to be fixed in a revision, but the underlying theorems appear sound.

major comments (2)

[Abstract; Theorem 5.10; Theorem 5.15; Assumptions 4.3(iv); Section 6.5.1] The abstract's unqualified statement that the paper provides 'necessary and sufficient conditions' on ε_n is not supported by the theorems as stated. The iff in Theorem 5.10 is proved only under Assumption 4.3(iv), and the same applies to Theorem 5.15. For sequences satisfying only (i)–(iii), Theorem 5.10 supplies a sufficient condition involving Λ_n^±, and Section 6.5.1 explicitly says it is not currently known whether the converse holds. Since the exact threshold ε_n |V_n|^{1/p−1/q} is the paper's headline contribution, the abstract and introduction should either attribute the necessity statement to the quasi-uniform case or otherwise qualify the claim.
[Section 6.5.1; Eq. (10)] The necessary condition reported in Section 6.5.1 is not the strongest one available from the same spike construction. Taking a spike at a vertex of minimal mass, μ_n(x_n)=Λ_n^-, gives ∥u_n∥_{L^p}=(Λ_n^-)^{1/p}, ∥u_n∥_{L^q}=(Λ_n^-)^{1/q}, and GE_p^n(u_n)≲Λ_n^-/ε_n^p, so a uniform inequality forces ε_n(Λ_n^-)^{1/q−1/p} to be bounded. This is stronger than the displayed ε_n(Λ_n^+)^{1/q−1/p} condition because Λ_n^-≤Λ_n^+. The ratio of the sufficient quantity in Theorem 5.10 to this necessary bound is (Λ_n^+/Λ_n^-)^{1/q+1/p}, which can be unbounded when (iv) fails. The open-problem discussion should acknowledge this sharper necessary condition and the resulting gap; as written, it understates the distance between the sufficient and necessary regimes.

minor comments (5)

[Assumptions 4.3(i), (i*)] The inequalities use 'sup_{x∈Ω}|T(x)-x|' where the map is T_n; replace T by T_n for consistency.
[Abstract and Section 1] There are repeated typos: 'vertices are take from from a Euclidean domain' and 'from from' later in Section 1.
[Remark 5.6] H(η) is said to be constructed in the proof of Lemma 5.5, but the relevant result is Proposition 5.5.
[Section 6.3, Example 4] The displayed definition of ε_n involving \tilde S is garbled/ambiguous; please rewrite the formula and explain the choice of \tilde S.
[Lemma 4.2 proof] The variable y is reused for both an element of V and a point in S^{-1}(y); use a different symbol, e.g. z, for the latter.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Sobolev thresholds are proved by explicit constructions; the abstract's unqualified 'iff' is a claim-scope caveat, not a circular step.

full rationale

The central derivation is self-contained. The sufficient direction of Theorem 5.10 is proved by an explicit mollification argument, the extension Lemmas 5.1 and 5.4, the k-friendly combinatorial Lemma 4.12, and Lemma 3.3; the necessary direction is proved with an explicit spike test function and Lemma 4.15. No fitted parameter is introduced, and no 'prediction' is equivalent by construction to the target inequality. The iff statement is conditional on Assumption 4.3(iv); absent (iv), Section 6.5.1 explicitly states 'we do not currently know if it is necessary,' so the abstract's unqualified 'necessary and sufficient conditions' overstates the proved scope but does not reduce the theorem to its inputs. Self-citations to [28] supply only auxiliary TL^p facts (Lemma 2.7, Proposition 5.17) used in Poincaré and compactness corollaries; these are not used to define the Sobolev inequality or to force the scaling threshold, and are therefore not load-bearing circularity. The derivation chain from Assumptions 4.3 to Theorems 5.10 and 5.15 is a genuine proof rather than a renaming or self-justification.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard continuum Sobolev theory, optimal transport, and three structural domain assumptions about the geometric graph sequence: the transport-map approximation property (i), the kernel conditions (ii)–(iii), and the quasi-uniform weight condition (iv). The only assumption whose absence changes the nature of the result is (iv): it converts the theorem from a sufficient condition into an if-and-only-if characterization. No free parameters are fitted to data, and no new physical or mathematical entities are postulated.

assumptions (8)

domain assumption Ω is open, bounded, with Lipschitz boundary; μ has density ρ with 1/D ≤ ρ ≤ D (Assumption 4.3(iii)).
Used for classical Sobolev embeddings, Rellich–Kondrachov compactness, transport-map existence, and local volume/density lower bounds in Lemmas 4.12, 4.14, 5.1.
domain assumption There exist Borel maps T_n: Ω → V_n with T_n#μ = μ_n and sup_x |T_n(x) − x| ≤ K ε_n (Assumption 4.3(i)).
Ensures the discrete measure approximates the continuum at scale ε_n; if ε_n is too small the geometric graph disconnects and the Sobolev inequality fails. Used throughout the local estimates and in Corollary 4.9.
domain assumption Kernel η is non-increasing, continuous at 0, η(0)>0, and ∫_0^∞ η(r) r^{d+t−1} dr < ∞ for some t > q (Assumption 4.3(ii) plus moment condition).
Finite moments control the boundary cross-term in Lemma 5.1 and allow passing from general kernels to indicator kernels via Remark 4.4.
domain assumption Quasi-uniform weights: 1/(D|V_n|) ≤ μ_n(x) ≤ D/|V_n| (Assumption 4.3(iv)).
This is the assumption under which the if-and-only-if characterization is proved. Without it, only the sufficient condition is established and necessity is open (Section 6.5.1).
standard math Classical Sobolev embedding and Rellich–Kondrachov compactness in W^{1,p}(Ω) ([22, Theorem 1.4.4.1], [25, Theorem 11.10]).
Used to pass from W^{1,p} bounds on mollified functions to L^q and L^∞ bounds in Theorems 5.10 and 5.15.
standard math Facts about TL^p(Ω) metric, including convergence of norms ([28, Proposition 5.17]) and interpolation Lemma 2.7 ([28, Lemma 5.18]).
Used in the Poincaré inequality proof and compactness corollaries. These are self-cited auxiliary results; they do not define the central inequality.
standard math Existence of optimal transport maps for the ∞-Wasserstein distance ([5, Theorems 3.2 and 5.5]).
Used to construct Borel maps T_n and to prove the connectivity-threshold upper bounds in Corollaries 4.9 and 4.10.
standard math Nonlocal-to-local liminf estimate for graph variations ([32, Lemma 4.6]).
Used in Lemma 5.7 to connect discrete variation limits to the continuum E^p functional in the Poincaré inequality proof.

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Pith. "Pith review of Uniform Sobolev inequalities on geometric graphs." pith.science (2026). https://pith.science/paper/HR6FZIT2

@misc{pith2026260716948,
  author       = {Pith},
  title        = {Pith review of: Uniform Sobolev inequalities on geometric graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HR6FZIT2}},
  note         = {Machine review of arXiv:2607.16948}
}

abstract

There is significant interest in the study of calculus on graphs, especially regarding the use of gradient-based methods for applications in data driven problems such as classification, clustering and regularisation for inverse problems. Geometric graphs, whose vertices are take from from a Euclidean domain and whose edge structure is determined by the distance between the nodes in the domain, have been central in theoretical studies. Typical approaches for analysis, such as studying consistency and the existence of continuum limits, rely on $\Gamma$-convergence. This technique has some limitations, as it requires the typical length scale which determines the connectivity structure of the graph to be much larger than the scales frequently used for applications. Moreover, it may fail to provide quantitative results. This paper provides necessary and sufficient conditions on the asymptotic behaviour of this length scale for the existence of a uniform collection of Sobolev inequalities on a sequence of geometric graphs. Furthermore, these inequalities hold when the length scales are much smaller than what is typically assumed for $\Gamma$-convergence results and within the range of what is used for data-driven problems. The Sobolev inequalities provide a quantitative estimate on the $L^q$-regularisation effect of discrete gradients.

Figures

Figures reproduced from arXiv: 2607.16948 by the authors.

**Figure 2.** Illustration of the friendly property for a simple graph [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** We shall take Tn : (0, 1)d → (0, 1)d to be the projection onto this discretization, again this is the same map as specified within example 4. of Section 6.3. Note that sup x∈(0,1)d |Tn(x) − x| ≤ 1 ⌈n1/d⌉ . Let ν be the probability measure as defined in the statement of the lemma. Construct a random variable X with law ν. Additionally enumerate the index set I as {1, . . . , l} where l is the cardinality of I; consid… view at source ↗

Figures from the paper (3 more)

**Figure 3.** Figure 3: Lattice discretization for d = 2, n = 6 We shall set q := pd d−p . Define 𝕕n to be the graph metric on 𝔾n: the distance between vertices x and y is the least number of edges that have to be traversed to go from x to y in the graph. Thus in the example of [PITH_FULL_IM…

**Figure 4.** Figure 4: Rectangular discretization for n = 2. We start by taking Ω := (0, 1)2 and, for each n ∈ ℕ, consider a rectangular discretization as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p049_4.png]

**Figure 5.** Figure 5: Discretizations that have an anisotropic continuum limit. [PITH_FULL_IMAGE:figures/full_fig_p052_5.png]

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Works this paper leans on

38 extracted references · 1 linked inside Pith

[32]

Analysis ofp-Laplacian regularization in semisupervised learning.SIAM J

Dejan Slepˇ cev and Matthew Thorpe. Analysis ofp-Laplacian regularization in semisupervised learning.SIAM J. Math. Anal., 51(3):2085–2120, 2019
[28]

An extension to Banach stackings of the Brezis–Pazy semigroup-convergence theorem, with applications toλ-convex gradient flows

Samuel Mercer and Yves van Gennip. An extension to Banach stackings of the Brezis–Pazy semigroup-convergence theorem, with applications toλ-convex gradient flows. arxiv.org/abs/2511.23233v1 [math.AP], 2025

arXiv 2025
[1]

Adams and John J

Robert A. Adams and John J. F. Fournier.Sobolev Spaces, volume 140 ofPure and applied mathematics. Elsevier, Oxford, Amsterdam, 2003

2003
[2]

A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies.European J

Giovanni Alberti and Giovanni Bellettini. A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies.European J. Appl. Math., 9(3):261–284, 1998

1998
[3]

Asymptotic behavior of the Dirichlet energy on Poisson point clouds.J

Andrea Braides and Marco Caroccia. Asymptotic behavior of the Dirichlet energy on Poisson point clouds.J. Nonlinear Sci., 33(5):Paper No. 80, 57, 2023

2023
[4]

Mumford–Shah functionals on graphs and their asymptotics.Nonlinearity, 33(8):3846–3888, 2020

Marco Caroccia, Antonin Chambolle, and Dejan Slepˇ cev. Mumford–Shah functionals on graphs and their asymptotics.Nonlinearity, 33(8):3846–3888, 2020

2020
[5]

The∞-Wasserstein distance: local solutions and existence of optimal transport maps.SIAM J

Thierry Champion, Luigi De Pascale, and Petri Juutinen. The∞-Wasserstein distance: local solutions and existence of optimal transport maps.SIAM J. Math. Anal., 40(1):1–20, 2008

2008
[6]

Fan R. K. Chung.Spectral graph theory, volume 92 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997

1997

Show all 38 references

[7]

F. H. Clarke. On the inverse function theorem.Pacific J. Math., 64(1):97–102, 1976

1976
[8]

Large data limit for a phase transition model with the p-Laplacian on point clouds.European J

Riccardo Cristoferi and Matthew Thorpe. Large data limit for a phase transition model with the p-Laplacian on point clouds.European J. Appl. Math., 31(2):185–231, 2020. 52

2020
[9]

Birkh¨ auser Boston, Inc., Boston, MA, 1993

Gianni Dal Maso.An introduction toΓ-convergence, volume 8 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1993

1993
[10]

Consistency of modularity clustering on random geometric graphs.Ann

Erik Davis and Sunder Sethuraman. Consistency of modularity clustering on random geometric graphs.Ann. Appl. Probab., 28(4):2003–2062, 2018

2003
[11]

Dunlop, Dejan Slepˇ cev, Andrew M

Matthew M. Dunlop, Dejan Slepˇ cev, Andrew M. Stuart, and Matthew Thorpe. Large data and zero noise limits of graph-based semi-supervised learning algorithms.Appl. Comput. Harmon. Anal., 49(2):655–697, 2020

2020
[12]

Evans and Ronald F

Lawrence C. Evans and Ronald F. Gariepy.Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992

1992
[13]

Variational analysis of discrete Dirichlet problems in periodically perforated domains.ESAIM Control Optim

Giuliana Fusco. Variational analysis of discrete Dirichlet problems in periodically perforated domains.ESAIM Control Optim. Calc. Var., 31:Paper No. 99, 28, 2025

2025
[14]

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace- Beltrami operator.Found

Nicol´ as Garc ´ ıa Trillos, Moritz Gerlach, Matthias Hein, and Dejan Slepˇ cev. Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace- Beltrami operator.Found. Comput. Math., 20(4):827–887, 2020

2020
[15]

On the consistency of graph-based Bayesian semi-supervised learning and the scalability of sampling algorithms.J

Nicol´ as Garc ´ ıa Trillos, Zachary Kaplan, Thabo Samakhoana, and Daniel Sanz-Alonso. On the consistency of graph-based Bayesian semi-supervised learning and the scalability of sampling algorithms.J. Mach. Learn. Res., 21:Paper No. 28, 47, 2020

2020
[16]

Continuum limits of posteriors in graph Bayesian inverse problems.SIAM J

Nicol´ as Garc ´ ıa Trillos and Daniel Sanz-Alonso. Continuum limits of posteriors in graph Bayesian inverse problems.SIAM J. Math. Anal., 50(4):4020–4040, 2018

2018
[17]

On the rate of convergence of empirical measures in ∞-transportation distance.Canad

Nicol´ as Garc ´ ıa Trillos and Dejan Slepˇ cev. On the rate of convergence of empirical measures in ∞-transportation distance.Canad. J. Math., 67(6):1358–1383, 2015

2015
[18]

Continuum limit of total variation on point clouds

Nicol´ as Garc ´ ıa Trillos and Dejan Slepˇ cev. Continuum limit of total variation on point clouds. Archive for Rational Mechanics and Analysis, 220:193–241, 2016

2016
[19]

A variational approach to the consistency of spectral clustering.Appl

Nicol´ as Garc ´ ıa Trillos and Dejan Slepˇ cev. A variational approach to the consistency of spectral clustering.Appl. Comput. Harmon. Anal., 45(2):239–281, 2018

2018
[20]

Estimating perimeter using graph cuts.Adv

Nicol´ as Garc ´ ıa Trillos, Dejan Slepˇ cev, and James von Brecht. Estimating perimeter using graph cuts.Adv. in Appl. Probab., 49(4):1067–1090, 2017

2017
[21]

Consistency of Cheeger and ratio graph cuts.J

Nicol´ as Garc ´ ıa Trillos, Dejan Slepˇ cev, James von Brecht, Thomas Laurent, and Xavier Bresson. Consistency of Cheeger and ratio graph cuts.J. Mach. Learn. Res., 17:Paper No. 181, 46, 2016

2016
[22]

Grisvard.Elliptic problems in nonsmooth domains, volume 24 ofMonographs and Studies in Mathematics

P. Grisvard.Elliptic problems in nonsmooth domains, volume 24 ofMonographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1985

1985
[23]

Piyush Gupta and P. R. Kumar. Critical power for asymptotic connectivity in wireless networks. InStochastic analysis, control, optimization and applications, Systems Control Found. Appl., pages 547–566. Birkh¨ auser Boston, Boston, MA, 1999

1999
[24]

Lee.Introduction to smooth manifolds, volume 218 ofGraduate Texts in Mathematics

John M. Lee.Introduction to smooth manifolds, volume 218 ofGraduate Texts in Mathematics. Springer, New York, second edition, 2013

2013
[25]

American Mathematical Society, Providence, RI, 2009

Giovanni Leoni.A first course in Sobolev spaces, volume 105 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009

2009
[26]

Martin W. Licht. Smoothed projections over weakly Lipschitz domains.Math. Comp., 88(315):179–210, 2019

2019
[27]

PhD thesis, Technische Universiteit Delft, Delft, Netherlands, in prep

Samuel Mercer.Novel approaches to the study of gradient flows: discrete-to-continuum limits and beyond. PhD thesis, Technische Universiteit Delft, Delft, Netherlands, in prep. 53
[29]

Uniform graph sobolev inequalities and graph gradient flows: convergence from discrete to continuum

Samuel Mercer and Yves van Gennip. Uniform graph sobolev inequalities and graph gradient flows: convergence from discrete to continuum. in prep
[30]

Consistency of Dirichlet partitions.SIAM J

Braxton Osting and Todd Harry Reeb. Consistency of Dirichlet partitions.SIAM J. Math. Anal., 49(5):4251–4274, 2017

2017
[31]

Oxford University Press, Oxford, 2003

Mathew Penrose.Random geometric graphs, volume 5 ofOxford Studies in Probability. Oxford University Press, Oxford, 2003

2003
[33]

Rohde, and Dejan Slepˇ cev

Matthew Thorpe, Serim Park, Soheil Kolouri, Gustavo K. Rohde, and Dejan Slepˇ cev. A trans- portationL p distance for signal analysis.J. Math. Imaging Vision, 59(2):187–210, 2017

2017
[34]

Asymptotic analysis of the Ginzburg–Landau functional on point clouds.Proc

Matthew Thorpe and Florian Theil. Asymptotic analysis of the Ginzburg–Landau functional on point clouds.Proc. Roy. Soc. Edinburgh Sect. A, 149(2):387–427, 2019

2019
[35]

Deep limits of residual neural networks.Res

Matthew Thorpe and Yves van Gennip. Deep limits of residual neural networks.Res. Math. Sci., 10(1):Paper No. 6, 44, 2023

2023
[36]

Minimax rates for the estimation of eigenpairs of weighted Laplace-Beltrami operators on manifolds

Nicol´ as Garc ´ ıa Trillos, Chenghui Li, and Raghavendra Venkatraman. Minimax rates for the estimation of eigenpairs of weighted Laplace-Beltrami operators on manifolds. arxiv.org/abs/2506.00171 [stat.ML], 2025

2025 arXiv
[37]

Bertozzi

Yves van Gennip and Andrea L. Bertozzi. Γ-convergence of graph Ginzburg-Landau functionals. Adv. Differential Equations, 17(11-12):1115–1180, 2012

2012
[38]

Springer, Berlin, Heidelberg, first edition, 2009

C´ edric Villani.Optimal Transport, volume 338 ofGrundlehren der mathematischen Wis- senschaften. Springer, Berlin, Heidelberg, first edition, 2009. 54

2009

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [32]

Analysis ofp-Laplacian regularization in semisupervised learning.SIAM J

Dejan Slepˇ cev and Matthew Thorpe. Analysis ofp-Laplacian regularization in semisupervised learning.SIAM J. Math. Anal., 51(3):2085–2120, 2019

[2] [28]

An extension to Banach stackings of the Brezis–Pazy semigroup-convergence theorem, with applications toλ-convex gradient flows

Samuel Mercer and Yves van Gennip. An extension to Banach stackings of the Brezis–Pazy semigroup-convergence theorem, with applications toλ-convex gradient flows. arxiv.org/abs/2511.23233v1 [math.AP], 2025

arXiv 2025

[3] [1]

Adams and John J

Robert A. Adams and John J. F. Fournier.Sobolev Spaces, volume 140 ofPure and applied mathematics. Elsevier, Oxford, Amsterdam, 2003

2003

[4] [2]

A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies.European J

Giovanni Alberti and Giovanni Bellettini. A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies.European J. Appl. Math., 9(3):261–284, 1998

1998

[5] [3]

Asymptotic behavior of the Dirichlet energy on Poisson point clouds.J

Andrea Braides and Marco Caroccia. Asymptotic behavior of the Dirichlet energy on Poisson point clouds.J. Nonlinear Sci., 33(5):Paper No. 80, 57, 2023

2023

[6] [4]

Mumford–Shah functionals on graphs and their asymptotics.Nonlinearity, 33(8):3846–3888, 2020

Marco Caroccia, Antonin Chambolle, and Dejan Slepˇ cev. Mumford–Shah functionals on graphs and their asymptotics.Nonlinearity, 33(8):3846–3888, 2020

2020

[7] [5]

The∞-Wasserstein distance: local solutions and existence of optimal transport maps.SIAM J

Thierry Champion, Luigi De Pascale, and Petri Juutinen. The∞-Wasserstein distance: local solutions and existence of optimal transport maps.SIAM J. Math. Anal., 40(1):1–20, 2008

2008

[8] [6]

Fan R. K. Chung.Spectral graph theory, volume 92 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997

1997