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Psup β∈Sε∥Yβ∥2≥nMR (ηt∨η2 t)+1(i) ≤C1 Sε exph −c2n·η2 ti (ii) ≤C1 3R ε!p exph −c2(C2 3p+t2)i (iii) ≤C1e−c2t2, where ( i) is via the fact that for any x≥0, we have (x∨x2)∧(x∨x2)2=x2, (ii) is via Corollary 4.2.13 of [78] bounding the cardinality of a minimal ε-net, and ( iii) is from the definition of ηta... | https://arxiv.org/abs/2502.15752v2 |
gives us our second condition, namely (38), with the constants ( R,K,C1,c2,C3) set as ( L,B,4,cm−1,C2). For the Gaussin case, the first condition holds for the exact same reason. For the second condition, we again use Lemma 33, which allows us to satisfy (38) with the constants ( R,K,C1,c2,C3) set as ( L,B,2,cS−1,KXc−1... | https://arxiv.org/abs/2502.15752v2 |
{z } (a), ˆRn(˜βγ;X)−ˆRγ n(˜βγ;X) | {z } (b)o . We bound the first term ( a) like so: ˆRn(˜β;X)−ˆRγ n(˜β;X) (i) ≤1 nnX i=1 log 1+e−yiX⊺ i˜β −log 1+e−ηiX⊺ i˜β (ii) ≤1 nnX i=1 X⊺ i˜β |yi−ηi| (iii) ≤1 n∥X˜β∥∥y−η∥ (iv) ≤1 n∥X∥Sp∥y−η∥, where ( i) uses that 0≤ωi≤1 for all i≤n, (ii) comes from treating the loss as a funct... | https://arxiv.org/abs/2502.15752v2 |
when we multiply by −nαthey will always be either zero or strictly negative. This means the sum inside of the logarithm lies in 1, ˜Sδ . By Proposition 4.2.12 of [78], we can say that since Sp⊆BRp(0,L√p), then ˜Sδ ≤ 3L√p δ√p!p = 3L δ!p , and so combining this with (44) we have D2≤p nαlog 3L δ! ≤C31 αlog 1 δ! fornsuf... | https://arxiv.org/abs/2502.15752v2 |
lim sup n→∞sup β0,...,βℓ ∈˜S E˜U⊺ iDi(UBi,β0)gϵ(|˜UT iβ0|,|UT iβ0|) exp −αℓX r=0X j∈Biωjℓ(ηj,U⊺ jβr) −E˜U⊺ iDi,L(UBi,β0) exp −αℓX r=0X j∈Biωjℓ(ηj,U⊺ jβr) ≤lim sup n→∞sup β0∈˜SE ˜U⊺ iDi(UBi,β0) I |UT iβ0|∈[L−ϵ,L]) +E ˜U⊺ iDi(UBi,β0) I |˜UT iβ0|∈[L−ϵ,L]) ≤lim sup n→∞sup β0∈˜SE... | https://arxiv.org/abs/2502.15752v2 |
j∈Biωjℓ(ηj,U⊺ jβ)E i,k≤k+X j∈BiED ∥β∥∥Uj∥E i,k ≤k+X j∈BiE∥Uj∥sup β∈˜Sδ∥β∥ ≤k+L√pX j∈BiEh ∥Uj∥i (i) ≤k+L√pX j∈BivtpX k=1E[U2 jk] (ii) ≤k+Lk√psup jq Tr(Σj) (iii) ≤C2k√p, where ( i) is via Jensen’s Inequality, ( ii) is because Eh U2 jki =Eh sin2(t)X2 jk+cos2(t)G2 jk+sin(t)cos(t)XjkGjki =(Σj)k,k sin2(t)+co... | https://arxiv.org/abs/2502.15752v2 |
µ:=E[|U⊺ jβ|] in the exponent, which satisfies µ≤q Eh (U⊺ jβ)2i =q β⊺Σjβ≤∥β∥∥Σj∥1/2 op≤CKXL√p √n≤C3 fornsufficiently large, by Jensen’s Inequality and Lemma 38, and ( iv) is via sub-Gaussianity of the centered version of U⊺ jβ, which is sub-Gaussian by Lemma 2.6.8 of [78], and thus satisfies Condition ( v) of Propositi... | https://arxiv.org/abs/2502.15752v2 |
(67) with the supremum over β0in (63), we conclude the result. □ The following lemma allows us to convert the statement about Gaussian approximation from Assumption 5, which involves kterms, to one that involves arbitrarily many, which will be im- portant when combined with the polynomial derived from the previous lemm... | https://arxiv.org/abs/2502.15752v2 |
eivec(˜GBiB)⊺˜s2 (ii) ≤2 E eivec(XBiB)⊺s −E eivec(GBiB)⊺s , (74) where ( i) is because the characteristic function factors due to ˜Gy(X,G), and ( ii) is because the characteristic function always has modulus in [0 ,1], and ( x−y)2≤2|x−y|forx,y∈[0,1]. From here, let us now decompose our vector s∈Rkℓintoksubvectors... | https://arxiv.org/abs/2502.15752v2 |
each Gi∼N 0,Var(Xi). Then if Assumption 8(i) is respected then there exists a constant Cdsuch that dH min βˆRn(β;X),min βˆRn′(β;XM)! ≤Cdmax( m,˜m) M√ M+˜m (76) dH min βˆRn(β;G),min βˆRn′(β;GM)! ≤Cdmax( m,˜m) M√ M+˜m (77) If instead Assumption 8(ii) is respected then there exists a constant C′ dsuch that dH min βˆRn(β;... | https://arxiv.org/abs/2502.15752v2 |
crop- ping method where a portion of the data is randomly set to 0. For a vector e:=(ei)∈{0,1}pand 57 x∈Rpwe write e·x:=(eixi). Let ( Ei) be an i.i.d sequence of random vectors in {0,1}pand define the random transformations ϕi(x)=Ei·x. We will prove that Assumption 5 holds for this type of data augmentation procedure u... | https://arxiv.org/abs/2502.15752v2 |
j,l)E(Z1,j,Z1,l) if i =m pj,lE(Z1,j,Z1,l)−(1−pj,l)E(Z1,j,Z1,l)if|i−m|≤k i,m 0otherwise where pj,l=P(E1,j=1)2+P(E1,j=−1)2and p∗ j,l=P(E1,j=1,E1,l=1)+P(E1,j=−1,E1,l= −1). Proof. Note that as the blocks ( Xmk+1,..., X(m+1)k) are identically distributed it is enough to prove Assumption 5 for m=0. Denote Bi,j=∪j∈Ni˜Nj×[|1,k... | https://arxiv.org/abs/2502.15752v2 |
m=0. For all (θi)∈Sk−1and all (βi)∈Sk pwe have X i≤kθiXT iβi=X i≤kθiX lXi,lβi,l=X lZ1,lX i≤kθiβi,πi(l). Now we notice that conditionally on ( πi) the random variables Z1,lP i≤kθiβi,πi(l)are independent. Moreover we observe that E Z1,lX i≤kθiβi,πi(l) (πi)=0 and for all l,m≤pwe have cov Z1,lX i≤kθiβi,πi(l),Z1,mX i≤kθiβ... | https://arxiv.org/abs/2502.15752v2 |
m2(r+2)2K2!# ≤2·exp" −cn m t 2K∧t2 4K2!# , where above we used the fact that r+2≥n m. We conclude by noting that this inequality also holds for the second term in (92) by the same reasoning. □ 62 K.2. Lemmas for mixing processes In this subsection, we present some useful lemmas to deal with β-mixing processes. The firs... | https://arxiv.org/abs/2502.15752v2 |
We observe that if we define ν:=(1,..., 1)Tthen we have cov( XT iν,XT jν)=P mci,mcj,mνTΣν. As we assumed λmin(Σ)>cwe obtain that νTΣν≥c.This implies that ∥CT MCM∥2 HS≤(c)−2X i,j≤nCov( XT iν,XT jν)2. To further bound this, we note that according to Lemma 26 of [6] we know that |cov(νTXi,νTXj)|≤4β(i−j)ϵ 2+ϵmax i∥νTXi∥2 2... | https://arxiv.org/abs/2502.15752v2 |
those latter are centered, we obtain that ( a)=0. Similarly, we notice for all j≤nthat E f′′ ϵ(X Xj,0(t))Xj 2√t−Zj 2√ 1−tXj√ t+Zj√ 1−t =E f′′ ϵ(X Xj,0(t)) EXj 2√t−Zj 2√ 1−tXj√ t+Zj√ 1−t ≤E f′′ ϵ(X Xj,0(t)) EX2 j 2−Z2 j 2 =0. Hence ( b)=0. 66 Finally we can note that ∥f(3) ϵ∥≤4 ϵ2. Hence, thanks to J... | https://arxiv.org/abs/2502.15752v2 |
ℓ≤pX (˜i,˜ℓ),(˜i2,˜ℓ2)∈Bi,ℓXi,l 2√t−Gi,l 2√ 1−t × X˜i,˜ℓ√ t+G˜i,˜ℓ√ 1−t X˜i2,˜ℓ2√ t+G˜i2,˜ℓ2√ 1−t(ail⊗a˜i˜ℓ⊗a˜i2,˜ℓ2) i =EhX s,˜s,˜s2≤qX i≤kX ℓ≤pX (˜i,˜ℓ),(˜i2,˜ℓ2)∈Bi,ℓθsiβsilXi,l 2√t−θsiβsilGi,l 2√ 1−t θ˜s˜iβ˜s˜i˜l X˜i,˜ℓ√ t+G˜i,˜ℓ√ 1−t θ˜s2˜i2β˜s2˜i2˜l2 X˜i2,˜ℓ2√ t+G˜i2,˜ℓ2√ 1−t21/2i (d3) ≤X s,˜s,˜s2≤qX ... | https://arxiv.org/abs/2502.15752v2 |
and block dependence, ( ii) is via (97) and the fact that QD(x)>0=⇒RD(x)=1 x−QD(x)<1 x forx∈(0,1), and ( iii) is via (62). Thus, if we choose tsufficiently large, namely t>s 1 clog 2C(k,α) τ! , then (100) yields that E(i,k)" RD e−αP j∈Biωjℓ(ηj,U⊺ jβ)2 IAc t# ≤τ 2. (101) For (98), we know that since the event Atoccurs... | https://arxiv.org/abs/2502.15752v2 |
have E[Yw,uYw′,u′]−E[Xw,uXw′,u′](a)=E[w⊺Hu(w′)⊺Hu′]+XM l=1∥w∥Σ(l)∥w′∥Σ(l)∥u∥˜Σ(l)∥u′∥˜Σ(l) −XM l=1 ∥w∥Σ(l)∥w′∥Σ(l)u⊺˜Σ(l)u′+w⊺Σ(l)w′∥u∥˜Σ(l)∥u′∥˜Σ(l) (b)=XM l=1 w⊺Σ(l)w′u⊺˜Σ(l)u′+∥w∥Σ(l)∥w′∥Σ(l)∥u∥˜Σ(l)∥u′∥˜Σ(l) −∥w∥Σ(l)∥w′∥Σ(l)u⊺˜Σ(l)u′−w⊺Σ(l)w′∥u∥˜Σ(l)∥u′∥˜Σ(l) =XM l=1 ∥w∥Σ(l)∥w′∥Σ(l)−w⊺Σ(l)w′ ∥u∥˜Σ(l)∥u′∥˜Σ(l)−u... | https://arxiv.org/abs/2502.15752v2 |
∥w2∥Σ(l)h⊺ l ˜Σ(l)1/2(u′ 2−u2)+w⊺ 2 Σ(l)1/2gl(∥u′ 2∥˜Σ(l)−∥u2∥˜Σ(l)) +(f(w2,u′ 2)−f(w2,u2)) ≥ −δS2KXM l=1(∥hl∥+∥gl∥)−ϵ . Applying (104) to each ∥hl∥and∥gl∥yields that, for any t>0 and 1≤l≤M, P(∥hl∥≥t)≤2n/2e−t2/4and P(∥gl∥≥t)≤2p/2e−t2/4. Taking another union bound, we get that for any t>0, P( minw∈Swmaxu∈SuLψ(w,u)≥c+... | https://arxiv.org/abs/2502.15752v2 |
set of deterministic equations whose solutions charac- terize the high-dimensional behavior of logistic regression estimate. This involves establishing the equivalence of a series of optimization problems, which are defined in this section. We also formally state all lemmas used to establish the equivalence. Original o... | https://arxiv.org/abs/2502.15752v2 |
for GandGΦwith i.i.d. standard normal entries and Σo= Σ = Id. This allows them to project GΦonto the subspace orthogonal to β∗, which is independent of Gβ∗, and apply CGMT. In our case, GandGΦare di fferent and have 77 non-trivial dependence. We instead make use of a projection P⊥ ∗adapted to the variance- covariance s... | https://arxiv.org/abs/2502.15752v2 |
also define PΣ=(Σ†)1/2Σ1/2, the projection onto the positive eigenspace of Σ, and the matrix ˜Σσ,τB1 2σ1τ1(PΣ−Σ∗)+1 2σ2τ2Σ∗. Also define the Gaussian random vectors qB1 κ∗√pGΦv(β∗)=GΦΣ∗Σ1/2 oβ∗ ∥Σ∗Σ1/2 oβ∗∥, ˜hα,σBκ∗αq−σ1h1−σ2√ kJmkh2, ˜gB−r1+r2√ mk(PΣ−Σ∗)g1−r2√mΣ∗g2. For a function f:S′→Rand someS′⊆Rmk, we define the ... | https://arxiv.org/abs/2502.15752v2 |
3exist for every r1,r2,θ,σ1,σ2,τ1,τ2. Then for S =Spand S =Sc ϵ, RSO S,Su,Sv(y,q,g,h)−RDO S P− →0. As with Salehi et al. [63], it remains to prove that the first order condition of (DO) for S= Spis equivalent to the system of 10 equations (EQs) in ( α,σ1,σ2,τ1,τ2,ν1,ν2,r1,r2,θ). This involves computing the derivative o... | https://arxiv.org/abs/2502.15752v2 |
prove the equivalence of (PO) and (AO) in Lemma 42, Assumption 11 is only critical for showing the independence of the di fferently projected data matrices, which hold even under the rescaling a1anda2in (108); see the proof of Lemma 50 below. As such, Meanwhile, a tedious extension of (EQs) also holds for random croppi... | https://arxiv.org/abs/2502.15752v2 |
positive semi-definite. Moreover, by another total law of variance, we get that Σ∗⪯(Σ†)1/2Var[ϕ11(X1)] (Σ†)1/2=Ip, where⪯denotes the Loewner partial order on positive semi-definite matrices. This implies that Ip−Σ∗is positive semi-definite and so is Σ1. □ We are now ready to prove the equivalence of (PO) and (AO). Proo... | https://arxiv.org/abs/2502.15752v2 |
1β+r2√ng⊺ 2Σ1/2 2β +r1√ mk P⊥ mk(u−GΦP∗Σ1/2β−h1∥β∥Σ1−1√ kJmkh2∥β∥Σ2) +r2√ mk Pmk(u−GΦP∗Σ1/2β−h1∥β∥Σ1−1√ kJmkh2∥β∥Σ2) . Minimizing over β∈S.As with (47) of Salehi et al. [63], we introduce new variables µ,w∈Rpto replaceβin the regularization term via the Lagrange multiplier method applied to the constraint PΣµ=PΣβ: min ... | https://arxiv.org/abs/2502.15752v2 |
di fferent projection matrices introduced so far: By the definition of Σ∗, we have Σ∗PΣ=(Σ†)1/2Cov[ϕ11(X1),ϕ11(X2)](Σ†)1/2PΣ= Σ∗. (113) Also by the definition of P∗and the idempotency of Σ∗, P∗Σ∗= Σ∗P∗=Σ∗(Σ∗Σ1/2 oβ∗)(Σ∗Σ1/2 oβ∗)⊺ ∥Σ∗Σ1/2 oβ∗∥2=P∗ifΣ∗Σ1/2 oβ∗,0 Σ∗×0=P∗ otherwise.(114) This implies that S⊥ Σ={√p... | https://arxiv.org/abs/2502.15752v2 |
Recall that by definition, y=Pmkysince yis a length- mkvector formed by k-fold repetitions of mentries. Then we can re-express the loss above as min u∈Su1 mk1⊺ mkρ(u)−1 mky⊺Pmku+r1ν1 2mk P⊥ mk(u−˜hα,σ) 2+r2ν2 2mk Pmk(u−˜hα,σ) 2 =min u∈Su1 mk1⊺ mkρ(Pmku+P⊥ mku)+r2ν2 2mk Pmk u−1 r2ν2y−˜hα,σ 2 +r1ν1 2mk P⊥ mk(u−˜hα,σ) 2 ... | https://arxiv.org/abs/2502.15752v2 |
that R′ 1is equivalent toR1in the sense that the CGMT inequalities of (124) hold also with R1replaced byR′ 1, therefore justifying the flipping of the minimum and the maximum. The remaining flipping of minimum and maximum over compact sets hold for the same reason, and any flipping that involves the Lagrange multiplier... | https://arxiv.org/abs/2502.15752v2 |
projections onto mutually orthogonal subspaces and that PΣΣ∗= Σ∗,PΣP∗=P∗. We can then express λ 2mΣ†+˜Σσ,τ†=λ 2mΣ†+1 2σ1τ1(PΣ−Σ∗)+1 2σ2τ2Σ∗†, which implies that λ 2mΣ†+˜Σσ,τ† ˜g+θ√pv(β∗)=−2(r1+r2)σ1τ1√ mk2σ1τ1λ 2mΣ†+Ip†(PΣ−Σ∗)g1 −2r2σ2τ2√m2σ2τ2λ 2mΣ†+Ip†Σ∗g2 +2σ2τ2θ√p2σ2τ2λ 2mΣ†+Ip†v(β∗) +O ∥m−1/2P∗g1∥+∥m−... | https://arxiv.org/abs/2502.15752v2 |
Also recall that y=yPmk=1 kyJmk. We can then express the last two terms of the loss above as −1 2r2ν2mk∥y∥2−1 mky⊺˜hα,σ=−1 2r2ν2mk∥y∥2−κ∗α mky⊺q+σ1 mky⊺h1+σ2 m√ ky⊺Pmkh2. (129) Since h1andh2are zero-mean and yis coordinate-wise bounded by one, by the weak law of large numbers, 1 mky⊺h1P− →0 and1 m√ ky⊺Pmkh2P− →0. (130)... | https://arxiv.org/abs/2502.15752v2 |
jvia¯Z0and ¯Z1in (132), we obtain that My,˜hα,σ,r,ν can be approximated by E min u′∈(Pmk(Su))1 u′′∈(P⊥ mk(Su))11 kXk j=1log 1+eu′ j+u′′ j +r2ν2 21 kXk j=1 u′ j−1 r2ν2I≥0{¯κo¯Z0+¯κ∗¯Z1−ε1}−α¯κ∗¯Z1+σ1ηj+σ2¯Z22 +r1ν1 21 kXk j=1(u′′ j−α¯κ∗¯Z1+σ1ηj)2−1 kXk j=1 u′′ j−α¯κ∗¯Z1+σ1ηj2 , whereη1,...,ηkand ¯Z2are i.i.d... | https://arxiv.org/abs/2502.15752v2 |
1, 0=σ2 2τ2 2+α2¯κ2 ∗ 2σ2τ2 2−∂τ2¯χr,θ,σ,τ 1, 0=−r1 2ν2 1+r1 2kEh Ik−1 k1k×k(u¯Z,ε1,η+σ1η) 2i , 0=−r2 2ν2 2+1 4r2ν2 2+r2 2kE 1 k1k×k u¯Z,ε1,η−1 r2ν2¯Y1k−α¯κ∗¯Z11k+σ1η+σ2¯Z21k 2 +1 ν2kE ¯Y 1⊺ ku¯Z,ε1,η−k r2ν2¯Y−kα¯κ∗¯Z1 , 0=1 2ν1−∂r1¯χr,θ,σ,τ 1+ν1 2kEh Ik−1 k1k×k(u¯Z,ε1,η+σ1η) 2i , 0=1 2ν2+1 4r2 2ν2−∂r2¯χr,θ... | https://arxiv.org/abs/2502.15752v2 |
, (143) where the last line is exactly the same as (83) of [63] via Stein’s lemma. Substituting (143) into the second equation of (137) gives 0=−1 2τ2+α2¯κ2 ∗ 2σ2 2τ2−1 σ2 2τ2r2 2κ+θ2¯κ2 ∗ 2(λκ+σ−1 2τ−1 2)2+2σ2 γE∂ρ(−¯κ∗¯Z1) 1+γ∂2ρ Proxγρ(•) α¯κ∗¯Z1+σ2¯Z2 +σ2 γ. Upon rearranging and a substitution of σ2 2+α2¯κ2 ∗=r... | https://arxiv.org/abs/2502.15752v2 |
COUNTING COMMUNITIES IN WEIGHTED STOCHASTIC BLOCK MODELS VIA SEMIDEFINITE PROGRAMMING DEBORAH OLIVEIRA, ANDRESSA CERQUEIRA, AND ROBERTO OLIVEIRA Abstract. We consider the problem of estimating the number of communities in a weighted balanced Stochastic Block Model. We construct hypothesis tests based on semidefinite pr... | https://arxiv.org/abs/2502.15891v1 |
we make use of the mentioned SDP relaxation. The use is justified by the fact that our main interest will be the more difficult case of sparse random graphs, and in this case, SDP-based methods have been shown to be successful compared with others, such as the spectral one [12]. Finally, we also deal with a balanced pl... | https://arxiv.org/abs/2502.15891v1 |
to estimate the entries of the community membership matrix Z0∈ {0,1}n×n, with Z0ij= 1 if i∼jandZ0ij= 0 if i̸∼j. Finally, the problem of estimating the true number of communities in our case is the one of estimating the number K. 1.1.Main results. We present here the main results of the paper. Before that, we need to de... | https://arxiv.org/abs/2502.15891v1 |
3 .2.2. Theorem C. In the previous setting, if (1.12)2n2 r2s2(Min−Mout)>4n(1 +δ)√w+ the Type I and Type II errors of the test Tn,s(X;δ) approach zero asymptotically. The main difficulty of this generalization will lie on the calculation of a lower bound for SDP( Mr−Ms). This requires a careful combinatorial analysis of... | https://arxiv.org/abs/2502.15891v1 |
also the one of estimating the communities themselves. It is subdivide such that in Subsection 3.1 we consider the distinction between a homogeneous Erd¨ os-Renyi graph and a graph with 2 communities and Subsection 3.2 is dedicated to the case of distinguishing between randscommunities with r > s ≥2. We then get to the... | https://arxiv.org/abs/2502.15891v1 |
using SDP. The first problem is known as distinguishability and the second is an instance of learnability as denoted in the famous review [15]. In this work, the semidefinite program approach is used in all of the analysis: to construct a useful statistic in the hypothesis tests and as a way to estimate the communities... | https://arxiv.org/abs/2502.15891v1 |
using their lemmas and theorems: SDP( ¯AG) OPT k(¯AG) Φ(β, k;¯AG) E[Φ(β, k;¯AG)] SDP( B) OPT k(B) Φ(β, k;B) E[Φ(β, k;B)]≈Teo. 4≈Lem. 3.3 + A.3 ≈Lem. A.1 ≈Teo. 4≈Lem. 3.3 + A.3 ≈Lem. A.2≈Lem. 3.3 (E.1 + E.2) where ¯Y:=Y−EYfor any random variable Y,AGis the adjacency matrix in their case and Btheir corresponding GOE matr... | https://arxiv.org/abs/2502.15891v1 |
number of communities. We subdivide this problem and the estimation of communities into two cases: distinguishing between 1 and 2 communities and distinguishing between randscommunities with r > s≥2. In [12] it was shown that it is possible to construct a hypothesis test to distinguish between a homogeneous Erd¨ os- Re... | https://arxiv.org/abs/2502.15891v1 |
the known semicircle distribution of eigenvalues of a GOE matrix. The idea, as is always the case for hypothesis tests, is to have a test statistic that is independent of the hypothesis that truly holds and we use the GOE matrix asymptotic behavior for this purpose. We see below how this equation appear on the error es... | https://arxiv.org/abs/2502.15891v1 |
is, we want to find X∈PSD 1(n) that gives the value of SDP( E[WG]). The following proposition shows that the only optimal solution to it is x0xT 0and for any other Xinside PSD 1(n) it gives a quantification of how good its top eigenvector v1(X) estimates the communities. Proposition 2. LetX∈PSD 1(n). Defining (3.20) ξ:... | https://arxiv.org/abs/2502.15891v1 |
same set we obtain by 3.32 (3.41) n− ⟨s(ˆv), x0⟩ ≤4(n− ⟨ˆv, x0⟩)≤8ξn as long as we have (3.42) ξ:=2 (Min−Mout)n2(tr(E[WG]x0xT 0)−tr(E[WG]ˆX))<1 2. Again, as ˆXattains the maximum of tr( WGX) in the viable set and we have (3.43) tr( WGx0xT 0)−tr(WGˆX)≤0 =⇒tr(E[WG]x0xT 0)−tr(E[WG]ˆX)≤2Err(WG) then (3.44) ξ≤4Err(WG) (Min−... | https://arxiv.org/abs/2502.15891v1 |
WGisMs. This is also upper bounded by λ0of the previous section. With that, we obtain again (3.56) E′∩E′′⊂ {SDP(√w+B)>2n(1 +δ)√w+−nλ0} such that P0(E′)≤P0(E′∩E′′) +P0(E′′c) ≤P0 SDP(√w+B)> 2n(1 +δ)−nλ0√w+√w+ +P0(E′′c) =o(1)(3.57) because for nlarge enough we haveλ0√w+≤δand then (3.58) P0 SDP(√w+B)≥2n 1 +δ 2√w+ =... | https://arxiv.org/abs/2502.15891v1 |
⟨Z0, Z⟩).(3.74) Defining (3.75) δ:=tr (E[WG]Z0)−tr (E[WG]Z) λ(Min−Mout) we obtain (3.76) λ− ⟨Z0, Z⟩=δλ. Let us define an estimator for Z0as (3.77) ˆZ∈arg max Z∈C⟨WG, Z⟩, in that case, we obtain (3.78) tr ( WGZ0)−tr (WGˆZ)≤0⇒tr (E[WG]Z0)−tr (E[WG]ˆZ)≤2Err(WG) Using Z=ˆZin the definition of δ, by the previous calculation... | https://arxiv.org/abs/2502.15891v1 |
have |{i:zi=a}|=n/K, fora= 1, . . . , K . ForK0∈N, a candidate for the number of communities, and ε >0 we create the hypothesis (4.1) H0,K0: #communities = K0 Ha,K 0: #communities > K 0 and statistical tests ˆTn,K 0(WG;ε), where ˆTn,K 0(WG;ε) = 1 means the hypothesis H0,K0is rejected and ˆTn,K 0(WG;ε) = 0 means H0,K0i... | https://arxiv.org/abs/2502.15891v1 |
(4.15)X ij|(MK−E[WG])ij| ≤n2max{|Yin|,|Yout|} and this fact is going to be used in the calculation below. Continuing, using the fact that for any matrix M∈Rn×nwe have ∥M∥∞→1≤P ij|Mij|we obtain P(ˆTn,K(WG, ε) = 1) ≤P(SDP( WG−E[WG]) +KG∥MK−E[WG]∥∞→1>2n(1 +ε)√w+) +o(1) ≤P SDP( WG−E[WG]) +KGX 1≤i≤j≤n|ˆMKij−E[WGij]|>2n(1 ... | https://arxiv.org/abs/2502.15891v1 |
MK−ˆDK0)≥2n2(Min−Mout) K2K2 0 we want to choose ϵ0such that SDP( MK−ˆDK0)−SDP( ˆCK0)≥2n2(Min−Mout) K2K2 0−ϵ0n2−2ϵ0n2 K0≥cn2(4.32) with c >0 to have a positive lower bound for SDP( MK−ˆMK0) of order O(n2). COUNTING COMMUNITIES IN WEIGHTED STOCHASTIC BLOCK MODELS VIA SEMIDEFINITE PROGRAMMING 21 Therefore, we need to have... | https://arxiv.org/abs/2502.15891v1 |
the entries ˆMK0,inare in the +1’s blocks and in the blocks −1’s it does not matter whether we put ˆMK0,inorˆMK0,outbecause in these cases we get entries −(Mout−ˆMK0,in) and −(Mout−ˆMK0,out) both greater than or equal to ˆMK0,out−Mout> ϵ0. COUNTING COMMUNITIES IN WEIGHTED STOCHASTIC BLOCK MODELS VIA SEMIDEFINITE PROGRA... | https://arxiv.org/abs/2502.15891v1 |
=an nX j≥0 jeven(2λ)j j!"jX k=0j k |µin|j−kτk 1E[Zk]#(5.7) where Z∼ N(0,1) and we used the fact that E[Bj] =an nfor all j∈N. Using that (5.8) E[Zk]∼ 0,ifkodd k! k 2!2k 2,ifkeven COUNTING COMMUNITIES IN WEIGHTED STOCHASTIC BLOCK MODELS VIA SEMIDEFINITE PROGRAMMING 25 and observing that this is increasing in kwe ob... | https://arxiv.org/abs/2502.15891v1 |
BLOCK MODELS VIA SEMIDEFINITE PROGRAMMING 27 and the relation also holds. Case podd: In this casep+1 2∈N, such that (5.24) Γp+ 1 2 =p−1 2 ! and then to have equation 5.15 holding it suffices to have (5.25) 2p+1p−1 2 !≤2p 2p!. Asp≥3 and is odd we can rewrite it as p= 2m+ 1 with m≥1, then we need 22m+2m!≤2m+1 2(2m+... | https://arxiv.org/abs/2502.15891v1 |
(DB), which discretizes the network weights before applying a spectral clustering algorithm, as proposed by [21]. The discretization level used in the DB approach is set to ⌊0.4(log log n)4⌋, and for the initialization of the Pseudo method, we used the SC. The metric used to compare the community detection methods eval... | https://arxiv.org/abs/2502.15891v1 |
explored the use of semidefinite programming in the context of community detection. Specifically, we addressed the problem of distinguishing between any two possible different numbers of communities present in a balanced weighted Stochastic Block Model and the recovery of community memberships. Our work is an extension... | https://arxiv.org/abs/2502.15891v1 |
Appendix A.Universality of the SDP function We prove the following theorem from the first section. Theorem 8. LetX∈Rn×nbe a symmetric random matrix with independent centered entries Xij∼subΓ(νij, cij) with variance σ2 ijand let D∈Rn×nbe independent of Xand the analogous Gaussian matrix, i.e., a symmetric random matrix ... | https://arxiv.org/abs/2502.15891v1 |
for ⟨x, My⟩for fixed x, y∈ {− 1,+1}n. Using the symmetry of Mand the independence of its entries we get ψ⟨x,My⟩(λ) =ψP i<jMij(xiyj+xjyi)+P iMiixiyi(λ) =X i<jψMij(λ(xiyj+xjyi)) +X iψMii(λ(xiyi))(A.11) then using that xiyj+xjyi∈[−2,2],xiyi∈[−1,1] and the definition of cwe obtain ψ⟨x,My⟩≤X i<j4λ2νij 2(1−c2|λ|)+X iλ2νii 2(... | https://arxiv.org/abs/2502.15891v1 |
of i. For i=jwe have Eη∼νσ,ϵ⟨ηi, ηi⟩=Eη∼νσ,ϵ[a2 i]⟨σi, σi⟩ 1 =Eη∼νσ,ϵ[a2 i]·1(A.29) such that Eη∼νσ,ϵ[a2 i] = 1 = (1 −α2) +α2. □ Proof of Lemma 2. Choose σ∈(Sk−1)nsuch that (A.30) OPT k(M) =nX i,j=1Mi,j⟨σi, σj⟩. By Proposition 6 and Jensen’s inequality, (A.31) α2OPT k(M)+(1−α2) tr (M) =Eη∼νσ,ϵ nX i,j=1Mi,j⟨ηi, ηj⟩ ... | https://arxiv.org/abs/2502.15891v1 |
i≤jE[|Mij−E[Mij] +E[Mij]−M′ ij|2] ≤4X i≤jV(Mij)≤CX i≤jνij(A.41) where the superscript ( ij) means that we are considering the same matrix but with a independent copy of the ij-th entry and the last inequality is due to the fact that for a distribution Y∼subΓ(ν, c) we have V(Y)≲ν. Now, for any ε >0 by the Chebyshev ineq... | https://arxiv.org/abs/2502.15891v1 |
E(|Dij|3) = (µ3 ij+ 2σ2 ij)2√ 2πσije−1 2θ2 ij −µij(1−2F(θij))−µijσ2 ij(1−2F(θij)) where θij=µij σij F(a) =1√ 2πZ∞ −∞e−1 2y2dy(A.66) then, considering we are dealing with the centered case we get (A.67) E(|Dij|3)≲σ3 ij. Remembering the Assumption 1 we are using, we conclude (A.68) |E[Φ(β, k;X)]−E[Φ(β, k;D)]|≲β2NX i≤j(... | https://arxiv.org/abs/2502.15891v1 |
in the calculations of the Type I and II errors in the text. We start with the case where sdoes not divide r. Case 1. s∤r:The idea is to fix a positive semidefinite matrix Zwith diagonal elements equal to 1, that is, a Z∈PSD 1(n), and use the definition of SDP function to find a configuration of scommunities such that ... | https://arxiv.org/abs/2502.15891v1 |
is the number of Minentries of Mr, the first term on the RHS is the number of Min entries of the C∗configuration of bigger matrices related to vertices inside the smaller communities, and the second term is the number of Minentries of C∗related to the vertices that are left out of these smaller communities but are unit... | https://arxiv.org/abs/2502.15891v1 |
rsmaller communities such that the sum (B.10) S1(C) :=A2 1+A2 2+...+A2 r is maximized, where Airepresents the number of vertices from s1in the i-th smaller community and Crepresents a configuration of vertices in scommunities (allowed or not). In other terms, by the previous claim, this sum represents the number of edg... | https://arxiv.org/abs/2502.15891v1 |
is in fact a bound, that is, from the rs2m2entries with MinofMr, we can subtract the Minentries of any allowed configuration of Msand obtain the positive lower bound rs2m2−max C′allowed(S1(C′) +...+Ss(C′))≥rs2m2−smax CS1(C)>0 =rs2m2−sS1(C∗)>0(B.18) for all valid configurations C′where the last inequality comes from Cla... | https://arxiv.org/abs/2502.15891v1 |
below illustrate this. 1 2 31r sr s+12r s(s−1)r s+1r . . . . . . . . . . . . Figure 8. We consider this enumeration without loss of generality Proof of the Claim. In fact, considering this relative numbering of the smaller communities inside the bigger ones (the communities themselves, not its elements) as the identity... | https://arxiv.org/abs/2502.15891v1 |
Let us consider what happens with the inner product of one of these big blocks with the corresponding block in ˜Z and only above the diagonal. (B.25) 0−1−1··· − 1 −10−1··· − 1 −1−10···... ......··· ··· − 1 −1−1··· ··· 0 · 1−11··· −11−1··· 1−11···... ......··· ··· − 1 ··· ··· 1 (Min−M... | https://arxiv.org/abs/2502.15891v1 |
In the region that matters to us of ˜Z̸= 0, the matrix Mr−Msis the same as in equation B.24 except by the 4 enumerated and not colored strips on Figure 11. Proof of the Claim. In fact, the colored region intersecting r1and ˜Z̸= 0 represents Mincoming from r1(because of the previous remark) and Mincoming from s1ors2(bec... | https://arxiv.org/abs/2502.15891v1 |
B.29 and subtracting equation B.30, we finally obtain r sodd (orr s−1even): (B.33) SDP( Mr−Ms)≥[rs2m2−s3m2+ 4k(sm−k)](Min−Mout) with k= 1,2, ..., m −1. The smaller bound is with k= 1 (B.34) SDP( Mr−Ms)≥[rs2m2−s3m2+ 4sm−4](Min−Mout) 50 DEBORAH OLIVEIRA, ANDRESSA CERQUEIRA, AND ROBERTO OLIVEIRA where 4 sm−4>0 because s, ... | https://arxiv.org/abs/2502.15891v1 |
is the minimum value for the first case, as we wanted. □ References [1] Paul W Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmodels: First steps. Social networks , 5(2):109– 137, 1983. [2] J-J Daudin, Franck Picard, and St´ ephane Robin. A mixture model for random graphs. Statistics and compu... | https://arxiv.org/abs/2502.15891v1 |
Journal of Complex Networks , 3(2):221–248, 2015. 52 DEBORAH OLIVEIRA, ANDRESSA CERQUEIRA, AND ROBERTO OLIVEIRA [20] Andressa Cerqueira and Elizaveta Levina. A pseudo-likelihood approach to community detection in weighted networks, 2023. [21] Min Xu, Varun Jog, and Po-Ling Loh. Optimal rates for community estimation in... | https://arxiv.org/abs/2502.15891v1 |
Concentration inequalities. In Summer school on machine learning , pages 208–240. Springer, 2003. [41] Anqi Fu, Balasubramanian Narasimhan, and Stephen Boyd. CVXR: An R package for disciplined convex optimization. Journal of Statistical Software , 94(14):1–34, 2020. [42] Song Mei, Theodor Misiakiewicz, Andrea Montanari... | https://arxiv.org/abs/2502.15891v1 |
A frequentist local false discovery rate Daniel Xiang1, Jake A. Soloff1, and William Fithian2 1Department of Statistics, University of Chicago 2Department of Statistics, University of California, Berkeley February 25, 2025 Abstract The local false discovery rate (lfdr) of Efron et al. (2001) enjoys major conceptual and... | https://arxiv.org/abs/2502.16005v1 |
scientific hypothesis is a random event whose probability rises and falls according to an observer’s prejudices (Goodman, 1999; Savage, 1972). Both frequentists and Bayesians are better equipped to answer the question when the hypothesis is one of many under consideration, provided that the other hypotheses are con- si... | https://arxiv.org/abs/2502.16005v1 |
work we introduce a new definition of the lfdr that addresses our motivating question within a fully frequentist model. Suppose that we observe test statistics z1, . . . , z m for hypotheses H1, . . . , H m, of which m0are true nulls. Let f(i)(t) denote the density of zi, and define the (frequentist) lfdr as the relati... | https://arxiv.org/abs/2502.16005v1 |
the mFDR and pFDR respectively, for a hypothetical local rejection rule that “rejects” only statistics in a small neighborhood of t. Although the frequentist lfdr depends on unknown quantities, it can be estimated effi- ciently from the data if mis reasonably large and the dependence between test statistics is not too ... | https://arxiv.org/abs/2502.16005v1 |
rule equates to rejecting Hi whenever ziis observed in a region with density f(t)≥(1 + λ)π0. For example, if a false positive is λ= 4 times as costly as a false negative, then we should reject when lfdr∗(zi)≤0.2, or equivalently when f(zi)≥5·π0. The intimate connection between the Bayesian lfdr and the marginal density... | https://arxiv.org/abs/2502.16005v1 |
any t, g(t) =E(lfdr( zI)|g(zI) =g(t)), (6) where I∼Uniform {1, . . . , m }. Interpretation 3: Optimal rejection rule. Thresholding lfdr( zi) at 1 /(1 +λ) gives the optimal separable rejection rule for testing H1, . . . , H munder the weighted classification loss with weight λ, defined in (5). For a decision rule δ(z1, ... | https://arxiv.org/abs/2502.16005v1 |
= E[FDP] . 7 0.000.250.500.751.00 0.00 0.25 0.50 0.75 1.00 Reported statisticFraction of nullsStatistic p-value q-value lfdr estimate lfdr Figure 2: Simulation example. The true means in (9) are µ1=···=µm1= 2 and µi= 0 for i=m1+ 1, . . . , m , where m= 3000 and m1= 150, so the true null proportion is 95%. The calibrati... | https://arxiv.org/abs/2502.16005v1 |
reject a null hypothesis Hiat level q= 25%, when our actual credence in the null hypothesis should be around 50%. By contrast, the true lfdr is exactly calibrated, and the estimated lfdr is nearly so: among t-statistics for which the estimated lfdr is close to 25%, close to a quarter correspond to true null hypotheses.... | https://arxiv.org/abs/2502.16005v1 |
expressed the frequentist lfdr as the intensity ratio ¯ π0f0(t)/¯f(t). Assuming f0is known, we may conservatively bound ¯ π0≤1 or estimate it via e.g. Storey’s method, reducing the problem of estimating lfdr to one of estimating the average density ¯f. Closely related is the classical problem of estimating a density f,... | https://arxiv.org/abs/2502.16005v1 |
m=¯f, it is sufficient that each density f(i)belongs to some base class of densities F0, and then we take F= conv( F0). For example if we knew that each observation was normally distributed with variance 1, then the mixture density ¯fis guaranteed to be in the set of Gaussian location mixtures. Given an estimate of the... | https://arxiv.org/abs/2502.16005v1 |
measuring the probability that our least promising rejection is a false discovery. 5.1 Comparison with FDR The usual FDR measures the null probability of a uniformly selected rejection: FDR(R) =P(H(I)is true) , I∼Uniform {1, . . . , R }. Figure 3 illustrates a numerical example in which the non-null p-values are highly... | https://arxiv.org/abs/2502.16005v1 |
}=[ i:Hiis true{p(Rα)=pi} which implies P(H(Rα)is true) =X i:Hiis trueP(p(Rα)=pi) =m0·α m. The last equality follows from Lemma 2 of Soloff et al. (2024), which states that for any configuration of the other p-values p1, . . . , p m−1, the probability that a null p-value pmachieves the optimum in (15) is equal toα m1. ... | https://arxiv.org/abs/2502.16005v1 |
, L/L }when Hiis true , for some large fixed grid length L, e.g. the number of permutations used to compute a p-value for a permutation test. In this case, the boundary FDR of the SL procedure is not controlled in finite samples (see Section B.0.1 for a counterexample). However, holding mfixed as L→ ∞ , or holding Lfix... | https://arxiv.org/abs/2502.16005v1 |
6.2 Example 2: Microbiome data analysis This section discusses a data set from Song et al. (2016) on storage techniques for biological samples in microbiome analysis. In scientific investigations with microbiome data, it can be necessary to store biological samples for some period of time after collection. A key questi... | https://arxiv.org/abs/2502.16005v1 |
psychological nudging on human behavior, Mertens et al. (2022a) collected data from 447 nudge experiments in the behavioral psychology literature. The formulation of this question and the authors’ conclusion was the subject of some debate (see e.g. Maier et al. (2022), Mertens et al. (2022b), Szaszi et al. (2022)). To ... | https://arxiv.org/abs/2502.16005v1 |
philosophical status. Frequentists have legitimate concerns about the Bayesian side of empirical Bayes. This paper introduces 19 [FDP([0 .01,0.27]) = ˆ π0m×0.26 87= 0.22 Slope =0.26 87 Slope =0.01 1150.27 0.01 115 202kp(k)Secant line FDP estimator Figure 6: Nudge example. The order statistics of the selection-adjusted ... | https://arxiv.org/abs/2502.16005v1 |
natural to require the decision rule to be symmetric with respect to the order in which the p-values are observed. This symmetry elicits another oracle function, called the compound lfdr, which plays a role parallel to that of the lfdr in characterizing the best permutation equivariant decision rule. We say that a deci... | https://arxiv.org/abs/2502.16005v1 |
problems (e.g. m≤20), but can be approximated numerically in larger problems (e.g. m≈1000) using a method developed by McCullagh (2014) for approximating a matrix permanent. Whereas the lfdr is a fixed function on [0 ,1], clfdr depends on the particular realization of p-values, as illustrated in Figure 8. The clfdr and... | https://arxiv.org/abs/2502.16005v1 |
Bayes estimate of compound lfdr. In this sense, we might say Bayesians with exchange- able priors are all in agreement that the compound lfdr is the right quantity to estimate. The same can nearly be said about the marginal lfdr, for the smaller subclass of Bayesians who look marginally at the data for each hypothesis.... | https://arxiv.org/abs/2502.16005v1 |
the realized boundary rejection is asymptotically no larger than α/m. 25 L m=1.8, m0 m=0.9, α=0.5 0.40.50.6 10 100 1000 10000 sample size m (log scale)bFDRbFDR for discrete nullsFigure 9: The simulation setting is ¯ π0= 0.9,α= 0.5, and L= 1.8m, and the non-nulls are fixed along the grid1 L,2 L, . . . ,m1 L . The sampl... | https://arxiv.org/abs/2502.16005v1 |
. , m }andℓλ(h, y) is the per-instance loss: ℓλ(h, y) =λ1{his true , y= reject }+ 1{his false , y= accept }. Since zI∼¯f= ¯π0f0+ (1−¯π0)¯f1follows a Bayesian two-groups model, the expected loss is minimized by the Bayes rule (Sun and Cai, 2007), which is characterized by the local fdr in this two-groups model, lfdr(t) ... | https://arxiv.org/abs/2502.16005v1 |
m−1, and note that pmachieves the maximum in (15) as the ( k+ 1)th order statistic if q(k)< pm< q(k+1)and α(k+ 1) m−pm> max j=k+1,...,m−1n ∆j+α mo ∨ max j=0,...,k∆j , 30 Figure 10: Each length of the interval range in which pmachieves the maximum in (15) is indicated by a vertical green bar, and the sum of these le... | https://arxiv.org/abs/2502.16005v1 |
testing between the following two hypotheses: Hyp0: Observe a random permutation of p−iwhen Hi= 1 Hyp1: Observe a random permutation of p−iwhen Hi= 0, where the permutations are drawn uniformly at random from Sm−1. A simpler testing prob- lem is: gHyp0:ep1, . . . ,epm0iid∼f0,and (epm0+1, . . . ,epm−1)iid∼f1 gHyp1:1 m0m... | https://arxiv.org/abs/2502.16005v1 |
is P(rank( pm) =R) =E 1 LLX ℓ=11{∆ℓ>[∆ℓ∗m−αL/m]∨maxj>ℓ∆j}·1 nℓ+ 1! , where nℓ:= #{i < m :pi=ℓ/L}and ∆ ℓare defined ∆ℓ:=αL mNℓ−ℓ, ℓ = 0, . . . , L, with Nℓ:=Pℓ k=1nkandℓ∗ m:= argmax ℓ=0,...,L∆ℓ. As m→ ∞ , we have the following convergence in probability, nℓ mf∗(ℓ/L)p→1,∆ℓp→αLℓX k=0f∗(k/L)−ℓ. 35 Since the maximizer of αL... | https://arxiv.org/abs/2502.16005v1 |
≤J−1for all twith |t−τ∗ α| ≤ε, where ε:= 24 αL21/3m−1/3log(2m/δ). Then P(ˆτα> τ∗ α+ε)≤δ, for any m≥C(α, J, δ ), a constant depending only on α, Jandδ. Proof of Lemma B.2. Applying Lemma B.1 with εdefined as above, we have P(ˆτα> τ∗ α+ε)≤X k>mε αP p(i∗+k)≤τ∗ α+αk m =X mε α<k≤mεlogm αP(Nk≥k) +X k>mεlogm αP(Nk≥k), (29)... | https://arxiv.org/abs/2502.16005v1 |
m−J(t−τ∗ α)2 2 τ∗ α+ε τ∗α−Jεαk m−ε# =k−Jmε2 2−Jαkε +Jmε2=k+Jmε2 2−Jαkε, which shows (30). For (31), note that the mean value theorem and the condition ¯f′≥ −J−1 on [τ∗ α, τ∗ α+ε] imply that ¯f(t)≥¯f(τ∗ α)−J−1(t−τ∗ α) for any t∈[τ∗ α, τ∗ α+ε]. Thus we have EeNk=mZτ∗ α+αk m τ∗α¯f(t)dt ≥mZτ∗ α+ε τ∗α ¯f(τ∗ α)−J−1(t−τ∗ α)... | https://arxiv.org/abs/2502.16005v1 |
with sample size mand average success probability ¯F τ∗ α)−¯F(τ∗ α−αk m because the p-values are independent. By the mean value theorem, for some ξ∈[τ∗ α−αk m, τ∗ α], we have ENk=m¯f(ξ)·αk m≥k, since ¯fis decreasing on (0 , τ∗ α) and ¯f(τ∗ α) =α−1. It thus follows from Theorem 5 in Hoeffding (1956) that P p(i∗−k)≤τ∗... | https://arxiv.org/abs/2502.16005v1 |
Asymptotic efficiency of simple decisions for the compound decision problem, Lecture Notes-Monograph Series pp. 266–275. Grenander, U. (1956). On the theory of mortality measurement: Part II, Scandinavian Actuarial Journal 1956 (2): 125–153. Gupta, C., Podkopaev, A. and Ramdas, A. (2020). Distribution-free binary class... | https://arxiv.org/abs/2502.16005v1 |
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