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the high-dimensional likelihood-ratio estimator to a corresponding low-dimensional surrogate. With this in mind, the following proposition states the correctness of (6) for these two cases. From this point forward, we regard the dimension pas a function of n(i.e., pgrows with n) without explicitly mentioning it. Consid... | https://arxiv.org/abs/2502.02580v2 |
bounded by inf bzsup (z,η)∈ΘE[h(bz,z∗)]≥exp −(1 +o(1))SNR2 0 2 . Proposition 2.1 together with Corollary 2.1 imply that (i) For the cases withSNR full SNR→1, a method that achieves exp( −(1 + o(1))SNR2 2) is minimax-optimal; (ii) In other cases, exp − (1 +o(1))SNR full2 2 could be significantly smaller than exp −(... | https://arxiv.org/abs/2502.02580v2 |
bound in Theorem 2.3 has an exponent related to SNR instead of SNR full. A straight- forward implication is that, log" inf bzsup (z∗,η)∈ΘαE[h(bz,z∗)]# ≫log" max η∈eΘαRBayes(η)# . As far as we know, this is the first result of proving the substantial discrepancy between the actual risk and the Bayesian oracle risk in ge... | https://arxiv.org/abs/2502.02580v2 |
the first sample based on bz, whose forms will be specified in the proof. The ez(y) is the likelihood ratio estimator. Step 3: Lower Bounding Lη(1)(bz) +Lη(2)(bz) In light of (12), establishing the lower bound amounts to lower bounding supη∈eΘαE Lη(bz) for an arbitrary bz. Heuristicly, the hardness of the anisotropic... | https://arxiv.org/abs/2502.02580v2 |
and adaptive to nonspherical and dependent noise. Algorithm 1: (Iterative) Covariance Projected Spectral Clustering (COPO) Input: Data matrix Y= (y1, . . . ,yn)⊤∈Rn×p, number of clusters K, an initial cluster estimate bz(0) Output: Cluster assignment vector bz(t)∈[K]n 1Perform top- KSVD of Yand obtain its top- Kright s... | https://arxiv.org/abs/2502.02580v2 |
exponent of our upper bound −SNR2 2is generally less than the exponent of the upper bound presented in [92], namely, −△2 8 max k∈[K]∥Σk∥(△was defined in (1)). •Iterative EM-type algorithms : The traditional EM algorithm and the hard-EM algorithm (the adjusted Lloyd’s algorithm in [24]) iteratively estimate the cluster ... | https://arxiv.org/abs/2502.02580v2 |
with entries Si,j̸= 0 if and only if (i, j)∈E. The local dependence structure is determined by the connected components of the graph G, in the sense that the noise is correlated within the same component but independent across different connected components. A natural extension is to consider the mixtures of Ising mode... | https://arxiv.org/abs/2502.02580v2 |
to UΛ. Canceling outΛ−1, the clustering criteria in the t-th step can be rewritten as (yi−bθ(t) k)⊤VbS(t) k−1V⊤(yi−bθ(t) k) (13) =(yi−bθ(t) k)⊤VΛ−1 Λ−1bS(t) kΛ−1−1Λ−1V⊤(yi−bθ(t) k) = Ui,:−¯u(t−1) k⊤1 n(t−1) kX j∈[n],z(t−1) j=k Uj,:−¯u(t−1) k Uj,:−¯u(t−1) k⊤−1 · Ui,:−¯u(t−1) k where ¯u(t−1) kis short forP l∈... | https://arxiv.org/abs/2502.02580v2 |
the projected vectors as rotated K-dimensional Gaussian random vectors, even if the original data are non-Gaussian. 4 Upper Bounds As indicated in Section 3.2 and validated in the later simulation studies in Section 5, our COPO clustering algorithm adapts to a wide range of noise distributions. We introduce two possibl... | https://arxiv.org/abs/2502.02580v2 |
independent and the projected covariance matrices S∗ 1, . . . ,S∗ Khave rank K; 2. Assume that the following conditions hold: σ∗ min=( ω ντ7 1τ2ςK1 2 eσ√n+σ√p ,under Assumption 4.1 ω ντ7 1τ2ςK1 2κ σm√n+σ√p ,under Assumption 4.2,(16) ν4τ16 1βK2(logd)4=o(n∧p),Under Assumption 4.1 o(n∧l),Under Assumption 4.2, (17... | https://arxiv.org/abs/2502.02580v2 |
then the block size mcan scale as the order O(pa) with a∈(0,1), which corresponds to cases with severely dependent entries in the noise matrix. •Adaptive to Spiked Noise. Our theory allows for some spiked directions of the covariance matrices Σkthat do not align with the subspace spanned by the cluster centers but lead... | https://arxiv.org/abs/2502.02580v2 |
arise from (a) estimating 21 the projected centers and projected covariance matrices in high dimensions, (b) dealing with the perturbation of the projection operator defined by the empirical singular subspace V, and (c) han- dling local dependence in the concentration treatments. We highlight the following parts as the... | https://arxiv.org/abs/2502.02580v2 |
can still go to infinity). With the aid of the matrix concentration universality recently developed in [12], we establish the concentration universality in the following two key aspects, which might be of independent interest: •Concentration on P i∈[n],z∗ i=kV∗⊤EiE⊤ iV∗/nk−S∗ k (Lemma B.20). The quantity of interest he... | https://arxiv.org/abs/2502.02580v2 |
the class with the largest posterior probability, based on the estimated parameters. Note that the empirical performance of the hard-EM algorithm proposed by [24] is similar to that of EM presented here, because they both require inverting p×psample covariance matrices. As shown in Table 1, the EM algorithm frequently ... | https://arxiv.org/abs/2502.02580v2 |
an autoregressive matrix Aρ=1ρ ρ1 . In each trial, we independently generate ρk,j(k∈[2], j∈[p/2]) and set two underlying covariance matrices to be eΣk= diag( Aρk,1,···,Aρk,p/2),k∈[2]. Then we draw an underlying Gaussian matrix ˇY= (ˇy1,···,ˇyn)⊤∈Rn×pwhere ˇyi∼ N(0,Σz∗ i). The binary data matrix is Y= (yi,j)i∈[n],j∈[p... | https://arxiv.org/abs/2502.02580v2 |
and the spectral clustering (accuracy 74.4%). We then look into the subset of the HapMap3 dataset composed of two subpopulations: CEU and MEX. Figure 4b demonstrates that these two subpopulations exhibit severe noise heterogeneity in terms of the projected covariance matrices. However, our method surprisingly achieves ... | https://arxiv.org/abs/2502.02580v2 |
. . . . . . . . . . . . . . . . 29 A.2 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.3 Proof of Proposition 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.4 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.02580v2 |
the homogeneous covariances. To apply [24, Lemma A.1] on testing error for Linear Discriminant Analysis to the Bayesian oracle risk, it suffices to verify that SNR full→ ∞ asngoes to infinity. By definition, we have SNR full({Σk,n}k∈[2],{θ∗ k,n}k∈[2]) = Σ−1 2 1,n(θ∗ 1,n−θ∗ 2,n) 2/2 = (θ∗ 1,n−θ∗ 2,n)⊤V∗ nV∗ n⊤Σ−1 1,nV∗ ... | https://arxiv.org/abs/2502.02580v2 |
for some suitably defined matrices BnandDn. For each k∈[2] and arbitrary y∈Ra= (y⊤ 1,y⊤ 2)⊤ where y1denotes the first two entries of yandy2denotes the remaining entries, we have ffull 1(y) = (y−eV⊤ nθ∗ 1,n)⊤(eV⊤ nΣ1,neVn)−1(y−eV⊤ nθ∗ 1,n) =(y1−w∗ 1,n)⊤ (S∗ 1,n)−1+B⊤ nD−1 nBn (y1−w∗ 1,n)−2y⊤ 2Bn(y1−V∗ n⊤θ∗ 1,n) +y⊤ 2D... | https://arxiv.org/abs/2502.02580v2 |
some constant ϵ >0. The proof consists of three main steps, detailed in Sections A.2.1, A.2.2, and A.2.3. Once these steps are established, the proof is concluded in Section A.2.4. A.2.1 Step 1: Reduction to a Subset of Θ z The first step is to reduce the Hamming distance under all possible permutations over [ K] to th... | https://arxiv.org/abs/2502.02580v2 |
order of the samples. For convenience, given a label vector zand a permutation πover [ n], we also introduce a permuted label vector zπby letting ( zπ)i=zπ−1(i). Given an arbitrary bz, the core step of the symmetrization argument lies in the randomized estimator bzsymthatP[bzsym=bzπ|Y] = 1 /(|B∁|!) for each π∈ΓB, where... | https://arxiv.org/abs/2502.02580v2 |
1 |B∁|X i∈B∁Pz∗[bzπ i̸=z∗ i] =1 |B∁|X i∈B∁Pzπ[bz(Yπ)i̸= (z∗ π)i] =Z 1{bz(Yπ)i̸= (z∗ π)i}dPz∗(Y) (i)=Z 1{bz(Yπ)i̸= (z∗ π)i}dPz∗π(Yπ) =1 |B∁|X i∈B∁Pz∗π[bzi̸= (z∗ π)i], where (i) holds since Pz∗(Y) =Pz∗π(Yπ). It follows that 1 |ZB|X z∗∈ZB1 |B∁|X i∈B∁Pz∗[bzπ i̸=z∗ i] =1 |ZB|X z∗∈ZB1 |B∁|X i∈B∁Pz∗π[bzi̸= (z∗ π)i] =1 |ZB|X z... | https://arxiv.org/abs/2502.02580v2 |
2 ϕ θ∗ 1,Σ(j1) 1≤1 2,ϕ θ∗ 1,Σ(j2) 1 ϕ θ∗ 2,Σ(j2) 2≤1 2o andnϕ θ∗ 1,Σ(j1) 1 ϕ θ∗ 2,Σ(j1) 2≤1 2,ϕ θ∗ 2,Σ(j2) 2 ϕ θ∗ 1,Σ(j2) 1≤1 2o . Instead of tackling these irregular regions directly, it is more practical to look for regions in regular shapes, satisfying that (i) they are contained within the integral region in the RH... | https://arxiv.org/abs/2502.02580v2 |
cluster, as depicted in Figure 6c. Reinterpreting the above in the context of (33), a neighborhood of ( x/√ 2, x/√ 2, z∗), the so-called critical region, will fall into the regionnϕ θ∗ 2,Σ(1) 2 ϕ θ∗ 1,Σ(1) 1≤1 2,ϕ θ∗ 1,Σ(2) 1 ϕ θ∗ 2,Σ(2) 2≤1 2o asx→ ∞ ; on the other hand, the quantity min{ϕθ∗ 1,Σ(1) 1(x), ϕθ∗ 2,Σ(2) 2(... | https://arxiv.org/abs/2502.02580v2 |
given j∈ {j1, j2},v⊤v(j)= 0 implies that v⊤Σ(j) kv=eσ2fork∈[2] by the formula of block matrix inverse. Intuitively, Σ(j)exhibits smaller variability along the directions of V∗andv(j), while showing a larger variability, eσ, in the orthogonal directions. Equipped with the above construction, we first notice that max z∈R... | https://arxiv.org/abs/2502.02580v2 |
summarize the notations used: •V∗is ap-by-2 orthonormal matrix representing the right singular space of E[Y]. •w∗ k=V∗⊤θ∗ k, and S∗ k=V∗⊤ΣkV∗. •The minimizer in the definition of SNR is defined as: w∗:= arg min x∈R2n (x−w∗ 1)⊤S∗ 1−1(x−w∗ 1) : (x−w∗ 1)⊤S∗ 1−1(x−w∗ 1) = (x−w∗ 2)⊤S∗ 2−1(x−w∗ 2) o . Additionally, we define... | https://arxiv.org/abs/2502.02580v2 |
same as theSNR fullof the Gaussian mixture N (V∗,V∗ ⊥v(j))⊤θ∗ k,(V∗,V∗ ⊥v(j))⊤Σk(V∗,V∗ ⊥v(j)) , k ={1,2} of dimension 3 for all j∈[M]. For ease of notation, we denote that w∗,(j) k:= (V∗,V∗ ⊥v(j))⊤θ∗ k, S∗,(j) k:= (V∗,V∗ ⊥v(j))⊤Σk(V∗,V∗ ⊥v(j)) = S∗ k−1+α′2 σ2ewew⊤,α′ σ2ew α′ σ2ew⊤,1 σ2 −1 ∈R3×3. (46) And we w... | https://arxiv.org/abs/2502.02580v2 |
Recall that the maximizer of ϕθ∗ k,Σ(j) k(V∗w∗+V∗ ⊥z) in terms of z∈Rp−2fork∈[2] and j∈[M] is written as zk,(j) ∗=− V∗ ⊥⊤Σ(j) k−1V∗ ⊥−1 V∗ ⊥⊤Σ(j) k−1V∗ V∗⊤ x∗−θ∗ k . (53) Plugging (36) and (38) into (53) yields that zk,(j) ∗=−α′ ew⊤(w∗−w∗ k) v(j). (54) Given j1̸=j2∈[M], we also introduced an orthonormal matrix d... | https://arxiv.org/abs/2502.02580v2 |
} C6, (57) where we make use of the property inferred from the definition of w∗in (35) that (w∗−w∗ 1)⊤S∗ 1−1V∗⊤(x∗−θ∗ 1)−(x∗−θ∗ 2)⊤V∗S∗ 2−1(w∗−w∗ 2) = 0 . To facilitate understanding, Cmain 1 andCmain 2 capture the substantial gap between two log- likelihood functions, while C1through C6collect the remnant effects infl... | https://arxiv.org/abs/2502.02580v2 |
and the fact (59). Second, recalling the definition of Σ(j1)in (43), the exponent of f(j1,j2) 1,essentialis decomposed as follows: −1 2(y(j1,j2) key−θ∗ 1)⊤Σ(j1)−1(y(j1,j2) key−θ∗ 1) =−1 2(w∗+△1−w∗ 1)⊤V∗⊤Σ(j1) 1−1V∗(w∗+△1−w∗ 1) − w∗+△1−w∗ 1⊤V∗⊤Σ(j1) 1−1V∗ ⊥ z1,(j1) ∗ +z2,(j2) ∗ +V(j1,j2)△2 −1 2 z1,(j1) ∗ +z2,(j2) ∗... | https://arxiv.org/abs/2502.02580v2 |
(2π)p−4 2eσp−4exp −∥△3∥2 2 2eσ2 ≥1 (2π)2¯σ4exp −1 2 1 +Cdensity 1 SNR 0+Cdensity 2 δ+Cdensity 3¯σ2 eσ2+ 2log eσ ¯σ SNR2 0 SNR2 0 ·1 (2π)p−4 2eσp−4exp −∥△3∥2 2 2eσ2 . Control Lη(j1)(bz)+Lη(j2)(bz) Now we are well prepared to lower-bound the “separation degree” Lη(j1)(bz) +Lη(j2)(bz) using Condition 1 andConditi... | https://arxiv.org/abs/2502.02580v2 |
thanks to (71). A.2.4 Putting All the Pieces Together We now summarize the preceding building blocks to derive the final minimax rate of the problem. We view the marginal distribution of eYunder1 2P∗,1,η(j)+1 2P∗,2,η(j)andLη(j)(bz) as the given distribution and the functions in Lemma A.1, respectively. Further, combini... | https://arxiv.org/abs/2502.02580v2 |
2 , ifeσ=ω(¯σ),¯σ/σ=O(1),log(eσ2/σ2) =o(SNR2 0), and neσ2(1+ϵ)=o(p¯σ2(1+ϵ))for some constant ϵ >0. The basic idea of the proof is to focus on the most hard-to-distinguish pair of clusters among the Kclusters. Reducing the problem into distinguishing these two components, the remaining parts follow a similar route in t... | https://arxiv.org/abs/2502.02580v2 |
of generality, we let S∗ k=σ2IK. Since p−K≥p 2for every sufficiently large n, we can always obtain a packing on Sp−K−1by appending zeroes to a packing on Sp 2for every sufficiently large n. Following the same way as in the proof of Theorem 2.3 (especially (40) and (41)), an almost-orthogonal packing on Sp−K−1is given a... | https://arxiv.org/abs/2502.02580v2 |
bounded by min{ϕθ∗ 1,Σ(j1) 1 x , ϕθ∗ 2,Σ(j2) 2 x } ≥1 (2π)2σ4exp −1 2 1 +Cdensity 1 SNR 0+Cdensity 2 δ+Cdensity 3σ2 eσ2+Cdensity 4log eσ/σ SNR2 0 SNR2 0 ·1 (2π)p−K−2 2eσp−K−2exp − V(j1,j2) ⊥⊤x 2 2 2eσ2 ·1 (2π)K−2 2σK−2exp(− V∗ −2⊤x 2 2 2σ2) for some constants Cdensity i >0,i∈[4]. We aim to verify the condit... | https://arxiv.org/abs/2502.02580v2 |
⊥⊤x 2 2 2eσ2 ·1 (2π)K−2 2σK−2exp(− V∗ −2⊤x 2 2 2σ2). We have thus verified Condition 2* . ByCondition 1* andCondition 2* , the right-hand side of (81) is lower bounded by Z R(j1,j2) K1 (2π)2σ4exp(−(1 +o(1))SNR2 2)·1 (2π)p−2 2eσp−2exp − V(j1,j2) ⊥⊤V∗ ⊥⊤x 2 2 2eσ2 ·1 (2π)K−2 2σK−2exp(− V∗ −2⊤x 2 2 2σ2)dx =π2ρ2 1ρ2 2 (... | https://arxiv.org/abs/2502.02580v2 |
Precisely, we notice that SNR full1,2({θ∗ k}k∈[K],{Σ(j) k}k∈[K]) =SNR full2,1({θ∗ k}k∈[K],{Σ(j) k}k∈[K]) =SNR full({θ∗,new 1,θ∗,new 2},{Σ(j),new 1 ,Σ(j),new 2}), where the second equality holds by an observation that the minimizer xin the definition of SNR full1,2({θ∗ k}k∈[K],{Σ(j) k}k∈[K]) must satisfy V∗ −2⊤x=0since ... | https://arxiv.org/abs/2502.02580v2 |
b},{Σa,Σb}))≥ α2SNR2 0 2foraorbin{3,···, K}and every sufficiently large nin the following: •Ifa̸=b∈ {3,···, K}, combining (90) with Proposition 2.1 directly leads to the conclusion. •Ifa∈ {1,2}, b∈ {3,···, K}, we look into the form of RBayesand have RBayes({θ∗ 1,θ∗ 3},{Σ1,Σ3})≤Ph ∥ϵ∥2≥min{SNR full1,3,SNR full3,1}i ≤exp... | https://arxiv.org/abs/2502.02580v2 |
oracle covariance matrices. We shall establish the exponential decay rate of l(bz(t),z∗) in the following steps. Step 1: Error Decomposition via a One-Step Analysis To begin with, a simple calculation tells us that a geometric decay of the alternative sequence {l(bz(t),z∗)}T t=0can exhibit geometric decay through the r... | https://arxiv.org/abs/2502.02580v2 |
derive that l(bz,z∗) 68 ≤X i∈[n]X k∈[K]\{z∗ i} V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i ·1 ζoracle ,i(k)≤δ 2 V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i +X i∈[n]X k∈[K]\{z∗ i} V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i · 1 Fi(k,z)≥δ 8 V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i +1 Gi(k,z)≥δ 8 V∗⊤ θ∗ k−θ∗ z... | https://arxiv.org/abs/2502.02580v2 |
i }i ≲nKν2ω max i∈[n],k∈[K]\{z∗ i}Ph ζoracle ,i(k)≤δ 2 V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i i ≲nKν2ω max i∈[n],k∈[K]\{z∗ i}Ph eζoracle ,i(k)≤δ V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i i + max i∈[n]P max k∈[K]\{z∗ i}Ξi,k≥δ 2 V∗⊤ θ∗ k−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ k−θ∗ z∗ i ≲nKν2ωh exp(−(1−eδ)SNR2 2) +O(d−1... | https://arxiv.org/abs/2502.02580v2 |
B.5 ω multiplied by some sufficiently large constant. Moreover, we define that Fgood:=n the inequalities in Lemma B.3, and Lemma B.4, Lemma B.5 hold , andl(bz(0),z∗)≤cn βK(logd)4is satisfiedo , Foracle :=n ξoracle <1 2cn βK(logd)4holdso . We note that P F∁ good ≤O(d−10) +o(n−2) =o(n−2) by Lemmas B.3, B.3, and B.5. I... | https://arxiv.org/abs/2502.02580v2 |
a,b≤λ−1 2 min∥θ∗ a−θ∗ b∥2. Moreover, with τ:=λ1 2max/λ1 2 min≥1we have λ−1 2 minτ−1 θ∗ a−θ∗ b 2≲SNR a,b≤λ−1 2 min θ∗ a−θ∗ b 2, 1 2τ−1ω1 2≤△ 2¯σ≤SNR≤ω1 2. B.2.2 Concentrations on Noise Matrices The following part comprises the concentration results for some linear forms of the noise matrix E, under the bounded noise cas... | https://arxiv.org/abs/2502.02580v2 |
2≲¯σp Klogdwith probability at least 1−O(d−10), and V(−i)Ei 2≲¯σ√ KSNR with probability at least 1−O exp(−SNR2 2)∨d−10 . Further, with probability at least 1−O(d−10)one has V⊤Ei 2≲ σξopp βrκ+ ¯σp Klogd. B.2.6 Center / Covariance Estimation Characterization The following four lemmas provide upper bounds on the fluct... | https://arxiv.org/abs/2502.02580v2 |
−O(exp(−SNR2/2)∨d−10) it holds that A1= V−V(−i)O(−i)⊤Ei 2· bSk(z∗)−1 V⊤ bθz∗ i(z∗)−bθk(z∗) 2 + bSz∗ i(z∗)−1−bSk(z∗)−1 O(−i)⊤V(−i)⊤Ei 2 +1 2 bSz∗ i(z∗)−1−bSk(z∗)−1 V−V(−i)O(−i)⊤Ei 2 ≲σξopςp βKrκ SNR1 σ2 ξop eσ+σp βKr + ¯σKr βlogd n+ν¯σω1 2 +σξopςp βKrκ SNR ≲Kς2ξ2 opτ2 2SNR2+Kςξ opτ2 τ1+ξopτ2)SNR+√ Kςξ op... | https://arxiv.org/abs/2502.02580v2 |
with probability at least 1 −O(d−10). Proceeding with the analysis of Ξ i,k, we combine (122), (123), (130) to derive that Ξi,k≲√ Kςξ opτ2ντ1ω+K τ2 1+ξ2 opτ2 2 80 +K τ2 1+ξ2 opτ2 2 +ντ1K1 2(τ1+ξopτ2)ω1 2 +ν2τ2 1τ2 2Kξ2 opω+ν2τ4 1r βK2 n(logd)ω = ν2τ4 1r βK2 nlogd+√ Kςξ opτ2ντ1+ν2τ2 1τ2 2Kξ2 op ω +ντ1K1 2(τ1+ξopτ2... | https://arxiv.org/abs/2502.02580v2 |
decision boundary in the following. To proceed, we focus on S∗ z∗ i−1 2O⊤O(−i)⊤V(−i)E=:eEi∈RKand define the perturbed signal noise ratio as follows: SNRperturbed z∗ i,k(δ):= arg min x∈RKn ∥x∥2: x,S1 2 z∗ iS∗ k−1V∗⊤ θ∗ z∗ i−θ∗ k +1 2 x, I−S∗ z∗ i1 2S∗ k−1S∗ z∗ i1 2 x + 1 2−δ) V∗⊤(θ∗ k−θ∗ z∗ i),S∗ k−1V∗⊤(θ∗ k−θ∗ z∗ i... | https://arxiv.org/abs/2502.02580v2 |
k +c2K1 2kl1 2kmB V(−i) 2,∞ V(−i)⊤Σz∗ iV(−i)−1 2 k2i ≤e−k, (138) 84 where ϱ:=K/2−1 kandc2>0is a constant. In the following, we only discuss the treatment of the case K≥3 since the case K= 2 similarly follows. Rewrite the bound in (138) as follows: √ 2c1 2k 2(1 +ϱ−1)ϱ/2+1 4k(1 +ϱ)1 2√ k +c2K1 2kl1 2kmB V(−i) 2 2,∞ ... | https://arxiv.org/abs/2502.02580v2 |
upper bounding the effect of misspecification of the cluster labels in the last step. B.3.1 Proof of Lemma B.3 Recall that Fi(b,z) =− V⊤Ei,bSb(z)−1V⊤ bθb(z∗)−bθb(z) + V⊤Ei,⊤bSz∗ i(z)−1V⊤ bθz∗ i(z∗)−bθz∗ i(z) − V⊤Ei, bSb(z)−1−bSb(z∗)−1 V⊤ θ∗ z∗ i−bθb(z∗) + V⊤Ei, bSz∗ i(z)−1−bSz∗ i(z∗)−1 V⊤ θ∗ z∗ i−bθz∗ i(z∗) . U... | https://arxiv.org/abs/2502.02580v2 |
+ ¯σ2ν2ω ¯σ2+ξ2 opeσ2+ξ2 oprσ2 ≲K3 ωnξ2 cov σ4 τ4 1+ξ4 opτ4 2 + ¯σ2ν2ω τ2 1+ξ2 opτ2 2 ≲1 ω2K8β3 τ8 1+ξ8 opτ8 2 l(z,z∗) +1 ων2K8β3τ2 1 τ6 1+ξ6 opτ6 2)l(z,z∗) ≲1 ων2K8β3τ2 1 τ6 1+ξ6 opτ6 2)l(z,z∗) with probability at least 1 −O(d−10), where ξcovis defined in (121), (i) holds from β2Klogd n≲1, 1 n ¯σ2n+ (eσ2... | https://arxiv.org/abs/2502.02580v2 |
2 V⊤ θ∗ z∗ i−bθz∗ i(z∗) ,bS−1 z∗ i(z∗)V⊤ θ∗ z∗ i−bθz∗ i(z∗) . To upper bound α1,1, we make the observation that α1,1=1 2 V⊤ θ∗ z∗ i−bθz∗ i(z) ,bS−1 z∗ i(z)V⊤ θ∗ z∗ i−bθz∗ i(z) −1 2 V⊤ θ∗ z∗ i−bθz∗ i(z∗) ,bS−1 z∗ i(z)V⊤ θ∗ z∗ i−bθz∗ i(z∗) 91 ≤ bSz∗ i(z)−1 V⊤ bθz∗ i(z∗)−bθz∗ i(z) · 1 2 V⊤ bθz∗ i(z∗)−bθz∗ i(z... | https://arxiv.org/abs/2502.02580v2 |
i−bθb(z∗) 93 +1 2 V⊤ θ∗ z∗ i−bθb(z∗) ,bS−1 b(z∗)V⊤ θ∗ z∗ i−bθb(z∗) ≲ V⊤ bθb(z∗)−θ∗ z∗ i 2 2 bSb(z)−1−bSb(z∗)−1 ≲ξcov σ4 ξ2 op eσ2+σ2βKr +¯σ2K2βlogd n+ν2¯σ2ω ≲K σ2·1√nωK3 2β5 2 ξ2 opβrκσ2+ξ2 opeσ2+ ¯σ2p l(z,z∗)· τ2 1+ν2τ2 1ω+ξ2 opτ2 2 ≲β2K2τ4 1+ξ4 opτ4 2 ω1 2+ω1 2ν2τ2 1 τ2 1+ξ2 opτ2 2 ≲β2K2ν2τ2 1(τ2 1+... | https://arxiv.org/abs/2502.02580v2 |
k 2 ≲¯σβK n S∗ k−1 2V∗⊤X i∈[n]1{zi=k} θ∗ z∗ i−θ∗ k 2 (i) ≲¯σβK nX i∈[n]1{z∗ i=k, zi̸=k}1 2 ·X i∈[n]1{z∗ i̸=k, zi=k} V∗⊤ θ∗ zi−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ zi−θ∗ z∗ i 1 2 (ii) ≲¯σβK nω1 2 X i∈[n]1{z∗ i=k, zi̸=k} V∗⊤ θ∗ zi−θ∗ z∗ i ,S∗ k−1V∗⊤ θ∗ zi−θ∗ z∗ i ≲¯σβK nω1 2l(z,z∗),(156) where (i) holds by the Holder’s... | https://arxiv.org/abs/2502.02580v2 |
i∈[n]1{zi=a}X i∈[n],zi=aV⊤ yi−bθa(z) yi−bθa(z) V −1P i∈[n]1{zi=a}X i∈[n],z∗ i=aV⊤ yi−bθa(z∗) yi−bθa(z∗) V + 1P i∈[n]1{zi=a}−1P i∈[n]1{z∗ i=a} X i∈[n],z∗ i=aV⊤ yi−bθa(z∗) yi−bθa(z∗) V ≤α1+α2+α3+α4,(162) where α1, α2, α3, and α4are defined as α1:=1P i∈[n]1{zi=a} V⊤ X i∈[n],z∗ i=zi=a yi−bθa(z) yi−bθa(z)⊤ −... | https://arxiv.org/abs/2502.02580v2 |
σ2ξ2 opβrκ+ ¯σ2 Klogd +βK n·l(z,z∗) ω ξ2 op eσ2+σ2βrK +¯σ2K2βlogd n|{z} ≲¯σ2 ≲1 nl(z,z∗) ωβ σ2ξ2 opβrκ+ξ2 opeσ2+ ¯σ2 K2logd ≲1 n1 2ωβ1 2K3 2 σ2ξ2 opβrκ+ξ2 opeσ2+ ¯σ2p l(z,z∗) uniformly holds for all possible zwith probability at least 1 −O(d−10). Here we make use of the facts thatl(z,z∗)βK(logd)2 n≲1 andβK2log... | https://arxiv.org/abs/2502.02580v2 |
4(bounded case)q βK n¯σ2√logd(Gaussian case)≲r βK2 n¯σ2logd,(168) where we use Assumption 4.2.3 that r1 4q µ2K pmB(logd)2≲σfor the bounded case. Putting (167) and (168) together, we conclude that O⊤bSk(z∗)O−S∗ k ≲ξ2 op(βKrσ2+eσ2) + ¯σ2r βK2 nlogd holds with probability at least 1 −O(d−10). For the second inequality, it... | https://arxiv.org/abs/2502.02580v2 |
1,j2Aj2A⊤ j3E(k) 1,j3E(k) 1,j4Aj4 +nk A⊤ΣkA 2 .(170) Regarding the first term in (170), it is related to its Gaussian analog that for every j1, j2, j3, j4∈[p]: Eh A⊤ j1E(k) 1,j1E(k) 1,j2Aj2A⊤ j3E(k) i,j3E(k) i,j4Aj4i =Eh A⊤ j1gj1gj2Aj2A⊤ j3gj3gj4Aj4i + Eh A⊤ j1E(k) 1,j1E(k) 1,j2Aj2A⊤ j3E(k) i,j3E(k) i,j4Aj4i −Eh A⊤ j... | https://arxiv.org/abs/2502.02580v2 |
1A)2 . and (b) holds by the fact that Tr( X)2≤KTr(X2) for a symmetric matrix X∈RK×K. To finish up, we invoke (172) again together with (174) to derive that v(S)2≲K2 nkK A⊤ΣkA 2+nkpm3B4∥A∥4 2,∞ | {z } upper bound in (173). 3. Finally, we make use of the modified logarithmic Sobolev inequality (Lemma C.2) to upper bou... | https://arxiv.org/abs/2502.02580v2 |
matrix concentration results. Proof of Lemma B.7 Throughout the proof, we will repetitively make use of the following lemma [12, Corollary 2.15] to upper bound the spectral norm of the matrices in interest: Lemma B.22 (Corollary 2.15 in [12]) .LetY=P i∈[n]Zi, where Z1,···,Znare independent (possibly not self-disjoint) ... | https://arxiv.org/abs/2502.02580v2 |
i 1 2. This combines with (178), (177), and (179), the concentration inequality could be further sim- plified as ∥E∥≲σ√p+eσ√n+eσ1 2(σ1 2p1 4+eσ1 2n1 4)(log d)3 4+eσp logd ≲σ√p+eσ√n with probability at least 1 −O(d−20). Similarly, the bounds for Ei,:,E:,k,EV∗, and E⊤U∗follow by the same argument: ∥Ei,:∥2≲∥(Ei,:)free∥+Cv... | https://arxiv.org/abs/2502.02580v2 |
begin with, let us introduce a fine-grained result in [92] that justifies the proximity of the leave-one-out estimate V(−i)and the original singular vector matrix Vto assist in the dependence decoupling part, as discussed in Section 4.2. Proposition B.23 (Theorem 2.2 in [92]) .Assume thatn βK2≥10andρ0:=σ∗ min ∥E∥>16. F... | https://arxiv.org/abs/2502.02580v2 |
2 σ∗ min SNR +mB ξopr βKκ n 1 + V(−i)⊤Ei 2 σ∗ min) SNR2∧logd +mBs µ2K p(logd)1 2 SNR2∧logd with probability at least 1 −O exp(−SNR2 2)∨d−10 . Rearranging the terms following the derivations of (188) and (189) gives that V(−i)⊤Ei 2≲K1 2¯σSNR with probability at least 1 −O exp(−SNR2 2)∨d−10 . For the term V... | https://arxiv.org/abs/2502.02580v2 |
b)⊤S∗ b−1(x−w∗ b)| {z } ≥1 ¯σ2∥x−w∗ b∥2 2 ≥min x∈RK1 ¯σ2∥x−w∗ a∥2 2:1 ¯σ2∥x−w∗ a∥2 2=1 ¯σ2∥x−w∗ b∥2 2 =1 4¯σ2△2≥1 4τ2ω, which leads to1 2τω1 2≤SNR a,b. Regarding the upper bound, without loss of generality, we suppose that ω1,2=ω. Then 118 SNR2 1,2≤ t0(w∗ 2−w∗ 1)⊤S∗ 1−1 t0(w∗ 2−w∗ 1) ≤ωwith t0∈(0,1) satisfying t... | https://arxiv.org/abs/2502.02580v2 |
2σ2SNR2 2σ2 subGby (16) ≲SNR4nexp −1 2σ2SNR2 2σ2 subG ≲σ4 σ4 subGSNR4nexp −1 2σ2SNR2 2σ2 subG+ 4 log σsubG σ =o(n βK(logd)4), provided the condition that SNR =(ω τ2√logτ2·log log d , under Assumption 4.1 ω mB σq log mB σ ·log log d ,under Assumption 4.2. C Some Other Auxiliary Lemmas The following lemma comes... | https://arxiv.org/abs/2502.02580v2 |
(2023). Matrix concentration inequal- ities and free probability. Inventiones Mathematicae , pages 1–69. [8] Belkin, M. and Sinha, K. (2010). Learning gaussian mixtures with arbitrary separation. In Proceedings of the 23rd Annual Conference on Learning Theory (COLT) . Omnipress. [9] Benaglia, T., Chauveau, D., Hunter, ... | https://arxiv.org/abs/2502.02580v2 |
(2010). Integrating common and rare genetic variation in diverse human populations. Nature , 467(7311):52. [29] Dasgupta, S. and Schulman, L. (2007). A probabilistic analysis of em for mixtures of separated, spherical gaussians. Journal of Machine Learning Research , 8(2). [30] Davis, C. and Kahan, W. M. (1970). The ro... | https://arxiv.org/abs/2502.02580v2 |
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Wu, Y. (2024). Information theory: From coding to learning. [72] Qin, T. and Rohe, K. (2013). Regularized spectral clustering under the degree-corrected stochastic blockmodel. Advances in neural information processing systems , 26. [73] Raiˇ c, M. (2019). A multivariate Berry–Esseen theorem with explicit constants. Ber... | https://arxiv.org/abs/2502.02580v2 |
Sample Complexity of Bias Detection with Subsampled Point-to-Subspace Distances German Martinez Matilla and Jakub Marecek February 6, 2025 Abstract Sample complexity of bias estimation is a lower bound on the runtime of any bias detection method. Many regulatory frameworks require the bias to be tested for all subgroup... | https://arxiv.org/abs/2502.02623v1 |
[4] claimed polynomial sample complexity, the more re- cent work explains the dependence on dimension [15], even under assumptions about smoothness coming from applying smooth kernels in a probabilistically approximately correct (PAC) setting. More broadly, while sample complexity may be lower under assumptions on the ... | https://arxiv.org/abs/2502.02623v1 |
Test measure, α0 Algorithm 1 Point-to-subspace query in the supremum norm Input : Test measure α0∈ M1 +(R),histogram h∈R10 + Parameter : ∆∈R. Output : True/False 1:a←normalise h. 2:fori=1 to 10 do 3:if|α0(xi)−ai| ≥∆then 4: return FALSE 5:end if 6:end for 7:return TRUE Next, we look at the normalised histograms in the l... | https://arxiv.org/abs/2502.02623v1 |
β] = sup ∥f∥H≤1Z Xf(x)dα−Z Xf(y)dβ ,∀f∈ H (2) 3.2 Wasserstein distance The origin of the Wasserstein distance [24] can be traced back to the work of Monge on Optimal Transportation (OT) [25]. For the discrete case, we consider discrete measures α∈ M (X) and β∈ M (Y),or equivalently, simplices a∈Σn andb∈Σm.Given a map... | https://arxiv.org/abs/2502.02623v1 |
i= 1, . . . , 10 j= 1, . . . , 10 Upon which we will compare the test measureP10 i=1P10 j=1a0 ijδ{xi,yj}=α0∈ M1 +(R2). As we have seen from the example, the number of hyperplanes |α0(xi, yj)− aij| ≤∆ scales very fast. Therefore, we consider a subsampling scheme, where the features are selected uniformly at random. In o... | https://arxiv.org/abs/2502.02623v1 |
is a matter of the next section. 5 Main Result Our main result shows that the subsampling scheme of Section 4 is a probabilis- tically approximately correct estimator. Specifically, we bound from below the 9 Algorithm 2 Subsampled point-to-subspace query in the supremum norm Input : Number of samples taken independentl... | https://arxiv.org/abs/2502.02623v1 |
taken. We thus have n half-spaces for each range. We choose one of the two, which we will denote for simplicity ( X,R),and proceed identically for the second. From this ( X,R),we can build the range space having as ranges not only the previous nhalf-spaces given by the nhyperplanes, but all possible half-spaces in Rn,(... | https://arxiv.org/abs/2502.02623v1 |
of an error drops substantially with an increase in sample size. However, as we decrease ϵ(= increase ∆), the error rate will increase, as the few pairs of histograms not fulfilling (10) will be increasingly difficult to catch by subsampling. In particular, where the fraction of these pairs of histograms is a thousandt... | https://arxiv.org/abs/2502.02623v1 |
and A. Smola, “A kernel two-sample test,” The Journal of Machine Learning Research , vol. 13, no. 1, pp. 723–773, 2012. [5] G. Wolfer and A. Kontorovich, “Statistical estimation of ergodic markov chain kernel over discrete state space,” 2021. [6] E. Hellinger, “Neue begr¨ undung der theorie quadratischer formen von un-... | https://arxiv.org/abs/2502.02623v1 |
and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics . Kluwer Academic Publishers, Boston, MA, 2004. With a preface by Persi Diaconis. 15 [24] G. Peyr´ e and M. Cuturi, “Computational optimal transport,” 2020. [25] G. Monge, M´ emoire sur la th´ eorie des d´ eblais et des remblais . Impr... | https://arxiv.org/abs/2502.02623v1 |
arXiv:2502.02726v1 [math.ST] 4 Feb 2025Multimarginal Schr ¨odinger Barycenter Pengtao Li and Xiaohui Chen Department of Mathematics University of Southern California {pengtaol, xiaohuic }@usc.edu Abstract The Wasserstein barycenter plays a fundamental role in aver aging measure-valued data under the framework of optima... | https://arxiv.org/abs/2502.02726v1 |
introduce the multimarginal Schr ¨odinger barycenter (MSB) based on the MOT formulation, which naturally extends the EOT problem in the two-marginal case and admits fast linear time complexity off-the-shelf algorithms to compute, and establish non-as ymptotic rates of convergence for estimating the cost functional, cou... | https://arxiv.org/abs/2502.02726v1 |
the Banach space of functions such that ∝⌊a∇d⌊lf∝⌊a∇d⌊lL∞(ν)<∞. We use P2(Rd)to denote the set of all probability measures onRdwith finite second moment. For a probability measure µand a measurable function f, we often write µ(f)to represent/integraltext fdµ. For a function class H,N(H,ε,∝⌊a∇d⌊l · ∝⌊a∇d⌊l)denotes the ε-... | https://arxiv.org/abs/2502.02726v1 |
in its margina l distributions. Theorem 3.3 (Lipschitz continuity) .Let¯νε(resp.˜νε) be the MSB for ν:= (ν1,...,νm)∈Πm j=1P2(X) (resp.˜ν:= (˜ν1,...,˜νm)∈Πm j=1P2(X)). We have W1(¯νε,˜νε)≤√mW2(ν,˜ν) +CW2(ν,˜ν)1/2,where W2(ν,˜ν) := (/summationtextm j=1W2(νj,˜νj)2)1/2andC >0is a constant depending on ε, and the second mom... | https://arxiv.org/abs/2502.02726v1 |
following problem ˆSα,ε(ˆνN 1,...,ˆνN m) := inf π∈Π(ˆνN 1,...,ˆνNm)/integraldisplay Xmcαdπ+εKL(π||ˆνN 1⊗···⊗ˆνN m). (12) The empirical dual objective function is defined as ˆΦα,ε(f) :=m/summationdisplay j=1/integraldisplay fjdˆνN j−ε/integraldisplay exp/parenleftBigg/summationtextm j=1fj−cα ε/parenrightBigg d(⊗m k=1ˆνN ... | https://arxiv.org/abs/2502.02726v1 |
one of the following is true: (i)k(j)∝\e}atio\slash=k′(j), for every j∈[m], which leverages the independence; 7 (ii)k(j0)=k′(j0)for some j0∈[m]andk(j)∝\e}atio\slash=k′(j), for every j∈[m]\{j0}, due to the marginal feasibility condition ( 9) of the multimarginal Schr¨ odinger system. Note that there are (N(N−1))mmany te... | https://arxiv.org/abs/2502.02726v1 |
least 1−2m(m−1)e−t, the first term (A) above satisfies ∝⌊a∇d⌊lp∗ α,ε(x)−ˆpα,ε(x)∝⌊a∇d⌊lL1(⊗m j=1ˆνN j)≤ ∝⌊a∇d⌊lp∗ α,ε(x)−ˆpα,ε(x)∝⌊a∇d⌊lL2(⊗m j=1ˆνN j)/lessorsimilarm,ε/radicalbigg t N. (31) The bound for the second term (B) follows from Lemma C.1and the boundedness of the potentials from Proposition 3.4. Indeed, with pr... | https://arxiv.org/abs/2502.02726v1 |
LetXh:=√ N/parenleftBig ⊗m j=1ˆνN j−⊗m j=1νj/parenrightBig (p∗ α,ε(h◦Tα))be a mean-zero process indexed by h∈ H. By Lemma C.1, we know that with probability more than 1−2e−t, /vextendsingle/vextendsingleXh−X˜h/vextendsingle/vextendsingle/lessorsimilarε∝⌊a∇d⌊lh−˜h∝⌊a∇d⌊l∞√ t. (37) Thus,Xhsatisfies the sub-Gaussian condit... | https://arxiv.org/abs/2502.02726v1 |
x, 1. for any sequence xεconverging to x,F(x)≤liminf ε→0Fε(xε), 2. there exists a sequence xεconverging to x,F(x)≥limsupε→0Fε(xε). For more information on Γ-convergence, please refer to [ Bra06 ]. Proposition A.1. The sequence (Cα,ε)ε>0Γ-converges to Cα,0w.r.t. the weak topology. Proof of Proposition A.1.The proof foll... | https://arxiv.org/abs/2502.02726v1 |
1×···×Anmm) νm(Anmm)/parenrightbigg +m−1/summationdisplay j=1/summationdisplay n∈/producttextm j=1[Lj]π∗ 0(An1 1×···×Anmm)log(1/νj(Aj)) =/summationdisplay n∈/producttextm j=1[Lj]π∗ 0(An1 1×···×Anmm)log/parenleftbiggπ∗ 0(An1 1×···×Anmm) νm(Anmm)/parenrightbigg +m−1/summationdisplay j=1/summationdisplay nj∈[Lj]π∗ 0 j−1... | https://arxiv.org/abs/2502.02726v1 |
(53) Moreover, the marginal feasibility constrain ( 9) indicates that E2,...,mZℓ1,...,ℓm= 0, (54) as well as E1,...,mZℓ1,...,ℓm= 0. (55) Combining ( 51), (52), (53), (54) and ( 55), the proof of the lemma is done. 18 B.2 Concentration of the empirical gradient norm Recall that as proved in Lemma B.2, K1≤2 Nm+1/summatio... | https://arxiv.org/abs/2502.02726v1 |
k=1φ(ξk)/bracerightBigg/bracketrightBigg . Applying Hoeffding’s Lemma and optimizing over λ >0yields PXm[(⊗m j=1ˆνN j)(φ)> t]≤exp/braceleftbigg −Nt2 2∝⌊a∇d⌊lφ∝⌊a∇d⌊l2∞/bracerightbigg . 21 The concentration of empirical potentials and joint densit y will be derived next. Recall the unique optimal dual potential identifie... | https://arxiv.org/abs/2502.02726v1 |
(67) Step 3. Note that the primal/dual problem has a nonnegative value, s o 0≤m/summationdisplay j=1νj(f∗ j)−ε/integraldisplay p∗ α,εd(⊗m k=1νk)+ε=m/summationdisplay j=1νj(f∗ j) =νm(f∗ m). 24 As a consequence, for ⊗m−1 k=1νk-a.e.(x1,...,x m−1), 1 =/integraldisplay exp/parenleftBig −cα ε/parenrightBig exp/parenleftBigg/... | https://arxiv.org/abs/2502.02726v1 |
x∈S, f(x)−inf y∈Hf(y)≤1 2β∝⌊a∇d⌊l∇f(v)∝⌊a∇d⌊l2 H. Proof. See [ KNS16 ]. 27 Lemma E.3 (Hoeffding’s inequality in Hilbert space) .LetX1,...,X nbe independent mean-zero random variables taking values in a Hilbert space (H,∝⌊a∇d⌊l · ∝⌊a∇d⌊lH). If∝⌊a∇d⌊lXi∝⌊a∇d⌊lH≤Cfor some constant C >0, then for everyt >0, we have P/paren... | https://arxiv.org/abs/2502.02726v1 |
2015. [GPRS19] Thibaut Le Gouic, Quentin Paris, Philippe Rigolle t, and Austin J. Stromme. Fast convergence of empirical barycenters in Alexandrov spaces and the Wasse rstein space. Journal of the Euro- pean Mathematical Society , 2019. [GS98] Wilfrid Gangbo and Andrzej ´Swie ¸ch. Optimal maps for the multidimensional ... | https://arxiv.org/abs/2502.02726v1 |
Highlights Sufficient dimension reduction for regression with spatially correlated errors: application to prediction Liliana Forzani, Rodrigo Garc ´ıa Arancibia, Antonella Gieco, Pamela Llop, Anne Yao • We introduce a Sufficient Dimension Reduction (SDR) methodology for spatial data. • Using a model-based inverse regre... | https://arxiv.org/abs/2502.02781v1 |
spatial econometrics (Hu et al., 2009; Goulard et al., 2017; Kopczewska, 2023, e.g.) and machine learning approaches (e.g. Heaton et al., 2019; Meyer and Pebesma, 2021; Nikparvar and Thill, 2021). Formally, spatial prediction involves a response (target) variable Ysto predict in some unob- served location s0∈D⊂R2given ... | https://arxiv.org/abs/2502.02781v1 |
Bura et al. (2022). In this context, ?obtained asymptotic results for the non-parametric regression es- timator of ˜ηregardless of whether the true R(.)or its estimator is used. Another SDR approaches are the moment-based ones, such as the Sliced Inverse Regression (SIR), (Li, 1991), the Sliced Average Variance Estimat... | https://arxiv.org/abs/2502.02781v1 |
spatial autoregressive error (SEM) models, as well as the maximum likelihood estimates of the sufficient dimension reductions. In Section 5 we present the two non-parametric strategies that we adopted for spatial prediction using our dimension reduction methodologies. Section 7 details a simu- lation experiment to comp... | https://arxiv.org/abs/2502.02781v1 |
the matrix form of model (4) is given by X|Y=NY+E, (7) where NY∈Rn×pis such that NT Y.= (µYs1, . . . ,µYsn). In this model, we will assume that the errors ϵsi,i= 1, . . . , n , follow a second-order stationary process, are independent of Ysi, and are normally distributed with mean E(ϵs) =0and positive-definite and cons... | https://arxiv.org/abs/2502.02781v1 |
given by X|Y=1nµT+F(AB)T+E, (13) with vec(ET)∼ N(0np,S⊗∆). Subsequently, from theorem 2.1 we establish that R(X) =X∆−1Aserves a sufficient dimen- sion reduction for the regression or YonX. Hence, we need to estimate ∆andA, both of which rely on the specification of the spatial association matrix Sfor the errors. In the... | https://arxiv.org/abs/2502.02781v1 |
this is, E=θWE+U, where θis the coefficient of the spatially lagged error, U∈Rn×pis such that UT.= (us1, . . . ,usn) withusi∼ N (0p,∆), i= 1, . . . , n so that, vec(UT)∼ N (0np,In⊗∆). Here 0ℓis the ℓ- dimensional vector of zeros. The matrix W∈Rn×nis the spatial weight matrix , which quantifies the structure of spatial ... | https://arxiv.org/abs/2502.02781v1 |
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