text string | source string |
|---|---|
steps in bounding the conditional expectation of the determinant appearing in the right-hand side of Eq. (70). We leave most technical details to Appendix B.4. First, note that since the mean of ζis in the column space of Σ(Θ,¯V) for any ( Θ,¯V), conditioning onz=0is equivalent to conditioning on ζ=0. The latter meanwh... | https://arxiv.org/abs/2502.01953v1 |
:=k 2αlog(α) +k αZ log(τ1)(λ)µ⋆(ν, µ)(dλ) +h0(µ, ν;α), and M(β)(A,B,Π) :={(µ, ν) :∃(µ0, ν0)∈M(A,B,Π)s.t.W2(ν, ν0)< β, W 2(µ, µ0)< β}, where M(A,B,Π) :=n (µ, ν)∈A×B:AR≻R(µ)≻σR,AV≻Eν[vvT]≻σV,Eν[∇ℓ∇ℓT]≻σL, Eν[∇ℓ(v,v0, w)(v,v0)T] +Eµ[ρ′(θ)(θ,θ0)T] =0k×(k+k0)o . Under Assumptions 1,2,3 and 4, there exists a constant C(AR,AV... | https://arxiv.org/abs/2502.01953v1 |
αKL(µ·|θ0∥N(0,Ik)), (98) where it is understood that the “outer” expectation in the conditional divergence is taken with respect to measure µ(θ0)=µ0. By maximum entropy property of Gaussian measures, we obtain that the minimizer is such that µ·|θ0=N(Aθ0,B) for certain matrices A,B. Enforcing the constraint R(µ) =Rwe ob... | https://arxiv.org/abs/2502.01953v1 |
S≻0Φcvx(µ, ν,R(µ),S) = Φ cvx,1(µ) + sup S≻0Φcvx,2(ν,R(µ),S) (109) = inf µ0:R(µ0)=R(µ) Φcvx,1(µ0) + sup S≻0Φcvx,2(νopt,R(µ),S) = inf µ0:R(µ0)=R(µ)Φcvx,1(µ0) + sup S≻0Φcvx,2(νopt,R(µ),S) (a) ≥0 + sup S≻0Φcvx,2(νopt,R(µ),S) (b) ≥0 + sup S≻0M(S;ν,R(µ)) = 0 . where in ( a) we used that for any R≻0, the Gaussian measure µo... | https://arxiv.org/abs/2502.01953v1 |
notation. For example, we write Hfor the matrix H(¯V). Additionally, we will sometimes write H(ˆν) when we want to emphasize the dependence of Hon ˆν= ˆν¯V. The main object of analysis will be the empirical matrix-valued Stieltjes transform defined for z∈H+ and ˆν∈Pn(Rk+k0+1) by Sn(z; ˆν) := (Ik⊗Tr)(H(ˆν)−znIdk)−1. (11... | https://arxiv.org/abs/2502.01953v1 |
s2 ∥M∥2 opd∧s ∥M∥op!) , where we used that ∥u⊗Id∥op≤ ∥u∥2≤1 (and same for v) to deduce ∥(u⊗Id)TM(v⊗Id)∥F≤√ d∥M∥op,∥(u⊗Id)TM(v⊗Id)∥op≤ ∥M∥op. A standard result [Ver18] gives that the size of Nis at most Ck 0for some C0>0. Then taking s= L1/2k+(d)1/2d1/2∥M∥op∨Lk+(d)∥M∥op , we obtain via a union bound P ∥ξTMξ−(Ik⊗Tr)M∥... | https://arxiv.org/abs/2502.01953v1 |
free probability background. Most of what follows can be found in [NS06, MS17]. Let ( A, τ) be a C∗-probability space. An element M∈ Ais said to have a free Poisson distribution with rate α0if the moments of Munder τcorrespond to the moments of the Marchenko- Pastur law with aspect ratio α0. For M, T∈ A, ifMis a free P... | https://arxiv.org/abs/2502.01953v1 |
a=0(−z)−(a+1)(Ik⊗τ)h (Ik⊗M1/2)T(Ik⊗M1/2)ai . (133) Then, Eq. (128) gives (Ik⊗τ)h (Ik⊗M1/2)T(Ik⊗M1/2)ai = lim m→∞(Ik⊗1 mTr)h (Ik⊗XT)¯K(m)(Ik⊗X)ai . Hence, there exists r0, possibly dependent on k, such that for |z|> r0, S⋆(z) = lim m→∞∞X a=0(−z)−(a+1)(Ik⊗1 mTr)h (Ik⊗XT)¯K(m)(Ik⊗X)ai = lim m→∞Sm(z,ˆν0,m) element-... | https://arxiv.org/abs/2502.01953v1 |
whose invertibility implies the uniqueness of the solution of (12). To define this operator, first introduce the notation η(S,W) := (I+WS)−1W. (139) For a given S∈Ck×kwithℑ(S)⪰0,z∈H+α >1, ν∈P(Rk+k0+1), define TS(·;z, α, ν ) :Ck×k→ Ck×k TS(∆;z, α, ν ) :=Fz(S⋆;ν)E[η(S⋆,W)∆η(S,W)]Fz(S;ν), (140) where S⋆=S⋆(z;α, ν), Now, t... | https://arxiv.org/abs/2502.01953v1 |
0F∗ 0B−1/2 0i op≤1−αnℑ(z) 2λmin(B−1/2 0S0S∗ 0B−1/2 0). Further, if 10(K2+|z|2) ℑ(z)2ErrFP(z;n, k)(1 + αnErrFP(z;n, k))≤1 2(144) then the following holds, for any L≥1, on the event Ω0∩Ω1(L)of Lemmas 12, 14 1 αn Eh B−1/2 nFnηnBnη∗ nF∗ nB−1/2 ni op≤1−ℑ(z) 2αnλmin B−1/2 nFnF∗ nB−1/2 n . (145) Proof. By Lemma 8, we have ℑ... | https://arxiv.org/abs/2502.01953v1 |
(150) Now we collect the bounds appearing on the right-hand side of this equation. On the event Ω 0of Lemma 12, we have Lemma 11 and Corollary 1 (recall that Bn=ℑ(Sn),B⋆=ℑ(S⋆)): B−1 n op≤C1 ℑ(z) K2+|z|2 ,and B−1 ⋆ op≤C2 ℑ(z) K2+|z|2 , respectively. Furthermore, by Lemma 8, then Lemma 11 and Corollary 1 respectively... | https://arxiv.org/abs/2502.01953v1 |
for any γ∈(0,1), denoting ˆIn(ˆν) :=1 dkTrf1 nH(ˆν) , I ⋆(ˆν) :=Z f(λ)µMP(ˆν, αn)(dλ), ˆIn(ˆν)−I⋆(ˆν) ≤C4(K) ∥f∥∞,A(K)sup x∈[−2A,2A]|sn(x+iγ; ˆν)−s⋆(x+iγ; ˆν, αn)|+γ ∥f∥Lip+∥f∥∞,A(K) . Meanwhile, on Ω 0∩Ω1(1) (choosing L= 1 in the definition of Ω 1), we have by Lemma 21, sup x∈[−2A,2A]|sn(x+iγ)−s⋆(x+iγ)| ≤ sup x∈... | https://arxiv.org/abs/2502.01953v1 |
i)j,axixT i(Ri)a,l j,l∈[k] = X a∈[k](AW i)j,aTr xixT i(Ri)a,l j,l∈[k] = X a∈[k](AW i)j,axT i(R)a,lxi j,l∈[k] = (AW i)ξT iRiξi. Using this for A:= Ik+WiξT iRiξi−1gives the result. To prove the identity of Eq. (117), we write (Ik⊗Tr) (Ri−R) = (Ik⊗Tr)RiξiWiξT iR = Tr X b,c∈[k](Ri)a,bxi(Wi)b,cxT iRc,d... | https://arxiv.org/abs/2502.01953v1 |
in Lemma 24, we have Z f(x0) (dµ1(x0)−dµ2(x0)) =ZZB −Bf(x)ρ(x;x0, γ)dx+ ∆ f,B,γ(x0) (dµ1(x0)−dµ2(x0)) =ZB −Bf(x)Z ρ(x;x0, γ) (dµ1(x0)−dµ2(x0)) dx +Z ∆f,B,γ(x0) (dµ1(x0)−dµ2(x0)) (162) where the change of order of integration is justified by integrability of the continuous fover [−B, B]. Noting that for j∈ {1,2},Z ρ... | https://arxiv.org/abs/2502.01953v1 |
of the nullspace of Σ).Let ai,j(Θ,¯V) := eT k,j⊗θT i,−eT k,i⊗ℓj(¯V)T,0T, . . . ,0T |{z} k0T, i, j ∈[k], (168) a0,i,j(¯V) := eT k,j⊗θT 0,i,0T, . . . ,0T |{z} k,−eT k0,i⊗ℓj(¯V)TT, i ∈[k0], j∈[k], (169) in which (ek,j)j∈[k]and(ek0,i)i∈[k0]are the canonical basis vectors for RkandRk0, respectively. Then for any(Θ,¯V)∈ ... | https://arxiv.org/abs/2502.01953v1 |
conditions of Theorem 5: Clearly, ψ(U) is an open subset of Rmby definition, and condition (1.)is by definition of the process. Condition (2.)holds by Assumptions 2 and 3. Meanwhile, condition (3.)is guaranteed by definition of BΣ(t) and Lemma 27. We move on to condition (4.). Let E0(σH) := ∃(Θ,¯V)∈ M(A,B,Π) :z(Θ,¯V) ... | https://arxiv.org/abs/2502.01953v1 |
Proof. Note that T(τ) is an open subset of the smooth manifold M×Rrk, and consequently is a smooth manifold embedded in the ambient space of dimension m+rkwith dim ( T(τ)) = dim( M) +rk=m.To lower bound the determinant, we will show that d φ(u,x) :T(u,x)T(τ)→Rmis a low-rank perturbation of a matrix that is approximatel... | https://arxiv.org/abs/2502.01953v1 |
τ∈(0,1), ∥u−u0∥2≤σG 2A(τ) G,2∧τ⇒ σmin(JuNg(u))≥σG 2. (201) Proof. That the Jacobian of gcan be decomposed as in (200) follows from the chain rule. Then JuTg(u0) =Jug(u0)B0=0by definition of M, and σmin(JuTg(u0)) = σmin(Jug(u0)B0)≥σGfollow immediately. Now to prove Eq. (201), an argument similar to the one in Lemma 30 g... | https://arxiv.org/abs/2502.01953v1 |
and by the choice of β, and similarly ∥w−u0∥2=∥Jug(u0)Tx0∥2≤AG,1∥x0∥2≤β 2. (216) Sou1∈Bm β(u0),w∈Bm β(u0). Now for any τN, τTas in Eq. (203), Corollary 3 furnishes the functions ˜ψ0 andψ0, defined therein, that are smooth on the set Bm−rkτT(u0,T). Since A(τ) G,1,A(τ) G,2≥1 and σG≤1 by definition, it is easy to check th... | https://arxiv.org/abs/2502.01953v1 |
the lemma. B.2.5 Proof of Lemma 2 By computing derivatives of the component of gjforj∈[rk] and using the Lipschitz assumptions of Assumption 2 on the partials of ℓand the continuity assumptions of Assumption 3 on the partials of ρ, one can check that there exists a constant C0(AV,AR)>0 depending only on AV,AR, such tha... | https://arxiv.org/abs/2502.01953v1 |
LTL+εIk ⊗Id−M1(R⊗In)MT 1 = det (LTL+εIk)⊗P⊥ R+ (LTL+εIk)⊗PR−LTL⊗PR = det (LTL+εIk)⊗P⊥ R+εIk⊗PR = det LTL+εIkd−rkεrk. So we conclude that for any ε >0, det(Σ+εI)≥det(R)ndet(LTL+εIk)d−rkεrk. (240) Using that the dimension of the nullspace of Σisrkby Lemma 27, we then have det∗(Σ) := lim ε→01 εrkdet(Σ+εI) ≥det(R)n... | https://arxiv.org/abs/2502.01953v1 |
∥ (S2+bΣ)−1∥op + 1 +∥(S2+bΣ)−1∥op·(∥SH∥op+∥bΣ∥op+∥S˜H∥op)2 ··· ∥(H+S)−2˜S∥op+ 1 +∥˜H∥op∥(H+S)−1∥op2 ∥bΣ−1∥op . 66 Recalling the definition K= supv,u,w ∇2ℓ(v,u, w) op, and ˜K= supv,u,w (∂i∂jℓ(v,u, w))i∈[k0],j∈[k] op from Appendix A, we can simplify the bounds as 1 +∥(S2+bΣ)−1∥op·(∥SH∥op+∥bΣ∥op+∥S˜H∥op)2 ≤C0 1 ... | https://arxiv.org/abs/2502.01953v1 |
that on {ζ=0}, we have X[Θ,Θ0] = ¯VandLTX=−PT. Letting PΘ,PLbe the projections on the columns spaces of [ Θ,Θ0],Lrespectively, we can write X=P⊥ LXP⊥ Θ−L(LTL)−1PTP⊥ Θ+¯V R−1ΘT. (258) Letting BΘ∈Rd×(k+k0)be a basis matrix for the columnspace of ( Θ,Θ0) and B⊥ Θ∈Rd×(d−k−k0)be a basis matrix for its complement, we define ... | https://arxiv.org/abs/2502.01953v1 |
above bound on the variation of gimplies that on Aγ, we have |g(X)−g(X′)| ≤C0K∥f∥Lip α1/2 nkγ1/2 X−X′ F. (269) We can apply Gaussian concentration on this event. For t >0, we have for some universal constant C2, c2, C1, c1>0, P 1 nTrf(H0)−1 nE[Trf(H0)] ≥t ≤P 1 nTrf(H0)−1 nE[Trf(H0)] ≥t ∩ Aγ +P Ac γ ≤C1exp( −c1t... | https://arxiv.org/abs/2502.01953v1 |
in (279) is small. To this ends, define the event Ω3:= {λ∈spec (H/n) :λ≤0} < r′ k , (295) Then by Eq. (282), {λmin(H+∆1,k)/n > 0} ⊆Ω3. We’ll bound the probability of Ω 3for¯Vsatisfy- ing (294). Define the Lipschitz test function fτ0:R→Ras fτ0(λ) = 1 if λ≤ −τ0, 1−1 τ0(λ+τ0) if −τ0< λ≤0 0 if 0 < λ.(296) This funct... | https://arxiv.org/abs/2502.01953v1 |
proposition. Once again, we suppress the indices (Θ,¯V) in the arguments. Step 1: obtaining the hard constraint on the support. First, we show that (I) := lim sup n→∞1 nlog EwZ ME[|det dz |1H≻nσH ζ= 0,w]pΘ,¯V(0) 1w∈G 1µ⋆(ˆν,ˆµ)((−∞,−τ0))≥τ0dMV (317) =−∞. (318) Directly by item (2.) of Lemma 5, followed by the bou... | https://arxiv.org/abs/2502.01953v1 |
0, (334) by the choice of βnand Assumption 7 that σG,n=e−o(n). Combining with Step 2 we conclude that (II)≤lim sup n→∞1 nlogEh Eh exp nFn,τ1(¯V,Θ) 1{µ⋆(ˆν,ˆµ)((−∞,−τ0])<τ0}∩M(βn) wi 1w∈Gi , (335) where the expectation is under p1(¯V), p2(Θ). Step 4: Concluding. Finally, we write the bound on (II) in terms of empirical... | https://arxiv.org/abs/2502.01953v1 |
that it is indeed closed, note that we have shown in Section A that if ( µ, ν)7→µ⋆is continuous in the topology of weak convergence, meanwhile, for a weakly converging sequence of random variables Xn→X, we have P(X∈ U)≤lim inf nP(Xn∈ U) for any open set U). Now note that S0(β, τ0, δ) is a compact subset of P(Rk+k0)×P(R... | https://arxiv.org/abs/2502.01953v1 |
We extend it below to complex z: for any ν∈P(Rk+k0+1), z∈C\supp( µ⋆,0(ν)) with ℑ(z)≥0,Q∈Hk +, Kz(Q;ν) :=−αzTr(Q) +αEν[log det( I+∇2ℓ(v,u, w)Q)]−log det( Q)−k(log(α) + 1) (354) where log denotes the complex logarithm (with a branch on the negative real axis). Lemma 39. Under Assumptions 1 to 7 of Section 2.2 along with ... | https://arxiv.org/abs/2502.01953v1 |
the positive part of log( ζ) is integrable because µ⋆,0is compactly supported, and lack of absolute integrability implies that the integral diverges to −∞. Under Assumption 8, we have supp( µ⋆,0(ν))⊆[0,∞),so for any λ≥0,Eq. (356) of Lemma 39 yields kZ log(ζ)µ⋆,λ(ν)(dζ)≤lim sup δ→0K−(λ+δ)(S⋆(−(λ+δ);ν);ν). (371) Now obse... | https://arxiv.org/abs/2502.01953v1 |
r/ 2, this then gives ∥β∥2 2≥1 2r2 r2+ 2R2. (386) Hence, for jbeing the index of maximum mass as above, we have |βj| ≥c0k−1/2. This allows us to conclude that Sδ,R⊆Sk j=1Vj(δ, R) where Vj(δ, R) :=( ∥Θ∥F≤R,∥P−jθj∥2≤√ k c0δ) . (387) 82 Meanwhile, for any j∈[k], vol(Vj(δ, R))≤(CR)dk √ kδ c0!d−k−k0+1 . (388) Bounding the v... | https://arxiv.org/abs/2502.01953v1 |
2αTr(S−1K2) +λ 2Tr(K2+MMT) (407) with the Moreau envelope defined in Eq. (27). Now, by straightforward differentiation of G(K,M,S) with respect to each of K,M,S, one can show that the critical points of G(K,M,S) are given by (K,M,S) = ( Ropt/R00,Ropt 10R−1 00,Sopt) by checking that the stationarity conditions correspon... | https://arxiv.org/abs/2502.01953v1 |
= 1 . This, along with Eq. (416) then implies that for i∈ Iβ, λmin(∇2ℓ(ΘTxi))> c3 (419) for some c3independent of n. Now noting that ∇2ˆRn,0(Θ) =1 nnX i=1∇2ℓ(ΘTxi)⊗(xixT i), (420) we can bound on the high probability event Ω 1,n∩Ω2,n, λmin(n∇2ˆRn,0(Θ))≥λmin X i∈Iβ∇2ℓ(ΘTxi)⊗(xixT i) ≥c3λmin X i∈IβIk⊗(xixT i) ≥c3... | https://arxiv.org/abs/2502.01953v1 |
any 0 ≤λ1< λ2, we have 1 2∥ˆΘλ1∥2 F≥r(λ2)−r(λ1) λ2−λ1≥1 2∥ˆΘλ2∥2 F, (437) where, for λ= 0,∥ˆΘ0∥Fis the norm of any minimizer when this exists. It follows in particular that ∥ˆΘλ∥F≤ ∥ˆΘ∥Ffor any λ >0, and therefore the claim follows. 2⇒3. Fix Cas in point 2,δ0>0 and we chose λ0>0 such that lim n→∞P(∥ˆΘλ∥F< C)≥1−δ0for al... | https://arxiv.org/abs/2502.01953v1 |
0, it is easy to see ∥ˆΘε,λ∥F≤C0/λfor some C0 independent of n, ε, since the multinomial loss is lower bounded by zero. This along with the assumption thatR00= lim n→∞ΘT 0Θ0is finite implies R(ˆµ√ d[ˆΘε,λ,Θ0])≺CIfor all fixed λ >0. 89 Lower bound on the Hessian ∇2 ΘˆRn,ε,λ(Θ).Clearly, since Θ7→ℓi,ε(Θ) is convex, we hav... | https://arxiv.org/abs/2502.01953v1 |
(Ropt(λ),Sopt(λ)) the unique solution for λ > 0, which corresponds to the unique minimizer of F(K,M) defined in Eq. (28), by Theorem 4. Since F(·) depends continuously on λand has a unique minimizer for λ= 0 (by the previous point), it follows that ( Ropt(λ),Sopt(λ))→(Ropt,Sopt) asλ→0. In particular, there exists C >0 ... | https://arxiv.org/abs/2502.01953v1 |
Local convexity of the tap free energy and amp convergence for 𭟋2-synchronization , The Annals of Statistics 51(2023), no. 2, 519–546. [C ¸LO24] Burak C ¸akmak, Yue M Lu, and Manfred Opper, A convergence analysis of approximate message passing with non-separable functions and applications to multi-class classification... | https://arxiv.org/abs/2502.01953v1 |
with generalized linear models: Precise asymp- totics in high-dimensions , Advances in Neural Information Processing Systems 34(2021), 10144–10157. [MAB20] Antoine Maillard, G´ erard Ben Arous, and Giulio Biroli, Landscape complexity for the empir- ical risk of generalized linear models , Mathematical and Scientific Ma... | https://arxiv.org/abs/2502.01953v1 |
dimensions , IEEE Transactions on Information Theory 64(2018), no. 8, 5592–5628. [TB24] Kai Tan and Pierre C Bellec, Multinomial logistic regression: Asymptotic normality on null covariates in high-dimensions , Advances in Neural Information Processing Systems 36 (2024). [TOH15] Christos Thrampoulidis, Samet Oymak, and... | https://arxiv.org/abs/2502.01953v1 |
Submitted to the Annals of Statistics SPECTRALLY ROBUST COV ARIANCE SHRINKAGE FOR HOTELLING’S T2 IN HIGH DIMENSIONS * BYBENJAMIN D. R OBINSON†,aAND VANLATIMER‡,b Air Force Office of Scientific Research 875 N. Randolph Rd Arlington, VA amachine.itel@us.af.mil Radial Research and Development 1210 E. Dayton-Yellow Springs... | https://arxiv.org/abs/2502.02006v2 |
the shrinkage-modified Hotelling T2statistic is delicate— even in the case of linear shrinkage [PZ11, LAP+20]. Finally, identifying and optimizing an effective detection criterion under fully general spectral conditions remains an open problem [NPW21]. In this paper, we develop a method of covariance shrinkage for Hote... | https://arxiv.org/abs/2502.02006v2 |
tection is the quadratic-form detector given by (2.1) (y−x)′Σ−1(y−x)H1≷ H0τ, where xis the sample mean of {xi},τis some detection threshold, and (·)′denotes the transpose operation. However, unless nis large— i.e., much larger than p—it is not realistic to assume that one can approximate all the p2entries of Σwith much... | https://arxiv.org/abs/2502.02006v2 |
(2.3) for a sub-Gaussian linear model (H1)-(H6) to be presented in the forthcoming sections. To make the problem well-posed, we assume a signal prior of the form N(0,Ω)and that the quantities u′ iΩuiw(λi)approximate h(λi)for some sufficiently regular function hin a manner to be described in Assumption (H6). Letting a(x... | https://arxiv.org/abs/2502.02006v2 |
at τ, the population spectral distri- bution πn=p−1Pp i=1δτniofΣndiffers in 1-Wasserstein distance by O(1/p)from some limiting distribution π∞with compact positive support, and (b) all population eigenvalues eventually lie in a positive compact interval. (H5) π∞isregular in the sense of [KY17, Definition 2.7]—for examp... | https://arxiv.org/abs/2502.02006v2 |
complex spectral parameters) not expressly deemed constant. We note that the sample covariance matrix can be expressed just as before, except with terms indexed by nto indicate their increasing dimension and dependence on n: Sn= (n−1)−1nX i=1(xni−xn)(xni−xn)′. Further, we will frequently make use of the sample eigen-de... | https://arxiv.org/abs/2502.02006v2 |
about these quantities are that Fis the union of finitely many disjoint closed intervals of nonzero length [BS+98], and that w(x)is a smooth function except near the edges of these intervals, where w(x)≍κ(x)1/2forx∈F[KY17]. Define the Stieltjies transform of a positive measure ρbyS[dρ](z) :=R (t−z)−1dρ(t), so thatmn(z)... | https://arxiv.org/abs/2502.02006v2 |
we will simplify notation by writing Θn(z)instead of Θid n(z). Then the following analogue of the classical Mar ˇcenko–Pastur theorem holds. THEOREM 2 (Theorem 4 of [LP11]). Assume (H1)-(H5) and z∈C+. Let Θn(z) :=p−1tr(ΣnRn(z)). Then Θn(z) =S[dνn](z), where νn:=1 ppX i=1u′ niΣnuniδλni, and there exists a nonrandom anal... | https://arxiv.org/abs/2502.02006v2 |
its Hilbert transform. Upon substitution in the expression for δ(x), these approximations give rise to a shrinkage function ˜dn(x)that exhibits uniform convergence in probability to δ(x)onF. As discussed, this mode of convergence is sufficient to establish asymptotic optimality in the applications of [LW20, RMH21]. Our... | https://arxiv.org/abs/2502.02006v2 |
notation Symbol Meaning / definition fn(·) Bounded spectral shrinkage function, possibly random fn(Sn) =Pp i=1fn(λni)uniu′ ni Precision-shrinkage estimator ( p×p) δ(x) (̸=measure δx) (2.4), Deterministic limit from Ledoit–Péché ˜dn(x) (4.8), Ledoit–Wolf observable approximation of δ(x) k(x) Semicircular kernel (2π)−1p ... | https://arxiv.org/abs/2502.02006v2 |
formulas in mind, we now present our choices of ˜σnand˜mn, as well as the resulting asymptotic equivalence of ˜ZnandZo n. LEMMA 5.2. Assume (H1)-(H5) and that fnis regular. Let ˜mn≡˜mn(fn) :=p−1pX i=1fn(λni)˜dn(λni) and ˜Γnf(x) :=f(x)−1 npX j=1(f(λnj)−f(x))˜dn(λnj)Knj(x) and ˜σ2 n≡˜σ2 n(fn) :=p−1pX i=1h ˜Γnfn(λni)i2 λn... | https://arxiv.org/abs/2502.02006v2 |
dimension is 200, the population covariance is the sum of a matrix with Unif [0,1]eigenvalues and rank-40 matrix with eigenvalues 10(40−j)/10forj= 0,...39. where ˜σn= ˜σn(fn), as defined in Lemma 5.2. The difference between ˜ZnunderH1andH0 is given by 2z′ nΣ1/2 nfn(Sn)µn ˜σn√p+Un, and the linear term in zncan be neglec... | https://arxiv.org/abs/2502.02006v2 |
following theorem (Theorem 7), we will show that under this new assumption (H6), the detection criterion Un(fn), like the variance σn(fn)from the previous subsection, can be approximated using a deterministic limit. Further, this deterministic limit can be optimized by solving a variational problem explicitly in terms ... | https://arxiv.org/abs/2502.02006v2 |
[HIPS22]. In the synthetic experiments, we choose the population covariance matrix and generate simulated data satisfying the moment conditions in (H1)-(H6). In the real-world set, a sensor network collects a time series of received signal strengths during periods of activity and inactivity in a lab setting. In this ca... | https://arxiv.org/abs/2502.02006v2 |
n2= 1) is replaced by a linear shrinkage estimator of the form Sn+λIp, for some data-dependent λ >0. The constant λis chosen to locally maximize an asymptotic detection criterion simi- lar to ours, where the mean-shift dispersion matrix is Ip,Σn,Σ2 n, or some linear combination thereof. (Higher-order polynomials in Σnc... | https://arxiv.org/abs/2502.02006v2 |
p= 200 , choose a mean-shift dispersion matrix Ωand non-centrality scale γ, choose the number of reference samples n, and generate a p×pcovariance matrix Σwith piece-wise log-linear eigenvalues {κi/40}40 i=1∪ {10(i−1)/(40(p−41))}p−40 i=1for some rough condition number κ=λmax(Σ)/λmin(Σ), not to be confused with a distan... | https://arxiv.org/abs/2502.02006v2 |
one [HIPS22], in which 14 Mica2 sensors were distributed throughout a lab space to detect whether a person was present and/or moving there. In order to do so, the sensor network collected and recorded re- ceived signal strength (RSS) measurements for each sender-receiver pair of sensors, totaling p= 14·13 = 182 measure... | https://arxiv.org/abs/2502.02006v2 |
for each using our algorithm and the comparators from the last section. We then generate an empirical ROC curve for each algorithm by plotting how many detection statistics for inactive and active time indices exceed a sliding threshold relative to the totals of 2800 and 327, respectively. Our results for n= 200 andn= ... | https://arxiv.org/abs/2502.02006v2 |
lower performance of CQ10 and LAPPW20 suggests that the reference samples are temporally dependent and/or do not have a common spiked or well-conditioned covariance matrix. COROLLARY 1.LetI= [a,b]⊆Rbe an interval. Then uniformly in I, (A.2) µ(I)−µ∞(I) =O≺1 n 26 ROBINSON AND LATIMER and ν(I)−ν∞(I) =O≺1 n PROOF OF CO... | https://arxiv.org/abs/2502.02006v2 |
the last line we used the holomorphy of ˆm, and then integrating over y, =−1 2πZ dxZ |y|>ηdyf′(x)χ(y)y∂yRe ˆm(x+ iy) =−1 2πZ dxf′(x) [yχ(y)Re ˆm(x+ iy)]−η η−Z |y|>ηdy(χ(y) +yχ′(y))Re ˆ m(x+ iy)! 28 ROBINSON AND LATIMER Continuing, |·|⪅Z dx f′(x) O(η)−Z |y|>ηdy(χ(y)Re ˆm(x+ iy) +yχ′(y)Re ˆm(x+ iy)) ⪅Z dx f′(x) O(η) +Z |... | https://arxiv.org/abs/2502.02006v2 |
we must adjust it. Let g:R→[0,1]beC∞, equal 0forx≥2, equal 1for x≤1and satisfy ∥g′∥∞+∥g′′∥∞≤C. Then define f(x) =ef(x)g((2∆)−1κ(x)), which is now C2and satisfies ∥f∥1+ f′ 1+ f′′ 1≤1. Therefore Lemma A.1 gives 1 pX κ(λi)>2∆|w(x)−w∆(x)|⪅1 pX κ(λi)>2∆∆2κ(λi)3/2 ⪅Z ef(x)dµ(x) =Z ef(x)dµ∞(x) +O≺(n−1) ⪅Z ef(x)κ(x)1/2dx+O≺(n−... | https://arxiv.org/abs/2502.02006v2 |
laws for sample covariance matrices [LP24], but we reproduce the proof here with the weaker O˜Pmode of convergence for completeness. Letck=n−1/2Σ1/2zkandR(k)(z) = (R(z)−1−ckc′ k)−1. First, we prove a lemma in- volving the average of the recurring estimation error ϵk, where ϵk:=c′ kR(k)(z)ck−tr(ΣR(z)) n. LEMMA C.2. Assu... | https://arxiv.org/abs/2502.02006v2 |
i=1u′ iΞui (λi−z)(λi−v)! =1 ptr(ΣR(z)ΞR(v)) 1 +1 ntr(ΣR(z)) 1 +1 ntr(ΣR(v))+O˜P1 ηzηvn so that, using the asymptotic boundedness of 1 +1 ntr(ΣR(z)), which follows from the boundedness clause in Theorem 2, we have 1 ptr(ΣR(z)ΞR(v)) = 1 +1 npX i=1u′ iΣui λi−z! 1 +1 npX i=1u′ iΣui λi−v! p−1pX i=1λiu′ iΞui (λi−z)(λi−... | https://arxiv.org/abs/2502.02006v2 |
differ in character. LEMMA D.1. Assume (H1)-(H5) and that fis regular. Then we have the following esti- mate ˜Hw[f]− H w[f] w q q≺∆2 for any finite q≥1. PROOF . Using the notation K∆(x) = ∆−1K(x/∆)and, we have ˜Hw[f] =K∆∗(f dµ n). Using the Helffer-Sjöstrand argument of Appendix A and the fact that ∥(d/dλ)K∆(x−λ)∥1⪅... | https://arxiv.org/abs/2502.02006v2 |
the analytic signal b+iBand its extension to the complex upper half-plane. Second, we invert T′onKby solving a singular integral equation. First we simplify b. Using the result from Theorem 2 that for x∈F, lim η→0+Im[Θ ∞(x+iη)] =πδ(x)w(x) Thus, by the properties of the Hilbert transform, πH[δw]is given by the limiting ... | https://arxiv.org/abs/2502.02006v2 |
just given: the other terms can be analyzed similarly. Thus, for brevity’s sake, we define f=ψ4Hw[Ψ]and˜f=˜ψ4˜Hw[Ψ](which overwrites our previous definiton of f). Since fand˜fare regular, Lemma D.1 gives ˜Hw˜f− H wf w 2 ≺ Hw˜f− H wf w 2, This inequality can be continued as ≺ ˜f−f w 2 = ˜ψ4˜HwΨ−ψ4HwΨ w 2 ≺ ˜... | https://arxiv.org/abs/2502.02006v2 |
2009. [Joh01] Iain M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Annals of Statistics , pages 295–327, 2001. [Kai15] Dong Kai. High-dimensional covariance matrix estimation with application to hotelling’s tests. 2015. [KY17] Antti Knowles and Jun Yin. Anisotropic local la... | https://arxiv.org/abs/2502.02006v2 |
[SB95] Jack W. Silverstein and Z. D. Bai. On the empirical distribution of eigenvalues of a class of large dimensional random matrices. Journal of Multivariate Analysis , 54(2):175–192, 1995. [SC+95] Jack W. Silverstein, Sang-Il Choi, et al. Analysis of the limiting spectral distribution of large dimensional random mat... | https://arxiv.org/abs/2502.02006v2 |
arXiv:2502.02160v1 [math.ST] 4 Feb 2025Information geometry of Bayes computations Giovanni Pistone Abstract Amari’s Information Geometry is a dually affine formalism for parametric probability models. The literature proposes various nonpa rametric functional ver- sions. Our approach uses classical Weyl’s axioms so that t... | https://arxiv.org/abs/2502.02160v1 |
the definition of two affine displacements associating to each couple of points a vector. If the first poi nt is the frame’s origin, the displacement becomes an affine chart. We define two atlas of charts: for all /u1D45D,/u1D45E∈ E (/u1D707), /u1D460/u1D45D(/u1D45E)=log/u1D45E /u1D45D−E/u1D45D/bracketleftbigg log/u1D45E /u1D... | https://arxiv.org/abs/2502.02160v1 |
(1+/u1D463(·,/u1D467))/u1D45D1(·)/u1D45D2(/u1D467)/u1D7072(/u1D451/u1D467)/u1D702/u1D45D1/uni∈1A6.endl−→/uni222B.dsp /u1D463(·,/u1D467)/u1D45D2(/u1D467)/u1D7072(/u1D451/u1D467). Now the derivative of the marginalization function /u1D453follows from eq. (8), /u1D451/u1D453(/u1D45E)[★/u1D45E]=mU/u1D45E1/u1D45D1/u1D451/u1... | https://arxiv.org/abs/2502.02160v1 |
equations we have derived depends on choosing an exponentia l family with suitable 8 Giovanni Pistone special characters. The literature on such applications is too extensive to present in a short conference note like this one and will be the object of a longer paper currently in progress. Acknowledgments De Castro Str... | https://arxiv.org/abs/2502.02160v1 |
arXiv:2502.02177v3 [math.ST] 4 Apr 2025AFFINE CALCULUS FOR CONSTRAINED MINIMA OF THE KULLBACK-LEIBLER DIVERGENCE GIOVANNI PISTONE ORCID Abstract. The non-parametric version of Amari’s dually affine Information Geom etry provides a practical calculus to perform computations of interest in statistical machine learning. The... | https://arxiv.org/abs/2502.02177v3 |
Kullback– Leibler divergence (KL-divergence). Indeed, our main result in sec tion 2 is a form of the totalgradientoftheKL-divergenceasexpressed intheduallyaffin egeometry. Namely, we consider symmetric divergences [18], generative adversarial net works [13], mixed entropy and transport optimization [4, 21], and variation... | https://arxiv.org/abs/2502.02177v3 |
of parallel transports below. The definition of the statistical bundle aims to capture an essential mechanism of Fisher’s approach to statistics ([12], Ch. 4). Suppose t∝mapsto→q(t)∈ E(Ω) is a one- dimensional statistical model. In that case, the Fisher’s score is t∝mapsto→d dtlogq(t) =⋆q(t), andt∝mapsto→(q(t),⋆q(t))∈SE... | https://arxiv.org/abs/2502.02177v3 |
. In conclusion, we review the derivation of a function fbetween two maximal exponen- tial models using the mixture charts eq. (7). The expressions of fand its derivative df 4 in the charts centered, respectively, at p1andp2, are E(µ1) E(µ2) Sp1E(µ1)Sp2E(µ2)f ηp2 fp1·p2n−1 p1andSq1E(µ1) Sf(q1)E(µ2) Sp1E(µ1) Sp2E(µ2)df(... | https://arxiv.org/abs/2502.02177v3 |
refers to the fact that Dis a function of two variables. Proposition 1. The total natural gradient of the KL-divergence is (20) ( q,r)∝mapsto→gradD(q∝ba∇dblr) = (−sq(r),−ηr(q))∈SE(Ω)×∗SE(Ω). Thatis, moreexplicitly, foreachsmooth coupleof curves t∝mapsto→q(t)andt∝mapsto→r(t), eq.(19)be- comes d dtD(q(t)∝ba∇dblr(t)) =−/a... | https://arxiv.org/abs/2502.02177v3 |
the total natural gradient of the KH-divergence of proposition 1, the derivative eq. (27), and the duality of parallel transports eq. (4): 8 grad(q∝mapsto→JS(q,r)) = 1 2/parenleftbigg −sq/parenleftbigg1 2(q+r)/parenrightbigg −1 2eUq 1 2(q+r)η1 2(q+r)(q)/parenrightbigg +1 2/parenleftbigg −1 2eUq 1 2(q+r)η1 2(q+r)(r)/par... | https://arxiv.org/abs/2502.02177v3 |
KL-divergence in eq. (34) is called mutual information. 10 Proposition 5. The natural gradients of the divergences of a joint distribu tionrand its mean-field approximation Π(r)are gradD(Π(r)∝ba∇dblr) =Er1⊗r2(sr(r1⊗r2)|Π1)+Er1⊗r2(sr(r1⊗r2)|Π2)−ηr(r1⊗r2). (33) gradD(r∝ba∇dblΠ(r)) =−sr(r1⊗r2)+Er(ηr(r1⊗r2)|Π1)+Er(ηr(r1⊗r2)... | https://arxiv.org/abs/2502.02177v3 |
=/braceleftbig q=q1|2·q2/vextendsingle/vextendsingleq1|2(·|y)∈ E(Ω1),y∈Ω2,q2∈ E(Ω2)/bracerightbig (37) =/braceleftbig q=q2|1·q1/vextendsingle/vextendsingleq2|1(·|x)∈ E(Ω2),x∈Ω1,q1∈ E(Ω1)/bracerightbig . (38) The two representations are (39) E(Ω1)⊗Ω2×E(Ω2)↔ E(Ω1×Ω2)↔ E(Ω1)×E(Ω2)⊗Ω1. Following the approach of [13], ([3],... | https://arxiv.org/abs/2502.02177v3 |
curve t∝mapsto→r(t)∈ Mis (43)d dtL(r(t);x) = /angbracketleftbig⋆r(t),sr(t)(q2)/angbracketrightbig r(t)+/angbracketleftbig⋆r(t),logq1|2(x|·)−Er(t)/bracketleftbig logq1|2(x|·)/bracketrightbig/angbracketrightbig r(t)= /angbracketleftbigg ⋆r(t),logq2 r(t)+logq1|2(x|·)−Er(t)/bracketleftBig logq2 r+logq1|2(x|·)/bracketrightB... | https://arxiv.org/abs/2502.02177v3 |
Shun-Ichi Amari, Natural gradient works efficiently in learning , Neural Computation 10(1998), no. 2, 251–276. 3. Shun-ichi Amari, Information geometry and its applications , Applied Mathematical Sciences, vol. 194, Springer, [Tokyo], 2016. MR 3495836 4. Shun-ichi Amari, Ryo Karakida, and Masafumi Oizumi, Information geo... | https://arxiv.org/abs/2502.02177v3 |
arXiv:2502.02213v1 [math.ST] 4 Feb 2025Sampling models for selective inference Daniel Garc´ ıa Rasines1*†and G. Alastair Young2† 1*Department of Quantitative Methods, CUNEF Universidad, Ca lle Almansa 101, Madrid, 28040, Spain. ORCID: 0000-0002-1558- 5860. 2Department of Mathematics, Imperial College London, Exhib itio... | https://arxiv.org/abs/2502.02213v1 |
right, this problem provides a simple t heoretical benchmark for the analysis of more complex scenarios. A general formulation of selection problems is as follows. Suppose th at we have dataY∈ Y, whose sampling distribution we model by some parametric family {F(y;θ):θ∈Θ}, whereF(y;θ) is the distribution function of Yun... | https://arxiv.org/abs/2502.02213v1 |
stylised example of what Rosenthal (1979) te rmed the ‘file drawer effect’, also known as publication bias, which occurs when t he decision of whether tomake ascientific finding public is influenced by the outcome ofthe analysis. Another important family of selection problems concerns inference for a regression model after ... | https://arxiv.org/abs/2502.02213v1 |
be of the form p(y) =/producttextn i=1w(yi). 2 The conditional approach to selective inference Different approaches to statistical inference lead to two opposing views as to the correct analysis of the data in the presence of selection. On the on e hand, frequentist methods evaluate the accuracy of inferential procedure... | https://arxiv.org/abs/2502.02213v1 |
can be relaxed in some circumstances (Barndorff -Nielsen and Cox, 1994, Chapter 2). In a similar vein, we argue that the conditional approach follows from arguments analogous to those underpinning the Conditionality Principle (Birnbau m, 1962), as both advocate conditioning on the random events that have occur red right... | https://arxiv.org/abs/2502.02213v1 |
if the trueθis smaller than zero, we have a higher chance of falsely concluding that θ>0if we have a less informative sample (i.e. if n1is small), and vice versa. Therefore, inference based on thi s sampling model would make use of the sample size distribution, and wou ld arrive to a different conclusion had the sample ... | https://arxiv.org/abs/2502.02213v1 |
identify a minima l sufficient statistic (T,A) such that the distribution of T|Ais independent of χand the distribu- tion ofAis independent of ψ, or depends on it in such a way that through observation ofAalone no information can be extracted about ψ(Barndorff-Nielsenand Cox, 1994, 7 Chapter 2). In such cases, a natural e... | https://arxiv.org/abs/2502.02213v1 |
the basis that the con- ditional distribution of Agiven the selection event depends very mildly on the parameterof interest (it is non-informative,in a certain sense), a nd leads to simpler inferences. We formalise this notion in the following section. Since the distribution of Y2,...,Ymdepends on θ1given selection, co... | https://arxiv.org/abs/2502.02213v1 |
to the best of our knowledge the only viable proposal to achieve validity in this setting is data splitting (Rinaldo et al., 2019). Scenario 2 . A different type of selection problem is as follows. In some cases, we may want to assess the effect of the selected covariates X(s) relative to the full set of covariates X. In... | https://arxiv.org/abs/2502.02213v1 |
function g, it must be the case that Pψ,χ{g(A) = 0}= 1 for allχ∈Θ2. We will refer to this type of ancillarity as G-ancillary. Godambe (1980) showed that if AisG-ancillary, then the distribution of Tcontains the same information about ψas the distribution of T|A, so that the latter can be used for inference about ψwitho... | https://arxiv.org/abs/2502.02213v1 |
g(x;λ)>(1−ε) sup ˜x∈Xg(˜x;λ). One drawback of this definition is that it is not invariant under transf ormations of the data. For example, the family of exponential distributions do es not provide a perfect fit for any positive observation, as all the densities have m ode 0. However, one can easily show that after applyi... | https://arxiv.org/abs/2502.02213v1 |
orthogonal to ηas columns, thenA=PTYcan be easily seen to be both Gand˜M-ancillary for ψwithout selec- tion, and therefore also after selection. On defining T=ηTY, one can easily show that the distribution of T|Agiven selection is a truncated one-dimensional Gaussian distribution, which admits an exact analytical charac... | https://arxiv.org/abs/2502.02213v1 |
(1): 155–162. https://doi.org/10.2307/2335328 . Harville, D.A. 2021. Bayesian inference is unaffected by selection: fa ct or fiction? Am. Stat. 00(0): 1–7. https://doi.org/10.1080/00031305.2020.1858963 . 14 Jørgensen, B. 1994. The rules of conditional inference: is there a uni- versal definition of non-information? J. Ita... | https://arxiv.org/abs/2502.02213v1 |
Minimax-Optimal Dimension-Reduced Clustering for High-Dimensional Nonspherical Mixtures Chengzhu Huang Yuqi Gu Department of Statistics, Columbia University Abstract In mixture models, nonspherical (anisotropic) noise within each cluster is widely present in real-world data. We study both the minimax rate and optimal s... | https://arxiv.org/abs/2502.02580v2 |
raised in [24], which studied anisotropic Gaussian mixtures with a fixed or slowly growing dimension. In this paper, we will resolve the above questions and uncover a surprising insight that, in the presence of unknown noise heteroskedasticity, the statistical limit of a high-dimensional anisotropic Gaussian mixture mo... | https://arxiv.org/abs/2502.02580v2 |
the signal-to-noise ratio defined in (3). Specifically, we consider an anisotropic Gaussian mixture model with Kcomponents. Denote a matrix collecting the top- Kright singular vectors of Y∗=E[Y] =Z∗Θ∗⊤byV∗, then V∗∈Rp×K. For every k∈[K], define w∗ k=V∗⊤θ∗ k∈RK,S∗ k=V∗⊤ΣkV∗∈RK×K, which represents the projected cluster c... | https://arxiv.org/abs/2502.02580v2 |
theory of clustering consistency for COPO applicable to flexible noise distri- butions. We focus on two high-dimensional settings: (i) general anisotropic Gaussian mixtures and (ii) mixtures of general distributions with local dependencies. These flexible local dependencies are defined by a latent block structure withi... | https://arxiv.org/abs/2502.02580v2 |
programming (SDP) methods as relaxed forms of the K-Means problem are studied in [37, 22, 70, 75]. Slightly deviating from our interest, there is a series of works focusing on estimating the population parameters rather than clustering the sample data points [88, 48, 93]. To understand how the unknown covariance matric... | https://arxiv.org/abs/2502.02580v2 |
understand the relation between the minimax rate and the subspace spanned by the cluster centers, we first clarify the distinction between our lower bound and the existing ones in terms of the Bayesian oracle risk. Bayesian Oracle Risk For ease of presentation, we consider two balanced Gaussian mixture components N(θ∗ ... | https://arxiv.org/abs/2502.02580v2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.