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uniformly at random. 9:foreach 1 ≤i≤tdo 10: For each H∈eH, compute XH(A, µ i) and XH(B, ν i) via Algorithm 2. 11:end for 12:Compute f(A, B) according to (5.5). 13:Output: f(A, B). It remains to prove Lemma 5.3. We now describe how to calculate XH(M, µ) via dynamical programming. For l≥1, denote by NB l(x, y) the collec... | https://arxiv.org/abs/2503.06464v1 |
invoke the value of L(x, y) when we compute Y(x,Ho, C, µ ) recursively. However, this can be done efficiently as we have already stored the value of all L(x, y) at Step 2, and we omit further details here for simplicity. A Preliminary results on graphs Lemma A.1. For any N≥log(ι−1)andT∈ ∪m≤NRm, we have # T′∈ ∪m≤NRm:T′... | https://arxiv.org/abs/2503.06464v1 |
our proof for k= 1. For the general case, it suffices to denote Hi=H∪L1∪. . .∪Li(letH0=Hfor convenience) and apply the case k= 1 to H=HiandS=Hi+1, and then a simple induction concludes the proof. The following lemma provides an upper bound on the number of subgraphs contained in a given connected graph J. Lemma A.5. Gi... | https://arxiv.org/abs/2503.06464v1 |
Definition 2.6. We have R(1,2) N ≥ RN ·exp −e−1 8log2(ι−1)N . Proof. LetL= log2(ι−1). Denote R(1) Nto be the set of T⊂ R Nsuch that each vertex in V(T) has at most Lchildren (i.e, satisfying Item (1) in Definition 2.6). It has been shown in [GS94, Theorem 1] that R(1) N ≥ RN ·exp −e−1 4LN . Thus, it suffices to sho... | https://arxiv.org/abs/2503.06464v1 |
γ a+b}, define κ=κ(γ1, . . . , γ a+b) =Y m>log2(ι−1)κm!,whereκm= #{a+ 1≤i≤a+b:γi=m}. Denote T1⊕. . .⊕Tato be the unlabeled rooted tree with children trees given by T1, . . . ,Ta. From the definition of µ∗andR(1,2,4) N+1, we then have (for two measures µandν, we define µ⊗νto be their product measure) µ∗ a,b;γ1,...,γa+b(... | https://arxiv.org/abs/2503.06464v1 |
. . . , (uk, vk) for k < K . Clearly there exists a vertex uk+1∈V(Tι) such that DistTι(uk+1, ui),DistTι(uk+1, vi)≥12(log log( ι−1))2for all 1 ≤i≤k , which yields a valid choice for ( uk+1, vk+1) and thus completes the proof by induction. Now we argue that we can choose W1, . . . , W Mfrom W0. Specifically, our choice r... | https://arxiv.org/abs/2503.06464v1 |
n λs n(1−λs n)1/2!# ⊜s(1 +ϵσiσj), EPσ,π" Ai,j−λs n λs n(1−λs n)1/2!2# =EPσ,π" Bπ(i),π(j)−λs n λs n(1−λs n)1/2!2# ⊜(1 +ϵσiσj). For the case where r≥3 orr+t≥3, it suffices to note that in this case EPσ,π" Ai,j−λs n λs n(1−λs n)1/2!r# =EPσ,π" Bi,j−λs n λs n(1−λs n)1/2!r# ⊜ n λsr 2·Pσ,π(Ai,j= 1)≤(1 +ϵσiσj)·(n/ϵ2λs)(r... | https://arxiv.org/abs/2503.06464v1 |
i)|EσW∼νW(Y u∈U∪∪i∈ΛEndP( P′ i)σu) =X Λ∈MY i∈Λϵ|E(P′ i)|, (C.3) where Mis the collection of Λ ⊂[t′] such that each vertex appears an even number of times in U∪ ∪ i∈ΛEndP( P′ i). Now we fix Λ ∈ M and denote H=∪i∈ΛP′ i⊂TU. Then the degree of any u∈V(TU)\UinHis even and the degree of any u∈UinHis odd. Thus, for all u∈Uthe... | https://arxiv.org/abs/2503.06464v1 |
with respect to G. We now divide fbadinto the following cases. For S1, S2⊢HandS1, S2∈ B(H), denote GS=eS1∪eS2. Let B1(H) :=n (S1, S2)∈ B(H) :τ(GS)>20ℵ2o ; (C.7) B2(H) :=n (S1, S2)∈ B(H) :τ(GS)≤20ℵ2, GScontains at least 40 ℵ3tangleso ; (C.8) B3(H) :=n (S1, S2)∈ B(H) :τ(GS)≤20ℵ2, GScontains at most 40 ℵ3tangleso . (C.9) ... | https://arxiv.org/abs/2503.06464v1 |
first inequality we use the fact that # (n1, . . . , n 2ιℵ) :n1+···+n2ιℵ≤v ≤v2ιℵv=|V(GS)| ≤ (ℓℵ)2ιℵ=no(1); # (r1, . . . , r 2ιℵ) :r1+···+r2ιℵ≤m+ 2 ≤(m+ 2)2ιℵm=τ(GS) ≤ (ℓℵ)2ιℵ=no(1), in the second inequality we use the fact that ℓ≤logn(see (2.6)), and the last inequality follows fromℵlogm≤10ℵ2+ 0.1(log m)2≤10ℵ2+ 0.1mf... | https://arxiv.org/abs/2503.06464v1 |
Gis with (C.15) and (C.16) if G∼=G∪for some G∪satisfying (C.15) and (C.16). Using this terminology, we get that (C.14) ≤n4ιℵ−2+o(1) (ϵ2λs)4ℓιℵX Gwith (C.15) ,(C.16) ϵ2λs n|E(G)| ·X G∪∼=G#{(S1, S2),(K1, K2)∈ B3:eS1∪eS2∪eK1∪eK2=G∪}. (C.17) We first need the following lemma. Lemma C.7. Given J⊂ K nwith at most z=O(1)conn... | https://arxiv.org/abs/2503.06464v1 |
tangled neighbor (of some vertex v) it either takes at least√lognsteps to go back to v, or traverses an edge in Tat least once. Therefore, our claim follows from |E(T)| ≤40ℵ3. Now we construct a mapping X: Tri,(v0, v1, vℓ), s(v),H(v) v∈V(i) ≥3, wv,j: 1≤j≤2ℓ√logn+ 80ℵ3, v∈V(i) ≥3 −→(v0, v1, . . . , v ℓ), (C.19) wh... | https://arxiv.org/abs/2503.06464v1 |
v1, s(v),H(v) v∈V(i) ≥3, wv,j: 1≤j≤2ℓ√logn+ 80ℵ3, v∈V(i) ≥3 to (v0, v1, . . . , v ℓ) =Li(S). Otherwise, suppose that the image is ( v′ 0, v′ 1, . . . , v′ ℓ)̸= (v0, v1, . . . , v ℓ) and there exists a minimal xsuch that v′ x̸=vx. Then by definition of our map we have x≥2; we also have dTri(vx−1)≥3 since both ( v′ ... | https://arxiv.org/abs/2503.06464v1 |
for the general stochastic block model. In Communications on Pure and Applied Mathematics , 71(7):1334–1406, 2018. [AYZ95] Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. In Journal of the ACM , 42(4):844–856, 1995. [Bab16] L´ aszl´ o Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the 48th ... | https://arxiv.org/abs/2503.06464v1 |
transition for detecting correlated stochastic block models by low-degree polynomials. Preprint, arXiv:2409.00966. [CS21+] Byron Chin and Allan Sly. Optimal reconstruction of general sparse stochastic block models. Preprint, arXiv:2111.00697. [CB81] Charles J. Colbourn and Kellogg S. Booth. Linear time automorphism alg... | https://arxiv.org/abs/2503.06464v1 |
circuits, low-degree polynomials, and Langevin dynamics. In SIAM Journal on Computing , 53(1):1–46, 2024. [GM20] Luca Ganassali and Laurent Massouli´ e. From tree matching to sparse graph alignment. In Proceed- ings of 33rd Conference on Learning Theory (COLT) , pages 1633–1665. PMLR, 2020. [GML21] Luca Ganassali, Laur... | https://arxiv.org/abs/2503.06464v1 |
matching with improved noise robustness. In Proceedings of 34th Conference on Learning Theory (COLT) , pages 3296–3329. PMLR, 2021. [MRT23] Cheng Mao, Mark Rudelson, and Konstantin Tikhomirov. Exact matching of random graphs with constant correlation. In Probability Theory and Related Fields , 186(2):327–389, 2023. [MW... | https://arxiv.org/abs/2503.06464v1 |
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EXTREMES OF STRUCTURAL CAUSAL MODELS SEBASTIAN ENGELKE1, NICOLA GNECCO2, AND FRANK R ¨OTTGER3 Abstract. The behavior of extreme observations is well-understood for time series or spatial data, but little is known if the data generating process is a structural causal model (SCM). We study the behavior of extremes in thi... | https://arxiv.org/abs/2503.06536v1 |
a limiting vector Y= (Yv:v∈V) describing the tail dependence. This limit is called a multivariate Pareto distribution and can be characterized as a new SCM on the extremal DAG Ge= (V, E e) Me= (Ge,{Ψv:v∈V},Pε), (2) where we call Ψ v:R|paGE(v)|×R→Rthe extremal structure functions, which satisfy the homogeneity Ψv(y+t1, ... | https://arxiv.org/abs/2503.06536v1 |
et al., 2000), relying on a new test for extremal conditional independence in the H¨ usler–Reiss model class. A second algorithm, called extremal pruning, leverages an estimate of the DAG Gof the initial SCM (1). Based on the fact that causal links can only disappear in the tails, the algorithm prunes suitable edges an... | https://arxiv.org/abs/2503.06536v1 |
(Zhang et al., 2018; Deuber et al., 2024). The treatment is binary in these cases, which fundamentally differs from our work where we consider continuous variables that can be extreme simultaneously. Interestingly, Wang and Miao (2024) observe that for light-tailed responses, confounding may become negligible in the ex... | https://arxiv.org/abs/2503.06536v1 |
Supplementary Material A for details on the directed global Markov property and the related notion of d-separation. Directed graphical models are closely connected to SCMs in (1) on the underlying DAG G= (V, E). We give here a formal definition including the behavior of the system under interventions. Definition 1. Let... | https://arxiv.org/abs/2503.06536v1 |
distribution functions. In order to describe the extremal dependence between large values of the components of this vector, a standard assumption is that the rescaled threshold exceedances converge in distribution to a (gener- alized) multivariate Pareto distribution Y, that is, P(Y≤y) = lim u→∞P(X∗−u1≤y|max i=1,...,dX... | https://arxiv.org/abs/2503.06536v1 |
matrix θ(I), called the H¨ usler–Reiss precision matrix (Hentschel et al., 2024), which we can obtain via the Fiedler–Bapat identity −1 2ΓII1 1⊤0−1 =θ(I)p(I) p(I)⊤σ2(I) , (10) where σ2(I) =1 2 1⊤Γ−1 II1−1andp(I) = 2 σ2(I)Γ−1 II1are the resistance radius and curvature, re- spectively (Devriendt, 2022). For the ful... | https://arxiv.org/abs/2503.06536v1 |
of view, in the sense that it forms a semi-graphoid under weak assumptions (R¨ ottger et al., 2023). Engelke et al. (2024) generalize the conditional independence notion in Definition 2 to infinite measures Λ, allowing for mass on sub-faces of Rd, asymptotic independence, and applications to L´ evy processes (Engelke e... | https://arxiv.org/abs/2503.06536v1 |
v∈V\{1}, the extremal structure function is homogeneous, that is, Ψv(x+s1, e) = Ψ v(x, e) +sfor all s∈R. Moreover, the distribution of X∗ is multivariate regularly varying and lim t→∞P(X∗−t1∈A|X∗ 1> t) =P(Y1∈A), (17) for any Borel subset A⊂ L1withP(Y1∈∂A) = 0 . The limiting random vector has the form Y1d=R1+W1for a sta... | https://arxiv.org/abs/2503.06536v1 |
examples for a random vector Xthat follows an SCM M:= (G,{fj:j∈V},Pε) on four nodes with Ggiven by the diamond graph in Figure 2b. To simplify the examples, we assume the following structure for the first three variables X1:=f1(ε1) =ε1, X 2:=f2(X1, ε2) =X1+ε2, X 3:=f3(X1, ε3) =X1+ε3, 10 S. ENGELKE, N. GNECCO, AND F. R ... | https://arxiv.org/abs/2503.06536v1 |
be convenient to parameterize qvthrough a common level t >0 that governs how extreme the interventions are. We set qv= 1−exp(−t− ξv) =H(t+ξv), where His the standard exponential distribution function and the ξv∈R,v∈V, quantify by how much the individual interventions deviate from the common level. For instance, if EXTR... | https://arxiv.org/abs/2503.06536v1 |
left-hand side of Figure 3. In extreme value theory it is natural to compare variables on a common scale. In particular, two variables are termed asymptotically dependent if they have joint exceedances over their respective high quantiles with common probabilistic level; see the definition of extremal correlation in Co... | https://arxiv.org/abs/2503.06536v1 |
agrees with the classical definition of intervened SCMs. All variables are extreme in this scenario as the intervention propagates from the root node. (ii) Consider the extremal intervention do( Y2:=ξ2, Y3:=ξ3) in Figure 4b for ξ2, ξ3∈R. The resulting intervened structure functions are ˜Ψ1(e) =−∞,˜Ψ2(y1, e) =ξ2,˜Ψ3(y1,... | https://arxiv.org/abs/2503.06536v1 |
G, that is, A⊥ ⊥GB|C⇒YA⊥eYB|YC, for all disjoint index sets A, B, C ⊂V. From the results in Engelke and Hitz (2020) we can directly derive a key property of the extremal graph structure G. Since a multivariate Pareto distribution Ywith positive and continuous exponent measure density can not exhibit extremal independen... | https://arxiv.org/abs/2503.06536v1 |
−αv for some constant Cdepending on the αi. Example 14. Recall the exponent measure density the H¨ usler–Reiss distribution in Example 5. The conditional density for node vwith parents pa( v) is λ(yv|ypa(v)) =1p 2πϑ−1vvexp −1 2ϑ−1vv(yv−µ∗)2 , where ϑ=θ({v} ∪pa(v)) and µ∗= −1 2Γv,pa(v),1 −1 2Γpa(v),pa(v)1 1 0−1y... | https://arxiv.org/abs/2503.06536v1 |
independent noise variables that admit densities andΦv:R|pa(v)|→Rare homogeneous functions satisfying Φv(x+t1) = Φ v(x)+tfor any x∈R|pa(v)| andt∈R. Assume the normalization condition E{exp(εv)}=1 E[exp{Φv(Xpa(v)−X11)}]. (29) Then, we obtain a multivariate Pareto vector Yby choosing the auxiliary vector Y1=X, which is a... | https://arxiv.org/abs/2503.06536v1 |
S⊆V\ {i, j}, we would like to perform the hypothesis test H0:Yi⊥eYj|YSversus H1:Yi̸⊥eYj|YS. (34) 18 S. ENGELKE, N. GNECCO, AND F. R ¨OTTGER Following (13) in Example 6 and Lemma 3 in the Supplementary Material, for H¨ usler–Reiss distribu- tions the null hypothesis H0can be equivalently expressed in terms of a vanishin... | https://arxiv.org/abs/2503.06536v1 |
max i=1,...dX∗ iforτ∈ {0.9,0.95,0.975}, resulting inm=⌊n(1−τ)⌋effective samples. To have a fair comparison, we use msamples of the exact multivariate Pareto distribution in (a). Given the generated data sets, for all pairs ( i, j)∈V×V, with i < j, and for all subsets S⊆V\{i, j} we test (5.1) with the following two appr... | https://arxiv.org/abs/2503.06536v1 |
When the CPDAG contains directed edges from itojand from jtoi, this usually is represented by an undirected edge between iandj. Based on the extremal Markov properties, we can deduce a set of conditional independence statements in Yby reading off d-separation statements in the graph Ge. In structure learning we usually... | https://arxiv.org/abs/2503.06536v1 |
i, j)∈Edo 4: ifthe edge cannot be removed, i.e., |adjC(i)\ {j}|< ℓ,then 5: continue thefor loop at line 3 6: end if 7: foreach separating set S⊆adjC(i)\ {j}with|S|=ℓdo 8: ifYi⊥eYj|YS,then 9: Update the edge set E←E\ {(i, j)}and undirected graph C←(V, E). 10: Append the separating set and edge pair S ← append([ S,(i, j)... | https://arxiv.org/abs/2503.06536v1 |
0.995 0.5 0.7 0.9 0.95 0.975 0.99 0.9950246 051015 0102030 τStructural Hamming DistanceMethod: (a) MPD (b) Max−stable (c) Extremal−SCM Figure 6. Performance of the extremal PC-algorithm (Algorithm 1) in terms of the average structural Hamming distance over 20 repetitions for different dimensions d∈ {5,10,15}, expected ... | https://arxiv.org/abs/2503.06536v1 |
2 has a computational bottleneck when it comes to computing the collection of separating setsSi,jfor the pair {i, j}(see Line 8). This computation requires evaluating whether each of the 2d−2subsets S⊆V\{i, j}separates ifrom jin the pruned graph G⋆ i,j. To speed up the computation, in place of Si,j, we consider the sin... | https://arxiv.org/abs/2503.06536v1 |
Agency ( https: //www.gkd.bayern.de ). The dataset comprises average daily discharge measurements collected across the upper Danube basin between 1960 and 2009. Following the preprocessing pipeline established by Gnecco et al. (2021), we retain n= 4,600 observations for the summer months June, July, and August. This da... | https://arxiv.org/abs/2503.06536v1 |
(2017). The HSIC test is conducted using a bootstrap procedure with B′= 100 repetitions and a Gaussian kernel, with the median heuristic for bandwidth selection (Pfister and Peters, 2017). The evaluation proceeds as follows. Each river branch is treated as an independent dataset, denoted dataset-i fori= 1, . . . , 6. W... | https://arxiv.org/abs/2503.06536v1 |
of Applied Probability 32 (1), 1–45. Asadi, P., A. C. Davison, and S. Engelke (2015). Extremes on river networks. The Annals of Applied Statistics 9 (4), 2023–2050. Asenova, S., G. Mazo, and J. Segers (2021). Inference on extremal dependence in the domain of at- traction of a structured H¨ usler–Reiss distribution moti... | https://arxiv.org/abs/2503.06536v1 |
Galles, D. and J. Pearl (1997). Axioms of causal relevance. Artificial Intelligence 97 (1-2), 9–43. Gissibl, N. and C. Kl¨ uppelberg (2018). Max-linear models on directed acyclic graphs. Bernoulli 24 (4A), 2693–2720. Gnecco, N., N. Meinshausen, J. Peters, and S. Engelke (2021). Causal discovery in heavy-tailed models. ... | https://arxiv.org/abs/2503.06536v1 |
Inference: Foundations and Learning Algorithms . Cambridge, MA, USA: MIT Press. Peters, J., J. M. Mooij, D. Janzing, and B. Sch¨ olkopf (2014). Causal discovery with continuous additive noise models. The Journal of Machine Learning Research 15 (1), 2009–2053. Pfister, N., P. B¨ uhlmann, B. Sch¨ olkopf, and J. Peters (2... | https://arxiv.org/abs/2503.06536v1 |
et al. (2018). Extremal quantile treatment effects. The Annals of Statistics 46 (6B), 3707–3740. Zhang, Z., D. Bolin, S. Engelke, and R. Huser (2023). Extremal dependence of moving average processes driven by exponential-tailed L´ evy noise. URL https://arxiv.org/abs/2307.15796 . 30 S. ENGELKE, N. GNECCO, AND F. R ¨OTT... | https://arxiv.org/abs/2503.06536v1 |
that we now normalize with the first component of X∗instead of the deterministic threshold t. Letπ:{1, . . . , d } → { 1, . . . , d }be a causal (or topological) ordering of the DAG G, that is, for all i∈an(j) it holds that π(i)< π(j),i, j∈V. Since the (only) root is node X∗ 1, we have π(1) = 1. We prove (18) and (39) ... | https://arxiv.org/abs/2503.06536v1 |
vis a function with values in R, we have that W(1) v>−∞ almost surely for all v∈V. Therefore, Segers (2020, Corollary 3) implies that Eexp{W(1) v}= 1; for details see also Proposition 1 in the Supplementary Material of Engelke and Volgushev (2022). □ B.2.Proof of Example 7. Proof. We need to show that the first term ha... | https://arxiv.org/abs/2503.06536v1 |
last equation follows from dominated convergence. Therefore, we have that Pdo(Y1:=ξ1) Y2= N(ξ1−σ2/2, σ2). □ B.4.Proof of Theorem 2. Proof. As in the proof of Theorem 1, let π:{1, . . . , d } → { 1, . . . , d }be a causal ordering of the DAG G, that is, for all i∈an(j) it holds that π(i)< π(j),i, j∈V. Since the only roo... | https://arxiv.org/abs/2503.06536v1 |
that also applies to sequences with possible values and limits −∞. We can now apply the extended continuous mapping theorem (van der Vaart, 1998, Theorem 18.11) to conclude weak convergence of the random variables inside the expectation in (44), where we use the induction assumption for m−1. Note that this also include... | https://arxiv.org/abs/2503.06536v1 |
Ykdefined in Section 2.2 and denote by fk=λits density onLk. By the definition of extremal conditional independence, the extremal local Markov property implies the classical local Markov property of Ykon the same graph G. Therefore, λ(y) =fk(y) =fk(yi|ypa(i))fk(y\i) =λ(yi|ypa(i))λ(y\i),y∈ Lk, where the second equation ... | https://arxiv.org/abs/2503.06536v1 |
of εv. Thus, f(x) =e−x1|V|Y v=2f(xv|xpa(v)) =e−x1|V|Y v=2fεv(xv−Φv(xpa(v))). If we select Y1d=X, we have λ(y) =f(y) and λ(yv|ypa(v)) =fεv(yv−Φv(ypa(v))). We need to check that the sufficient conditions for valid conditional exponent measure densities are satisfied. First, we observe that λ(· |ypa(v)) is a probability d... | https://arxiv.org/abs/2503.06536v1 |
usual sense for all k∈V. Since GeandHeare Markov equivalent in the classical sense as they have the same skeleton and v-structures (Pearl, 2009, Theorem 1.2.8), we conclude that Ykis Markov to Hefor all k∈V. This implies that Ysatisfies the extremal global Markov property with respect to He. □ B.13. Proof of Propositio... | https://arxiv.org/abs/2503.06536v1 |
define the DAG Gi,j:= (V, E m\{(i, j)}) and the collection of sets separating iandjinGi,jasS(m+1) i,j:={S⊆V\ {i, j}:i⊥ ⊥Gi,jj|S}. By defining S:=paGi,j(j)\ {i}, and since |paGm(j)| ≥2, using the same argument as in the base case, the collection of separating sets S(m+1) i,j ̸=∅. We now consider two cases. (a) Suppose (... | https://arxiv.org/abs/2503.06536v1 |
collection of nodes S⊆V\ {i, j}, i.e., Yi⊥eYj|YS, if and only if Wk i⊥ ⊥Wk j|Wk Sfor some k∈S(Engelke and Hitz, 2020). The marginal H¨ usler–Reiss exponent measure density λ(yA) for some A⊆Vis again H¨ usler–Reiss with parameter matrix Γ A,A. We define θ(A),p(A) and σ2(A) as in identity (10). Proof of Example 5. Leted=... | https://arxiv.org/abs/2503.06536v1 |
H¨ usler–Reiss precision matrix corre- sponding to the variogram Γ {i,j}∪C,{i,j}∪C, see (10). As this is the variogram of the {i, j}∪C-th H¨ usler–Reiss marginal, we obtain the result from Hentschel et al. (2024). (2) From Lemma 2 it follows that λ(yi, yj|yC) is the density of a bivariate Gaussian with conditional cova... | https://arxiv.org/abs/2503.06536v1 |
directed graphical model with respect to Ge. EXTREMES OF STRUCTURAL CAUSAL MODELS 45 Appendix D.Graphical representation of the river network True DAG τ=0.9 τ=0.95 τ=0.975 dHSIC PCMdataset−1 dataset−2 dataset−3 dataset−4 dataset−5 dataset−6 10111213101112131011121310111213101112131011121347.548.048.549.0 47.548.048.549... | https://arxiv.org/abs/2503.06536v1 |
arXiv:2503.06691v1 [math.PR] 9 Mar 2025Limit Theorems for One-Dimensional Homogenized Diffusion P rocesses Jaroslav I. Borodavka, Sebastian Krumscheid March 7, 2025 Abstract We present two limit theorems, a mean ergodic and a central li mit theorem, for a specific class of one- dimensional diffusion processes that depend ... | https://arxiv.org/abs/2503.06691v1 |
emergence of parametric and nonparamet- ric estimation procedures for homogenized equations like ( 1.1), but with observations coming from perturbed equations like (1.4). While this is an instance of model-mis specification, the laws of the processes are sim- ilar in a weak sense when ε→0; and, yet, it turns out that ma... | https://arxiv.org/abs/2503.06691v1 |
small parameter ε>0, dXε(t) =bε(Xε(t))dt+σ(Xε(t))dW(t), t>0, Xε(0) =x0, (2.1) 2 with suitable Borel-measurable functions bε:R→R,σ:R→R, and assume, for the moment, that there exists a unique, strong solution to this SDE on the given prob ability space. We suppose a setting where Xε converges weakly to another stochastic... | https://arxiv.org/abs/2503.06691v1 |
explode as ε→0. This makes the analysis substantially more difficult. Observe that an ordinary integral transformation leads to 1 Tε/integraldisplayTε 0ϕε(Xε(t))dt−/parenleftbigg/integraldisplay R1 σ(x)2f′ε(x)dx/parenrightbigg−1/integraldisplay Rϕε(x) σ(x)2f′ε(x)dx =1 Tε/integraldisplayTε 0ϕε(gε(ξε(t)))dt−1 Cρε/integrald... | https://arxiv.org/abs/2503.06691v1 |
to be modified. 5 Lemma 2.3. Letε>0be fixed. If/integraltext∞ 0ρε(x)2dx<∞, then for any x,y∈Rwithy<x Eτy(ξx ε) = 2(x−y)/integraldisplay∞ xρε(z)2dz+2/integraldisplayx y(z−y)ρε(z)2dz. (2.14) If/integraltext0 −∞ρε(x)2dx<∞, then for any x,y∈Rwithx<y Eτy(ξx ε) = 2(y−x)/integraldisplayx −∞ρε(z)2dz+2/integraldisplayy x(y−z)ρε(z... | https://arxiv.org/abs/2503.06691v1 |
0Eϕε(ξx ε(t))tdt−2 T2ε/integraldisplayTε 0Eϕε(ξy ε(t))tdt=2 T2ε/integraldisplayTε 0/integraldisplayTε sEϕε(ξx ε(t))−Eϕε(ξy ε(t))dtds =2 Tε/integraldisplayTε 0Eϕε(ξx ε(t))−Eϕε(ξy ε(t))dt−2 T2ε/integraldisplayTε 0E/integraldisplays 0ϕε(ξx ε(t))−ϕε(ξy ε(t))dtds. Define the sequence of stopping times τn:=τϕε,x n∧τϕε,y nthro... | https://arxiv.org/abs/2503.06691v1 |
Therefore, for sufficiently small ε>0we have with (2.13) /integraldisplay |xε−y|>Tερε(y)2dy≤/integraldisplay |y|>Tε−|xε|ρε(y)2dy≤/integraldisplay |y|>Tε−(cf∨Cf)|f(x0)|ρε(y)2dy<δ, and 1 Tε/integraldisplay |xε−y|≤Tε|xε−y|ρε(y)2dy=1 Tε/bracketleftBigg/integraldisplay |xε−y|≤Tε,|y|≤M|xε−y|ρε(y)2dy+/integraldisplay |xε−y|≤Tε,... | https://arxiv.org/abs/2503.06691v1 |
. Let us analyze the second term in detail. Equation (2.33), Fu bini’s theorem, and the triangle inequality enable us to estimate /integraldisplay |y|≤Mρε(y)2dyE/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayTε∧τn 0ϕε(ξxεε(t))dt/vextendsingle/vextendsingle/vextendsingle/vextends... | https://arxiv.org/abs/2503.06691v1 |
the following adjustment to the Assumptions (C). Assumptions (C*). i) The functions bandσare locally Lipschitz-continuous on Randσis strictly positive. ii) The function σis locally Lipschitz-continuous on R, and satisfies m≤σ≤MonRfor someM,m>0. For eachε>0andN∈Nthere exists a constant LN>0such that for all x,y∈[−N,N] |b... | https://arxiv.org/abs/2503.06691v1 |
process Xx εfromU. With these facts at our disposal, we may now state and prove the following lemma. Lemma 2.12. Assume (C*) ii). Then, for any fixed A∈B(R), the function (t,x)/ma√sto→Pε(x,t,A)is a solution of ∂tuε(t,x)−Aεuε(t,x) = 0,(t,x)∈D, (2.45) whereD⊂(0,T)×Uis an arbitrary subdomain with closure in (0,T]×U. Moreov... | https://arxiv.org/abs/2503.06691v1 |
of all relevant orders for any exponent α∈(0,1). At last, we can derive (2.46) which is a consequence of (2.41), since sup (t,x)∈(t0,T)×C|∂xPε(x,t,A)| ≤|d1,∂xPε(·,·,A)|α dist(∂C,∂U)∧√t0≤Kε−(10+η) dist(∂C,∂U)∧√t0. For the remainder of this subsection and the next subsection , we will need the following additional assump... | https://arxiv.org/abs/2503.06691v1 |
again and (2.65) as follows P/parenleftbig |Xx0ε(Tε)|>R(ε),τy−1(Xx0ε)≤Tε/parenrightbig =/integraldisplayTε 0Pε/parenleftbig y−1,Tε−s,BR(ε)(0)c/parenrightbig dPτy−1(Xx0ε)(s) ≤Kε+C(b,σ,M,γ,µ )exp(−γR(ε)) r(ε)µ(x0). Hence, asε→0 lim ε→0P(|Xx0ε(Tε)|>R(ε))≤δ, which yields the claim for the case x0∈Ic, as well. Remark 2.14. ... | https://arxiv.org/abs/2503.06691v1 |
(2.4). 21 Theorem 2.18. Let Assumptions (C*), (MET), and (CLT), and Tε=O(ε−η)asε→0withη >0hold. Assume that the functions hε∈L1(µε)are continuous and satisfy /integraldisplay Rhε(x)µε(x)dx= 0,sup ε>0/bardblhε/bardbl∞<∞. (2.77) Additionally, assume that τ2 ε:=/integraldisplay Rσ(x)2Φ′ ε(x)2µε(x)dx→/integraldisplay Rσ(x)... | https://arxiv.org/abs/2503.06691v1 |
dx. Observe that Zε−ZZ+=/integraldisplay Rdexp/parenleftBig −α 2σx2/parenrightBig/bracketleftbigg exp/parenleftbiggcos(x/ε) σ/parenrightbigg −1 2π/integraldisplay2π 0exp/parenleftbiggcos(y) σ/parenrightbigg dy/bracketrightbigg dx. 23 We will prove that |Zε−ZZ+| ≤C(α,σ)exp/parenleftbig −1/ε2/parenrightbig . (3.9) For th... | https://arxiv.org/abs/2503.06691v1 |
asymptotic proper ties of the minimum distance estimator forε→0. Proposition 3.2. LetTε=O(ε−η)asε→0for someη >0. Then, under the true parameter ϑ0=α0K withα0>0, asε→0 ˆϑTε(Xε)P−→ϑ0, (3.15) /radicalbig Tε(ˆϑTε(Xε)−ϑ0)D−→ N/parenleftbigg 0,4τ2 σ2ϑ2 0/parenrightbigg , (3.16) where τ2= 2σ/integraldisplay RΦ′(x)2µ(x)dx. (3.... | https://arxiv.org/abs/2503.06691v1 |
away the attenti on from the mean ergodic theorem and pursue an almost sure ergodic theorem. After all, if the ultimate ai m is the consistency or asymptotic normality of estimators, then it is irrelevant if one has the mean ergod ic or the almost sure ergodic theorem – even a stochastic ergodic theorem would suffice. 27... | https://arxiv.org/abs/2503.06691v1 |
diffusions. Multiscale Model. Simul. , 11(2):442–473, January 2013. doi:10.1137/110854485 . [11] Yury A. Kutoyants. Statistical inference for ergodic diffusion processes . Springer series in statistics. Springer, London, New York, 2004. doi:10.1007/978-1-4471-3866-2 . 28 [12] Theodoros Manikas and Anastasia Papavasiliou.... | https://arxiv.org/abs/2503.06691v1 |
Journal of Machine Learning Research 23 (2025) 1-31 Submitted -; Revised -; Published - BASIC: Bipartite Assisted Spectral-clustering for Identifying Communities in Large-scale Networks Tianchen Gao gaotianchen@stu.xmu.edu.cn Department of Statistics and Data Science, School of Economics, Xiamen University, Xiamen, Fuj... | https://arxiv.org/abs/2503.06889v1 |
(SupSBM) (Huang et al., 2020; Paul et al., 2023), motif tensor block model (MoTBM) (Yu and Zhu, 2024), and so forth. During network analysis, community detection is a crucial tool that focuses on iden- tifying closely connected groups within networks (Girvan and Newman, 2002; Newman, 2012; Jin, 2015). Jin (2015) propos... | https://arxiv.org/abs/2503.06889v1 |
DCBM and its bipartite modification BiDCBM. We handily integrate the adjacency matrices of both the primary and bipartite networks, then conduct an eigen- value decomposition toward the aggregated matrix, and apply the SCORE normalization to obtain a new ratio matrix. This crucial step automatically corrects for degree... | https://arxiv.org/abs/2503.06889v1 |
( i, j) if node iand node jare connected. Define the adjacency matrix of GtasA= (A(i1, i2))∈Rn×n, where i1, i2= 1, . . . , n , that is, A(i1, i2) =I((i1, i2)∈ Et). Without loss of generality, setA(i, i) = 0 for i= 1, . . . , n , indicating the absence of self-connections, although the proposed BASIC method can be easil... | https://arxiv.org/abs/2503.06889v1 |
primary network GtandQbipartite networks defined in section 2.1, the target is to identify the community structure of primary network, hence, it is important to find an efficient way to aggregate information from the primary and all bipartite networks, even if its signal strength is weak. Solving this problem is not st... | https://arxiv.org/abs/2503.06889v1 |
Step 2 : Apply the eigenvalue decomposition to Mand obtain the first Kleading eigenvectors ˆU= [ˆu1,ˆu2, . . . , ˆuK]∈Rn×K. Step 3 : Compute the ratio matrix ˆR∈Rn×(K−1)in (2). Step 4 : Apply the k-means algorithm to the columns of ˆR, and solve for N∗= argmin N∈Nn,K−1,K∥N−ˆR∥2 F, where Nn,K−1,Krepresents the set of n×... | https://arxiv.org/abs/2503.06889v1 |
ΩMcan be expressed as U¯i=θi ∥θ(li)∥J¯lifor 1⩽i⩽n, (6) 7 BASIC and∥U¯i∥ ≍θi/∥θ∥. Proposition 1 implies that the rank of original aggregated matrix ΩM∈Rn×nis at most K, and connects this n×nmatrix with a low-dimensional matrix ¯S∈RK×K. It also explains the rationale of applying the SCORE-type normalization (2) to ΩMupon... | https://arxiv.org/abs/2503.06889v1 |
(2020) and define W={1⩽i⩽n: N∗¯i−R¯i ⩽1/2 as the set of nodes that are accurately clustered by BASIC, and thus Vt\Wconsists of nodes that are mis-clustered. Theorem 1 establishes a non-asymptotic bound for the mis-clustering rate of BASIC. Theorem 1 Under Assumptions 1, 2 and 3, with probability at least 1−o n−4 , th... | https://arxiv.org/abs/2503.06889v1 |
combinations of node sizes and the number of communities, addressing both balanced and imbalanced community structures. To evaluate the clustering accuracy of BASIC, we calculate the Adjusted Rand Index (ARI) (Hubert and Arabie, 1985) that reflects the consistency between the clustering results and the inherent true la... | https://arxiv.org/abs/2503.06889v1 |
Recall that a larger βleads to a weaker signal. Then, we vary the out-in ratios of the 5 bipartite networks in the following four cases: •Case 1: 0 .5,0.5,0.5,0.5,0.5 (5 weak signals) •Case 2: 0 .1,0.5,0.5,0.5,0.5 (1 strong and 4 weak signals) •Case 3: 0 .1,0.1,0.5,0.5,0.5 (2 strong and 3 weak signals) •Case 4: 0 .1,0.... | https://arxiv.org/abs/2503.06889v1 |
BA- SIC does not deteriorate the community detection. These phenomena are observed in both balanced and imbalanced cases and are consistent with the theoretical results in Theorem 1. 12 BASIC 0.750.800.850.900.951.00 Baseline Case 1 Case 2 Case 3 Case 4 Balanced Community SizeAdjusted Rand Indexn = 600, m = 300, K = 3 ... | https://arxiv.org/abs/2503.06889v1 |
extract the 4-core of the collaboration network and obtain the largest connected component, resulting in a core network with 737 nodes and 2,453 edges, with a network density of 0.904%. This core network serves as the primary network for community detection. In addition, author information can help us obtain the corres... | https://arxiv.org/abs/2503.06889v1 |
and Ghent Univer- sity. The top-ranked author is Professor Peter J. Rousseeuw, a renowned statistician from KU Leuven, whose research focuses on robust statistics and cluster analysis. Professors Geert Molenberghs and Christophe Croux are among his doctoral students. 5.3 Community Structure and Collaboration Patterns I... | https://arxiv.org/abs/2503.06889v1 |
Robust estimation, Smoothing Mammen, Enno Bandwidth, Kernel estimator 10 29Ibrahim, Joseph G. Gibbs sampling, Missing data Lipsitz, Stuart R. EM algorithm, Generalized estimating equations Zeng, Donglin Markov Chain Monte Carlo, Semiparametric efficiency Ryan, Louise M. Longitudinal data, Missing at random Zhu, Hongtu ... | https://arxiv.org/abs/2503.06889v1 |
The density of this community is 21.2%, much higher than that of Community 2 (5.5%) and Community 5 (9.3%). This suggests that the collaboration among the authors in this community is more close. The representative authors of this community, Professors Ibrahim, Joseph G., Zhu, Hongtu, and Styner, Martin A. are all from... | https://arxiv.org/abs/2503.06889v1 |
the same community. In contrast, BASIC shows more balanced communities than SCORE. To provide a clearer comparison of the differences be- tween BASIC and SCORE, the subplot on the top right shows the community structure of the selected nodes. Visually, these nodes exhibit distinct community structures, but the SCORE me... | https://arxiv.org/abs/2503.06889v1 |
(10) can be rewritten as ( θi/∥θ(li)∥)J¯li, giv- ing (6). Moreover, recall Jis a square and orthogonal matrix, and we have ∥U¯i∥= (θi/∥θ(li)∥)J¯li =θi/∥θ(li)∥, implying ∥U¯i∥ ≍θi/∥θ∥. B.2 Proof of Lemma 1 Proof Denote the empirical version of the aggregated adjacency matrix as M=AA⊤+PQ q=1B(q)B(q)⊤. For notation simpli... | https://arxiv.org/abs/2503.06889v1 |
that for ¯i1,¯i2∈ W, N∗¯i1−N∗¯i2 = N∗¯i1−R¯i1+R¯i1−R¯i2+R¯i2−N∗¯i2 ⩾ R¯i1−R¯i2 − N∗¯i1−R¯i1+R¯i2−N∗¯i2 ⩾ R¯i1−R¯i2 − N∗¯i1−R¯i1 − N∗¯i2−R¯i2 ⩾2−1 2−1 2= 1 (by the definition of W). Then, by the same deduction as Theorem 1 in Wang et al. (2020), we can show |V\W| ≲log(n)ZT2 n θ2 min∥θ∥4 maxn maxQ q=1σmax F(q) , λmax... | https://arxiv.org/abs/2503.06889v1 |
show that the elements in the leading eigenvector of U are all positive. To see this, we only need to show C⩽j1(i)⩽1 for a constant C > 0. Then by the Expression (10), we can get the desired results. Note that j1is the leading eigenvector of ¯S, based on Lemma 7 in Wang et al. (2020), what we need to show is that ¯S is... | https://arxiv.org/abs/2503.06889v1 |
X i∈VU ˆR¯i−R¯i 2 ⩽CX i∈VU ˆU2∼K ¯i ˆu1(i)−(U2∼KOU′)¯i CUu1(i) 2 (by the definition of ˆR,Rand Lemma 5) ⩽CX i∈VU 1 (ˆu1(i))2 ˆU2∼K ¯i−(U2∼KOU′)¯i 2 +(CVu1(i)−ˆu1(i))2 (ˆu1(i)CUu1(i))2∥(U2∼KOU′)¯i∥2! ⩽CX i∈VU∥θ∥2 θ2 i ˆU2∼K ¯i−(U2∼KOU′)¯i 2 +∥θ∥2 θ2 i(CUu1(i)−ˆu1(i))2 (by Lemma 4) ⩽C∥θ∥2 θ2 min X i∈VU ˆU2... | https://arxiv.org/abs/2503.06889v1 |
and Abigail Z Jacobs. Efficiently inferring community structure in bipartite networks. Physical Review E , 90(1):012805, 2014. Jing Lei and Alessandro Rinaldo. Consistency of spectral clustering in stochastic block models. The Annals of Statistics , 43(1):215–237, 2015. Tianxi Li, Elizaveta Levina, and Ji Zhu. Network ... | https://arxiv.org/abs/2503.06889v1 |
THE LEVEL OF SELF-ORGANIZED CRITICALITY IN OSCILLATING BROWNIAN MOTION: n-CONSISTENCY AND STABLE POISSON-TYPE CONVERGENCE OF THE MLE BYJOHANNES BRUTSCHE1,aAND ANGELIKA ROHDE1,b 1Mathematical Institute, University of Freiburg,ajohannes.brutsche@stochastik.uni-freiburg.de, bangelika.rohde@stochastik.uni-freiburg.de For s... | https://arxiv.org/abs/2503.07022v1 |
is known. They especially establish stable convergence of their estimators towards Gaussian mixtures at√n-convergence rate. Likewise, when the level of self-organized crit- icality is known to the statistician, Mazzonetto (2024) proved very recently the convergence of suitable statistics to the local time. For so-calle... | https://arxiv.org/abs/2503.07022v1 |
1{z<0}log(α2/β2)N′((−z/α2)−) ,(1.5) where the constants bα,β,b′ α,β<0are given explicitly in (2.8). Note that ℓ(z)is the sum of a two-sided compensated Poisson process and a negative drift, where we have used the left- continuous version N′(•−)to ensure ℓbeing c `adl`ag. For statistical purposes, we even prove F-stabl... | https://arxiv.org/abs/2503.07022v1 |
have (ℓn(z/n))z∈[−K,K]F−st−→ (ℓ(zLρ0 1(X)))z∈[−K,K] in the Skorohod space D([−K,K ]). n-CONSISTENCY AND STABLE POISSON-TYPE CONVERGENCE OF THE MLE 5 Based on those results, the proof of Theorem 1.1 is then completed with argsup-continuous mapping type arguments. The proof of Proposition 1.2 adopts the M-estimation appr... | https://arxiv.org/abs/2503.07022v1 |
current formulation, this result does not apply to pro- cesses defined on discretized filtrations. We therefore adapt this result by combining it with the first one to bridge the gap between those two, see Proposition G.1. Based on this, prov- ing stable convergence of fidis is traced back to proving (uniform) stochast... | https://arxiv.org/abs/2503.07022v1 |
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