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is studied for both the empirical and population spectral features, with mn,τ=⌈τ−d(logn)d+1⌉features. Corollary 4 Suppose the prior basis (φj)j∈Nin(24) satisfies condition (11) for some γ > 0. If f0∈Sβ(L)for some β≥β−, then the population and empirical spectral features inducing variables variational posteriors corresp... | https://arxiv.org/abs/2504.03321v1 |
.88,99.59,59.17). By eye, the variational approximation can hardly be distinguished from the empirical Bayes posterior. 4.2 Truncated series priors As third example, consider a prior of the form f∼D−1/2DX j=1Zjφj, Z j∼i.i.d.N(0,1), (26) for a uniformly bounded basis φ1, φ2, . . .(that is, satisfying condition (11) for ... | https://arxiv.org/abs/2504.03321v1 |
a first step we study the asymptotic behaviour of the maximum marginal likelihood estimator (MMLE). A.1 Asymptotic behavior of the MMLE We show that with P(n) f0-probability tending to one the maximum marginal likelihood esti- mator ˆλndefined in (17) belongs to the set of good hyperparameters Λ n,0defined in (15), whe... | https://arxiv.org/abs/2504.03321v1 |
function φf0,λ(ϵ) := inf h∈Hλ: ∥h−f0∥≤ϵ1 2∥h∥2 Hλ−log Π λ(∥f∥ ≤ϵ), (36) where Hλis the reproducing kernel Hilbert space of the prior kernel kλ. By Lemma 5.3 in van der Vaart and van Zanten (2008b) and by definition of ϵn(λ), it follows that φ0,λ(Kϵn(λ))≤φf0,λ(Kϵn(λ))≤ −log Π λ(∥f−f0∥ ≤Kϵn(λ)) =nϵ2 n(λ). Together with a... | https://arxiv.org/abs/2504.03321v1 |
bounded from above by X λ∈Λn\Λn,0E 1BnEf0h (1−ϕn(λ))Z F′n(λ)p(n) f p(n) f0(y|x)dΠλ(f) xi ≤X λ∈Λn\Λn,0EZ F′n(λ)1BnEf[1−ϕn(λ)|x]dΠλ(f) ≤ |Λn|exp(−nϵ2 n,0), and so the full term (43) vanishes. Finally, using (41), the term in (44) is bounded from above by exp(nϵ2 n,0/2)|Λn\Λn,0|max λ∈Λn\Λn,0Πλ(F′ n(λ)c)≲|Λn|exp(−nϵ2 n,0... | https://arxiv.org/abs/2504.03321v1 |
al. (2016); Rousseau and Szabo (2017) for similar derivations. One of the key difference is the explicit exponential up- per bound for the posterior contraction, needed to obtain frequentist guarantees for the variational approximation. 22 Let us split the probability to the cases when λ∈Λn,0andλ∈Λn\Λn,0, i.e. Π(∥f−f0∥... | https://arxiv.org/abs/2504.03321v1 |
ϵn(λ) solves φ0,λ(Kϵn(λ))≥nϵ2 n(λ), it is a lower bound for ϵn(λ). In each corollary we verify the conditions of Theorem 2. First we show the tail condition (12) on the signal f0∈Sβ(L) holds with δn=cnn−β/(d+2β)for sufficiently smooth signals. Note that for β≥d+ 2γthe Cauchy-Schwarz inequality gives ∥f>Jγ 0∥∞≲X j>Jγjβ/... | https://arxiv.org/abs/2504.03321v1 |
It was already shown above that ϵn(λn)≤ϵn(λn)≲n−β/(d+2β). The concentration inequality implies that there exists h∈Hλnsuch that ∥f0−h∥ ≤ϵn(λn) and ∥h∥2 Hλn≤nϵ2 n(λn). Combining this with Lemma 3 in Nieman et al. (2022) it follows that Ef0DKL(˜Πλn(·|x,y)∥Πλn(·|x,y))≲ (1 +Ex∥Kff−Qff∥op)n¯ϵ2 n(λn) +Extr(Kff−Qff). Hence it... | https://arxiv.org/abs/2504.03321v1 |
γ/(2d+4α)is uniformly bounded away from 0 for τ∈Λn,n∈N, and the preceding display implies that there exists a positive constant Csuch that e−τj1/d≤Cj−1−2α/d 28 for all n∈N,j≥Jγandτ∈Λn. Hence the tail bound (13) follows from its counterpart for polynomially decaying eigenvalues in the proof of Corollary 3. Lemma 10 also... | https://arxiv.org/abs/2504.03321v1 |
jbetween 1 and D |ν−n| ≤ ν−nX i=1φj(xi)2 + nX i=1φj(xi)2−n ≤X ℓ̸=j nX i=1φℓ(xi)φj(xi) + nX i=1φj(xi)2−n =DX ℓ=1 nX i=1φℓ(xi)φj(xi)−nδℓj ≤CDp nlogn. For the optimal Dof the order ( n/logn)d/(d+2β), it follows from the assumption β > d/ 2 that the upper bound is o(n) and 1Hlog|ˆΣ| |Σ|≤DlogD+σ−2n D+σ−2n−CD√nlogn=o(D) =o(n... | https://arxiv.org/abs/2504.03321v1 |
inf h∈Hτ: ∥h−f0∥≤ϵ∥h∥2 Hτ≤τ−d L2eτ∨ϵ2exp(2 τ(L/ϵ)1/β) . Proof Forϵsufficiently small, let Jbe a positive integer such that ϵJβ/d≥L > ϵ (J−1)β/d, and define the function h=P j≤J⟨f0, φj⟩φj∈Hτ. Note that for this Jandh, ∥h−f0∥2=X j>J⟨f0, φj⟩2≤J−2β/dL2≤ϵ2. By convexity of the function x7→eτxx−2βfor positive x, it follows... | https://arxiv.org/abs/2504.03321v1 |
2020. Julyan Arbel, Ghislaine Gayraud, and Judith Rousseau. Bayesian Optimal Adaptive Esti- mation Using a Sieve Prior. Scandinavian Journal of Statistics , 40(3):549–570, 2013. Christer Borell. The Brunn-Minkowski Inequality in Gauss Space. Inventiones mathemati- cae, 30:207–216, 1975. David R. Burt, Carl Edward Rasmu... | https://arxiv.org/abs/2504.03321v1 |
B.T. Szab´ o, A.W. van der Vaart, and J.H. van Zanten. Empirical Bayes scaling of Gaussian priors in the white noise model. Electronic Journal of Statistics , 7:991–1018, 2013. Michalis Titsias. Variational model selection for sparse Gaussian process regression. Report, University of Manchester, UK , 2009a. Michalis K.... | https://arxiv.org/abs/2504.03321v1 |
Existence and non-existence of consistent estimators in supercritical controlled branching processes Peter Braunsteins∗, Sophie Hautphenne†, and James Kerlidis‡ Abstract We consider the problem of estimating the parameters of a supercritical controlled branching process consistently from a single observed trajectory of... | https://arxiv.org/abs/2504.03389v1 |
N1, ∀ε >0,lim n→∞P(|ˆθn−θ|> ε|Z0=z0) = 0 ( weak consistency ) (3) P lim n→∞ˆθn=θ Z0=z0 = 1 ( strong consistency ). (4) If{ˆθn}is consistent, then, as more data become available, the sequence converges to the true parameter valueθ. On the other hand, if {ˆθn}is not consistent, then we may question whether a consistent... | https://arxiv.org/abs/2504.03389v1 |
directly extends the BGWP case: when the distribution of the control function is known, only the first two moments of the 2 offspring distribution can be estimated. In S2andS3, however, the extension is no longer direct; indeed, the parameters of the control function must now be estimated, and since the control functio... | https://arxiv.org/abs/2504.03389v1 |
size counts. In the context of CBPs, this principle translates to the idea that both the parameters of ξ(demographic stochasticity) and those of ϕ(·) (environ- mental stochasticity) should not be estimated simultaneously from a single trajectory of population sizes. Our results provide theoretical support for this prin... | https://arxiv.org/abs/2504.03389v1 |
display the mean squared error (MSE) of the MLEs—again computed using parametric bootstrap— for different trajectory lengths. We observe that, for Model (i), the MSE for each estimate appears to be converging steadily to zero, whereas for Model (ii) this does not seem to be the case. For a supercritical BGWP, it has be... | https://arxiv.org/abs/2504.03389v1 |
mean and variance for their offspring distributions. Then, by Theorem 2.1, it is not possible to consistently estimate θ. This provides a theoretical justification for why the MSE of the estimates in Model (ii) appears notto converge to zero in Figure 2. To understand the intuition behind Theorem 2.1, we note that the ... | https://arxiv.org/abs/2504.03389v1 |
{Zn, z0}supercritical if lim inf z→∞τ(z)>1. (5) 6 Recall that {Zn, z0}is said to grow unboundedly if Zn→ ∞ asn→ ∞ . Unlike BGWPs, supercritical CBPs do not necessarily have a positive probability of unbounded growth (see [8, Example 3.1]). Theorem 3.2 of [8] provides a sufficient condition for P(Zn→ ∞ )>0, namely that ... | https://arxiv.org/abs/2504.03389v1 |
the above relationship between the total variation distance and the non-existence of consistent estimators, we obtain the next result, which we will further refine in Sections 3.2–3.4 (see Theorems 3.4, 3.8, and 3.12). Proposition 3.2. If there exist two CBPs {Zn, z0},{Xn, z0} ∈Πsatisfying (7)but with θZ̸=θX, then no w... | https://arxiv.org/abs/2504.03389v1 |
3.3 CBPs with an unknown control function In most cases, we would notexpect that the distribution of the control function is known beforehand; hence, it needs to be estimated alongside the offspring distribution. Recall that a control function {ϕ(z)}z∈ N0is specified by a countably infinite set of random variables, one... | https://arxiv.org/abs/2504.03389v1 |
case, E(Z1|Z0=z) =m(z+azq) and Var( Z1|Z0=z) = (m2+σ2)(z+azq). (i) When q <1/2, (9) holds for any r∈(2q,1) if, for example, we let {Zn, z0}be the process with a= 0 and{Xn, z0}be the process with a= 1. By Theorem 3.8, this implies that θ=acannot be consistently estimated when q <1/2. (ii) When q >1/2, (9) does not hold ... | https://arxiv.org/abs/2504.03389v1 |
Π(u)in Section 3.3, plus the additional assumptions that the processes in Π(p)satisfy lim inf z→∞ν2(z)/z > 0, and that they have linearly divisible control functions, ϕ(z)d=Pl(z) i=1χ(z) i, such that there exists a constant η >0 and a sequence {xz}z∈ N1with P(χ(z)=xz)∧ P(χ(z)=xz+ 1)≥η (13) for all z∈ N1. Equation (13) ... | https://arxiv.org/abs/2504.03389v1 |
αandβ, as supported by Theorem 3.10 and commentary below); (iii) Collect additional data beyond a single trajectory of population size counts, as in setting S3(Theorem 3.13). We propose two future research directions. First, consider the variance decomposition: Var(Z1|Z0=z0) = E(ϕ(z0))·σ2+ Var( ϕ(z0))·m2, where the fir... | https://arxiv.org/abs/2504.03389v1 |
result follows by letting z→ ∞ . ■ Given the above lemmas, we now proceed to prove Lemma 3.1, from which Proposition 3.2 follows. Proof of Lemma 3.1. Since {Zn, z0}is assumed to be supercritical, lim inf z→∞τZ(z)>1. Hence for any tsuch that 1 < t < lim inf z→∞τZ(z), there exists M1∈ N1such that for all z≥M1,ε(z)·m > t ... | https://arxiv.org/abs/2504.03389v1 |
, while P(ˆθ({Xn, z0}) =θX, Xn→ ∞ ) = P(Xn→ ∞ ). Therefore, given that ˆθis strongly consistent, we can rewrite (17) as ||L{Zn,z0}− L{Xn,z0}||TV≥ P(Xn→ ∞ ). However, we note the following two facts: (i) Since ||LZ1|Z0=z0− LX1|X0=z0||TV=O(z−q 0), by Lemma 3.1, lim z0→∞||L{Zn,z0}− L{Xn,z0}||TV= 0, (ii) Given (6), [8, The... | https://arxiv.org/abs/2504.03389v1 |
the estimators. (i) Strong consistency of ˆ mn:LetUn:= ˜mn−m, so that E(Un|Fn−1) = 0 and E|Un| ≤2m, while E(U2 n|Fn−1) =ε−2(Zn−1)·Var(Zn|Fn−1) =σ2 ε(Zn−1)+m2ν2(Zn−1) ε2(Zn−1). Since ε(z) = Θ( z) and ν2(z) =O(z), there exists a positive constant C1such that E(U2 n|Fn−1)≤C1, so that∞X n=11 n2EU2 n<∞X n=1C1 n2<∞. Then by ... | https://arxiv.org/abs/2504.03389v1 |
Var E2(m−˜mn) nnX k=1 Zk−m·ε(Zk−1) Fn−1 n→∞−→0. Consequently, since Var2(m−˜mn) nnX k=1 Zk−m·ε(Zk−1) = E Var2(m−˜mn) nnX k=1 Zk−m·ε(Zk−1) Fn−1 + Var E2(m−˜mn) nnX k=1 Zk−m·ε(Zk−1) Fn−1 , 19 (21) will follow if we show that E Var2(m−˜mn) nnX k=1 Zk−m·ε(Zk−1) Fn−1 n→∞−→0. (22) Indeed, Var2(m−... | https://arxiv.org/abs/2504.03389v1 |
for Section 3.3 The proof of Lemma 3.7 relies on the control functions of our CBPs converging to a discretised normal distribution as the population size gets large. We use the following convention to describe this distribution: Definition 5.4. We say that a random variable Whas a discretised normal distribution with p... | https://arxiv.org/abs/2504.03389v1 |
we are ready to prove Lemma 3.7. Proof of Lemma 3.7. Suppose that {Zn, z0},{Xn, z0} ∈Π(u), such that their respective control functions ϕZ(·) and ϕX(·) are both linearly-divisible and satisfy (8), their offspring distributions ξZandξXboth have finite third moments and lattice size one, and that there exists r <1 such t... | https://arxiv.org/abs/2504.03389v1 |
2p lZ(z)·˜σ2 Z(z) =3|Var(Z1|Z0=z)−Var(X1|X0=z)| 2(lZ(z)·˜σ2 Z(z))+| E(Z1|Z0=z)− E(X1|X0=z)| 2p lZ(z)·˜σ2 Z(z) ≤3λ2 ZmZ 2tZσ2 Z·|Var(Z1|Z0=z)−Var(X1|X0=z)| z+λZ√mZ 2σZ√tZ·| E(Z1|Z0=z)− E(X1|X0=z)|√z forz≥M3, and since there exists r < 1 such that | E(Z1|Z0=z)− E(X1|X0=z)|=O(zr/2) and |Var(Z1|Z0=z)−Var(X1|X0=z)|=O(zr), w... | https://arxiv.org/abs/2504.03389v1 |
we will have from Lemma 5.1 that, for any N∈ N0, ||L(ϕZ(Z0), Z1)|Z0=z0− L (ϕX(X0), X1)|X0=z0||TV≤ ||L ϕZ(z0)− LϕX(z0)||TV+ P(ϕZ(z0)≤N) +c√ N+ 1. Then, taking N:=⌊α·ε(u)⌋forα∈(0,1), we can use Chebyshev’s inequality to further bound ||L(ϕZ(Z0), Z1)|Z0=z0− L (ϕX(X0), X1)|X0=z0||TV≤ ||L ϕZ(z0)− LϕX(z0)||TV+ν2 Z(z0) εZ(z... | https://arxiv.org/abs/2504.03389v1 |
Branching processes: variation, growth, and extinction of populations . Cambridge University Press, 2005. [16] T.E. Harris. Branching processes. The Annals of Mathematical Statistics , pages 474–494, 1948. [17] C.C. Heyde. Extension of a result of seneta for the super-critical galton-watson process. The Annals of Mathe... | https://arxiv.org/abs/2504.03389v1 |
Eigen-inference by Marchenko-Pastur inversion Ben Deitmar Department of Mathematical Stochastics, ALU Freiburg Ernst-Zermelo-Str. 1, 79104 Freiburg, Germany E-mail: ben.deitmar@stochastik.uni-freiburg.de A new formula for Marchenko-Pastur inversion is derived and used for in- ference of population linear spectral stati... | https://arxiv.org/abs/2504.03390v2 |
This paper describes a new method of Marchenko-Pastur inversion , i.e. reconstruc- tion ofH(orsH) fromνandc. Specifically, we in Lemma 2.2 show that sH(z) is for allz∈C+with Im/parenleftbig(1−czsH(z)−c)z/parenrightbig>0 and/vextendsingle/vextendsingle/vextendsingleczIm(zsH(z)) Im((1−czsH(z)−c)z)/vextendsingle/vextendsi... | https://arxiv.org/abs/2504.03390v2 |
Zhou, 2008) and in the isotropic case Bn= Idd(Fleermann and Heiny, 2023) even allows correlations between rows and columns of Xnprovided they go to zero sufficiently quickly with n→∞ . A series of papers (Yaskov, 2016), (D¨ ornemann and Heiny, 2022) and (Dong and Yao, 2025) deals with necessary and sufficient condi- ti... | https://arxiv.org/abs/2504.03390v2 |
j=1δλj(1 nX∗nΣnXn)= (1−cn)δ0+cnˆνnandνn= (1−cn)δ0+cnνn.(1.11) The corresponding Stieltjes transforms clearly satisfy sˆνn(z) =1−cn −z+cnsˆνn(z) and sνn(z) =1−cn −z+cnsνn(z). (1.12) 1.3. Eigen-inference In practice, one is often interested in estimating the underlying population covariance Σn, but only has access to the... | https://arxiv.org/abs/2504.03390v2 |
optimal shrinkers are examined for almost arbitrary bulk population distribution. The quality of their results also depends on the quality of available eigen-inference, since the population eigenvalues contribute to their formulas for optimal shrinkers. As such, optimal shrinkage methods are less contenders in the fiel... | https://arxiv.org/abs/2504.03390v2 |
Coming to our main contribution to the field of Marchenko-Pastur inversion, we will first need to define some sets. 9 Definition 2.1 (Domains on C+). Dependent on Handcfor anyε,θ> 0define the sets DH,c(ε,θ) :=/braceleftig z∈C+/vextendsingle/vextendsingle/vextendsingleIm/parenleftbig(1−czsH(z)−c)z/parenrightbig≥ε,/vext... | https://arxiv.org/abs/2504.03390v2 |
(2.14) in the set Qz,c(0+,1−). Moreover, this solution will be close enough to sH(z)such that/vextendsingle/vextendsingleˆs(z)−sH(z)/vextendsingle/vextendsingle≤δz |z|. Proof. 12 •Uniqueness: In complete analogy to the proof of uniqueness in Lemma 2.2 it follows that there can be at most one solution to (2.14) in the s... | https://arxiv.org/abs/2504.03390v2 |
>0define D(τ,n) :=/braceleftbig˜z∈C+/vextendsingle/vextendsingle0<Im(˜z)≤τ−1,|Re(˜z)|≤τ−1, τ≤|z|/bracerightbig S(τ,n) :=/braceleftbig˜z∈D(τ,n)/vextendsingle/vextendsingledist(˜z,[0,σ2(1 +√cn)2])≥τ/bracerightbig. For every ˜ε,D,τ > 0there exists a constant C=C(˜ε,D,τ )>0such that P/parenleftig ∃˜z∈S(τ,n) :/vextendsingl... | https://arxiv.org/abs/2504.03390v2 |
that surround [0,σ2] while mostly staying in Gn. Lemma 3.6 (Shape ofGnand good curves) . Suppose (1.16) and (1.18)-(1.20) hold. For anyθ∈(0,1)and small ˜ε,τ > 0all complex z∈C+that satisfy Im(z)≥2εn≡2n4˜ε−1(3.17) |z|≤n2˜ε(3.18) dist(z,[0,σ2])≥4σ2 θ(1 +cn) + 8τ (3.19) will be inGn(θ,τ,˜ε)as defined in (3.5). It easily f... | https://arxiv.org/abs/2504.03390v2 |
|v|nin the event (4.7)+5εn||f||U πτ |v|≥τ ≤1 πτ/parenleftbigC′′(θ,U) + 5/parenrightbig||f||Un4˜ε−1(4.8) where we needed the trivial bound ∀v∈γn((a,b)) :|sHn(v)|≤/integraldisplay R1 |λ−v|dHn(λ)(1.20) ≤1 dist(v,[0,σ2])(3.21) ≤1 τ.(4.9) We can again without loss of generality assume nto be large enough that 1 πτ/parenleft... | https://arxiv.org/abs/2504.03390v2 |
For this, we iterate the operator ˆTz,cn,n(w) :=/integraldisplay Rλ λ−(1−cnw)zdˆνn(λ) =1 dd/summationdisplay j=1λj(Sn) λj(Sn)−(1−cnw)z(5.2) until an approximate fixed point w0∈C+is found. If w0lies inQz,cn(0+,1−) as de- fined in (2.15), we proceed by setting ˆ sn(z) :=w0−1 z. This method is thus not guaranteed to arriv... | https://arxiv.org/abs/2504.03390v2 |
the curve ˆ γnis used to approximate the integral by Gauss-Legendre quadrature (see Figure 5 for the support points). We see virtually no difference between the two lines. As demonstrated in Figure 6, the exact choice of curve does not greatly influence the PLSS-estimators. Figure 4 however highlights the importance of... | https://arxiv.org/abs/2504.03390v2 |
our estimator with distances ˆd1 (orange) and ˆd2(dashed orange) as well as El Karoui’s estimator (blue). Performance correlates negatively with cnand positively with d×n. 28 5.5. Comparison to the eigen value estimation method by Ledoit-Wolf Ledoit-Wolf’s estimating algorithm, often referred to as the QuEst algorithm,... | https://arxiv.org/abs/2504.03390v2 |
same property of Hnand the assumption Hnn→∞= = = =⇒H∞with a test-function f∈Cb(R) that satis- fiesf[0,σ2]= 0 andf|[0,σ2]c>0. The second statement of (d) is follows immediately from (2.3) and an analogous argument. e) We show that (a) and supp( Hn)⊂[0,σ2] already implies Hn(f)n→∞−−−→H∞(f) for every holomorphic function ... | https://arxiv.org/abs/2504.03390v2 |
3.20 of (Knowles and Yin, 2017). Theorem 3.22 of (Knowles and Yin, 2017) would, if applicable, directly show (A.6).We thus check its conditions. Conditions (2.1), (2.4), (2.5), (2.7) and (2.8) of (Knowles and Yin, 2017) are easily seen to follow from the assumptions of this theorem. For condition (3.20) of (Knowles and... | https://arxiv.org/abs/2504.03390v2 |
=2 πτdist(γ,˜z)/parenleftbigg/integraldisplayn˜ε 2n 02dt+/integraldisplayτ n˜ε 2nn˜ε ntdt/parenrightbigg +σ2(1 +√cn)2+ 3τ πτdist(γ,˜z)n˜ε n =2 πτdist(γ,˜z)/parenleftbiggn˜ε n+n˜ε n/bracketleftbiglog(t)/bracketrightbigτ n˜ε 2n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ≤n˜εfor largen/parenrigh... | https://arxiv.org/abs/2504.03390v2 |
( ˆw(p) n)j−z/parenrightig dz −/contintegraldisplay γnf(z)/parenleftig sHn(z)−1 dd/summationdisplay j=11 ( ˆw(p) n)j−z/parenrightig dz/vextendsingle/vextendsingle/vextendsingle/vextendsingle. The assumptions (4.11) and (4.13) together bound the arc-length of γnbym n≤1 τ and we can refine the above bound to /vextends... | https://arxiv.org/abs/2504.03390v2 |
MR3449395 Ding, Xiucai, Yun Li, and Fan Yang. 2024. Eigenvector distributions and optimal shrinkage estimators for large covariance and precision matrices . Ding, Yi and Xinghua Zheng. 2024. High-dimensional covariance matrices under dynamic volatility models: asymptotics and shrinkage estimation , Ann. Statist. 52, no... | https://arxiv.org/abs/2504.03390v2 |
random matrices , Matematicheskii Sbornik 114(4) , 507–536. Mei, Tianxing, Chen Wang, and Jianfeng Yao. 2023. On singular values of data matrices with general independent columns , Ann. Statist. 51, no. 2, 624–645. MR4600995 Najim, Jamal and Jianfeng Yao. 2016. Gaussian fluctuations for linear spectral statistics of la... | https://arxiv.org/abs/2504.03390v2 |
arXiv:2504.03405v1 [math.ST] 4 Apr 2025On the rate of convergence of an over-parametrized deep neural network regression estimate learned by gradient descent∗ Michael Kohler Fachbereich Mathematik, Technische Universität Darmstad t, Schlossgartenstr. 7, 64289 Darmstadt, Germany, email: kohler@mathematik.tu-darmst adt.d... | https://arxiv.org/abs/2504.03405v1 |
Dn={(X1,Y1),...,(Xn,Yn)} (3) is given, and the task is to construct an estimate mn(·) =mn(·,Dn) :Rd→R such that its L2error /integraldisplay |mn(x)−m(x)|2PX(dx) is small. A systematic introduction to nonparametric regre ssion, its estimates and known results can be found, e.g., in Györfi et al. (2002). 1.3 Rate of conve... | https://arxiv.org/abs/2504.03405v1 |
of neurons, and the weights αk∈R(k= 0,...,K),βk,j∈R (k= 1,...,K,j = 0,...,d)are chosen by the principle of least squares. The rate of convergence of shallow neural networks regressi on estimates has been an- alyzed in Barron (1994) and McCaffrey and Gallant (1994). Barro n (1994) proved a dimensionless rate of n−1/2(up ... | https://arxiv.org/abs/2504.03405v1 |
points, so in principle it is possible to choose the w eights such that the data points are interpolated (at least, if the x values are all dis tinct). In practice it has been observed, that this procedure leads t o estimates which predict well on new independent test data. There have been various at temps to explain t... | https://arxiv.org/abs/2504.03405v1 |
remain bounded and consequently th e derivatives of the esti- mate stay bounded, which enables us to bound the covering num ber using metric entropy bounds. Secondly, we derive new approximation results for n eural networks with bounded weights, where the bounds fit the upper bounds on the covering number derived by usi... | https://arxiv.org/abs/2504.03405v1 |
of smooth functions which do either depend o nly on a few components or are rather smooth. This is due to the network structure of d eep networks, which implies that the composition of networks is itself a deep net work. Consequently, any approximation result of some kind of functions by deep netwo rks can be extended ... | https://arxiv.org/abs/2504.03405v1 |
it was only shown that t he results hold if this number of neurons is sufficiently large, and it is not clear whe ther it must grow, e.g., exponentially in the sample size or not. Another approach wh ere the estimate is studied in some asymptotically equivalent model is the mean field app roach, cf., Mei, Montanari, and N... | https://arxiv.org/abs/2504.03405v1 |
defined by f(l) k,i(x) =f(l) w,k,i(x) =σ r/summationdisplay j=1w(l−1) k,i,j·f(l−1) k,j(x)+w(l−1) k,i,0 (9) for some w(l−1) k,i,0,...,w(l−1) k,i,r∈R(l= 2,...,L)and f(1) k,i(x) =f(1) w,k,i(x) =σ d/summationdisplay j=1w(0) k,i,j·x(j)+w(0) k,i,0 (10) for some w(0) k,i,0,...,w(0) k,i,d∈R. This means that we conside... | https://arxiv.org/abs/2504.03405v1 |
the proof concerning optimizat ion, approximation and gener- alization. 3.1 Neural network optimization Our first lemma is our main tool to analyze gradient descent. I n it we relate the gradient descent of our deep neural network to the gradient descent of the linear Taylor polynomial of the deep network, and use metho... | https://arxiv.org/abs/2504.03405v1 |
2·λn·/ba∇⌈bl∇wFn(w(t))/ba∇⌈bl2 =1 2·λn·/parenleftBig /ba∇⌈blw(t)−w∗/ba∇⌈bl2−/ba∇⌈blw(t+1)−w∗/ba∇⌈bl2/parenrightBig +1 2·λn·/ba∇⌈bl∇wFn(w(t))/ba∇⌈bl2. This together with (17) implies T1,n≤1 tntn−1/summationdisplay t=0/parenleftbigg1 2·λn·/parenleftBig /ba∇⌈blw(t)−w∗/ba∇⌈bl2−/ba∇⌈blw(t+1)−w∗/ba∇⌈bl2/parenrightBig +1 2·λn... | https://arxiv.org/abs/2504.03405v1 |
(24) for alla,b∈RKsatisfying /ba∇⌈bla−a0/ba∇⌈bl ≤/radicalbigg 8·t L·max{F(a0),1}and/ba∇⌈blb−a0/ba∇⌈bl ≤/radicalbigg 8·t L·max{F(a0),1}.(25) Then we have /ba∇⌈blak−a0/ba∇⌈bl ≤/radicalbigg 2·k L·(F(a0)−F(ak))for allk∈ {1,...,t} and F(ak)≤F(ak−1)for allk∈ {1,...,t}. Proof. See Lemma A.1 in Braun et al. (2023) /square Our ... | https://arxiv.org/abs/2504.03405v1 |
we construct a piecewise Taylor polynomial. To do this, letA≥1,K∈Nand subdivide [−A,A]dintoKdmany cubes of sidelength δ=2A K. Set uk=−A+k·2A K(k= 0,...,K−1). Then uk= (uk(1),...,uk(d)) (k∈I:={0,1,...,K−1}d) denote the lower left corners of these cubes. 22 Fora,b∈Rdwe write a≤bifa(l)≤b(l)for alll∈ {1,...,d} and a<bifa≤b... | https://arxiv.org/abs/2504.03405v1 |
≤d·e−(logK)2 (which follows as above) bounded from above by c51/Kp. Hence it suffices to show that /summationdisplay k∈I,k/\e}atio\slash=0 :u(i) k≤u(i) r−δfor all i∈{1,...,d}\{j1,...,js}, u(jt) k=u(jt) r+lt·δfor all t∈{1,...,s}Pk(x)·1[uk,∞)(x) (29) and /summationdisplay k∈I,k/\e}atio\slash=0 :u(i) k≤u(i) r−δfor all i∈{1,... | https://arxiv.org/abs/2504.03405v1 |
network from the previous lemma in such a way that it can multiply a finite number of real values simultaneo usly. Lemma 11 Letσ(x) = 1/(1+exp( −x)), let0< A≤1, letN∈NwithN >2and let d∈N. Assume c56·4d·N·AN−1≤1, (31) wherec56is the constant from Lemma 10. Then there exists a neural netw ork fmult,d with at most ⌈log2d⌉m... | https://arxiv.org/abs/2504.03405v1 |
if suffices to show that for all i1,...,iq∈ {0,...,d}, allu∈[−A,A]dand z0= 1,zj=x(j)−u(j)(j= 1,...,d)there exists f1,f2∈ Fsuch that sup x∈[−A,A]d|q/productdisplay s=1zis−f1(x)| ≤c65 Kp+d(34) and sup x∈[−A,A]d|q/productdisplay s=1zis·d/productdisplay j=1σ(M·(x(j)−u(j) k))−f2(x)| ≤c66 Kp+d. (35) Letfid=fnet,xbe the network... | https://arxiv.org/abs/2504.03405v1 |
k−1which vanish outside of [−α,α]dandΠis the family of all partitions of Rdwhich consist of a partition of[−α,α]dinto K=/parenleftBigg/ceilingleftBigg 2·α /parenleftbig c87·ǫ c/parenrightbig1/k/ceilingrightBigg/parenrightBiggd many cubes of sidelenght at most/parenleftBig c87·ǫ c/parenrightBig1/k wherec87=c87(d,k)>0is ... | https://arxiv.org/abs/2504.03405v1 |
same construction for all of Nn·˜Knweights and we can conclude: The probability that there exists k∈ {1,...,N n·˜Kn}such that none of the Knweight vectors of the fully connected neural network differs by at most ǫnfrom((w∗)(l) i,j,k)i,j,lis for large nbounded from above by Nn·˜Kn·exp(−n0.5)≤c98·n5·exp(−n0.5)≤c99 n2. Thi... | https://arxiv.org/abs/2504.03405v1 |
by gradient descent. Bernoulli 30, pp. 475-502. [6] Chizat, L. and Bach, F. (2018). On the global convergence o f gradient descent for over-parameterized models using optimal transport. Prepr int,arXiv: 1805.09545. [7] Choromanska, Anna, Henaff, Mikael, Mathieu, Michael, Ar ous, Gerard Ben, and LeCun, Yann (2015). The l... | https://arxiv.org/abs/2504.03405v1 |
rate of convergence of an over- parametrized deep neural network estimate learned by gradi ent descent. Preprint, arXiv: 2210.01443. [26] Kohler, M., and Krzyżak, A. (2023). On the rate of conver gence of an over- parametrized deep neural network regression estimate with ReLU activation function learned by gradient des... | https://arxiv.org/abs/2504.03405v1 |
T., et al. (2017). Mastering the game of go without hum an knowledge. Nature 550, pp. 354-359. [45] Stone, C. J. (1982). Optimal global rates of convergenc e for nonparametric regres- sion.Annals of Statistics ,10, pp. 1040-1053. [46] Stone, C. J. (1985). Additive regression and other nonp arametric models, Annals of S... | https://arxiv.org/abs/2504.03405v1 |
arXiv:2504.03427v1 [math.ST] 4 Apr 2025On empirical Hodge Laplacians under the manifold hypothesis Jan-Paul Lerch∗Martin Wahl† Abstract Given i.i.d. observations uniformly distributed on a closed subman- ifold of the Euclidean space, we study higher-order generalizations of graph Laplacians, so-called Hodge Laplacians ... | https://arxiv.org/abs/2504.03427v1 |
metric measure spaces [ 2,24] and weighted graphs [33]. A relationship between random walks on simplicial comple xes and higher-order (combinatorial) Laplacians has been esta blished in [ 37,38]. In a complementary but related line of research, topologica l data anal- ysis aims to provide statistical and algorithmic me... | https://arxiv.org/abs/2504.03427v1 |
introduced in the context of simplicial complexes. In this section, we summarize some basic elements and formulas of this theory in a form suitable for our study. S imilar treat- ments can be found in [ 2,33]. 3 LetV={X1,...,X n}be a finite set of data points (in a Euclidean space). We call a function ω:Vℓ+1→Ranℓ-form i... | https://arxiv.org/abs/2504.03427v1 |
weighted graph ( V,K). Note that dim(ker( D−K)) = 1 if all weights are non-zero, while it might be strictly larger for non-negativ e weights. More- over, the first nonzero eigenvalue is related to the Cheeger c onstant and satisfies the so-called Cheeger inequality. For more detail s see [12,32]. Similar statements for ℓ... | https://arxiv.org/abs/2504.03427v1 |
0 in the second sum cancel with the negative part of the first s um. Substituting σbyσ◦(0,1,...,a), we arrive at (δ0f1∧η+f1δℓ−1η)(Xi0,...,X iℓ) (5) = (ℓ+1)1 (ℓ+1)!/summationdisplay σ∈Sℓ+1sgn(σ)f1(Xiσ(1))η(Xiσ(1),...,X iσ(ℓ)) =1 ℓ!/summationdisplay σ∈Sℓ+1 σ(0)=0sgn(σ)f1(Xiσ(1))η(Xiσ(1),...,X iσ(ℓ)) +1 ℓ!ℓ/summationdispla... | https://arxiv.org/abs/2504.03427v1 |
Similarly, the inner product induces an inner product on Ωℓ(M), which we denote by /a\}b∇acketle{t·,·/a\}b∇acket∇i}ht. In what follows it is important that if f1,...,fℓ∈C∞(M), then /a\}b∇acketle{tdf1∧···∧dfℓ,df1∧···∧dfℓ/a\}b∇acket∇i}ht=/integraldisplay Mdet /a\}b∇acketle{tdf1,df1/a\}b∇acket∇i}htx.../a\}b∇acketle{tdf... | https://arxiv.org/abs/2504.03427v1 |
weights (ki0···iℓ)introduced in (11). Furthermore, let f1,...,fℓ∈C∞(M)and letω=f1(df2∧···∧dfℓ)andω=f1(δf2∧···∧δfℓ) as introduced in (10). Finally, let ∆up ℓ−1be the up Laplace-Beltrami opera- tor onMand suppose that Assumptions 2and3are satisfied. Then, with probability at least 1−n−A, we have /vextendsingle/vextendsing... | https://arxiv.org/abs/2504.03427v1 |
The bias term: continuous Hodge theory 6.1 Main approximation bound In this section we relate /a\}b∇acketle{tω,∆up ℓ−1ω/a\}b∇acket∇i}htto E/a\}b∇acketle{tω,Lup ℓ−1ω/a\}b∇acket∇i}htn =E1/parenleftbign ℓ+1/parenrightbig/summationdisplay 1≤i0<···<iℓ≤n1 ℓ!(2t)ℓ/parenleftig1 ℓ+1ℓ/summationdisplay a=0ℓ/productdisplay b=0 b/... | https://arxiv.org/abs/2504.03427v1 |
Inserting ( 14) into (13), the claim follows from integrating with respect to xand the triangle inequality. 7 The variance term: concentration of U-statistics 7.1 Main concentration bound In this section we study the stochastic error /a\}b∇acketle{tω,Lup ℓ−1ω/a\}b∇acket∇i}htn−E/a\}b∇acketle{tω,Lup ℓ−1ω/a\}b∇acket∇i}htn... | https://arxiv.org/abs/2504.03427v1 |
Fubini theorem, we get /integraldisplay M|J∁|/parenleftig1 tℓℓ/productdisplay b=1kt(x0,xb)/parenrightig ·D2(f1,...,fℓ,x0,...,x ℓ)dxJ∁(21) ≤ℓ!/summationdisplay σ∈Sℓ/integraldisplay M|J∁|ℓ/productdisplay b=11 tkt(x0,xb)(fσ(b)(xb)−fσ(b)(x0))2dxJ∁ 20 =ℓ!/summationdisplay σ∈Sℓ/productdisplay b∈J b/\e}atio\slash=01 tkt(x0,... | https://arxiv.org/abs/2504.03427v1 |
dx0/vextendsingle/vextendsingle/vextendsinglep ≤Cℓ/summationdisplay j=0/parenleftigpj+1 2 tdj 4nj+1+pj+2 2 tdj 2nj+2 2/parenrightig . Inserting this into Markov’s inequality P/parenleftig/vextendsingle/vextendsingle/vextendsingleUn(ℓ,t)−/integraldisplay Mℓ+1/parenleftig det ℓ×ℓ(fa(xb)−fa(x0))/parenrightig2/parenle... | https://arxiv.org/abs/2504.03427v1 |
Harmonische Funktionen und Randwertauf gaben in einem Komplex. Comment. Math. Helv. , 17:240–255, 1945. [18] Herbert Edelsbrunner, David Letscher, and AfraZomoro dian. Topolog- ical persistenceand simplification. Discrete Comput. Geom. , 28(4):511– 533, 2002. Discrete and computational geometry and graph dr awing (Colum... | https://arxiv.org/abs/2504.03427v1 |
1997. An int ro- duction to analysis on manifolds. [40] Nat Smale and Steve Smale. Abstract and classical Hodge –de Rham theory.Anal. Appl. (Singap.) , 10(1):91–111, 2012. [41] Yanglei Song, Xiaohui Chen, and Kengo Kato. Approximat ing high- dimensional infinite-order U-statistics: statistical and computational guarante... | https://arxiv.org/abs/2504.03427v1 |
Identifiability of VAR(1) model in a stationary setting Bixuan Liu∗ Sorbonne Universit´ e and Universit´ e Paris Cit´ e, CNRS, Laboratoire de Probabilit´ es, Statistique et Mod´ elisation, F-75005 Paris, France May 16, 2025 Abstract We consider a classical First-order Vector AutoRegressive (VAR(1)) model, where we inte... | https://arxiv.org/abs/2504.03466v2 |
the underlying graph structure), that is, we seek a way to locate its nonzero elements. Importantly, conclusions from Gaussian Graphical Models (GGMs) (see [10] for an introduction) are inapplicable to our settings because the graphical model of the stationary distribution, i.e. the graph corresponding to the support o... | https://arxiv.org/abs/2504.03466v2 |
the methods from algebraic statistics (see [14] for an introduction). Specifically, each graph is associated with a set of possible stationary distributions, and we represent them by the set of corresponding covariance matrices Σ, denoted as MG. We study the naturally associated algebraic variety defined 2 as the Zaris... | https://arxiv.org/abs/2504.03466v2 |
space includes the case where ( i, j)∈EGandλij= 0 so that if the graphs G1= (V,EG1) and G2= (V,EG2) are nested, i.e. EG1⊆EG2, then the sets of possible stationary distributions generated from the two models are also nested. Next, we formally define the model. Definition 1. For the stationary VAR(1) model corresponding ... | https://arxiv.org/abs/2504.03466v2 |
0 .67−0.01 0.00 0 .94 0 .02 0.00 0 .00 0 .38 and if we let ωin both cases equals to 1, then by (3), both models yield the following covariance matrix: Σ = 1.33 0 .85 0 .00 0.85 22 .85 0 .60 0.00 0 .60 1 .19 . Therefore, we focus instead on a less restrictive definition of identifiability, called generic identifi... | https://arxiv.org/abs/2504.03466v2 |
Jacobian matrix of the parametrization map. Definition 6. LetMGbe a stationary VAR(1) model, and ϕGin Definition 1 parametrizes MG. Then the Jacobian matrix of the model is JG=∂ϕj ∂θi ,1≤i≤EG+ 1,1≤j≤n(1 +n) 2, where EGis the number of edges in G,ϕj’s are the distinct entries of Σ, and θi’s are the entries of Λ and ω.... | https://arxiv.org/abs/2504.03466v2 |
focus specially on the identifiability of models with the same dimension, in which case the role of the two models are equivalent. Proposition 4. Let{Mk}K k=1be a finite set of stationary VAR(1) models corresponding to graphs {Gk}K i=1with the same set of nodes and the same dimension. These models are generically ident... | https://arxiv.org/abs/2504.03466v2 |
( i, j)∈EGorω, and each column corresponds to σabwhere a≤b∈[n]. Note that we only consider a≤bbecause σab=σbafor all a, b∈[n]. Consider a stationary VAR(1) model MGwith G= (V,EG) complete, i.e. EG={(i, j)|i, j∈[n]}. Denote the Jacobian matrix of this model as J, which is of size n2+ 1 ×(n(n+ 1)) /2. Define an ”extend... | https://arxiv.org/abs/2504.03466v2 |
the extended Jacobian matrix JGsatisfies: JG=JGA+ψG(B), where A= Λ⊗Λ = λ2 11 0 λ11λ13 λ11λ21λ11λ22 0 0 0 λ11λ330λ11λ13 0 λ2 13 λ13λ21λ13λ22 0 0 0 λ13λ33 λ21λ11 0 λ21λ13 λ2 21 λ21λ22 0 0 0 λ21λ33λ22λ11 0λ22λ13 λ22λ21λ2 22 0 0 0 λ22λ330 0 0λ33λ11 0 λ33λ13 λ33λ21λ33λ22 0 0 0 λ2 33 , and ψG(B) =... | https://arxiv.org/abs/2504.03466v2 |
independent. The next section presents a new concept called ”maximal classes”, for any directed graph, which is highly related to the linear independence of the Jacobian matrix, and is at the core of this study. 4 Maximal classes In this section, we introduce the definition and properties of maximal classes and highlig... | https://arxiv.org/abs/2504.03466v2 |
the set and that is maximal with respect to inclusion, i.e. the set of all reachable nodes from a source node i.iis defined as the source node of the maximal class. The source node ior the SCC containing ithat is a source is defined as the source of the maximal class. By definition, a maximal class might have multiple ... | https://arxiv.org/abs/2504.03466v2 |
3: for all nodes w∈Vs.t. (i, w)∈EGdo ▷Do the same thing to all children of i 4: DFS( G, w, j ) 5:procedure MAXC (G) ▷Search for the maximal classes of graph G 6: Perform Kosaraju’s algorithm, return L, a list of SCCs 7: Construct the condensed graph G′= (L,EG′) 8: j←1 9: maxc←an empty list of lists 10: for all i∈ |L|s.... | https://arxiv.org/abs/2504.03466v2 |
individually. In other words, we avoid cancellations among elements of different powers of Λ in this set. In this paper, because we consider generic identifiability, and hence generic properties of the parameters, we assume that for all stationary VAR(1) models, the associated interaction matrices satisfy: Λ ∈MS∩MP. It... | https://arxiv.org/abs/2504.03466v2 |
same maximal class. Theorem 2. LetMGbe a stationary VAR(1) model. Then the set of maximal classes MCsatisfies MC=C\C′, where C={Ci|i∈[n]}, C′={Ci∈C|i∈[n],and∃k, l∈ Cis.t. σ kl= 0}. Proof. First, we prove that MC⊆C. Consider MC ∈ MC, there exists a source node i∈[n], s.t. MC=MC [i]. By Proposition 8, for all j∈ MC [i], ... | https://arxiv.org/abs/2504.03466v2 |
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