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=Pt∼P/parenleftbigg t∈/bracketleftbigg1 2∝⌊a∇⌈⌊lθ∝⌊a∇⌈⌊l,3 2∝⌊a∇⌈⌊lθ∝⌊a∇⌈⌊l/bracketrightbigg/parenrightbigg ≤1 ∝⌊a∇⌈⌊lθ∝⌊a∇⌈⌊l·O/parenleftBig√ d/parenrightBig/parenleftbigg 1−4 ∝⌊a∇⌈⌊lθ∝⌊a∇⌈⌊l2/parenrightbigg(d−3)/2 ≤O(1)√ d Cexp/parenleftbigg −d−3 2C2/parenrightbigg , Therefore, as long as C≤c0√ d, we have inf µ∈∆(X),... | https://arxiv.org/abs/2501.14928v1 |
Private Minimum Hellinger Distance Estimation via Hellinger Distance Differential Privacy Fengnan Deng Department of Statistics George Mason University Fairfax, VA 22030Anand N. Vidyashankar Department of Statistics George Mason University Fairfax, VA 22030 Abstract Objective functions based on Hellinger distance yield... | https://arxiv.org/abs/2501.14974v2 |
noise added for privacy (referred to as a mechanism). In a non-interactive setting, the data warehouses offer a dataset with added noise, and users can apply any models and methods to this data. Controlling privacy breaches is challenging in the non-interactive setting (see Dwork et al. (2006), for instance). In contra... | https://arxiv.org/abs/2501.14974v2 |
(Bun and Steinke (2016); Dwork and Rothblum (2016)) can also be used for privacy guarantees. Some of these are described in Section 2. The sensitivity of the query function plays a central role in DP investigations. In applications in point estimation, the query concerns the gradient of the loss function under appropri... | https://arxiv.org/abs/2501.14974v2 |
Mis a measurable mapping from ( Rm×Rn,B(Rm)× B(Rn))7→(Rm,B(Rm)).Mis said to be an additive mechanism, if M(w, D) =f(w, D)+Y, where Y= [Y1,···, Ym]∈Rm, is a random vector (with i.i.d. components) representing the noise and independent of (w, D). In here, M(·,·) represents a private version of f(·,·). Continuing with the... | https://arxiv.org/abs/2501.14974v2 |
in parametric models (see also Basu et al. (2011)). Let P1andP2be two probability distributions possess- ing densities p1(·) and p2(·) onRm. The power divergence Dλ(P1, P2) between P1andP2, denoted by Dλ(P1, P2) is defined as follows: for λ̸=−1,0 Dλ(P1, P2) =1 λ(λ+ 1)EX∼p2"p1(X) p2(X)λ+1 −1# . D0(P1, P2) and D−1(P1, ... | https://arxiv.org/abs/2501.14974v2 |
[M1(M(0)(w), D)]; for n≥2 M(n)(w, D) = [M(n−1)(w, D), Mn(⟨en, M(n−1)(w)⟩, D)], where en= (0,0,···,1)1×(n−1)is the unit vector. We note here that the trajectory is useful for describing the composition property and for calculating the mechanisms. However, only the nthcomponent of M(n) is released. Next, turning to the s... | https://arxiv.org/abs/2501.14974v2 |
in λand is minimized at λ=−1 2,σ2is minimized at λ=−1 2. 3. For both adaptive and sequential composition, the privacy is maximized in the power divergence class at λ=−1 2. 4. When λ=−1 2, PDP is a symmetric divergence and has a simpler group privacy representation (see Theorem 2.4 below). We now turn to a more detailed... | https://arxiv.org/abs/2501.14974v2 |
ϵ−HDP, σ2=(∆L2W)2 8 log(1 1−0.5ϵ). 2. IfY= (Y1,···, Ym),Yi∼Lap(0, b), then to achieve ϵ−HDP, b=∆L1W 2 log(1 1−0.5ϵ). Ifm= 1, then a sharper value of bfor the Laplace mechanism can be obtained by using Lemma B.3 in Appendix B and solving −2 e−∆L1w 2b+∆L1w 2be−∆L1w 2b−1 =ϵ. The exact value of bfor multidimensional para... | https://arxiv.org/abs/2501.14974v2 |
proposed in Beran Beran (1977), has been extended to several general statistical models, including dependent data (see Basu et al. (2011); Cheng and Vidyashankar (2006); Li et al. (2019)). A useful feature of these estimators is that they 10 are, like maximum likelihood estimators (MLEs), first-order efficient when the... | https://arxiv.org/abs/2501.14974v2 |
this end, we need a few additional notations. (U1). Letuθ,i(x) =∂ ∂θilogfθ(x). Assume that for all 1≤j≤m,0≤kj≤6andk1+k2+···km≤6, Eθ"mY i=1|uθ,i(X)|ki# <∞. Additionally, assume that the expectation above is continuous in θ. (U2). Assume that all the partial and cross-partial derivatives of fθup to order three exist and ... | https://arxiv.org/abs/2501.14974v2 |
at θg,Brϵ(θg), such that for all θ∈Brϵ(θg),H∞(θ)is strictly positive definite. Furthermore, λmax(H∞(θ))≤C, where 0< C < ∞is independent of θ. Proof : First notice using equation (D.2) in Appendix D that Di,j(θ)≤c·HD(g, fθ) where c >0 is independent of θ, which implies that D(θ)≤C′HD(g, fθ)Jmwhere Jmis am×mmatrix of one... | https://arxiv.org/abs/2501.14974v2 |
The idea is to update the estimator ˆθ(k) nuntil it reaches the zero of ∇Ln(θ). Letting kincrease without bound ensures that ˆθ(k) nis close to ˆθn, where ˆθnis the stationary point of Ln(θ). It is known that ˆθnis not guaranteed to be the global minimizer of the loss function (see Agarwal et al. (2009)). However, unde... | https://arxiv.org/abs/2501.14974v2 |
As explained before, sensitivity is defined on a pair of adjacent datasets with an unbounded range. In the HD setting, the sensitivity appears through the integrals of kernel densities of adjacent datasets, yielding a weak upper bound. The disadvantage of this weak-upper bound is that it does not yield asymptotic norma... | https://arxiv.org/abs/2501.14974v2 |
Input: MHDE loss function Ln(θ), number of iteration K, learning rate η, MHDE privacy level ϵ, each iteration privacy level ϵ′from Proposition 3.8, initial point ˆθ(0) n. Output: Private MHDE ˆθ(K) n. k= 1. while k≤Kdo Generate Zkfrom N(0, I), with same dimension of ˆθ(0) n. Calculate Hessian matrix of Ln(θ) atˆθ(k−1) ... | https://arxiv.org/abs/2501.14974v2 |
associated with ¯ gn(·). (U3). LetAn= (−bn−βcn, bn+βcn)denote the support of ¯gn. Let δn= inf x∈An¯gn(x). Let p∈(1,2) and satisfy1 p+1 q= 1.We assume that c1 pn(ncn)−(1−1 p)≤δ1 2n→0. Additionally, assume that Eθ ∥uθ(X)f1 2−1 q θ(X)∥q 1 ,Eθ ∥uθ(X)uT θ(X)f1 2−1 q θ(X)∥q 1 ,andEθ ∥˙ uθ(X)f1 2−1 q θ(X)∥q 1 are all fi... | https://arxiv.org/abs/2501.14974v2 |
(3.10) with probability at least 1−ξ, where c2∈(0,∞)is a constant depending on fθandm. That is, ||ˆθ(Kn) n− ˆθn||2=Op n−1 p(Knlog(Kn))1 2 . Remark 3.2. 1. The calculations show that the upper bound in the above theorem is ||ˆθ(Kn) n−ˆθn||2≤C·rnoi, where rnoi∼∆(H) n· 2mlog4Knm ξ1 2 −8 log(1 −ϵ 4Kn)1 2. The specif... | https://arxiv.org/abs/2501.14974v2 |
described in Proposition 2.3. Then, both satisfy the ϵ−HDP using the post-processing property. Now, since ˆθ(Kn) n isϵ−HDP, we obtain, using Theorem 4 in Wang et al. (2018), that the resulting confidence interval is 3 ϵ−HDP. To derive the private version of Σ g, it is convenient to use an alternative expression frequen... | https://arxiv.org/abs/2501.14974v2 |
loss function is approximated using the Monte-Carlo approach; that is, Ln(θ) =2·Z R f1/2 θ(x)−g1/2 n(x)2 dx≈2" 2−21 nrnX i=1s fθ(Xn,i) gn(Xn,i)# , 22 where rnis the number of Monte Carlo samples and {Xn,i···Xn,rn}|(X1,···Xn)i.i.d.∼gn(·).The algorithm in Cheng and Vidyashankar (2006) is used to generate data from gn(·... | https://arxiv.org/abs/2501.14974v2 |
poor in contrast to the HDP setting. 23 ϵ 2.00 0.60 0.20 Estimatorµ: Mean (Std. Error) 4.991 (0.083) 4.989 (0.2) 4.996 (0.349) σ: Mean (Std. Error) 1.984 (0.058) 2.002 (0.144) 2.043 (0.256) CI coverage for µCorrected 0.861 0.836 0.824 Uncorrected 0.861 0.487 0.327 CI coverage for σCorrected 0.819 0.933 0.927 Uncorrecte... | https://arxiv.org/abs/2501.14974v2 |
feasible. In Appendix E, we provide numerical experiments illustrating the behavior of private estimators for different sample sizes (ranging from 200 to 500) and privacy levels. Turning to Table 6 last row ( ϵ= 0.2), we note that the standard error for PMHDE is larger due to aberrant values of the private estimate of ... | https://arxiv.org/abs/2501.14974v2 |
change. A detailed analysis of this case with a concentration inequality for the Laplace random variables and other probabilistic properties of compositions, especially when the number of queries diverges, will be discussed elsewhere. It is also possible to extend the results to other minimum divergence estimators, suc... | https://arxiv.org/abs/2501.14974v2 |
D) and M2(w, D′) respectively and Vhas the density qY2. To start the proof of part (3), we first notice that adjacent D(2)andD(2)′can be decomposed into two distinct cases. By definition, ||D(2)−D(2)′||H=2X i=1||Di−D′ i||H= 1. Since the Hamming distance is a non-negative integer, the above equation holds if either: Cas... | https://arxiv.org/abs/2501.14974v2 |
λ=−1 2 and replacing ϵby 2ϵto obtain ϵ−HDP. 6.3 Proof of Corollary 2.1 The proof is based on the following iterative argument for adaptive and sequential compositions. Setting ϵ1=ϵandϵ2=h1(ϵ) it follows from Theorem 2.3 part 1. and part 2., that h2(ϵ) =ϵ+h1(ϵ)−1 2ϵh1(ϵ). Now iterating, we obtain hj+1(ϵ) =ϵ+hj(ϵ)−1 2ϵhj... | https://arxiv.org/abs/2501.14974v2 |
the proof. Before we prove the theorem, we recall that gn(x) =1 ncnnX i=1Kx−Xi cn . and for the neighboring i.i.d. observations {X′ 1, X2,···, Xn}, the corresponding density estimator is ˜gn(x) =1 ncnnX i=2Kx−Xi cn +1 ncnKx−X′ 1 cn . The corresponding loss functions are given by Ln(θ) = 2 HD2(gn, fθ) and ˜Ln(θ) =... | https://arxiv.org/abs/2501.14974v2 |
upper bound of the last term on RHS. By assumption (A2) ,K(·) has compact support, say [ −β, β]. In the calculation, fix any X1=x1,X′ 1=x′ 1andx1, x′ 1∈Bn. Then write S=supp x K(x−x1 cn) ∪supp x K(x−x′ 1 cn) = [x1−βcn, x1+βcn]∪[x′ 1−βcn, x′ 1+βcn] andλ(S)≤4βcn, where λ(S) is the Lebesgue measure of S. Notice that h... | https://arxiv.org/abs/2501.14974v2 |
6.9 Proof of Theorem 3.2 The proof of the theorem relies on the behavior of the Hellinger loss function at private estimates. Intuitively, we show that under ASLSC and τ2−smoothness, the closeness of the loss functions implies the closeness of the parameter estimates and vice-versa. This is achieved via Lemma 6.1-Lemma... | https://arxiv.org/abs/2501.14974v2 |
result verifying the validity of the conditions in Lemma 6.1 above. Lemma 6.3. Under assumptions (A1) -(A8) and(U1) -(U2) , for η≤1 τ2, assume that for n≥N, ˆθn∈Br/c(θg)⊂Br/2(θg), where c >2 τ2 τ11 2, then there exists ˆθ(0) n, such that Ln(ˆθ(k) n)−Ln(ˆθn)≤τ1r2 4 and||ˆθ(k) n−ˆθn||2≤r 2hold with probability 1−kξ Kfo... | https://arxiv.org/abs/2501.14974v2 |
that Clog2(n)converges to C∞(γ):= 2(ητ2+ 1)( γτ1)−1. Also, notice that r1 n noiconverges to 1. Now choosing γ∈(0,2ητ1) and C∞>1 (such aγexists) it follows that lim sup n→∞r−2 noi||ˆθ(kn) n−ˆθn||2=C∞. (6.15) This requires K≥k0+k1+···+klog2(n)∼(logn)·(logrnoi) which implies K≥clogn, since rnoiis bounded by a constant by ... | https://arxiv.org/abs/2501.14974v2 |
n(ˆθ(k) n)∇Ln(ˆθ(k) n) +˜Nn,k, that ∇Ln(ˆθ(k) n) +Hn(ˆθ(k) n)·[ˆθ(k) n−ˆθ(k+1) n−˜Nn,k] = 0. We now rewrite ||∇Ln(ˆθ(k+1) n )||2as ||∇Ln(ˆθ(k+1) n )||2=||T1−T2+Hn(ˆθ(k) n)˜Nn,K||2,where T1=∇Ln(ˆθ(k+1) n )− ∇Ln(ˆθ(k) n) and T2=Hn(ˆθ(k) n)(ˆθ(k+1) n−ˆθ(k) n). 43 Notice that T1−T2can be written as T1−T2=Z1 0Hn(ˆθ(k) n+t(ˆ... | https://arxiv.org/abs/2501.14974v2 |
is for n > N such that P(||˜Nn,k||2≤rnoi)≥1−ξ Kforrnoi∼n−1 p(Klog(K/ξ))1 2. We will use Lemma 6.5 and Lemma 6.6 to obtain the following claim: Claim : For ˆθn∈Br/2(θg) and ||∇Ln(ˆθ(0) n)||2≤min{τ1r 2,τ2 1 α}, the inequality α 2τ2 1||∇Ln(ˆθ(K) n)||2≤α 2τ2 1||∇Ln(ˆθ(0) n)||22K + 3C·rnoi holds for some constant C∈(0,∞) ... | https://arxiv.org/abs/2501.14974v2 |
of the difference between the square root of density functions, that is, HD2(f, g) =||f1 2(x)−g1 2(x)||2 2=Zh f1 2(x)−g1 2(x)i2 dx. Let{X1, X2,···, Xn}be i.i.d. real-valued random variables with density g(·), and postulated to belong to a parametric family {fθ:θ∈Θ⊂Rm}. The minimum Hellinger distance estimator in the po... | https://arxiv.org/abs/2501.14974v2 |
=1 (√ 2πσ)me−||x−w2||2 2 2σ2. Lety=x−w2,v=w1−w2, and denote by yi, vitheithelement of yandv. For the case λ(λ+ 1)̸= 0, the power divergence with parameter λbetween XandYis given by Dλ(X,Y) =1 λ(λ+ 1)Z Rmpλ+1(x) qλ+1(x)·q(x)−q(x) dx =1 λ(λ+ 1)Z Rm1 (√ 2πσ)me−(λ+1)||x−w1||2 2−λ||x−w2||2 2 2σ2 dx−1 =1 λ(λ+ 1)Z Rm1 (√... | https://arxiv.org/abs/2501.14974v2 |
For each i, we remove the absolute sign by studying case vi<0 and vi≥0. If vi<0, Z R1 2be−(λ+1)|yi−vi|−λ|yi| b dyi =1 2bZvi −∞e−−(λ+1)(yi−vi)+λyi b dyi+Z0 vie−(λ+1)(yi−vi)+λyi b dyi+Z∞ 0e−(λ+1)(yi−vi)−λyi b dyi =1 2b e−λvi b b+b 2λ+ 1 +e(λ+1)vi b b−b 2λ+ 1 . Ifvi≥0, Z R1 2be−(λ+1)|yi−vi|−λ|yi| b dyi =1 2bZ0 −∞... | https://arxiv.org/abs/2501.14974v2 |
δ)−DP. Ifλ <−1, write λ′=−λ−1>0, Dλ(f2, f1) =1 λ(λ+ 1)"Z Rmfλ+1 2(x) fλ 1(x)dx−1# ≤ϵ ⇐⇒Z Rmfλ+1 2(x) fλ 1(x)dx≤λ(λ+ 1)ϵ+ 1 = elog(λ(λ+1)ϵ+1) ⇐⇒Z Rmfλ′+1 1(x) fλ′ 2(x)dx≤λ′(λ′+ 1)ϵ+ 1 = elog(λ′(λ′+1)ϵ+1). Applying Holder inequality for p=λ′+ 1>1,q=λ′+1 λ′>1, by the same method, we obtain PX∼f1(X∈A)≤h e1 λ′log(λ′(λ′+1)ϵ+... | https://arxiv.org/abs/2501.14974v2 |
that |Di,j(θ)|= Z R g1 2(x)−f1 2 θ(x) f1 2 θ(x)[uθ,i(x)uθ,j(x) + 2uθ,i,j(x)]dx . Now, splitting the RHS of the above equation, we see that it is bounded above by Z R g1 2(x)−f1 2 θ(x) f1 2 θ(x)uθ,i(x)uθ,j(x)dx + 2 Z R g1 2(x)−f1 2 θ(x) f1 2 θ(x)uθ,i,j(x)dx . Now, applying Cauchy-Schwarz inequality, we get |Di,j(θ... | https://arxiv.org/abs/2501.14974v2 |
where the last inequality follows by taking nlarge. This is equivalent to choosing nsuch that ∆n 4√m+ 2q 2 log(K ξ) p 8 log(1 −0.5ϵ K)≤ru. Finally, the inequality Ln(ˆθ(k+1))−Ln(ˆθn)≤τ1r2 4follows using Lemma 6.2 and ||ˆθn−ˆθ(k+1) n||2≤ τ1 τ21 2r 2. This completes the induction. To complete the proof of the Lemma, ... | https://arxiv.org/abs/2501.14974v2 |
Uncorrected 0.819 0.379 0.277 Table 9: Results for different values of ϵ(Gradient descent). Sample size is 1000, K= 50. λ= 0.5,ϵ Non-private 1.20 0.40 Estimatorµ: Mean (Std. Error) 5 (0.08) 4.942 (0.483) 4.808 (2.064) σ: Mean (Std. Error) 1.975 (0.076) 2.03 (0.523) 2.297 (2.313) CI coverage for µCorrected 0.883 0.972 0... | https://arxiv.org/abs/2501.14974v2 |
results, Newton, sample size is 200, µ= 5. 64 Sample size 300: ϵ 2.00 0.60 0.20 Estimatorµ: Mean (Std. Error) 4.989 (0.128) 4.979 (0.405) 4.926 (0.866) σ: Mean (Std. Error) 1.962 (0.086) 2.023 (0.338) 1.952 (1.971) CI coverage for µCorrected 0.918 0.837 0.733 Uncorrected 0.918 0.493 0.317 CI coverage for σCorrected 0.8... | https://arxiv.org/abs/2501.14974v2 |
(0.08) 6.424 (0.076) PMHDE ϵ= 2 (Std. Error) 4.995 (0.103) 5.169 (0.105) 5.309 (0.11) 5.553 (0.113) 5.793 (0.117) PMHDE ϵ= 0.6 (Std. Error) 4.915 (0.631) 5.069 (0.606) 5.2 (0.629) 5.383 (0.659) 5.546 (0.716) PMHDE ϵ= 0.2 (Std. Error) 4.777 (1.316) 4.923 (1.306) 4.995 (1.384) 5.151 (1.386) 5.259 (1.419) Table 22: Contam... | https://arxiv.org/abs/2501.14974v2 |
data analysis. InTheory of Cryptography: Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006. Proceedings 3 , pages 265–284. Springer, 2006. C. Dwork, K. Talwar, A. Thakurta, and L. Zhang. Analyze gauss: optimal bounds for privacy-preserving principal component analysis. In Proceedings... | https://arxiv.org/abs/2501.14974v2 |
arXiv:2501.15680v1 [math.PR] 26 Jan 2025RANDOM PROCESSES WITH STATIONARY INCREMENTS AND INTRINSIC RANDOM FUNCTIONS ON THE REAL LINE Jongwook Kim Indiana University Bloomington jki5@iu.eduChunfeng Huang Indiana University Bloomington huang48@iu.edu ABSTRACT Random processes with stationary increments and intrinsic rando... | https://arxiv.org/abs/2501.15680v1 |
1.1 Random Processes with Stationary Increments To identify random processes with stationary increments, i t is necessary to define the (weakly) stationarity and its associated properties. Definition 1.1. Suppose that X(·)is a random process on the real line R. Then, the process is called (weakly) stationary if E/parenle... | https://arxiv.org/abs/2501.15680v1 |
) is satisfied. We also assume that the random variables X0,X1,···,Xd−1are zero. Remark 5. (Chiles 1999) [11] The differencing operator ∆hin Definition 1.2 reduces the degree of the polynomial by one level. That is, if we suppose any polynomial function w ith any degree n∈Nsuch as pn(x) =n/summationdisplay i=0aixi Then, ... | https://arxiv.org/abs/2501.15680v1 |
is a constant. Since pd−1(λ∆d h,t) = ∆d hpd−1(x) = ∆h/braceleftbigg ∆d−1 hpd−1(x)/bracerightbigg , we can conclude that ∆d hpd−1(x) = 0 To sum up, ∆d hannihilates polynomials of degree d−1; thus, it is an allowable measure of order d. Lemma 2. SupposeY(t)is stationary for t∈R. Then,Y(λ)is stationary for any allowable m... | https://arxiv.org/abs/2501.15680v1 |
of allowable measures that ann ihilate the polynomial components of an intrinsic random function. Theorem 1 also allows us to use the spectral repres entation of I(d) in Remark 3 to explore the structure of an intrinsic random function on the real line. 3 Applications and Discussions So far, we have demonstrated the eq... | https://arxiv.org/abs/2501.15680v1 |
on the allowable measure λt0. As a result, we can obtain the coefficient for the kriging predictor such t hat η ∼= (Ψ+σ2I)−1/bracketleftBigg φ ∼+Q/braceleftbigg QT(Ψ+σ2I)−1Q/bracerightbigg−1/braceleftbigg q ∼−QT(Ψ+σ2I)−1φ ∼/bracerightbigg/bracketrightBigg . This demonstrates that the kriging estimate for intrinsic r and... | https://arxiv.org/abs/2501.15680v1 |
Eigenvector fluctuations and limit results for random graphs with infinite rank kernels Minh Tang Department of Statistics, North Carolina State University Joshua Cape Department of Statistics, University of Wisconsin-Madison January 28, 2025 Abstract This paper systematically studies the behavior of the leading eigenv... | https://arxiv.org/abs/2501.15725v1 |
estimation and inference problems for both single and multiple networks can be formulated and tackled via the spectral embeddings given by α∈ {0,1/2,1}. Examples include community detection via spectral clustering, vertex nomination, two-sample hypothesis testing, and more; see Section 4 for additional discussion and r... | https://arxiv.org/abs/2501.15725v1 |
∥M∥F, its maximum absolute row sum norm by ∥M∥∞, its nuclear norm by ∥M∥∗, and its maximum absolute entrywise norm by ∥M∥max. 2 We denote the two-to-infinity norm (2 → ∞ norm) of the matrix Mby ∥M∥2→∞= max ∥x∥=1∥Mx∥∞≡max i∥mi∥, where ∥x∥denotes the Euclidean norm of the vector x, and midenotes the i-th row vector of M.... | https://arxiv.org/abs/2501.15725v1 |
integer r≥1, letUbe the n×rmatrix whose columns are orthonormal eigenvectors corre- sponding to the rlargest-in-magnitude eigenvalues of P. Let Λbe the diagonal matrix containing the corresponding eigenvalues of P. Then, with probability one, ∥U|Λ|1/2∥2→∞=∥U|Λ|U⊤∥1/2 max≤ ∥P∥1/2 max≤ρ1/2 n. (2.2) Note that the above bo... | https://arxiv.org/abs/2501.15725v1 |
Above, the notation ≲andO(·)hides universal constants that depend only on νbut not on r,n,ρn, orκ. Asnρn≥λ1≥λr≥δr, Eq. (3.1) through Eq. (3.4) automatically imply a lower bound of nρn= Ω(log n) as typically seen in the literature on spectral inference for random graphs. Furthermore, it holds that ∥Q∥2→∞=o(λ−1/2 rρ1/2 n... | https://arxiv.org/abs/2501.15725v1 |
with high probability. ♢ For ease of exposition, the conditions for rin Theorem 1 are stated in terms of the eigenvalues λrand λr+1of the edge probability matrix P. As Pis unknown, the next result, Corollary 1, replaces these conditions with those based on the eigenvalues bλrandbλr+1ofA. For simplicity, ρnis assumed to... | https://arxiv.org/abs/2501.15725v1 |
function is κ(x, y) = exp( −∥x−y∥) and the latent positions are sampled i.i.d. from the uniform distribution on the unit sphere in R3. While the eigenvalues of Pcan be fitted quite well by a curve of the form λr/n∝r−3/2, the gaps between consecutive eigenvalues are generally near-zero except for visible jumps at a few ... | https://arxiv.org/abs/2501.15725v1 |
1−O(n−ν). Here, ≲andO(·)hide universal constants that depend only on νbut not on r,n,ρn, orκ. Remark 8 (Residual analysis for positive semidefinite versus possibly indefinite kernels) .We now compare the terms appearing in the upper bound for Qin Theorems 1 and 2. To begin, observe that the second terms in each of Eqs.... | https://arxiv.org/abs/2501.15725v1 |
this bound is qualitatively similar to Eq. (2.5), there are two significant differences. Firstly, Eq. (3.22) is difficult to verify in practice, since we only observe Awhile κis unknown. Secondly, the rate O(n−1/2) only holds with probability 1 −o(1), so we cannot guarantee that δ2((nρn)−1{λk}k≥1,{µk}k≥1) =O(n−1/2) asy... | https://arxiv.org/abs/2501.15725v1 |
all terms depending on Qin the limiting distribution. Eq. (3.28) then follows from applying the Lindeberg–Feller central limit theorem [57, Proposition 2.27] to Σ−1/2 iEiU|Λ|−1/2. Finally, ris fixed in Corollary 2 as otherwise ifr→ ∞ then the convergence in distribution of Eq. (3.28) is possibly not well-defined. Never... | https://arxiv.org/abs/2501.15725v1 |
norm bounds for bU−UW(n)have appeared in numerous publications to date; see [1, 15, 16, 22, 39, 42, 45, 60] for an incomplete list of references. However, these works overwhelmingly either focus only on first order upper bounds for bU−UW(n), which may not be sufficiently refined for inference purpose, or are restricted... | https://arxiv.org/abs/2501.15725v1 |
author considered the notion of graph root distribution. More specifically, let κbe a graphon (equivalently, a symmetric measurable function from [0,1]2to [0,1]), and suppose the integral operator associated with κ(see Eq. (2.1)) satisfies Eq. (3.23). Let ξ1≥ξ2≥ ··· >0 be the enumeration of the positive eigenvalues of ... | https://arxiv.org/abs/2501.15725v1 |
Indeed, as Remark 7 clearly shows, it is only meaningful to make assumptions about the rate of decay of eigenvalues but not their gaps. 4 Implications for inference 4.1 Entrywise bound for P We now apply the results in Section 3 to obtain entrywise error bounds for estimating the edge probability matrix P. Note that, f... | https://arxiv.org/abs/2501.15725v1 |
Appendix A]. These estimators, however, can be computationally infeasible as their running time can be exponential in n. In contrast, estimators based on singular value thresholding, and thus (low rank) matrix factorizations, have running time of order O(n3). Furthermore, while the Frobenius norm error of USVT estimate... | https://arxiv.org/abs/2501.15725v1 |
Let ζijbe the ij-th element of EUU⊤+UU⊤E. Then, for i̸=j, we have ρ−1 n(ζij+ζji) =ρ−1 nX k̸=jeikmkj+ρ−1 nX k̸=iekjmik+ρ−1 neij(mii+mjj), which, conditioning on P, is a sum of independent mean zero random variables. Next, define σ2 ij=ρ−2 n X k[pik(1−pik)m2 kj+pkj(1−pkj)m2 ik] + 2pij(1−pij)miimjj! . (4.5) Then, for a fi... | https://arxiv.org/abs/2501.15725v1 |
the i-th andj-th row of A−P, respectively. We then have ∥bξi−bξj∥2=∥W⊤(bξi−bξj)∥2= (1 + o(1))∥(Ei−Ej)U∥2= (1 + o(1))ζ⊤DUU⊤Dζ, (4.8) where ζis now a vector whose components are independent sub-Gaussian random variables with mean zero and variance one, and Dis an×ndiagonal matrix whose diagonal entries are dk= (2pik(1−pi... | https://arxiv.org/abs/2501.15725v1 |
1 yields the following testing procedure for the hypothesis testing problem in Eq. (4.7) Notably, the proposed test statistic only involves A. Corollary 4 (Testing equality of latent positions with data-driven rank selection) .Consider the setting in Theorem 4. Let br=br(n)be defined as br= arg maxn j:|bλj| − |bλj+1| ≥... | https://arxiv.org/abs/2501.15725v1 |
consider pik= 1/2, so pik(1−pjk) +pjk(1−pik) = 1 /2, which does not depend on the value of pjk. Therefore, if pik≡1/2 for all kthen the distribution of T1(bXi,bXj) does not depend on {pjk}and we cannot use T1to construct a consistent test procedure. ForT2, by using the same arguments as for Theorem 4, we have that r−1/... | https://arxiv.org/abs/2501.15725v1 |
converges to a central (non-central, resp.) χ2 runder the null (resp. local alternative) hypothesis. A precise statement of this result is left to the interested reader. We note, however, that a fixed value of rdoes not lead to a consistent test procedure as it can fail to reject the null hypothesis when Xi̸=Xjif their... | https://arxiv.org/abs/2501.15725v1 |
kernel. bivariate normal distribution with mean zero and identity covariance matrix. We set Xn=X1and let Pbe the matrix whose entries are pij=ρexp(−∥Xi−Xj∥2/σ2), where ρ= 0.4 and σ2= 0.4. Given P, we sample 500 independent realizations of Afrom P. For each realization Awe obtain bU, the matrix formed by the r leading e... | https://arxiv.org/abs/2501.15725v1 |
and Graduate Education, with funding from the Wisconsin Alumni Research Foundation. References [1] E. Abbe, J. Fan, K. Wang, and Y. Zhong. Entrywise eigenvector analysis of random matrices with low expected rank. Annals of Statistics , 48:1452–1474, 2020. [2] J. Agterberg, Z. Lubberts, and C. E. Priebe. Entrywise estim... | https://arxiv.org/abs/2501.15725v1 |
Information Theory , 67:7380–7419, 2021. [22] A. Damle and Y.Sun. Uniform bounds for invariant subspace perturbations. SIAM Journal on Matrix Analysis and Its Applications , 41:1208–1236, 2020. [23] C. Davis and W. Kahan. The rotation of eigenvectors by a pertubation. III. Siam Journal on Numerical Analysis , 7:1–46, 1... | https://arxiv.org/abs/2501.15725v1 |
Davis-Kahan theorem in the two-to-infinity norm and its application to perfect clustering. arXiv preprint #2411.11728, 2024. [46] Y. Qin, L. Yu, and Y. Li. Iterative connecting probability estimation for networks. Advances in Neural Information Processing Systems , 34:1155–1166, 2021. [47] L. Rosasco, M. Belkin, and E.... | https://arxiv.org/abs/2501.15725v1 |
define the eigenvalue gap δs=λs−λs+1≥0. Let E0be the event E0={δr≥max{4∥E∥,8C2(ν) logn}}, (A.1) where C2(ν) =2 3(ν+ 2). By Weyl’s inequality and the stated hypotheses, on E0it holds that bλr≥λr− ∥E∥ ≥λr−1 4δr≥1 2λr>0. (A.2) Furthermore, by the discussion in the main text, since κis positive semidefinite, for any r≥1, ∥... | https://arxiv.org/abs/2501.15725v1 |
nψ0 exp(−9n1/2) +4∥E∥ δr +nρ1/2 nX k=1ψ(k) 3|bλr|−(k+1/2).(A.10) Bounding EbUbΛ−1/2 We begin with the expansion EbUbΛ−1/2=EUU⊤bUbΛ−1/2+E(I−UU⊤)bUbΛ−1/2 =EUΛ−1/2h W(n)+ (U⊤bU−W(n)) + (Λ1/2U⊤bU−U⊤bUbΛ1/2)bΛ−1/2i +E(I−UU⊤)bUbΛ−1/2, where W(n)denotes the orthogonal matrix mentioned above that solves the Frobenius norm Pr... | https://arxiv.org/abs/2501.15725v1 |
constant ν >0 such that all of the following events simultaneously hold. E0={δr≥max{6∥E∥,8C2(ν) logn}}, E1={∀h∈[n],∥e⊤ hEV[h]∥ ≤C1(ν)p ρnlogn∥V[h]∥F+C2(ν)∥V[h]∥2→∞logn}, E2={λr≥24C2(ν) logn+ 64δ−1 r∥E∥2)}. Recall that V[h]= (I−UU⊤)bU[h],C1(ν) =p 2(ν+ 2), and C2(ν) =2 3(ν+ 2). We therefore have bUbΛ1/2−UΛ1/2W(n)=EUΛ−1/2... | https://arxiv.org/abs/2501.15725v1 |
(A.19) with probability at least 1 −2n−(ν+1). By taking a union over all h∈[n], we conclude that E1holds with probability at least 1 −2n−ν, where C1(ν) =p 2(ν+ 2) and C2(ν) =2 3(ν+ 2). For ease of presentation, we shall now drop explicit constants from our derivations. Readers who are inter- ested in keeping track of t... | https://arxiv.org/abs/2501.15725v1 |
the above expressions for ∥V[h]∥Fand∥V[h]∥2→∞into Eq. (A.13) yields ∥e⊤ hEV[h]∥ ≤4C1(ν)√rρnlogn∥E∥ δr +C2(ν)(∥(I−UU⊤)bU∥2→∞+∥(I−bU[h]bU[h]⊤)bU∥) logn.(A.29) Substituting Eq. (A.29) into Eq. (A.28) and then rearranging terms yields ∥(I−bU[h]bU[h]⊤)bU∥ ≤8∥E∥(∥U∥2→∞+∥(I−UU⊤)bU∥2→∞) δr +8∥EU∥2→∞ δr+32C1(ν)√rρnlogn∥E∥ δ2r +... | https://arxiv.org/abs/2501.15725v1 |
Davis–Kahan theorem, we have the bound ∥U⊤ +bU+−W(+)∥ ≤ ∥ (I−U+U⊤ +)bU+)∥2≤2∥E∥ δr2 =ψ2 0, which likewise applies to ∥U⊤ −bU−−W(−)∥. Next, by the general form of the Davis–Kahan theorem [7, Theorem VII.3.1], we obtain ∥U⊤ +bU−∥ ≤∥U⊤ +(A−P)bU−∥ |λr| ≤∥U⊤ +EU−∥+∥U⊤ +E(I−U−U⊤ −)bU−∥ |λr| ≤∥U⊤EU∥+∥E∥ × ∥ (I−U−U⊤ −)bU−∥ |... | https://arxiv.org/abs/2501.15725v1 |
· ∥U∥2→∞+∥EU∥2→∞)(C2(ν) logn+ 2∥E∥) δr|λr|1/2, ∥Y2∥2→∞≤24C2(ν) logn+ 64δ−1 r∥E∥2 |λr| × ∥T∗∥2→∞+∥R2∥2→∞+∥Y0∥2→∞+∥Y1∥2→∞ . We now bound the quantities appearing in the above expressions. First recall that ϵ0=∥U|Λ|1/2∥2→∞≤n1/2ρn|λr|−1/2, ϵ 1=∥U⊥Λ⊥∥2→∞≤n1/2ρn. (A.45) Next, recall that, according to the conditions in Eq.... | https://arxiv.org/abs/2501.15725v1 |
|λr|1/2δr. Recall that T∗=EU|Λ|−1/2JW(n). Then, by Eq. (A.56), we have ∥Y2∥2→∞≤24C2(ν) logn+ 64δ−1 r∥E∥2 |λr| ∥T∗∥2→∞+∥R2∥2→∞+∥Y0∥2→∞+∥Y1∥2→∞ ≲ρ1/2 n(r1/2+ log1/2n) |λr|1/2logn |λr|+nρn δr|λr| +∥R2∥2→∞+∥Y0∥2→∞+∥Y1∥2→∞ ≲n3/2ρ2 n δ2r|λr|1/2+(ρnlogn)1/2((rnρn)1/2+ log n) |λr|1/2δr, where the second inequality follows ... | https://arxiv.org/abs/2501.15725v1 |
the same derivations as for Theorem 2, we have ∞X k=1Π⊥ UPkEbUbΛ−k 2→∞≤4ϵ1∥E∥ψ0 δr+nρ1/2 nX k=1ψ(k) 3|bλr|−k. Next, write EbU=EU[W(n+ (U⊤bU−W(n))] +E(I−UU⊤)bU. Lemma 2 then implies ∥E(I−UU⊤)bU∥2→∞≤16C1(ν)(rρnlogn)1/2∥E∥ δr +8(∥E∥ · ∥U∥2→∞+∥EU∥2→∞)(Cνlogn+ 2∥E∥) δr +∥(I−UU⊤)bUbΛ∥2→∞12C2(ν) logn |λr|+32∥E∥2 δr|λr| . No... | https://arxiv.org/abs/2501.15725v1 |
Theorem 3. Next, if E1holds, then we can apply the same arguments as that in the proof of Lemma 2 to obtain ∥E(I−UU⊤)bU∥ ≤16αr1/2∥E∥ δr +(∥E∥ · ∥U∥2→∞+∥EU∥2→∞)(8β+ 16∥E∥) δr +∥(I−UU⊤)bU∥2→∞ 6β+16∥E∥2 δr .(A.62) Given Eq. (A.62), the terms y(0) 2,∞andy(1) 2,∞are derived the same way as the upper bounds for ∥Y0∥2→∞and ... | https://arxiv.org/abs/2501.15725v1 |
nλ−1/2 r(r1/2+ log1/2n) (A.66) with high probability. As ∥U∥2→∞≤ρ1/2 nλ−1/2 r, combining Eqs. (A.65) and (A.66) yields ρ−1 n∥bPr−Pr∥max≲(r1/2+ log1/2n) λ1/2 r(A.67) 50 with high probability. Hence, under the aforementioned assumptions, we arrive at the entrywise bound ρ−1 n∥bPr−P∥max≤ρ−1 n∥Pr−P∥max+O λ−1/2 r(r1/2+ log... | https://arxiv.org/abs/2501.15725v1 |
term is either Uor (I−UU⊤) are the same. Hence bξi−bξj= (Ei−Ej)U[W(n)+ (U⊤bU−W(n)] + (Ei−Ej)(I−UU⊤)bU, where EiandEjare the i-th and j-th row of E, respectively. Finally, by following the remaining steps in the proof of Theorem 3, we obtain bξi−bξj= (Ei−Ej)UW(n)+γij,∥γij∥=Oh (ρnlogn)1/2logn δr+(rnρn)1/2 δri (A.79) wi... | https://arxiv.org/abs/2501.15725v1 |
n |λr|, (A.86) ∥M(0)∥2 F=∥M∗∥2 F−X k(m∗ kk)2≤ ∥M∗∥2 F−16nρ4 n λ2r= (1−o(1))∥M∗∥2 F. (A.87) The quadratic form ζ⊤DUU⊤Dζin Eq. (4.8) can be written as ζ⊤M∗ζ=ζ⊤M(0)ζ+X kζ2 km∗ kk=∥M(0)∥F×ζ⊤Mζ+X kζ2 km∗ kk. (A.88) For the condition in Eq. (A.84), let m(2) kkandm(2∗) kkdenote the k-th diagonal element of M2and ( M∗)2. Then,... | https://arxiv.org/abs/2501.15725v1 |
Z2, . . .) is a vector of independent N(0,1) random variables and λs(M) are the eigenvalues of the matrix Mordered in decreasing magnitudes. As the diagonal entries of Mare all zeroes and ∥M∥2 F= 1, we have E X s≥1Z2 sλs(M) =X s≥1E[Z2 s]λs(M) =X s≥1λs(M) = 0 , Var X s≥1Z2 sλs(M) =X s≥1Var[Z2 s]×λ2 s(M) = 2∥M∥2 ... | https://arxiv.org/abs/2501.15725v1 |
k], where urkis the rk-th entry of U. The above is, conditioned on P, also a sum of independent mean zero random variables. Hence, by another application of Bernstein’s inequality, we have X kurkusk[(aik−ajk)2−d2 k]≲(nρn)1/2× ∥U∥2 2→∞×log1/2n ≲n3/2ρ5/2 nlog1/2n λ2r(A.98) with high probability, and thus ∥bU⊤bD2bU−U⊤D2U∥... | https://arxiv.org/abs/2501.15725v1 |
1, and let ξ=Ux, so x⊤U⊤EUx = 2X i<jeijξiξj+X ieiiξ2 i. Letting uidenote the i-th row of U, we have max i|ξi| ≤max i∥ui∥ × ∥ x∥ ≤ ∥ U∥2→∞. In particular,P i<jeijξiξjis a sum of independent mean zero random variables satisfying 2 max i,j|eijξiξj| ≤2∥U∥2 2→∞. Define σ2=P i<jVar[2 eijξiξj] +P iVar[eiiξ2 i]. Since ∥ξ∥=∥Ux∥... | https://arxiv.org/abs/2501.15725v1 |
for bλ↓ i. Letting eSbe the 2 r×2r matrix with entries esij= 1/(γi+γj) and eHbe the 2 r×2rmatrix with entries ehij= 2γ1/2 iγ1/2 j/(γi+γj), we have ∥P(1)◦S◦H∥=∥P(2)◦eS◦eH∥. Here, eHis the Hadamard product of a positive semidefinite matrix with entries 2 γ1/2 iγ1/2 jand a Cauchy matrix with entries 1 /(γi+γj). Hence, by ... | https://arxiv.org/abs/2501.15725v1 |
n ) +8 3∥U∥2→∞ϑ(ν, r, n ) with probability at least 1 −n−(ν−1), where the second inequality follows from Eq. (1.1). This completes the proof of Lemma 7. Lastly, we present a technical lemma for bounding ∥EU∥2→∞. The result here is a slight improvement compared to a direct application of matrix Bernstein’s inequality. ... | https://arxiv.org/abs/2501.15725v1 |
Hard edge asymptotics of correlation functions between singular values and one eigenvalue Matthias Allard1,2* 1School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville, Melbourne, 3010, Victoria, Australia. 2Department of Mathematics, KU Leuven, Celestijnenlaan 200 B bus 2400, Leuve... | https://arxiv.org/abs/2501.15765v2 |
. . . . . 33 6 Discussion 45 A Consistency check of assumptions 48 1 Introduction When studying a square complex matrix X∈Cn×n, one is usually interested in itseigenvalues z= diag( z1, . . . , z n)∈Cn, which are in general complex for non- Hermitian matrices, as they encode important information about the associated li... | https://arxiv.org/abs/2501.15765v2 |
Gaussian unitary ensemble, when looking at the singular values) fLag(X)∝det X†Xαexp −Tr X†X , α > −1, (1.3) as well as the induced Jacobi ensemble [1, 17] (also known as the ensemble of truncated unitary matrices) fJac(X)∝det X†Xαdet 1n−X†XβΘ(1n−X†X), α > −1, β > 0, (1.4) with1nthen×nidentity matrix and Θ the... | https://arxiv.org/abs/2501.15765v2 |
by ∆n(x) = det xk−1 jn j,k=1=Y 1⩽j<k⩽n(xk−xj). (1.10) 4 We introduced, here, the Mellin transform MonR+, Mf(s) :=ˆ∞ 0dx xs−1f(x) (1.11) for an L1(R+)-function fands∈Csuch that the integral converges absolutely. Note that the structure of polynomial ensembles (1.9) appears very naturally in random matrices as the Jaco... | https://arxiv.org/abs/2501.15765v2 |
can choose Knto be polynomial of degree n−1 in the second entry, which thus makes it unique. It is important to stress that not only does the kernel play a crucial role in the point process of the singular values but also in the point process of the eigenradii and, a fortiori, in the combined point process as it can be... | https://arxiv.org/abs/2501.15765v2 |
(1.26) according to [25, Lemma 4.2]. The goal of this article is to exploit the results of [5] where everything has been done for fixed matrix size nand study, now, the large nlimit. Let us stress that all the results of this article are completely new as the results of [5] were new. The article is organized as follows... | https://arxiv.org/abs/2501.15765v2 |
a k)7→ a1 nνn, . . . ,ak nνn . This surprising difference between the scaling of the squared eigenradius and squared singular values is discussed later. Accounting for the change of measure and multiplying by the appropriate global scaling, the correct scaling of the 1 , k-point correlation function between one squar... | https://arxiv.org/abs/2501.15765v2 |
guarantees that the contribution of the polynomials to the integral in (2.3) remains in a bounded interval near 0, i.e. near the hard edge. Remark 2.3 While finding a set of bi-orthonormal functions pj,qjis not easy in general, once found for any n, the conditions are rather easy to check. Note that imposing the condit... | https://arxiv.org/abs/2501.15765v2 |
lim n→∞n νnρEV r νn =ˆ∞ 0dtˆ∞ 0dv vφ(∞) v r, t K(∞)(v,−rt).(2.18) Remark 2.6 Rewriting the result as follows 1 νn(nνn)kf1,kr νn;a1 nνn, . . . ,ak nνn = n→∞1 n1+kh f(∞) 1,k(r;a1, . . . , a k) +o(1)i ,(2.19) one can see that the scaling limit of the 1, k-point function at the origin is of order O(1/n1+k)asn→ ∞ . The... | https://arxiv.org/abs/2501.15765v2 |
functions composing the kernel is not uniquely given. Example 2.10 For Laguerre, Jacobi and Cauchy-Lorentz ensembles, the respective P´ olya weights are given by (1.13) where α,βare fixed, i.e. independent of n. The Laguerre weight being independent of n, there is no need to rescale, thus νn= 1and ξn= 1. For Jacobi and... | https://arxiv.org/abs/2501.15765v2 |
This last assumption could possibly be lifted due to some property of the P´ olya frequency functions such as their log- concavity [15, 19, 44]. However, due to assumption (2.27), it is not entirely clear if the set of P´ olya ensembles verifying Assumptions 2.8 is strictly included in the set of polynomial ensembles v... | https://arxiv.org/abs/2501.15765v2 |
xt)Jα(√yt) =√yJ′ α(√y)Jα(√x)−√xJ′ α(√x)Jα(√y) 2(x−y), (2.50) withJαthe usual Bessel function of the first kind of order α. Remark 2.16 Technically, Jαis analytic on C\]−∞,0]. For negative arguments one can use the relation with modified Bessel function Iα ∀x∈R+, J α(ix) =iαIα(x), J′ α(ix) =iα−1I′ α(x). (2.51) For insta... | https://arxiv.org/abs/2501.15765v2 |
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