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equivalence results are limited by incom- pleteness of existing SpWig→SpCovreductions and any improvement in this direction would immediately yield even stronger equivalence results. Theorem 1.3.(Main Result 2: Equivalence between SpWigandSpCov) 1. (Bijective Equivalence )SpWig(ν)andSpCov(µ)are bijectively equivalent w... | https://arxiv.org/abs/2503.02802v1 |
Mixtures of Isotropic Gaussians) .LetX∼N(0,Id)and A∈Rda random variable independent of X. Then 1+χ2(X+A/⌊ar⌈⌊lX) = IEexp(/an}⌊ra⌋k⌉tl⌉{tA,A′/an}⌊ra⌋k⌉tri}ht), whereA′is an i.i.d. copy of A. Proof.The proof is a simple application of the so-called “Ingster t rick”. Note that X+A∼IE APA, wherePAis the lawN(A,Id). We star... | https://arxiv.org/abs/2503.02802v1 |
mappi ng guarantees for Rad(2p),Rad(0)in place ofBern(1/2+p),Bern(1/2). We will denote the corresponding map RKRad G. 5.4 Gram-Schmidt One of our subroutines is the classic Gram-Schmidt Process d escribed in Algorithm 3below. Given vectorsX1,...,X N∈RMit outputs an orthonormal basis /tildewideX1,...,/tildewideXNinRM. A... | https://arxiv.org/abs/2503.02802v1 |
quantity in the lemma is upper bounded by O(√ θ)IE g|/⌊ar⌈⌊lg/⌊ar⌈⌊l−√n|=O(√ θ)≤O(√ θcomp) =o(1), which concludes the proof. 7 Total Variation Equivalence between SpCovandSpWigforn≫d3 In this section, we obtain a two-way equivalence result for t he regimen≫d3, building on the noise distribution convergence [ BDER16 ,JL... | https://arxiv.org/abs/2503.02802v1 |
before the last reflection step and G∼N(0,Id)⊗d. Note that given this bound the statement for Ysymimmediately follows from the DPI for total variation (Lemma 5.1). For each column i∈[d],Zi=Xi+√ θuig∈Rn, whereXi∼N(0,In) andg∼N(0,In). By Lemma 6.2and the triangle inequality for total variation we may assum e that/⌊ar⌈⌊lg/... | https://arxiv.org/abs/2503.02802v1 |
ed class of potential recovery algorithms for SpWig, which we believe to be a very mild restriction. The recovery algorithm in this case also finishes inO(d2+γ) and given a recovery algorithm for SpWigwith loss < ℓ′⋆, is guaranteed to output an estimate for SpCov with loss at most ℓ′⋆+o(1). 27 Algorithm 6: SpCovToSpWig ... | https://arxiv.org/abs/2503.02802v1 |
output of SpCovToSpWig . Note that we can rewrite IEYdenoised ij=M−1(1 2Cα,ǫθ2nk−2)M1i,j∈S=λ′u′ iu′ j, where the signal u′∈Bd(k) is such that u′ i=|ui|,∀iandλ′=M−1(1 2Cθ2nk−2)Mk. Finally, from the guarantees for RKRad G(Lemma 5.9), we have dTV(RKRad G(Ydenoised),W+λu′u′⊤)≤O(n−3), where λ=λ′ 4/radicalbig 6logn+2logλ′−1.... | https://arxiv.org/abs/2503.02802v1 |
/an}⌊ra⌋k⌉tl⌉{tg,/tildewideZi/an}⌊ra⌋k⌉tri}htin Lemmas 10.10,10.1. This is the crucial consequence of the whole perturbation t heorem and it allows us to explicitly quantify the spike in the basis vectors/tildewideZ. The high probability bounds in Lemmas 10.11,10.12and10.10are expressed in terms of both d,k,θ,n anda−,a... | https://arxiv.org/abs/2503.02802v1 |
the statement of the theorem by induction on i. 10.3.3 Base Case i= 1. Note thats1/tildewideX1=X1. Hence, writing /tildewideZ1in the form given in the theorem statement, /tildewideZ1=1 r1Z1=1 r1(s1/tildewideX1+√ θu1g) =s1 r1(/tildewideX1+√ θn−1/2(u1+(n1/2s−1 1−1)u1)g) =s1 r1(/tildewideX1+√ θn−1/2(u1+α1)g+W1), where we ... | https://arxiv.org/abs/2503.02802v1 |
E/a\}brack⌉tl⌉{tg,/tildewideX/a\}brack⌉tri}ht j−1,Enorm j, Pr[E/a\}brack⌉tl⌉{tg,/tildewideX/a\}brack⌉tri}ht j]≥1−jn−K+1−2jn−K≥1−n−K+2. Lemma 10.10 (Bound on/an}⌊ra⌋k⌉tl⌉{tg,/tildewideZj/an}⌊ra⌋k⌉tri}ht).GivenE/a\}brack⌉tl⌉{tg,/tildewideX/a\}brack⌉tri}ht j, |/an}⌊ra⌋k⌉tl⌉{tg,/tildewideZj/an}⌊ra⌋k⌉tri}ht−(/an}⌊ra⌋k⌉tl⌉{t... | https://arxiv.org/abs/2503.02802v1 |
into the expression for |II|, we get |II|≤2cu√ θn−1/2k−1/21i∈S/parenleftBig/summationdisplay j∈[i−1]\S18c′ Ka− +/summationdisplay j∈[i−1]∩S2(7cu√ θn1/2k−1/2+2wsp)·2cuk−1/2/parenrightBig = 2cu√ θn−1/2k−1/21i∈S/parenleftBig 18c′ Kda−+28(cu)2√ θn1/2+8cuk1/2wsp/parenrightBig =/parenleftBig 36cuc′ K√ θn−1/2k−1/2da−+56(cu)3θ... | https://arxiv.org/abs/2503.02802v1 |
variable with variance σ2 i=s−2 ii−1/summationdisplay j=1((sjr−1 j−1)/tildewideXj+sjr−1 j(√ θn−1/2(uj+αj)g+Wj))2/an}⌊ra⌋k⌉tl⌉{tg,/tildewideXj/an}⌊ra⌋k⌉tri}ht2. Similarly,A3is ann-dimensional Gaussian random variable with covariance Σ i, where Tr(Σi) :=s−2 ii−1/summationdisplay j=1((sjr−1 j−1)/tildewideXj+sjr−1 j(√ θn−1... | https://arxiv.org/abs/2503.02802v1 |
such collections, as parameters µ,νvary. Each fixed point of µorνcorresponds to a specific planted problem, whose complexity might differ from the one of a different planted problem with parameters µ′orν′. Moreover, every value of the dimensionality parameterdimplies a separate collection of distributions. For simplicity, ... | https://arxiv.org/abs/2503.02802v1 |
total variation distance, /vextendsingle/vextendsingle/vextendsingleIPX′∼Qν,N u′,H0[B(X′) = 1]−IPX∼Pµ,N u,H0[B(Ared(X))= 1]/vextendsingle/vextendsingle/vextendsingle≤dTV(Qν,N u′,H0,Ared(X)) ≤ǫN,forX∼Pµ,N u,H0/vextendsingle/vextendsingle/vextendsingleIPX′∼Qν,N u′,H1[B(X′) = 0]−IPX∼Pµ,N u,H1[B(Ared(X))= 0]/vextendsingle/... | https://arxiv.org/abs/2503.02802v1 |
phase diagrams. 50 A.4 Average-Case Reductions on the Computational Threshol d As motivated in Sec. 1.4and1.5, hardness results for points arbitrarily close to the compu tational thresholds of SpCovandSpWighave implications for the whole hard regimes ΩCovand ΩWig. To capture this idea, we introduce the following definit... | https://arxiv.org/abs/2503.02802v1 |
that Asolves the recovery problem if sup u⋆∈UdIE X∼Pµ u⋆ℓd(u⋆,/hatwideru⋆=A(X))≤ǫd≤1−k−1/2d2δ. This choice of ℓdaligns with the most common approach to planted rank-1 signa l estimation - solving argmax u:/bar⌈blu/bar⌈bl2=1,/bar⌈blu/bar⌈bl1=kuTΣu, where Σ comes from either SpWigor the empirical covariance matrix of SpC... | https://arxiv.org/abs/2503.02802v1 |
N. We say that an O(NC)-time algorithm A={AN: RDN→{0,1}}is anaverage-case reduction for recovery from Pµ UtoQν U′if for any sequence {µi}∞ i=1, such that lim i→∞µi=µ, there exist a sequences {νi,Ni}∞ i=1and an index i0, such that 1. limi→∞νi=νand lim i→∞Ni=∞; 2. for any i > i0,ANiis an average-case reduction for recove... | https://arxiv.org/abs/2503.02802v1 |
we choose any small η>0 and have θ′≈θ. As established above, since γ+η>2,CloneCov is an average-case reduction for both detection and recover y for the second step in the chain. We now establish a simple average-case reduction that achie ves the first step - that is, slightly increases the number of samples nrelative to... | https://arxiv.org/abs/2503.02802v1 |
the last halves of the rows of Z. Note that Z(1),Z(2)are two independent instances of SpCov(d,k,θ,n/ 2). Letα′=α,ǫ′= logd(n/2)−1,δ′=θcomp(d,k,n/2)/θand define new constants Aα′,ǫ′,Cα′,ǫ′,Kα′,ǫ′,ψ′as in Theorem 9.2. We let ARed recovery(Z) :=SpCovToSpWig (Z(1),2Kα′,ǫ′,ψ′), whereZ(1)consists of the first n/2 rows ofZas des... | https://arxiv.org/abs/2503.02802v1 |
with planted sparse structure. In Conference on Learning Theory (COLT) , pages 48–166, 2018. [BBH19] Matthew Brennan, Guy Bresler, and Wasim Huleihel. U niversality of computational lower bounds for submatrix detection. In Conference on Learning Theory , pages 417– 468. PMLR, 2019. [BBH21] Matthew Brennan, Guy Bresler,... | https://arxiv.org/abs/2503.02802v1 |
i. Asymptotic mutual infor- mation for the binary stochastic block model. In 2016 IEEE International Symposium on Information Theory (ISIT) , pages 185–189. IEEE, 2016. [DKS17] Ilias Diakonikolas, Daniel M Kane, and Alistair Ste wart. Statistical query lower bounds for robust estimation of high-dimensional gaussian s a... | https://arxiv.org/abs/2503.02802v1 |
manuscript , 2004. [JL09] Iain M Johnstone and Arthur Yu Lu. On consistency and s parsity for principal com- ponents analysis in high dimensions. Journal of the American Statistical Association , 104(486):682–693, 2009. [JL15] Tiefeng Jiang and Danning Li. Approximation of recta ngular beta-laguerre ensembles and large... | https://arxiv.org/abs/2503.02802v1 |
Annals of Statistics , 46(5):2416–2451, 2018. [RM14] Emile Richard and Andrea Montanari. A statistical mo del for tensorPCA. Advances in neural information processing systems , 27, 2014. [Spo23] Vladimir Spokoiny. Concentration of a high dimensi onal sub-gaussian vector. arXiv preprint arXiv:2305.07885 , 2023. [Ver18] ... | https://arxiv.org/abs/2503.02802v1 |
Asymmetric Cross-Correlation in Multivariate Spatial Stochastic Processes: A Primer Xiaoqing Chen Department of Mathematics and Statistics, University of Exeter, Exeter, EX4 4PY, U.K. Email: xiaoqing.a.chen@gmail.com Abstract Multivariate spatial phenomena are ubiquitous, spanning domains such as climate, pandemics, ai... | https://arxiv.org/abs/2503.02903v1 |
(a) Lon:[ −180◦,−90◦) (b) Lon:[ −90◦,0◦) (c) Lon:[0◦,90◦) (d) Lon:[90◦,180◦] Figure 1: Empirical same-component auto-correlation matrix plots for PM2.5 across four longitude strips [ −180◦,−90◦), [−90◦,0◦), [0◦,90◦), [90◦,180◦]. All symmetric about y = x. Figure 2 is the empirical cross-correlation plots, displaying co... | https://arxiv.org/abs/2503.02903v1 |
,i, j= 1, . . . , n . 4 The joint covariance matrix for the random vector [Y1(s1), . . . , Y p(s1), . . . , Y 1(sn), . . . , Y p(sn)]T∈Rnp, consisting of the multivariate spa- tial stochastic process, is Σnp×np= [cov(Y(s1),Y(s1))]p×p[cov(Y(s1),Y(s2))]p×p··· [cov(Y(s1),Y(sn))]p×p [cov(Y(s2),Y(s1))]p×p[cov(Y(s2)... | https://arxiv.org/abs/2503.02903v1 |
sj)/p Cll(si, si)p Ckk(sj, sj). 3.1 Auto-correlation •Symmetric, see Eq. (5), (6), and Fig. 1. 7 •The main diagonal values Corrll(si, si) of the auto-correlation matrix have the largest magnitude of 1, where l= 1, . . . p , and i= 1, . . . , n , see Fig. 1. •The magnitude of the off-diagonal values in the auto-correlat... | https://arxiv.org/abs/2503.02903v1 |
matrix Vp×p, making the off-diagonal block [C(si, sj)]p×psymmetric as a whole. 9 To illustrate clearly, we substitute the generic Eq. (3) with Eq. (7), [cov(Y(s1),Y(s2))]p×p= C11(s1, s2)C12(s1, s2)··· C1p(s1, s2) C21(s1, s2)C22(s1, s2)··· C2p(s1, s2) ............ Cp1(s1, s2)Cp2(s1, s2)··· Cpp(s1, s2) ... | https://arxiv.org/abs/2503.02903v1 |
4.4 Conditional Modeling Approach The main idea of this class of model is to construct the desired joint covariance matrix Σnp×npor the joint precision matrix Σ−1 np×npthrough the specification of conditional mean and conditional variance. The idea of obtaining the joint covariance from conditional mean and covariance ... | https://arxiv.org/abs/2503.02903v1 |
matrix expands its dimension by encoding the correlation at different spatial locations for a given pair of components. By inducing a shifting parameter to the spatial separation leg, the ( i, j)thelement of the n×ncross-covariance matrix is not the same as the ( j, i)thelement, where i, j= 1, . . . , n . Therefore, th... | https://arxiv.org/abs/2503.02903v1 |
the conditional mean in Eq. (12) only regresses on the values within a neighborhood of si, the resulting precision matrix Σ−1 np×npnaturally embodies structural sparsity, enhancing computational efficiency. Asymmetric cross-covariance matrix blocks reside in the joint covariance matrix Σnp×np, 15 while the sparsity is ... | https://arxiv.org/abs/2503.02903v1 |
Adaptive monotonicity testing in sublinear time Housen Li Zhi Liu Axel Munk Institute for Mathematical Stochastics, University of G¨ ottingen, Germany Abstract Modern large-scale data analysis increasingly faces the challenge of achieving computational efficiency as well as statistical accuracy, as classical statistica... | https://arxiv.org/abs/2503.03020v2 |
Gaussian noise, where the observations are given by Yi=f(xi) +εi, i∈[n] :={1, . . . , n } (1) for equidistant sampling points xi≡i/n. Here f: [0,1]→Ris an unknown function and the random errors εiarei.i.d. Gaussian distributed with mean zero and known variance σ2, for simplicity. We stress, however, that all our result... | https://arxiv.org/abs/2503.03020v2 |
regression splines with a prior over the regression coefficients. Salomond [40] put a posterior distribution on the largest absolute discrepancy between the parameter and the null model. The resulting test is shown to attain asymptotic frequentist optimality, being adaptively minimax optimal (up to a log-factor) for H¨... | https://arxiv.org/abs/2503.03020v2 |
properties of γ-exceedance fraction (incl. the existence of minimizers in (3a)) are provided in Section A in the appendix. The γ-exceedance fraction allows to display computational and statistical efficiency in a phase diagram of FOMT in Figure 1. More precisely, we show that FOMT has the following three favorable prop... | https://arxiv.org/abs/2503.03020v2 |
DS, C and BHL, the factor Rcorresponds to the number of repetitions in Monte–Carlo or bootstrap procedures. We consider the statistical optimality in terms of minimax separation rates over H¨ older smooth functions of order β. In this regard, the established statistical guarantees for different methods are summarized i... | https://arxiv.org/abs/2503.03020v2 |
j≡j/nfor some fixed i, j∈[n]. To this 6 end, we will use the LPE ˆfnto estimate f(xi) and f(xj) by ˆfn(xi) and ˆfn(xj), respectively, and then to check whether the difference Ti,j:=ˆfn(xi)−ˆfn(xj), (4) is significantly larger than zero. In particular, we employ the LPE ˆfnoffwith an optimally chosen bandwidth hn≍ log(... | https://arxiv.org/abs/2503.03020v2 |
(see (5) below), and for each xI, we repeat the left and right searches for xJbyO(logn) times. The resulting procedure is called FOMT (Fast and O ptimal Monotonicity T est), see Algorithm 1. An illustration is given in Figure 2. 7 Algorithm 1 FOMT: Fast and Optimal Monotonicity Test Φ Input: data Y1, . . . , Y n, and s... | https://arxiv.org/abs/2503.03020v2 |
model. Ifσ2>0 is unknown in the nonparametric regression model (1), then it can be replaced by any estimators ˆ σ2 nthat are uniformly consistent over eΣ(β, L) with β∈(0,2] and L >0, given by eΣ(β, L) := Σ(β, L), ifβ∈(0,1], {f∈Σ(β, L)| ∥f′∥∞≤L},ifβ∈(1,2]. Corollary 2.4. Consider the nonparametric regression model i... | https://arxiv.org/abs/2503.03020v2 |
A.3 (ii)). To ensure with high probability that at least one such index is sampled, we repeatedly generate O ε0,γn(f)−1 uniform indices. By Corollary A.4, we have ε0,γn(f)≥hnfor all f∈ Fβ(C∆0,n), as defined in Theorem 2.3 (see also the green region above h−1 n in Figure 1). Consequently, choosing Cn(α)≍h−1 n≳ ε0,γ... | https://arxiv.org/abs/2503.03020v2 |
(47) in the appendix) such that Bf,A(h)≤c1hβ=:G1(h), for all h∈(0,1/2], E ρ2 A(h) ≤E ρ2 [n](h) ≤C2 ρlogn nh=:G2 2(h),for all h∈(0,1/2] and A ⊆[n]. Namely, G1andG2 2serve as upper bounds for the bias and variance, respectively. We consider an increasing sequence of hm=n−14m−1form∈[M] as bandwidth candidates with M≡M... | https://arxiv.org/abs/2503.03020v2 |
constant bandwidth. The condition in (11) is shown to be crucial for the existence and construction of adaptive confidence bands of density functions. It is also known that (11) is a rather weak requirement as it remains valid in H¨ older classes, except for a “topologically small” subset. More precisely, the exception... | https://arxiv.org/abs/2503.03020v2 |
Φ A Input: data Y1, . . . , Y n, a constant κ >1 and significance level α∈(0,1) Parameters: standard deviation σ, kernel function Kand radius L 1:Cn(α) =−log(α/2)·(n/log(n)) 2:for1≤l≤Cn(α)do 3: Generate I∼Unif([ n]) 4: Generate ( P,A) with Algorithm 5 in the appendix with parameters ( n, I) 5: Compute estimates ( ˆfn,¯... | https://arxiv.org/abs/2503.03020v2 |
9) and A-FOMT (cf. Algorithm 5 in lines 3 and 11) is due to technical reasons and overly conservative in practice; We instead use 0 .1 in both algorithms. Besides, we use LPE of order one with the Epanechnikov kernel and bandwidth hn= 0.3 log(n)/n1/3in FOMT. In A-FOMT, we replace the base of exponential bandwidth gri... | https://arxiv.org/abs/2503.03020v2 |
range of sample sizes. The impact of sample size on computational time is twofold: On 17 500100015002000250030001e−03 1e−01 1e+01 1e+03Computation time [s] 500100015002000250030001e−03 1e−01 1e+01 1e+03Computation time [s] 500100015002000250030001e−03 1e−01 1e+01 1e+03Computation time [s] 500100015002000250030001e−03 1... | https://arxiv.org/abs/2503.03020v2 |
Carothers, N. L. (2000). Real Analysis . Cambridge University Press. [9] Chakraborty, M. and Ghosal, S. (2021). Convergence rates for Bayesian estimation and testing in monotone regression. Electron. J. Stat. , 15(1):3478–3503. [10] Chernozhukov, V., Chetverikov, D., and Kato, K. (2014). Anti-concentration and honest, ... | https://arxiv.org/abs/2503.03020v2 |
K., and Sugasawa, S. (2024). Locally adaptive Bayesian isotonic regression using half shrinkage priors. Scand. J. Stat. , 51(1):109–141. [34] Petrov, V. V. (1995). Limit Theorems of Probability Theory . Oxford University Press, New York. [35] Ramsay, J. O. (1998). Estimating smooth monotone functions. J. R. Stat. Soc. ... | https://arxiv.org/abs/2503.03020v2 |
/k. By the pointwise convergence of (˜ gk)k∈N, we have D∗⊆ ∪∞ m=1Amwith Am:=∩k≥m˜Dk. Then λ(Am)≤λ(˜Dk)≤ε0,γ(f) + 1 /kfor all k≥m, which implies that λ(Am)≤ε0,γ(f) and further λ(D∗)≤λ(∪∞ m=1Am) = lim m→∞λ(Am) =ε0,γ(f). By definition, λ(D∗)≥ε0,γ(f). Thus, λ(D∗) =ε0,γ(f). Definition A.2. Letf: [0,1]→Rbe a continuous funct... | https://arxiv.org/abs/2503.03020v2 |
construct a monotone increasing function gonHc f(γ) such that |f(x)−g(x)| ≤γforx∈Hc f(γ), in the same way as in Part (i). We 23 further extend gto ˜g: [0,1]→Rby ˜g(x) = sup t∈[0,x]∩Hc f(γ)g(t) if Hc f(γ)∩[0, x)̸=∅, inf t∈Hc f(γ)g(t) otherwise . Then ˜ gis monotone on [0 ,1] and ˜ g|Hc f(γ)=g. Thus, ε0,γ(f)≤λ {... | https://arxiv.org/abs/2503.03020v2 |
but requiring additional technicalities, to Section G. For convenience, we restate this result from Theorem 2.3 below. Theorem B.1 (Separation rate for 0 < β≤1).Under the model (1), suppose that Assumptions (M1) and (K1)–(K3) hold with β∈(0,1]andL > 0. Let (fn)n≥1be a sequence of functions in Σ(β, L) such that ε0,γn(fn... | https://arxiv.org/abs/2503.03020v2 |
xi+j∈A1/n, there exists c∈Afulfilling |c−xi+j| ≤1/nandfn(a)−fn(c)≥ 26 γn, by definition. Then, we have Di,i+j=nX k=1Wnk(xi)fn(xk)−nX l=1Wnk(xi+j)fn(xk) =nX k=1Wnk(xi) fn(xk)−fn(xi) +fn(xi)−fn(a) +fn(a)−fn(c) +fn(c)−fn(xi+j) +nX k=1Wnk(xi+j) fn(xi+j)−fn(xk) ≥ −Lhβ n−L1 nβ +γn−L1 nβ −Lhβ n≥γn−4Lhβ n, which shows ... | https://arxiv.org/abs/2503.03020v2 |
. . . , 0.4}. See Table 2 for details. We set λ0= 0.5, ifβ∈(0,1], 0.0228,ifβ∈(1,2].(22) Lemma C.2 ([47, Lemma 1.3 and Proposition 1.12]) .Suppose that xk≡k/nfor all k= 1, . . . , n and Assumption (K1) holds. Let x∈[0,1]andh≥1/(2n). Then the weights (Wnk)n k=1of the LPE(⌈β⌉ −1) satisfy, for sufficiently large n, (i)... | https://arxiv.org/abs/2503.03020v2 |
linear estimator (LPE( 1)), coincides with the Nadaraya–Watson estimator (LPE( 0)). Proposition C.7. Suppose that Assumptions (M1), (K1) and (K3) hold. Let f∈H≡ M ∩ Σ(β, L) and0≤a < b≤1andnbe sufficiently large. Then Da,b:=nX k=1Wnk(a)f(xk)−nX k=1Wnk(b)f(xk)≤16Kmax λ0∨2 Lhβ. (26) If additionally a=xiandb=xjfor some i... | https://arxiv.org/abs/2503.03020v2 |
(33) we have the upper bound B−1 na− B−1 nb op Fxk−b h ≤√eKmax8L2 λ2 0|a−b| h. Thus, |Wnk(a)−Wnk(b)| ≤ L1/λ0+ 8L2√eKmax/λ2 0 · |a−b|/(nh2). Step 4: Since Wnkis Lipschitz continuous, we have nX k=1(Wnk(a)−Wnk(b))2=X k∈Ia∪Ib|Wnk(a)−Wnk(b)|2≤4L1 λ0+√eKmax8L2 λ2 02|a−b|2 nh3. This finishes the proof of (30), which im... | https://arxiv.org/abs/2503.03020v2 |
=δ. Ifbi−ai< δ for all i∈N, we obtain ( ai, bi)⊆(ai, ai+δ)⊆(Aδ∩[m.M ])\A, and [ m, a)∪(b, M]⊆ Aδ∩[m, M ] \A. LetB=∪∞ i=1(ai, bi)∪[m, a)∪(b, M], then by (35), λ(B) =∞X i=1(bi−ai) +a−m+M−b=M−m−λ(A)≥δ. Summerizing all cases, we can always find a set Bwith Lebesgue measure λ(B)≥δandB⊆ Aδ∩[m, M ] \A. Therefore, λ Aδ∩[m... | https://arxiv.org/abs/2503.03020v2 |
1)≤α. Lemma G.1. Suppose that I∼Unif({i,(i+ 1), . . . , j })/nfor some 1≤i≤j≤nand let Abe a non-empty measurable subset of [0,1]. Then P I∈A1/n ≥n j−i+ 1λ A∩i−1 n,j n , where Aδdenotes the δ-expansion of Ain Definition D.1. Proof. LetJbe a uniformly distributed random variable on (( i−1)/n, j/n ] and set I′=⌈nJ⌉/... | https://arxiv.org/abs/2503.03020v2 |
P(Ei)≤exp (−2 logn) = n−2. Thus, for all large n, P(E)≤n−1X i=1P(Ei)≤nmax iP(Ei)≤1 n≤α 2. For the second term in (17), consider an arbitrary pair ( i, i+ 1)∈ I. On event Ec, we have Ri,i+1>−8L2σ λ0s 4W C2β+1 hn−1hβ−1 n. Consequently, Ti,i+1=Di,i+1+Ri,i+1 >(Cβ−4L)hβ−1 n n−8L2σ λ0s 4W C2β+1 hhβ−1 n n =8L2σ λ0s 4W C2β+1 h... | https://arxiv.org/abs/2503.03020v2 |
to see the following facts: •The support of gn,jis [(2j−1)hn,(2j+ 1)hn]. •The cardinality of Inis|In| ≈1/(2hn) =O(h−1 n). Furthermore, we have gn,j∈Σ(β, L) for all j∈In, as |gn,j(x)−gn,j(y)|=Lhβ n|ψn,j(x)−ψn,j(y)|=Lhβ nh−β n|ψ(x)−ψ(y)| ≤L|x−y|β. Moreover, gn,j(2jhn)−gn,j((2j+ 1)hn) =Lhβ n, which implies that {gn,j|j∈In... | https://arxiv.org/abs/2503.03020v2 |
1. The support of gn,jis [(4j−2)hn,(4j+ 2)hn]. 2. The cardinality of Inis|In| ≈1/(4hn) =O(h−1 n). Furthermore, |g′ n,j(x)−g′ n,j(y)|=Lhβ nh−1 n ψ′x−4jhn hn −ψ′y−4jhn hn ≤Lhβ nh−1 n x−y hn β−1 =L|x−y|β−1. Namely, gn,j∈Σ(β, L) for all j∈In. Moreover, g′ n,j((4j+ 1)hn) =Lhβ nh−1 nψ′(1) =−Lhβ−1 n, 44 which implies that... | https://arxiv.org/abs/2503.03020v2 |
from Mill’s ratio that Pfn(Ei,j∩Fc δ) =P Ri,j(S)≤ −σs 6W C2β+1 hhβ n, S∈[1−δ,1 +δ]! ≤max s∈[1−δ,1+δ]P Ri,j(s)≤ −σs 6W C2β+1 hhβ n! ≤exp −1 2 σq6W C2β+1 hhβ n2 max s∈[1−δ,1+δ]V(Ri,j(s)) 46 ≤exp −1 2 σq6W C2β+1 hhβ n2 σ2Wn−1h−1n(1−δ)−1 2β+1 =n−3(1−δ)1 2β+1 where the last inequality follows from (... | https://arxiv.org/abs/2503.03020v2 |
4W C2β+1 h−16Kmax λ0∨2 L! hβ n=σs 6W C2β+1 nhβ n, which implies ( i, i+j)∈ I. Thus, given Il=iand the γn-right-heaviness of xi, we have Pfn(Pl∩ I ̸=∅ Il=i, xi∈Hfn,R(γn)) ≥Pfn there exists j∈ J+ lsatisfying xi+j∈A1/n Il=i, xi∈Hfn,R(γn) ≥1−1 n. Analogously, we can achieve the same upper bound with any γn-left-heavy p... | https://arxiv.org/abs/2503.03020v2 |
and Spokoiny [13] investigated the Gaussian white noise model and introduce two multiscale test statistics, denoted by TDS,1andTDS,2, to examine the monotonicity of regression in Σ(1 , L) and Σ(2 , L), respectively. The first test employing TDS,1detects violations of monotonicity by comparing multiscale regression esti... | https://arxiv.org/abs/2503.03020v2 |
same order, more precisely, O(Rn2). (ii) The test in [19] has computational complexity O(n3h2 n)with bandwidth hnsatisfying n−1/3≪ hn≪1. (iii) The computational complexity of test proposed by [11] is O(Rn3). Proof. The computational cost of all aforementioned procedures is independent of the choice of K, so we use S(l,... | https://arxiv.org/abs/2503.03020v2 |
the quantityP j∈Jvi/|J|, where |J|denote the cardinality of J. 3. Let 1denote the Rn-vector (1 , . . . , n )⊤. Moreover, we define Vnas the linear span of {1J|J∈ Jln}. Note that the dimension of Vnisln. LetIandJbe two disjoint subsets of [ n] such that Iis on the left of Jin the sense that every element 53 inIis smalle... | https://arxiv.org/abs/2503.03020v2 |
the values TBHL(ε, uj) and the probabilities p(uj) =P(TBHL(ε, uj)>0) for all uj∈U. This requires O(mRl n) additional steps. Thus, computing TBHL(Y, uα) takes O(l3 n∨nln) steps. Combining all the computational costs, the total complexity is O(mR(ln+ log( R))) + O(R(l3 n∨nln)) =O(R(l3 n∨nln)) with a user-specified consta... | https://arxiv.org/abs/2503.03020v2 |
Theorem 4.1. Letκ >1 and define Π(ω):= max m∈[M]ρn(m)(ω) G2(m),and Eκ:={ω|Π(ω)≥κ}. (49) Note that for any fixed ω∈Ec κ,ρn(m)(ω) is fixed and satisfies ρn(m)(ω)≤κG2(m) for all m∈[M]. Thus, by Lemma G.9, we have dA(f,ˆfn,¯m)≤12 min m∈[M]{G1(m) +κG2(m)} ≤Cmaxlogn nβ 2β+1 , for some constant Cmaxdepends only on G1,G2andκ... | https://arxiv.org/abs/2503.03020v2 |
independent of both fandA. Consequently, there exists a constant Cm∗>0, independent of fandA, such that hm∗=4m∗−1 n=Cm∗logn n 1 2β+1 . By applying Proposition G.11, we obtain that the CALM procedure yields a bandwidth h¯m=h¯m(f,A) 59 satisfying inf A⊆[n]inf f∈Σ(β,L)P h¯m(f,A)≥Cm∗logn n 1 2β+1! ≥1−n−(κ−1)2µ2/(4K2 ma... | https://arxiv.org/abs/2503.03020v2 |
min h≥hm∗ Ti,j(h)−Cn,α,i,j (h) = min h≥hm∗ ˆfn(xi;h)−fn(xi) +fn(xi)−fn(xj) +fn(xj)−ˆfn(xj;h)−Cn,α,i,j (h) ≥ −Cmax·∆β,n+ 2Cmax+σr 4W Cm∗! ·∆β,n−Cmax·∆β,n−max h≥hm∗Cn,α,i,j (h) ≥σr 4W Cm∗logn nβ 2β+1 −Cn,α,i,j (hm∗)≥0, where Cn,α,i,j (h) is given in (37b) with hnreplaced by h. The first inequality follows from Ecan... | https://arxiv.org/abs/2503.03020v2 |
j)∈ Pε−1 n, Gc∩Fc) into Pfn(Φi,j= 0,for all ( i, j)∈ Pε−1 n, Gc∩Fc) =Pfn(Φi,j= 0,for all ( i, j)∈ Pε−1 n,Pε−1 n∩ I ̸=∅, Gc∩Fc) +Pfn(Φi,j= 0,for all ( i, j)∈ Pε−1 n,Pε−1 n∩ I=∅, Gc∩Fc), (63) with I=In= (i, j) hm∗≤xi< xi+1≤1−hm∗,min h≥hm∗Di,i+1(h)−Cn,α,i,i +1(h)−σr 6W Cm∗·1 n∆β,n≥0 . Then, for any pair ( i, j)∈ Pε−1 n∩... | https://arxiv.org/abs/2503.03020v2 |
97% 99% DS 76% 99% 100% 97% ABD 81% 96% 100% 96% C 72% 95% 100% 99% n= 1200FOMT 100% 100% 100% 99% A-FOMT 96% 100% 100% 100% DS 98% 100% 100% 100% ABD 83% 99% 100% 99% C 94% 100% 100% 99% n= 1600FOMT 100% 100% 100% 100% A-FOMT 100% 100% 100% 100% DS 99% 100% 100% 100% ABD 100% 99% 100% 99% C 95% 95% 100% 99% n= 2000FOM... | https://arxiv.org/abs/2503.03020v2 |
.752 n= 2000 361.777 14 830 .578 242 213 .872 n= 2400 611.162 24 424 .565 381 613 .095 n= 2800 783.221 44 426 .914 655 026 .810 n= 3200 1012.866 68 112 .276 767 809 .500 Table 7: The median of computational times (in seconds) of FOMT and A-FOMT for large scale datasets over 100 and 10 repetitions, respectively, with si... | https://arxiv.org/abs/2503.03020v2 |
arXiv:2503.03047v1 [math.PR] 4 Mar 2025Stochastic block models with many communities and the Kesten–Stigum bound Byron Chin∗Elchanan Mossel†Youngtak Sohn‡Alexander S. Wein§ Abstract We study the inference of communities in stochastic block mo dels with a growing number of communities. For block models with nvertices an... | https://arxiv.org/abs/2503.03047v1 |
relabeling, and (ii)the sparsity of the model, which, even up to these symmetries, allows for recovery only up to constant co rrelation with the true communities. The formal definition of the recovery notion we consider, com monly referred to as weak recovery , is provided in Definition 1.1below. The paper of Decelle, Kr... | https://arxiv.org/abs/2503.03047v1 |
below the KS bound when χ<1/2. Moreover, we establish efficient recovery above the KS bound whenq→ ∞. Perhaps surprisingly, we establish that for χ >1/2, efficient recovery is achievable using non- backtracking walks even below the usual KS bound dλ2= 1. Thus, there is a transition in feasibility of weak recovery with resp... | https://arxiv.org/abs/2503.03047v1 |
difference between 1/q+δ andδbecomes negligible, and we use the simpler δthreshold. Theorem 1.2. LetG∼SBM(n,q,d,λ). Then, the following holds. 1. Suppose q=nχwhere1/2 +ε≤χ <1for someε >0. Then, there exists a universal constantC >0such that if dλ1/χ>C(logd)2then Algorithm 1achievesδ-recovery for some δ≡δ(ε,d)>0. 2. Supp... | https://arxiv.org/abs/2503.03047v1 |
andCorr≤D=o(1)indicates failure to correlate with x. It is a standard fact (see e.g. [ 52, Fact 1.1]) thatCorr≤Dis directly related to the minimum mean squared error in esti mating 1{σ⋆ 1=σ⋆ 2}by a degree-Dpolynomial. Bounds on Corr≤Dhave a conjectural interpretation for time complexity: if Corr≤D=o(1)for some super-lo... | https://arxiv.org/abs/2503.03047v1 |
KS threshold when q≪√n. Combining our methods to prove Theorem 1.2(2) with Theorem 1.3yields the following result. Corollary 1.5. Fixε,γ,η > 0. LetG∼SBM(n,q,d,λ)whereq→ ∞ andq≤n1/2−ε. Then we have the following: 1. Ifdλ2= 1 +η >1, then there exist constants C≡C(η)>0andδ≡δ(η,d)>0such that Corr≤Clogn≥δfor large enough n.... | https://arxiv.org/abs/2503.03047v1 |
remains an open problem. 1.1.5 Information-theoretic thresholds and detection To complement the algorithmic picture we study the informat ion-theoretic behavior as well. We show that the scaling for the threshold is determined by the S NRdλwhenever the number of communities increases with the size of the graph. Theorem... | https://arxiv.org/abs/2503.03047v1 |
concentration to the same quantitative strength as abov e the KS bound. This is also the source of the(logd)2term in our bound, as partitioning the graph reduces the SNR w hich we require to be large enough on each piece. We next estimate the performance of the local estimators, wh ich are effectively a weighted count o... | https://arxiv.org/abs/2503.03047v1 |
overlapping regimes. Lei and Rinaldo [ 36] use spectral clustering to achieve weak recovery. Their re sults are about the semi-sparse regime where a≥logn, and their error rate is meaningful when dλ2>C(1+λ(q−1)). This matches our result up to constant factors when λqis rather small, corresponding to the case whereq≪√ d.... | https://arxiv.org/abs/2503.03047v1 |
de nser block models with q≫√n communities. This conjecture corresponds to the yellow reg ions in the leftmost diagram in Figure 1 being low-degree hard. Conversely, handling the logarithm ic factors in the achievability result seems to require a new idea. Our analysis leaves open the “critical ” regime where q≍√n. It ... | https://arxiv.org/abs/2503.03047v1 |
Proposition 1.10. For any two fixed vertices u,v, the weighted self-avoiding walk count between u andvisSu,v=/summationtext γXγwhereγsums over all self-avoiding walks of length k=⌊β(d,s)logn⌋from utov. Letu,v,u′,v′be distinct vertices. Then the following holds: E[Su,v] =/braceleftBigg (1+o(1))(q−1)sk nσ⋆ u=σ⋆ v, (1+o(1)... | https://arxiv.org/abs/2503.03047v1 |
section introduces Algorithms 1and2in detail and is devoted to combining the ingredients to prove Theorem 1.2and Corollary 1.5(1). 2.1 Algorithm preliminaries Our efficient algorithms will all follow the same structure. T he specific algorithm used in each regime will differ slightly, so we postpone more detailed des cript... | https://arxiv.org/abs/2503.03047v1 |
and for any vector z, the vector Nzcan be computed in timeO((m+n)k). 2.2 Path counting In this subsection, we recall the combinatorial path bounds from [ 45, Section 4] that control the number of paths of fixed length in the complete graph with cond itions on the number of self- intersections. These will be a crucial in... | https://arxiv.org/abs/2503.03047v1 |
=σ⋆(v)}−1. The first lemma describes the moments for the weight of a self- avoiding walk or simple cycle. This will serve as the building block for our more involved co mputations. Lemma 2.11. Letγbe a self-avoiding walk of length kfromutov. Suppose that dmk≤n1−ε. Then E/bracketleftbig Xm γ/vextendsingle/vextendsingleσ⋆... | https://arxiv.org/abs/2503.03047v1 |
t are fixed, we need to ensure that the vertices appear only at the endpoints of the walks, and no t in the interior. Lemma 2.13 ([45, Section 5.2]) .For any set of vertices Uand any path γfromutov, there exists a decomposition, which we refer to as the U-canonical SAW-decomposition of γwith Vend=U∪V≥3∪{u,v}∪{w∈γ:γbackt... | https://arxiv.org/abs/2503.03047v1 |
our sample fro m SBM(n,q,d,λ)involving deleting some vertices and choosing representat ives. This slightly biases the community distribution of the vertices in the remaining part of the graph. However, and importantly, conditioned on the communities edges are stil l drawn independently and at random with probabilitiesa... | https://arxiv.org/abs/2503.03047v1 |
the following proofs, we will refer only todandnwith the understanding that the1+o(1)factors are absorbed into the leading constants. In essence , the moments are computed according to SBM(n,q,a,b), but the result holds for /tildewideSBM(n′,(si)q i=1,a n,b n)as the bias is negligible. Proof of ( 2.1).The number of walk... | https://arxiv.org/abs/2503.03047v1 |
is of total length 2k. Thus, there can be at most kof these edges. In the average over σ⋆ Vend\U, the exponent of uis at mostkrby the following argument. Each returning edge that has an en dpoint inVendintroduces at most one factor of qon average, by an analogous argument to the Γ1case above. Since σ⋆ u/\e}atio\slash=σ... | https://arxiv.org/abs/2503.03047v1 |
since γis the concatenation of two walks that are non-backtracking but not self-avoiding, we must have kr(γ)≥1. Thus, Lemma 2.6implies that the number of such paths is at most nkn(γ1)+kn(γ2)−2+o(1). Thus, the total contribution is at most /summationdisplay kr≥1nkn−2+o(1)·qkr+r+1+ /BD{σ⋆ u=σ⋆ v}s2kn−kn−kr= q1+ /BD{σ⋆ u=... | https://arxiv.org/abs/2503.03047v1 |
Letτw=u∗if Zloc i(w,u∗)>q·/parenleftbigg1 3a a+(q−1)b/parenrightbigg |N(u∗)∩Vi| ∀1≤i≤M and Zloc i(w,u)<q·/parenleftbigg1 6a a+(q−1)b/parenrightbigg |N(u∗)∩Vi| ∀1≤i≤M, u/\e}atio\slash=u∗∈U∗. Otherwise assign it randomly. We first claim that Propositio n2.17applies to our setup. Claim 2.19. With high probability G|Visatis... | https://arxiv.org/abs/2503.03047v1 |
a union bound, each of these N(i) u,wcan be replaced with S(i) u,wat a cost ofo(qsk i ni)with probability logn/parenleftBig q ni/parenrightBig1/3 =o(1). Union bounding over the remaining pairs, for which the two ve rtices are in different communities, eachN(i) u,wcan be replaced by S(i) u,wwith probability qlogn·q−2/3n−... | https://arxiv.org/abs/2503.03047v1 |
in Ti. Note that the number of neighbors of viis distributed identically with the number of children Ci, so we can couple the vertices perfectly. Applying Bayes’ rul e, conditioned on σ⋆ vithe probability that its neighbor has the same label isa a+(q−1)b=1+(q−1)λ q. Thus, the broadcast process on the tree is chosen so ... | https://arxiv.org/abs/2503.03047v1 |
over σ⋆ Uwe obtain P/parenleftbigg |Z(w,u∗)−E[Z(w,u∗)]|>1 12qsk n|N(u∗)|/vextendsingle/vextendsingle/vextendsingle/vextendsingleσ⋆ w/\e}atio\slash=σ⋆ u∗/parenrightbigg ≤o/parenleftBigg E[P] |N(u∗)|+E/bracketleftbig P2/bracketrightbig |N(u∗)|2+1 q/parenrightBigg . Recall that P∼Bin(|N(u∗)|,1−λ q)so E[P] =|N(u∗)|1−λ qand... | https://arxiv.org/abs/2503.03047v1 |
3.1. 29 For the rest of this section, we prove Theorem 3.1. Following the framework of [ 55], our proof will take the following steps. We will choose a basis {φα}α∈IforR[Y]≤D, so that an arbitrary degree-Dpolynomial (in Y) can be expanded as f(Y) =/summationdisplay αˆfαφα(Y). We will choose a collection of random varia... | https://arxiv.org/abs/2503.03047v1 |
for α /∈ˆIthen follow automatically. As a result, it suffices to verify ( 3.5) below. Proof. Considerα /∈ˆI. We divide into two cases. First, suppose that {1,2}/notsubseteqlV(α), and w.l.o.g. assume2/∈V(α). By setting ˆα=∅, Lemma 3.3shows that Mβγ,α= 0 =Mβγ,ˆαfor allβ/\e}atio\slash=∅. Moreover, since 2/∈V(α), cα=E/bracke... | https://arxiv.org/abs/2503.03047v1 |
compute cα≡E[φα(Y)·x]forα∈ˆI. Note that E[Yij−b n|σ⋆] =a−b n·1{σ⋆=σ⋆ j}, so E/bracketleftbig φα(Y)·x/vextendsingle/vextendsingleσ⋆/bracketrightbig =/parenleftbigga−b n/parenrightbigg|α| ·/parenleftbigg 1{σ⋆ 1=σ⋆ 2}−1 q/parenrightbigg ·1{σ⋆ i=σ⋆ j,∀i,j∈V(α)}. Sinceα∈ˆIis connected, it follows that cα= (q−1)·q−|V(α)|·/pa... | https://arxiv.org/abs/2503.03047v1 |
where the inequality holds by the induction hypothesis. Thu s, it follows that /summationdisplay βγ∈ˆJ:β/lessnotequalα|uβγ|·|Mβγ,α| ≤q−|V(α)|+1/parenleftbigg|a−b| n/parenrightbigg|α|/summationdisplay β∈ˆI:∅/lessnotequalβ/lessnotequalαf(β). Plugging this estimate into ( 3.12) and using the expression ( 3.9) for|cα|, it ... | https://arxiv.org/abs/2503.03047v1 |
, where the last equality holds because M1,2(p) =p1+(p2−p1)1{zi=zj}. By Theorem 1.3, as long asp1≤p2≤qp1and n(p2−p1)2 q/parenleftbig p1(1−p1)+(q−1)p2(1−p2)/parenrightbig≤1, D≤nc thenCorr≤D≤C/radicalBig Dq nwherec,C >0depend on ε. Note that we can choose p1,p2. The choice p2=1 2andp1=1 2−1 4/radicalBig q2 nsatisfies the ... | https://arxiv.org/abs/2503.03047v1 |
ρ)≥/parenleftBig λq1−2 dλ/parenrightBigdλ 2 /parenleftBig λq1−2 dλ/parenrightBigdλ 2+q/parenleftBig λq1−2 dλ/parenrightBig2≍qdλ 2−1 qdλ 2−1+q3−4 dλ. Whendλ>7we havedλ 2−1>3−4 dλso in particular Xρ(σ⋆ ρ) = 1−o(1). Thus, conditioned on this good event in the neighborhood of a vertex, one iteration of b elief propagation ... | https://arxiv.org/abs/2503.03047v1 |
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