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=fY Y p AA f using associativity of copying, (22), almost sure determin ism off, and explicitly copying and discarding. Lemma A.2. LetCbe an a.s.-compatibly representable Markov category with c ondi- tionals. Ifp∈C(I,X), thensampXis(δX◦p)-a.s. deterministic. Proof.We want to show that the following equality holds pX X ... | https://arxiv.org/abs/2502.14941v1 |
Symmetric observations without symmetric causal explanations Christian William,1Patrick Remy,1Jean-Daniel Bancal,2Yu Cai,3Nicolas Brunner,1and Alejandro Pozas-Kerstjens1,∗ 1Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland 2Universit´ e Paris-Saclay, CEA, CNRS, Institut de physique th´ eor... | https://arxiv.org/abs/2502.14950v1 |
set of two out of the three visible variables A,BandC. This structure has received considerable attention in the context of nonlocal- ity in quantum networks. (b) Pictorial representation of the relations between realizations and observations. In this work we demonstrate that ?=✗. Namely, there exist probability distri... | https://arxiv.org/abs/2502.14950v1 |
on, we will use the shorthand notation p(a, b, c ) =p(A=a, B=b, C=c). Consider now the symmetric version of the scenario, where the three latent nodes now distribute copies of the same state, ω, and all the parties perform the same effect, e(see Fig. 2a). The model for distributions observed in this situation has the f... | https://arxiv.org/abs/2502.14950v1 |
given in Eq. (3) with E1=Ec 1≈0.1753 (the exact form of Ec 1is given in Appendix A) and E2= −1/3. For these values of E1andE2, positivity of Eq. (3) for any a, b, c fixes E3=Ec 3≈ −0.5260. However, our proof technique does not use this information, so when we consider different values of E1andE2later on, the proofs wil... | https://arxiv.org/abs/2502.14950v1 |
implies that X a0,a2pinf(a0, a1, a2, a3, a4, a5, a6) =1 2(1 +a1Ec 1)X a′ 0,a′ 1,a′ 2pinf(a′ 0, a′ 1, a′ 2, a3, a4, a5, a6) ∀a1, a3, a4, a5, a6, X a0,a3pinf(a0, a1, a2, a3, a4, a5, a6) =1 4 1 + (a1+a2)Ec 1−1 3a1a2 ×X a′ 0,a′ 1,a′ 2,a′ 3pinf(a′ 0, a′ 1, a′ 2, a′ 3, a4, a5, a6) ∀a1, a2, a4, a5, a6. (5) Equations (4) and... | https://arxiv.org/abs/2502.14950v1 |
part due to the fact that it relaxes the sets of correlations under study to forms amenable to linear or semidefinite programming [30], that can be solved in many cases with standard com- putational resources [30, 31]. Moreover, it is possible to define hierarchies of inflation relaxations [26, 32, 33], that in some ca... | https://arxiv.org/abs/2502.14950v1 |
(7) and running the respective linear programs, we are able to determine the infeasibility of distributions for E2=−1/3 only for E1≥ 0.1656 using the 15th level of the hierarchy. In exchange, we obtain a witness of incompatibility with symmetric realizations in the triangle, which is the upper boundary to the dark gree... | https://arxiv.org/abs/2502.14950v1 |
GB of RAM and one day of compute we are able to identify as incompatible also the point denoted with a red circle ( E1= 0.1580 and E2=−1/3) using the 15th level of the hierarchy. The dark green area contains the distributions that are identified as incompatible by the certificate obtained by substituting the constraint... | https://arxiv.org/abs/2502.14950v1 |
f a= 1 0 1 0 0 0 1 1 1 , f b= 1 1 0 0 1 0 0 0 0 , f c= 0 1 0 1 1 0 0 0 0 , where xis the root between 0 and 1 for 3 x4−9x3+ 9x2−5x+ 1 = 0, i.e., x=3 4−1 2vuuuut1 2+2 33s 2√ 41 + 3−1 33r 1 2√ 41 + 3 +13 2r 3 3−83q 2√ 41+3+ 25/33p√ 41 + 3 +1 4r 3 3−83q 2√ 41+3+25/33√√ 41+3 andy=1 3(2x2−2x+1). Append... | https://arxiv.org/abs/2502.14950v1 |
minimal triangle scenario, New J. Phys. 25, 113037 (2023), arXiv:2305.03745. [12] C. J. Wood and R. W. Spekkens, The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning, New J. Phys. 17, 033002 (2015), arXiv:1208.4119. [13] J. Henson, R. ... | https://arxiv.org/abs/2502.14950v1 |
and D. Gross, A convergent inflation hierarchy for quantum causal structures, Commun. Math. Phys. 401, 2673 (2023), arXiv:2110.14659. [34] L. T. Ligthart and D. Gross, The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario, J. Math. Phys. 64, 072201 (2023), arXiv:2212.11299... | https://arxiv.org/abs/2502.14950v1 |
arXiv:2502.14988v1 [math.CO] 20 Feb 2025Partial and Exact Recovery of a Random Hypergraph from its Graph Projection Guy Bresler∗, Chenghao Guo†, Yury Polyanskiy‡, Andrew Yao§ February 24, 2025 Abstract Consider a d-uniform random hypergraph on nvertices in which hyperedges are included iid so that the average degree is... | https://arxiv.org/abs/2502.14988v1 |
posterior distribution given the projected grap h [21, 14]. In [6] a large foundational model is used to recover a weighted hypergrap h from a sample of its hyperedges, where each hyperedge is assigned a probability propo rtional to its weight. Theoretically, interest in reconstructing a hypergraph from its pr ojection... | https://arxiv.org/abs/2502.14988v1 |
2)×(n d)is a fixed (non-random) matrix encoding the incidence relation between edges and hyperedg es,H∈ {0,1}(n d)×1is theiidBernoulliplantedhypergraph, and W∈Z(n 2)×1 ≥0istheobservedweightedprojection. (For the unweighted projection, we have W∈ {0,1}(n 2)×1and the problem is a generalized linear inverse problem: min( A... | https://arxiv.org/abs/2502.14988v1 |
the definition of ambiguous graphs in Subsection 2.6. The contributions of this paper as compared to previously establish ed exact recovery thresholds for d≥4 are summarized in Table 1. dPrevious work (exact recovery) New (exact recovery) New (partial recovery) 3 2/5 2/5 1/2 4 [1/2,4/7] 4/7 3/5 5 [1/2,2/3] 2/3 2/3 ≥6 [d... | https://arxiv.org/abs/2502.14988v1 |
the loss of BWis ℓW(BW)/definesE[|BW(ProjW(H))∆E(H)|] p/parenleftbign d/parenrightbig . The optimal weighted partial recovery algorithm B∗ Wis defined analogously to B∗. 5 Definition 1.6. Thepartial recovery loss isℓ(B∗). Ifℓ(B∗) =on(1), then almost exact recovery is possible. Theweighted partial recovery loss isℓW(B∗ W)... | https://arxiv.org/abs/2502.14988v1 |
H. Theorem 2.2. q= (1+on(1))(p+/parenleftbiggcnδ−1 (d−2)!/parenrightbigg(d 2) ). Corollary 2.3. q= (1+on(1))pifδ <d−1 d+1, p+Θn(p)ifδ=d−1 d+1, ωn(p) ifδ >d−1 d+1. The following theorem tells us that BCachieves a partial recovery loss of on(1) when- everδis belowd−1 d+1, because BCfalsely recoversq p−1 =on(1) of t... | https://arxiv.org/abs/2502.14988v1 |
the case, we must show that Pe= (1−on(1))psinceℓ(B∗) =Pe p; then, using theprevious chainof inequalities, we requirethat H(H|Ψ) = (1−on(1))H(H). Wewill see in Theorem F.3 or Lemma F.5 that H(Ψ) = Ω n(H(H)), i.e.H(H|Ψ) = (1−Ωn(1))H(H). So we cannot extend the argument. In the next section, we explain how we circumvent t... | https://arxiv.org/abs/2502.14988v1 |
defi ne an important ambiguous graph which the paper discovers, see Definition 2.11. Definition 2.10. ForG∈ G, we refer to a preimage HofGthat minimizes e(H) as a minimal preimage ofG. The graph Gisambiguous if it has at least two distinct minimal preimages. Also, the graph Gisweighted-ambiguous if it has at least two dis... | https://arxiv.org/abs/2502.14988v1 |
Suppose d≥3andhis a preimage of an ambiguous graph. Then, d−1−1 m(h)≥2d−4 2d−1. Proof.Thed= 3 case is resolved in [5, Appendix D]. The result is proved in a slightly more general setting for d= 4 and d≥5 in the sense that the projected graph does not need to be ambiguous in Theorem E.4 and Theorem E.5, respectively. Es... | https://arxiv.org/abs/2502.14988v1 |
Projections. In Proceedings of Thirty Seventh Conference on Learning Theory , volume 247 of Proceedings of Machine Learning Research, pages 632–647. PMLR, 2024. [6] Yang Chen, Cong Fang, Zhouchen Lin, and Bing Liu. Relational lear ning in pre- trained models: a theory from hypergraph recovery perspective . InProceeding... | https://arxiv.org/abs/2502.14988v1 |
edges in some G∈ G′is (1+ǫ)/parenleftbigd 2/parenrightbig p/parenleftbign d/parenrightbig . Therefore, |G′| ≤(1+ǫ)/parenleftbiggd 2/parenrightbigg p/parenleftbiggn d/parenrightbigg/parenleftbigg/parenleftbign 2/parenrightbig (1+ǫ)/parenleftbigd 2/parenrightbig p/parenleftbign d/parenrightbig/parenrightbigg , ifnis suffic... | https://arxiv.org/abs/2502.14988v1 |
in at least khyperedges ison(1). 18 Suppose kis sufficiently large. Then, set Z′to be the set of H∈ Zsuch that each edge{i,j}is contained in less than khyperedges. We have that Pr[ H ∈ Z′] = 1−on(1), so we can use Z′in place of Zby Lemma 2.8. We can now essentially use the same proof as Theorem 2.7, of course after repla... | https://arxiv.org/abs/2502.14988v1 |
The partial recovery loss is 1−on(1). Proof.This follows from Lemma 2.1 and Theorem B.1. B.2 Correlation inequality ItisinfactpossibletoextendtheresultsfromTheorem2.7fromone hyperedgetomultiple hyperedges. As explained in the proof of the following result, the ca sek= 1 is equivalent to Theorem 2.7. Theorem B.3. Suppos... | https://arxiv.org/abs/2502.14988v1 |
Usatisfies the conditions of Lemma C.2; the lemma implies that an upper bound on p|U|/producttext u∈U/parenleftbign−k d−|u|/parenrightbig is an upper bound on the probability that E ⊂ E(Proj(H)) after scaling by some constant since the number of Uis finite. First observe that |E| ≤/summationtext u∈U,|u|≥2/parenleftbig|u|... | https://arxiv.org/abs/2502.14988v1 |
vertex yi= 1 for 1 ≤i≤M−1 andyM=/parenleftbigd 2/parenrightbig −M+1. The value offat this vertex is g(M) :=M(1−δ)+2−1+/radicalig 8(/parenleftbigd 2/parenrightbig −M+1)+1 2 and max y∈RMf(y) =g(M). Observe that gis convex in Mover [1,/parenleftbigd 2/parenrightbig ]. Hence, the maximum value of gforM∈[1,/parenleftbigd 2... | https://arxiv.org/abs/2502.14988v1 |
Sof[1,2d−k] such that 2≤ |S| ≤d. Assume that M/uniondisplay i=1/parenleftbiggKi 2/parenrightbigg ⊃/parenleftbigg/parenleftbigg[d] 2/parenrightbigg/uniondisplay/parenleftbigg[k]∪{i:d+1≤i≤2d−k} 2/parenrightbigg/parenrightbigg \/parenleftbigg[m] 2/parenrightbigg Then (1+δ)M−M/summationdisplay i=1|Ki| ≤k−m+(d(d−1)−m(m−1))(... | https://arxiv.org/abs/2502.14988v1 |
prove that −4 d+1+4−d−1+√ 1+4d2−4d−4k2+4k 2≤k−2d(d−1) d+1 ⇔2d2−2d−4 d+1+4−d−k−1 2≤√ 1+4d2−4d−4k2+4k 2 ⇔2d−2k−1≤√ 1+4d2−4d−4k2+4k⇔8k2≤8dk, which follows from 0 ≤k≤d−1. Next, suppose M=/parenleftbigd 2/parenrightbig −/parenleftbigk 2/parenrightbig −/parenleftbigm 2/parenrightbig +1. Then, yM= 1 so the required inequality... | https://arxiv.org/abs/2502.14988v1 |
decrease but all of the edges of Ewill remain covered. Hence, we can assume that no element of [ d−1] is contained in two elements of Sa. We can similarly assume that no element of [ d−1] is contained in two elements of Sband that no element of [ d−1] is contained in two elements of Sab. Suppose k∈Saand|k| ≥3. Suppose ... | https://arxiv.org/abs/2502.14988v1 |
of such UbyP. Suppose k > m. From Lemma C.7, for U ∈ Pwe have that (1+δ)|U|−/summationdisplay u∈U|u|< k−m+(d(d−1)−m(m−1))(δ−1), where the equality case of the lemma cannot occur. Using (11) then gives that Pr[a,b∈X] =on(nk−m+(d(d−1)−m(m−1))(δ−1)). Suppose k=m. From Lemma C.7, for U ∈ Pwe have that (1+δ)|U|−/summationdi... | https://arxiv.org/abs/2502.14988v1 |
using two-connected components. Furthermore, when d≥5 the probability of exact recovery is on(1) ifδ >2d−4 2d−1≥d−1 d+1from Theorem 1.1. Hence the main contribution of this result is proving tha t the probability of exact recovery is on(1) ifδ >2d−4 2d−1andd= 3,4. Proof of Theorem D.4. Suppose δ≥2d−4 2d−1. It suffices to... | https://arxiv.org/abs/2502.14988v1 |
each hyperedge in Ehis a subset of U. Suppose Pis a set of edges such that for all {a,b} ∈ P,{a,b} ⊂Uand there exists i∈Isuch that {a,b} ⊂i. If (d−1)|Eh|−|U|+|P| |Eh|+|P|≥γ, thend−1−1 m(h)≥min(γ,1). 37 Proof.Assume that (d−1)|Eh|−|U|+|P| |Eh|+|P|≥γ. For alli∈I, letxi=|i∩U|. LetZbe the set of i∈Isuch that xi≤1. LetVbe t... | https://arxiv.org/abs/2502.14988v1 |
e(h)≥e(g). Note that Gis not necessarily ambiguous (despite the title of this section) andgis not necessarily minimal. In this section we derive lower bounds for d−1−1 m(h) ford≥4. LetEhbe the set of hyperedges in hbut notgandEgbe the set of hyperedges in g but noth. Furthermore, let Ibe the set of hyperedges in both h... | https://arxiv.org/abs/2502.14988v1 |
w∈isuch that v∝\e}a⊔io\slash=w,{v,w} ∈ Eh\Egbecausev∈Vh. Therefore, d∗ h(v)≥d−1. Similarly, if v∈Vg, d∗ g(v)≥d−1. Hence, (15) gives that 2|P| ≥(d−1)|Vh|+(d−1)|Vg|+/summationdisplay v∈VI(2−1{nh(v) = 0}−1{ng(v) = 0}) = (d−1)(|Vh|+|Vg|)+2|VI|−Nh−Ng.(19) We have that d(d−1)|Eh|+2|P| ≥(d−1)|U| 2+(d+3)|VI| 2+(d−1)(|Vh|+|Vg|)... | https://arxiv.org/abs/2502.14988v1 |
Ifκis the subgraph of hinduced by the vertices of a,b,c,e, andi,e(κ)≥5 andv(κ)≤12 so e(κ) v(κ)≥5 12>7 17. Assume that w /∈ V. Then, no element of Sis covered by an element of {a,b,c,i}. By (∗2), eachoftheelements of Smust becovered byadistinct element of( Eh∪I)\{a,b,c,i}, otherwise two elements of {u1,u2,u3}will be con... | https://arxiv.org/abs/2502.14988v1 |
there exists e,f,j∈Eh∪Isuch that either {w1,w2,u1} ⊂e, {w1,u2} ⊂f, and{w2,u2} ⊂jor{w1,w2,u2} ⊂e,{w1,u1} ⊂f, and{w2,u1} ⊂j. Withoutlossofgenerality, suppose e,f,j∈Eh∪Isatisfytheconditionthat {w1,w2,u1} ⊂ e,{w1,u2} ⊂f, and{w2,u2} ⊂j. Ifκis the subgraph of hinduced by the vertices of a, b,c,e,f,j, andi, thene(κ)≥7 andv(κ)... | https://arxiv.org/abs/2502.14988v1 |
v(κ)≥13 31>7 17. Assume that w′ 1,w′ 2/∈ V′. Then, e′,f′,j′,k′,l′/∈ {a,b,c,e,f,j,k,l,i }. Thus, if κis the subgraph of hinduced by the vertices of a,b,c,e,e′,f,f′,j,j′,k,k′,l,l′, andi, e(κ)≥14 andv(κ)≤34 so e(κ) v(κ)≥14 34=7 17. This proves that m(h)≥7 17. Theorem E.5. Supposed≥5. Then,d−1−1 m(h)≥2d−4 2d−1. Proof.For t... | https://arxiv.org/abs/2502.14988v1 |
We have that |t(v)| ≤/summationdisplay ih∈Eh:v∈ih|ih∩t(v)| ≤(d−1)dh(v). Similarly, |t(v)| ≤(d−1)dg(v). Hence, (d−1)(dh(v)+dg(v))≥2|t(v)|. We then have that (d−1)/summationdisplay v∈T(dh(v)+dg(v))≥2/summationdisplay v∈T|t(v)|. Suppose u∈ V∗∗. We have that ih(u)∩ig(u)∩Tis nonempty by the definition of V∗∗. Ifv∈ih(u)∩ig(u)... | https://arxiv.org/abs/2502.14988v1 |
Leth′be the subgraph of hinduced by V′. Furthermore, let h′′be the graph with vertex set V′and edge set E′ h∪I′. We have that d−1−1 m(h)≥d−1−1 α(h′)≥d−1−1 α(h′′) so it suffices to prove that d−1−1 α(h′′)≥2d−4 2d−1. Using Lemma E.1 gives that to prove that d−1−1 α(h′′)≥2d−4 2d−1, it suffices to prove that (d−1)|E′ h|−|U′|+|... | https://arxiv.org/abs/2502.14988v1 |
i∝\e}a⊔io\slash=j, letX{i,j}=1{{i,j} ∈E(Ψ)}. We have that H(Ψ) =H(X{i,j}: 1≤i < j≤n)≤/summationdisplay 1≤i<j≤nH(X{i,j}) =/parenleftbiggn 2/parenrightbigg HB(1−(1−p)(n−2 d−2)).(29) Bernoulli’s inequality implies that 1−(1−p)(n−2 d−2)≤/parenleftbiggn−2 d−2/parenrightbigg p and for sufficiently large nwe have that/parenleft... | https://arxiv.org/abs/2502.14988v1 |
a= 1 toa=/parenleftbign 2/parenrightbig gives H(Ψ)≥(1−/parenleftbign d−2/parenrightbig p 1−/parenleftbign d−2/parenrightbig p)n−d+2/summationdisplay i=2(i−1)HB(1−(1−p)(n−i d−2)). Observe that we use the fact that if i > n−d+ 2 then HB(1−(1−p)(n−i d−2)) = 0. It is straightforward to check that this lower bound on H(Ψ) i... | https://arxiv.org/abs/2502.14988v1 |
arXiv:2502.15024v2 [cs.CC] 28 Apr 2025Low degree conjecture implies sharp computational thresholds in stochastic block model/asterisk.math Jingqiu Ding†Yiding Hua†Lucas Slot†David Steurer† April 29, 2025 Abstract We investigate implications of the (extended) low-degree conjecture (recently formal- ized in [ MW23 ]) in ... | https://arxiv.org/abs/2502.15024v2 |
. . . . . . . . . . . . . . . . . . . . . . . . 18 A.2 Proof of Lemma A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 A.3 Proof of Lemma A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 A.4 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.15024v2 |
,Mas14 ,AS15 ], while no polynomial-time algorithms have been discovered b elow this threshold despite significant research effort. This computational threshold i s known as the Kesten-Stigum threshold (KS threshold) in the statistical physics litera ture [ DKMZ11 ], and it is an impor- tant topic in both probability the... | https://arxiv.org/abs/2502.15024v2 |
[ BHK+19, KMOW17 ,JPR+22,MRX20 ], statistical queries [ BKW03 ,Fel17 ,BBH+20], or low-degree polynomials [ HKP+17,HS17 ,Hop18 ,KWB19 ]. As it appears that significant technical barriers have to be overcome to prove lower bounds against th e former two classes for average-case problems with sharp statistical threshold, w... | https://arxiv.org/abs/2502.15024v2 |
computational-statistical gap similar to weak recovery can also be observed in the close ly related problem of learning the parameters of the stochastic block model [ Xu17 ]. Although [ LG24 ] established a low- degree recovery lower bound for learning the probability ma trix in the symmetric SBM, their result does not... | https://arxiv.org/abs/2502.15024v2 |
ecovery (which is to say, achieving constant recovery rate), their lower bound direc tly rules out estimators based on degree /u1D45B/u1D6FFpolynomials, which captures many natural candidates of alg orithms. Such lower bound is beyond reach with techniques from [ Hop18 ,BBK+21a]. For this, they introduce significantly n... | https://arxiv.org/abs/2502.15024v2 |
give evidence of hardness for polynomial-time algorit hms, the polynomial degree ℓneeds to be taken as large as log (/u1D45B). Therefore, their lower bound on the error rate can only match the guarantees of existing al gorithm up to logarithmic factors. As a result, they cannot give evidence of a computat ional-statist... | https://arxiv.org/abs/2502.15024v2 |
the low-degree lower bound for the stochastic block model: Theorem 3.1 (Low-degree lower bound for SBM, Thm. 2.20 in [ BBK+21a]).Let/u1D451=/u1D45C(/u1D45B),/u1D458= /u1D45B/u1D45C(1)and/u1D>00∈ [0,1]. Let/u1D>0>:{0,1}/u1D45B×/u1D45B→ℝbe the relative density of SSBM (/u1D45B,/u1D451,/u1D>00,/u1D458)with respect to/u1D4... | https://arxiv.org/abs/2502.15024v2 |
for a contradiction that we have a polynomial-time r ecovery algorithm for SSBM(/u1D45B,/u1D451 /u1D45B,/u1D>00,/u1D458),/u1D>002/u1D451/lessorequalslant0.99/u1D4582, that achieves recovery rate /u1D6FF/greaterorequalslantΩ(/u1D45B−0.49), in the sense ofDefinition 1.2 . Using this algorithm, we will construct a function... | https://arxiv.org/abs/2502.15024v2 |
/u1D4A6defined in ( 4.1) using the correlation preserving projection. 4. Return /u1D453(/u1D44C)=/u1D7D9{/a\}brack⌉tl⌉{tˆ/u1D440,/u1D44C 2−/u1D>02/u1D451 /u1D45B/u1D7D9/u1D7D9⊤/a\}brack⌉tri}ht/greaterorequalslant/u1D45B0.51}, where /u1D44C2is the adjacency matrix of /u1D43A\/u1D438(/u1D43A1). Lower bound on the expectat... | https://arxiv.org/abs/2502.15024v2 |
probability, the error rate /bar⌈blˆ/u1D>03−/u1D>03◦/bar⌈bl2 F/lessorequalslant0.99/u1D458/u1D451, then an algorithm with running time exp/parenleftbig/u1D45B0.99/parenrightbigcan achieve weak recovery when /u1D>002/u1D451/greaterorequalslant0.99/u1D4582. The key observation is that, for the symmetric stochastic block ... | https://arxiv.org/abs/2502.15024v2 |
equivalent , ArXiv abs/2009.06107 (2020). 2 13 [BBK+21a] Afonso S Bandeira, Jess Banks, Dmitriy Kunisky, Christ opher Moore, and Alex Wein, Spectral planting and the hardness of refuting cuts, colora bility, and communities in random graphs , Conference on Learning Theory, PMLR, 2021, pp. 410–473. 1,2,3,4,5,7,8 [BBK+21... | https://arxiv.org/abs/2502.15024v2 |
and David Steurer, The power of sum-of-squares for detecting hidden structures , 2017 IEEE 58th Annual Symposium on Foundations of Com- puter Science (FOCS), IEEE, 2017, pp. 720–731. 2 [HLL83] Paul W Holland, Kathryn Blackmond Laskey, and Samue l Leinhardt, Stochastic blockmodels: First steps , Social networks 5(1983),... | https://arxiv.org/abs/2502.15024v2 |
Wein, Sharp phase transitions in estimation with low-degree polynomials , arXiv preprint (accepted to STOC 2025) (2025). 4 16 [Tie24] Stefan Tiegel, Improved hardness results for learning intersections of ha lfspaces , arXiv preprint arXiv:2402.15995 (2024). 2 [Xu17] Jiaming Xu, Rates of convergence of spectral methods... | https://arxiv.org/abs/2502.15024v2 |
we can project the estimator into t he set of matrices with bounded entries and bounded nuclear norm, while preserving correlation. Lemma A.5. Let/u1D440◦∈ {−1//u1D458,1−1//u1D458}/u1D45B×/u1D45Bbe a symmetric matrix with rank- (/u1D458+1). For any /u1D6FF/lessorequalslant/u1D442(1), given matrix ˆ/u1D4400such that /a\... | https://arxiv.org/abs/2502.15024v2 |
bounded by /u1D442(1//u1D6FF)for/u1D456/lessorequalslant/u1D457. Moreover, we have /summationdisplay.1 /u1D456,/u1D457ˆ/u1D440(/u1D456, /u1D457)2/u1D53C/bracketleftbig˜/u1D44A2(/u1D456, /u1D457)2/bracketrightbig/lessorsimilar/u1D451 /u1D45B/summationdisplay.1 /u1D456,/u1D457ˆ/u1D440(/u1D456, /u1D457)2/lessorequalslant/... | https://arxiv.org/abs/2502.15024v2 |
we have /a\}brack⌉tl⌉{tˆ/u1D>03−/u1D451 /u1D45B1 1⊤,/u1D440◦/a\}brack⌉tri}ht/greaterorequalslant/u1D>00/u1D451 2√ /u1D458/bar⌈bl/u1D440◦/bar⌈blF− /bar⌈bl/u1D440◦/bar⌈blF√ 0.99/u1D458/u1D451 2/greaterorequalslantΩ/parenleftbigg /u1D>00/u1D451/bar⌈bl/u1D440◦/bar⌈blF√ /u1D458/parenrightbigg . On the other hand, by triangl... | https://arxiv.org/abs/2502.15024v2 |
GW(ˆ/u1D44A,/u1D44A 0)=/uniB.dsp1 0/uniB.dsp1 0(ˆ/u1D44A(/u1D465,/u1D466)−/u1D44A0(/u1D465,/u1D466))2/u1D451/u1D465/u1D451/u1D466. Therefore, the function /u1D453(·)can be evaluated in polynomial time. This contradicts the low-degree lower bound ( Theorem 3.1 ) assuming Conjecture 1.3 . /square C Low-degree recov... | https://arxiv.org/abs/2502.15024v2 |
/u1D44B◦ 12(/u1D456, /u1D457)=1 if/u1D456, /u1D457belongs to the same community and /u1D44B◦ 12(/u1D456, /u1D457)=−1 if/u1D456, /u1D457belongs to different communities. Moreover, we let /u1D44B◦ 1 be the community matrix for the induced subgraph on /u1D4461and let/u1D44B◦ 2be the community matrix for the induced subgrap... | https://arxiv.org/abs/2502.15024v2 |
universal constant. Taking /u1D461/greaterorequalslant1000 log(/u1D45B), we have the claim. /square D.2 Decoupling edge partition In this section, we give a lemma that describes the approxima te independence between edge sets of the subsampling process. Lemma D.3. Let/u1D44B∼Ber(/u1D45D), and let /u1D44B1be obtained fr... | https://arxiv.org/abs/2502.15024v2 |
we can get a more general bound (with respect to /u1D458) on low degree likelihood ratio of k-SBM. The proof of the exte nded result follows trivially from the proof of Theorem 2.20 of [ BBK+21b]. Therefore, we only provide a proof sketch by pointing out the simple modifications that we need from the original proof. The... | https://arxiv.org/abs/2502.15024v2 |
Extreme Value Analysis based on Blockwise Top-Two Order Statistics Axel B¨ ucher∗Erik Haufs† February 24, 2025 Abstract Extreme value analysis for time series is often based on the block maxima method, in particular for environmental applications. In the classical univariate case, the latter is based on fitting an extr... | https://arxiv.org/abs/2502.15036v1 |
. 19 6.3. Bootstrap approximations for the top-two estimator . . . . . . . . . . . . . . 20 7. Case Study 21 8. Conclusion 22 A. Proofs for Section 3 24 B. Proofs for Section 4 30 B.1. Disjoint Blocks: Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . 30 B.2. Sliding Blocks: Proof of Theorem 4.8 . . . . ... | https://arxiv.org/abs/2502.15036v1 |
found that estimators based on block maxima may be made more efficient by considering sliding rather than disjoint block maxima, both in the univariate [BS18a; BZ23] and in the multivariate case [ZVB21; BS24b], or, in the iid case, by even considering all block maxima [OZ20]. The current paper is motivated by yet anoth... | https://arxiv.org/abs/2502.15036v1 |
of this paper are organized as follows: some mathematical prelimi- naries on limit results for large order statistics are provided in Section 2. The limit results give rise to a pseudo maximum likelihood estimator, which is studied mathematically in Section 3 for general observation schemes. The theory is then speciali... | https://arxiv.org/abs/2502.15036v1 |
later sections. We collect some important properties of the Fr´ echet-Welsch-distribution. Remark 2.2 (The Fr´ echet-Welsch-distribution) . [a] Marginal distributions. The first marginal distribution of Hρ,α,σ is the Fr´ echet( α, σ)- distribution, that is, its cdf is given by H(1) ρ,α,σ(x) := exp −x σ−α . (2.5) Th... | https://arxiv.org/abs/2502.15036v1 |
dz=−c̸=−1, the associated Welsch-distribution does not have a Lebesgue density. One can show that this ρ-function appears in the classical ARMAX(1)- model, defined by the recursion ξt= max {(1−c)ξt−1, cZt}with ( Zt)tiid standard Fr´ echet. We will reconsider a version of this example in the simulation study. 3. Maximum... | https://arxiv.org/abs/2502.15036v1 |
Υρ0(x) :=ρ0Γ(x+ 2) + (1 −ρ0)Γ(x+ 1). (3.7) Moreover, for each ρ0∈[0,1], the function y7→Πρ0(y)is a continuous decreasing bijection from (0,∞)toRwith Πρ0(1)≤0, which allows to define ϖρ0:= UniqueZero( y7→Πρ0(y))∈(0,1]; (3.8) see Figure 2 for the graph of ρ07→ϖρ0. We have ϖρ0= 1if and only if ρ∈ {ρ⊥ ⊥,0}. Addi- tionally,... | https://arxiv.org/abs/2502.15036v1 |
using the models described in Section 6. It was found that the estimator did not perform better than the bias-corrected version of (ˆ αn,ˆσn) proposed in Section 3.3. We are therefore not pursuing this any further. 3.2. Asymptotic Distribution of the ML Estimator We formulate conditions under which (ˆ αn,ˆσn/σn), after... | https://arxiv.org/abs/2502.15036v1 |
+oP(1)⇝Mbc ρ0(α0)W, (3.21) where, recalling s1from (3.13) , Mbc ρ0(α0) = 1/ϖρ0 0 −(α1s1)−1log(s1) 1/s1! Mρ0(α0)∈R2×4(3.22) withMρ0(α0)from Theorem 3.7 and from (3.13) . Ifρ0= 1, we have Mbc 1(α0) =M1(α0)as in(3.19) . 10 4. Top-Two Order Statistics Extracted from a Stationary Time Series Throughout this section, we ... | https://arxiv.org/abs/2502.15036v1 |
the smallest Srn,imay be smaller than c, which we will prevent from hap- pening with the following condition. As shown in Lemma B.1, the condition, together with the max-domain of attraction condition, will also imply the no-tie condition in Lemma 3.1. 11 Condition 4.2 (All second largest order statistics diverge) .For... | https://arxiv.org/abs/2502.15036v1 |
(4.7) withα1from (3.9) ands1from (3.13) . Here, Mρ0(α0)∈R2×4is as in Theorem 3.7, W(db) n= (G(db) nf1,G(db) nf2,G(db) nf3,G(db) nf4)⊤,B= (B(f1), B(f2), B(f3), B(f4))⊤, withfjfrom (3.16) , and Σ(db) ρ,α0= (σ(db) ij)4 i,j=1has entries σ(db) ij= Cov (X,Y)∼W(ρ,α0,1)(fi(X, Y), fj(X, Y)). Ifρ=ρ⊥ ⊥, we have α1=α0, s1= 1,σ(db)... | https://arxiv.org/abs/2502.15036v1 |
whose bivariate cdfs needed for evaluating the covari- ance are given by Kρ,α0,ζfrom (B.12) . Ifρ=ρ⊥ ⊥, we have α1=α0, s1= 1,σ(db) ij= 2sij(α0) withsij(α)from Lemma E.2, and Mρ0(α0) =M1(α0)is explicitly given in (3.19) . 4.3. Bias-corrected estimation The inconsistency of the disjoint and sliding blocks MLE can be reso... | https://arxiv.org/abs/2502.15036v1 |
convergence of the largest order statistic only [Col01, Theorem 3.5]. In addition, the mean vector of the asymptotic normal distributions in Section 4 can be made explicit provided a standard second order condition on the weak convergence of affinely standardized maxima is met. More specifically, recall that Fis in the... | https://arxiv.org/abs/2502.15036v1 |
(5.6) is inherent to the top-two estimator: it results from a Taylor expansion of the logarithm that is needed within the proofs when dealing with empirical means of the second largest order statistics. If A(ar) =o(1/r), the second condition with λ2̸= 0 implies the first convergence with λ1= 0. For A(ar) of the exact o... | https://arxiv.org/abs/2502.15036v1 |
where ( Zt)tis iid standard Fr´ echet, and let ξt= F−1 α(−1/log˜ξt). It can be shown that ξthas cdf Fα, and that Condition 4.1 is met with ρ(η) = min(1 −β,1−η) and α0=α; see also Example 2.3 [d]. (3) The AR-Pareto-model: for β∈(0,1], let ˜ξtbe a stationary solution of the recursion ˜ξt=β˜ξt−1+Zt, where ( Zt)tis iid sta... | https://arxiv.org/abs/2502.15036v1 |
following insights: first, the sliding blocks top-two estimator is the best estimator in all scenarios under consideration. Second, each of the sliding blocks versions consistently outperforms its disjoint blocks counterpart. Third, the top-two estimators are consistently better than their max-only counterparts. Finall... | https://arxiv.org/abs/2502.15036v1 |
blocks. The situation is more complicated for sliding block maxima, where the simple disjoint blocks solution is inconsistent but where a certain ‘circular block bootstrap’ can be shown to be consistent [BS24a]. In this section, we apply that circular block bootstrap to our sliding top-two estimators and provide some i... | https://arxiv.org/abs/2502.15036v1 |
5 10 ErrorCircular Bootstrap, RL Figure 6: Histograms of estimation error (blue) and (circular block) bootstrap estimation errors (green) together with associated kernel density estimates. Left: shape esti- mation. Right: RL(100,100)-estimation. Fitting the Fr´ echet distribution to the annual maxima using the botw-met... | https://arxiv.org/abs/2502.15036v1 |
the two-parametric Fr´ echet case. For more flex- ibility, it would be worthwhile to additionally include a location parameter µ, or to even fit the three-parametric GEV distribution to allow for non-positive shape parameters. A particular challenge would then be to derive a suitable bias correction. (3) Asymptotic the... | https://arxiv.org/abs/2502.15036v1 |
In particular, for some ε∈(0,1), we have ϖan(k)<1−εalong a subsequence an(k), for all k∈N. Hence, by monotonicity of Π an(k)and continuity of ρ07→Πρ0(1−ε), 0 = Π an(k)(ϖan(k))>Πan(k)(1−ε)→Π0(1−ε)>0 ( k→ ∞ ), which is a contradiction. A similar argument shows continuity at 1. Finally, since the derivative of ρ07→ϖρ0was ... | https://arxiv.org/abs/2502.15036v1 |
∞ . Lemma A.2 (Asymptotics of vnΨn).Assume Condition 3.6. Then, as n→ ∞ , vnΨn(α1) =2 Υρ0(ϖρ0)Gnf1+2Υ′ ρ0(ϖρ0) α0Υρ0(ϖρ0)2Gnf2−Gnf3−Gnf4+oPr(1), withfjas defined in (3.16) . The expression on the right converges weakly to W=2 Υρ0(ϖρ0)W1+2Υ′ ρ0(ϖρ0) α0Υρ0(ϖρ0)2W2−W3−W4 26 Proof. Recall that, from the definition of Ψ kin... | https://arxiv.org/abs/2502.15036v1 |
A.3 and Lemma A.2, it follows that, as n→ ∞ , Sn1=vn(ˆαn−α1)Υ′ ρ0(ϖρ0) α0+oPr(1) =−Υ′ ρ0(ϖρ0) α0˙Ψρ,α0(α1)vnΨn(α1) +oPr(1) =−Υ′ ρ0(ϖρ0) α0˙Ψρ,α0(α1)2 Υρ0(ϖρ0)Gnf1+2Υ′ ρ0(ϖρ0) α0Υρ0(ϖρ0)2Gnf2−Gnf3−Gnf4 +oPr(1). 28 =−Υ′ ρ0(ϖρ0) α0˙Ψρ,α0(α1)1 z0Gnf1+Υ′ ρ0(ϖρ0) 2α0z2 0Gnf2−Gnf3−Gnf4 +oPr(1), where we used z0=1 2Υρ0(ϖρ0... | https://arxiv.org/abs/2502.15036v1 |
of Theorem 4.6 needs some lemmas as preparation. 30 Lemma B.1 (Largest two order statistics rarely show ties) .Under Conditions 4.1 and 4.3, for every c∈(0,∞), we have lim n→∞Pr (Mrn,1∨c, Srn,1∨c) = (Mrn,3∨c, Srn,3∨c) = 0. Proof. Since the event in question is contained in the event {Mrn,1∨c=Mrn,3∨c}, the result is a... | https://arxiv.org/abs/2502.15036v1 |
+ρ0ε−α0 /2 . Now apply Lemma B.3 for u= exp −ε−α0 1 +ρ0ε−α0 /2 to arrive at the claim. Lemma B.5 (Clipping doesn’t hurt) .Assume Condition 4.1. If ℓn=o(rn)and if α(ℓn) = o(ℓn/rn)asn→ ∞ , then Pr {Mrn> M rn−ℓn} ∪ {Srn> Srn−ℓn} →0, n → ∞ . Proof. Throughout all convergences are for n→ ∞ . Since Pr Mrn> M rn−ℓn ... | https://arxiv.org/abs/2502.15036v1 |
that purpose, we will in fact show that the first equality in (B.6) is met for any f∈ F2:=F2(α−, α+) and that the finite-dimensional distributions of ( ˜G[ℓn] nf)f∈F2converge weakly to the finite-dimensional distributions of ( Gf)f∈F2, where Gis aP-Brownian bridge; that is, a zero-mean Gaussian process with covariance ... | https://arxiv.org/abs/2502.15036v1 |
tsuch that 1 ≤s≤t≤n, define (Xs:t, Ys:t) = (X(n,c) s:t, Y(n,c) s:t) =Ms:t∨c σrn,Ss:t∨c σrn . Forζ∈[0,1], define Fn,ζ,c(x, y,˜x,˜y) = Pr X(n,c) 1:rn≤x, Y(n,c) 1:rn≤y, X(n,c) ⌊rnζ⌋+1:⌊rnζ⌋+rn≤˜x, Y(n,c) ⌊rnζ⌋+1:⌊rnζ⌋+rn˜y . We are interested in weak convergence of the bivariate margins. For that purpose, define Fα,ζ(... | https://arxiv.org/abs/2502.15036v1 |
An,ζ(y,˜x) = Pr X1:⌊rζ⌋> y, Y 1:⌊rζ⌋≤y, X⌊rζ⌋+1:r≤˜x. 36 We may now use asymptotic independence to obtain that, for ˜ x≤y, An,ζ(y,˜x) = Pr Y1:⌊rζ⌋≤y < X 1:⌊rζ⌋ Pr X⌊rζ⌋+1:r≤˜x +o(1). (B.18) Next, if ˜ x > y , (B.17) yields An,ζ(y,˜x) = Pr X1:⌊rζ⌋> y, Y 1:⌊rζ⌋≤y, X⌊rζ⌋+1:r≤y ∪ ˜x≥X⌊rζ⌋+1:r> y, Y ⌊rζ⌋+1:r≤y, X ... | https://arxiv.org/abs/2502.15036v1 |
where Fn,ζ,c(∞, y,˜y,∞) has been calculated in part [b] and where p1= Pr X1:r≤y, Y⌊rζ⌋+1:⌊rζ⌋+r≤˜y < X ⌊rζ⌋+1:⌊rζ⌋+r , p2= Pr Y1:r≤y < X 1:r, Y⌊rζ⌋+1:⌊rζ⌋+r≤˜y < X ⌊rζ⌋+1:⌊rζ⌋+r . Regarding p1, we have p1= Pr X1:r≤y, Y⌊rζ⌋+1:⌊rζ⌋+r≤˜y −Pr X1:r≤y, X⌊rζ⌋+1:⌊rζ⌋+r≤˜y =Fn,ζ,c(y,∞,∞,˜y)−Fn,ζ,c(y,∞,˜y,∞) =Fα,ζ(y,˜y)... | https://arxiv.org/abs/2502.15036v1 |
Bn,ζ(˜y, y) has been calculated in (B.26), and we obtain, using (B.24), p21= exp −ζy−α ρ0ζy−α·exp −(1−ζ)˜y−α (1−ζ)˜y−αρ (˜y/y)α + exp( −ζy−α)·exp −(1−ζ)˜y−α ˜y−α(1−ζ) ρ0−ρ (˜y/y)α +o(1), = exp −ζy−α−(1−ζ)˜y−α ˜y−α(1−ζ) ζρ0y−αρ (˜y/y)α +ρ0−ρ (˜y/y)α +o(1). (B.34) Finally, we need to assemble terms. Fir... | https://arxiv.org/abs/2502.15036v1 |
of Theorem 3.7, we need to show the following three properties: (1) lim n→∞Pr(Zn,1=···=Zn,n−rn+1) = 0. (2) There exist constants 0 < α−< α 1< α +<∞such that Pnf⇝Pffor all f∈ F2(α−, α+), where F2(α−, α+) is as in (3.14). (3) We have Wn= (Gnf1, . . . ,Gnf4)⊤⇝N4(B,Σ(sb) ρ,α0), where Band Σ(sb) ρ,α0are as in Theo- rem 4.8.... | https://arxiv.org/abs/2502.15036v1 |
A(u)L(ux) L(u)−1 =hτ(x), x ∈(0,∞). (C.2) As argued in the proof of Theorem 4.2 in [BS18b] (beginning of the proof of Condition 3.5), we can find, for any fixed δ∈(0, α0) , constants x(δ)≥1 and c(δ)>0 such that, for all u≥x(δ) and x≥x(δ)/u, L(u) L(ux)≤(1 +δ) max{x−δ, xδ}, L(ux)−L(u) g(u) ≤c(δ) max{xτ−δ, xτ+δ}, (C.3) w... | https://arxiv.org/abs/2502.15036v1 |
, where, for some 1 ≤ξn,y≤1/F(arny), Rn(y) =1 3ξ2n,yn1 F(arny)−1o3 . We have |Rn(y)| ≤1 3nF(arny)−1 F(arny)o3 =O(r−3 n), where the last bound follows from F(arny) = 1 + o(1) and rn{F(arny)−1}=y−α0+o(1). As a consequence, since√kn/rn=λ1+o(1) by (5.5), p kngn(y) =√kn rnhr2 n{F(arny)−1}2 2F(arny)2+r2 nRn(y)i ={λ1+o(1)} ... | https://arxiv.org/abs/2502.15036v1 |
L(arn)o ≤(1 +δ) max yδ, y−δ . Hence, in view of the bounds on f′ j, we conclude that there exists a finite constant c′(δ) such that, for 1 ≥y≥c/arn fn(y)≤c′(δ) exp −(1 +δ)−1y−α0+δ yτ−3α0−3δ−1 and the function is integrable since δ < α 0. On the other hand, for y≥1 we find the bound fn(y)≤c′′(δ)yτ+2δ−2α0−1 which is ea... | https://arxiv.org/abs/2502.15036v1 |
then u=x−α/z with d u=−αx−α−1/zdx, we obtain that Z R2hρ,α,1(x, y) d(x, y) =Z∞ 0Zx 0hρ,α,1(x, y) dydx =−αZ∞ 0Z1 0exp(−x−αz−1)n x−2α−1z−2ρ′(z) +x−α−1ρ′′(z)o dzdx =−Z∞ 0Z1 0exp(−u) uρ′(z) +zρ′′(z) dzdu =−Z1 0ρ′(z) +zρ′′(z) dz≥0. 48 Hence, B7→µ(B) :=R Bhρ,α,1(x, y) d(x, y) defines a finite Borel measure on R2. It is a pr... | https://arxiv.org/abs/2502.15036v1 |
6−1π2 6−1 π2 3−1−γ −απ2 6−1π2 6 . 50 Proof. This follows from tedious but straightforward calculations. Lemma E.2 (Asymptotic covariance for the sliding block maxima estimator under inde- pendence) .Suppose (X, Y, ˜X,˜Y)is a random vector whose bivariate cdfs are needed for evaluating the following covariances are... | https://arxiv.org/abs/2502.15036v1 |
the previous two displays, and observing that supr∈Nr{1−F(ar)}<∞as argued in the proof of Lemma C.1 in [BS18b], we find that, for sufficiently large rand all y∈(c/ar,1], r(1−F(ary)) =r(1−F(ar))1−F(ary) 1−F(ar)≤Kc,δy−α0+δ, where Kc,δis a positive constant. Altogether we now have, for sufficiently large r, Z1 c/arr(1−F(a... | https://arxiv.org/abs/2502.15036v1 |
AR-model with β̸= 0.5 and the ARMAX-model, both with fixed block size r= 100. The results are presented in Figure 12 (shape estimation) and Figure 13 (return level estimation with T= 100). The results are mostly consistent with those presented in Section 6.1: unless the serial dependence is very strong, the top-two sli... | https://arxiv.org/abs/2502.15036v1 |
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