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distribution with the s ame average degree. Our main result is that the detection problem is efficiently solvable w heneverqis increasing and λ/\e}atio\slash= 0. While the conditions of the proposition may be difficult to dig est at first, consider the setting in whichd,λare fixed while n→ ∞. Our result says that in this reg... | https://arxiv.org/abs/2503.03047v1 |
ran- dom graphs: community detection and non-regular ramanujan graphs , Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, IEEE, 20 15, pp. 1347–1357. 2 [9] Stanley Chan and Edoardo Airoldi, A consistent histogram estimator for exchangeable graph models , International Conference on Machine Lea... | https://arxiv.org/abs/2503.03047v1 |
B Hopkins, Pravesh K Kothari, Aaron Potechin, Pr asad Raghavendra, Tselil Schramm, and David Steurer, The power of sum-of-squares for detecting hidden structure s, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (F OCS), IEEE, 2017, pp. 720– 731.4 [29] Samuel B Hopkins and David Steurer, Efficient baye... | https://arxiv.org/abs/2503.03047v1 |
[47] Elchanan Mossel, Allan Sly, and Youngtak Sohn, Weak recovery, hypothesis testing, and mutual information in stochastic block models and planted f actor graphs , arXiv preprint arXiv:2406.15957 (2024). 4 [48] Sebastien Roch and Allan Sly, Phase transition in the sample complexity of likelihood-ba sed phylogeny infe... | https://arxiv.org/abs/2503.03047v1 |
Generating Networks to Target Assortativity via Archimedean Copula Graphons Victory Idowu February 2025 Abstract We develop an approach to generate random graphs to a target level of assortativity by using copula structures in graphons. Unlike existing random graph generators, we do not use rewiring or binning approach... | https://arxiv.org/abs/2503.03061v1 |
real world networks exhibit proprieties like: dif- fering levels of resilience and robustness [22, 23, 24, 10]; triangles, clustering and connectivity patterns [25, 26, 27, 28]; and influence information diffusion [4, 29, 6, 5]. Popular network generation methods like the configuration model [30, 31, 32, 33], Chung-Lu ... | https://arxiv.org/abs/2503.03061v1 |
graphon and copula density graphon; and another algorithm for their tensor product. This paper is organized as follows. First, we provide a brief overview of graphs, graphon theory, homomorphism density, copulas and assortativity. Next, we show how degree assortativity coefficient can be rewritten in terms of homo- mor... | https://arxiv.org/abs/2503.03061v1 |
graphs hom( F1∪F2, G) = hom ( F1, G) hom ( F2, G). 1.2 Graphons and their Densities The graphon [45, 46] is a graph-function that is defined over the space of all bounded symmetric measurable functions, W. A graphon W∈ W is an inte- grable function W: [0,1]2→[0,1]. The probability of an edge between points ui, ujisW(ui... | https://arxiv.org/abs/2503.03061v1 |
For every bivariate copula, C(u1, u2)the upper and lower bounds are given by: C−(u1.u2)≤C(u1.u2)≤C+(u1.u2) (7) where C−:= max {u1+u2−1,0}andC+:= min {u1, u2}. We will drop the u1, u2when clear. The limits of the Fr´ echet-H¨ offding bounds are called the countermonotonic (maximum) copula C−and the comonotonic (minimum)... | https://arxiv.org/abs/2503.03061v1 |
distribution. As Newman’s degree assortativity coefficient is a network analogue of the Pearson correlation coefficient −1≤r≤1. Clearly, assortative networks have r >0, disassortative networks have r <0, and networks which are neither assortative or disassortative are r= 0. Perfectly disassortative networks are closer ... | https://arxiv.org/abs/2503.03061v1 |
rG=(n−3)tinj(P3, G) +3nt(C3,G) (n−1)(n−2)−(n−2)tinj(P2,G)2 tinj(P1,G) (n−3)3tinj(S3, G) +tinj(P2, G)−(n−2)tinj(P2,G)2 tinj(P1,G)(13) 9 Indeed, this is immediate as that for graph Fand simple G tinj(F, G)̸= t(F, G). Next, we establish that graphs sampled from graphons have assortativity coefficients that converge to tru... | https://arxiv.org/abs/2503.03061v1 |
+φ−1(u2))du1du2 11 Theorem 3.2 (Degree operator of the Copula Graphon) .Under the copula graphon framework for dense graphs, the edge density is degree operator becomes: λ(x) =−Z∞ xφ−1(s) φ′(φ−1(s−x))ds (22) where x=φ(u1). Moreover, λ(x)is bounded above by φ−1(x). Proof. The degree operator is given by, Z1 0φ[−1](φ(u1)... | https://arxiv.org/abs/2503.03061v1 |
Z1 0φ′(φ[−1](x) +φ[−1](1)) φ′(φ[−1](1))−φ′(φ[−1](x) + 1) φ′(1)k dx (29) Like with the Copula Graphon, the densities t(C3, W) and t(Pi, W) for i= 2,3 can be further simplified for specific copula graphons as shown in Section 5. 3.3 Tensor Copula Graphon Real world networks often have more heterogeneous mixing patterns... | https://arxiv.org/abs/2503.03061v1 |
to implement as the solver may face stability issues near the parameter boundary of the copulas. Note there are often many different possible combinations of Wandθthat can create the a target level of assortativity, the flexibility of these methods in reflecting the wide range of mixing patterns that create target asso... | https://arxiv.org/abs/2503.03061v1 |
using Π or copula graphons that will approximate Π for some θin their parameter space. Note that Π does not induce any dependency structure between the nodes. For W∼Π,C≈0.04 and thus |P3/2| ≈ |P2/1|. Several copula graphons have parameter ranges within their parameter space or can approach the independence copula in th... | https://arxiv.org/abs/2503.03061v1 |
generate a net- work to meet a target level of assortativity without using rewiring. First, we showed how the degree assortativity coefficient can be rewritten in terms of the homomorphism densities of a graphon and a simple random graph. Then, we showed how a graphon can be represented within a copula framework. Next,... | https://arxiv.org/abs/2503.03061v1 |
for Signal Processing (MLSP) . IEEE, 2020, pp. 1–6. [14] S. Suresh, V. Budde, J. Neville, P. Li, and J. Ma, “Breaking the limit of graph neural networks by improving the assortativity of graphs with local mixing patterns,” in Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining , 2021, pp.... | https://arxiv.org/abs/2503.03061v1 |
scale-free networks: From random to assortative,” Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , vol. 70, no. 6, p. 066102, 2004. [30] E. A. Bender and E. R. Canfield, “The asymptotic number of labeled graphs with given degree sequences,” Journal of Combinatorial Theory, Series A , vol. 24, no. 3, ... | https://arxiv.org/abs/2503.03061v1 |
Topics in Discrete Mathematics: Dedicated to Jarik Neˇ setˇ ril on the Occasion of his 60th Birthday . Springer, 2006, pp. 315–371. [47] P. J. Bickel, A. Chen, and E. Levina, “The method of moments and degree distributions for network models,” The Annals of Statistics , vol. 39, no. 5, pp. 2280–2301, 2011. [48] Z. Liu,... | https://arxiv.org/abs/2503.03061v1 |
An Analytical Theory of Power Law Spectral Bias in the Learning Dynamics of Diffusion Models Binxu Wang1 2 Abstract We developed an analytical framework for un- derstanding how the learned distribution evolves during diffusion model training. Leveraging the Gaussian equivalence principle, we derived ex- act solutions f... | https://arxiv.org/abs/2503.03206v1 |
training is truncated early: high-variance modes dominate, leaving low-variance modes undertrained and producing inaccuracies in fine-grained de- tails. All proofs and extended derivations are deferred to the appendices. 1arXiv:2503.03206v1 [cs.LG] 5 Mar 2025 Spectral Bias in Diffusion Learning Dynamics 2. Related Work... | https://arxiv.org/abs/2503.03206v1 |
follow the convention and notations in (Karras et al., 2022) throughout this paper. Consider a data distri- bution p(x0)to model, the score function ∇logp(x, σ)is the gradient of the log data density convolved by Gaussian N(0, σ2I). Diffusion model leverages the probability flow Ordinary Differential Equation (ODE), dx... | https://arxiv.org/abs/2503.03206v1 |
component. This eigenbasis will be critical to the analysis below. Gradient Differentiating this quadratic loss yields gradi- ents that depend linearly on the parameters, ∇bLσ= 2(b−(I−W)µ) (7) ∇WLσ=−2Σ+ 2W(σ2I+Σ) +∇bLσµ⊺(8) Global optimum Setting the gradients to zero yields the unique global optimum, which recovers th... | https://arxiv.org/abs/2503.03206v1 |
σ)/Φ(σT) (16) Bk(σ, σT) =Zσ σT−bk(λ) λΦ(σ) Φ(λ)dλ Which is a linear function over the initial condition xT, thus the sampling distribution is Gaussian with mean and covariance p(˜x)∼ N(˜µ,˜Σ), with variance ˜λkalong k-th eigenvector. (derivation in Sec.B.2) ˜µ=X kBk(σ0, σT)uk (17) ˜Σ=X k˜λkuku⊺ k=X kσ2 TΦ(σ0)2 Φ(σT)2uk... | https://arxiv.org/abs/2503.03206v1 |
found efficiently by numerical algebra, with eigenvectors ex- pressed by Bunch–Nielsen–Sorensen formula (Bunch et al., 1978; Gu & Eisenstat, 1994). Without closed form formula, we’d resort to qualitative analysis and low dimensional ex- amples to gain further insights. We can see the coupling of Wandbdynamics comes fro... | https://arxiv.org/abs/2503.03206v1 |
lawwith the target variance, τ∗ k∝λ−α k,α≈1(Fig. 2C). Here we quantified the convergence time as the first time that the generated variance ˜λreached the harmonic or geo- metric mean between the initial and target variance. When weight initialization Qkis larger, the initial sampling vari- ance will be higher, the mode... | https://arxiv.org/abs/2503.03206v1 |
when the magnitude of the mode ∥qk(τ)∥2reached the har- monic mean between the initial and asymptotic value. Remark 5.3 (Qualitative analysis of general initialization) . Even when the weights are initialized from small random values, the analytical solution above will be good approxima- tion to the actual gradient flo... | https://arxiv.org/abs/2503.03206v1 |
˜Στ. To evaluate convergence, we measure the deviation of the generated sample mean from the true mean, ˜µτ−µ, and project their deviation onto the eigenbasis u⊺ k(˜µτ−µ). Finally, we compute the variance of the generated samples along the eigenbasis of training data, ˜λτ k=u⊺ k˜Στuk. To stress test our theory and maxi... | https://arxiv.org/abs/2503.03206v1 |
Quantitatively, the emergence time τ∗of top eigenval- ues follow a power law, and modes with smaller variance follow a different power law, with smallest eigenmodes con- verged at last (Fig. 4D). This shows, even when the point cloud to be modeled is not Gaussian, the learning dynamics of MLP based diffusion model are ... | https://arxiv.org/abs/2503.03206v1 |
in flow matching models (Lipman et al., 2022; 2024). Implication for model training Further, the demon- strated spectral bias in diffusion learning may be taken into consideration to better precondition the data or the model to accelerate convergence. Our theory predicts to fully learn features of 1/100smaller variance... | https://arxiv.org/abs/2503.03206v1 |
curves in kernel regression and wide neural networks. In III, H. D. and Singh, A. (eds.), Pro- ceedings of the 37th International Conference on Ma- chine Learning , volume 119 of Proceedings of Machine Learning Research , pp. 1024–1034. PMLR, 13–18 Jul 2020. URL https://proceedings.mlr.press/ v119/bordelon20a.html . Bu... | https://arxiv.org/abs/2503.03206v1 |
P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition , pp. 10684–10695, 2022. Ronneberger, O., Fischer, P., and Brox, T. U-net: Con- volutional networks for biomedical image segmenta- tion. In Medical image... | https://arxiv.org/abs/2503.03206v1 |
were quantified via different criterions, via harmonic mean in A, and geometric mean in B. Within each panel, red markers and lines denote the modes where their variance increases; blue markers and lines denote modes that “decrease” their variance. The solid lines show least-squares fits on log-log scale, giving rise t... | https://arxiv.org/abs/2503.03206v1 |
images. 17 Spectral Bias in Diffusion Learning Dynamics Figure 14. Spectral Bias in CNN-Based Diffusion Learning: Variance Dynamics in Image Patches — AFHQv2 (64 pixel resolution). Left, Raw variance of generated patches along true eigenbases during training. Right, Scaling relationship between the target variance of e... | https://arxiv.org/abs/2503.03206v1 |
0ψ(λ)−1 λdλ) Then the integration functions can be expressed as Ak(σ;σT) = Φ( σ)/Φ(σT) Bk(σ;σT) =Zσ σT−bk(λ) λΦ(σ)/Φ(λ)dλ Since ck(σT)∼ N(0, σ2 T)by initial noise, variance of ck(σ)can be easily estimated V ar[ck(σ)] =σ2 Texp −2Zσ σTψ(λ)−1 λ =σ2 T(Φ(σ) Φ(σT))2 21 Spectral Bias in Diffusion Learning Dynamics SinceE[ck... | https://arxiv.org/abs/2503.03206v1 |
dynamics Consider a target quantity of interest, i.e. difference of the score approximator from the true score. First under the µ= 0assumption, we have Es=Ex∥s(x)−s∗(x)∥2=1 σ4 ∥b−b∗∥2+Tr[(W−W∗)T(W−W∗)(Σ + σ2I)] Using the deviations we have b−b∗=b0exp(−2ητ) W−W∗=X k v0 k−λk (σ2+λk)uk uT ke−2η(σ2+λk)τ with the initia... | https://arxiv.org/abs/2503.03206v1 |
now the dynamic variables are {vk, ..., b} ˙vk−(µTuk)˙¯b= 2η[λkuk−(σ2+λk)vk] ˙¯b=−2η(¯b+X k(µTuk)vk) ˙vk= 2η[λkuk−(σ2+λk)vk]−2η(µTuk)[¯b+X l(µTul)vl] = 2η[λkuk−(σ2+λk)vk−(µTuk)¯b−X l(µTuk)(µTul)vl] ˙¯b=−2η(¯b+X k(µTuk)vk) The whole dynamics is linear and solvable, but now the dynamics in each component vkbecomes entang... | https://arxiv.org/abs/2503.03206v1 |
or entrained by w b∗(w) = (1 −w)µ C.3. Sampling ODE and Generated Distribution For simplicity consider the zero mean case where W(τ;σ) =W∗+X k W(0;σ)uk−λk (σ2+λk)uk uT ke−2η(σ2+λk)τ =X kλk (σ2+λk)ukuT k(1−e−2η(σ2+λk)τ) +W(0;σ)X kukuT ke−2η(σ2+λk)τ b(τ;σ) =b(0) exp( −2ητ) To let it decompose mode by mode in the sampli... | https://arxiv.org/abs/2503.03206v1 |
each qT kqm= 0, q̸=mat network initialization. i.e. each qare orthogonal to each other. Then it’s easy to show thatd dt(qT kqm) = 0 at the start and throughout training. Thus we know orthogonally initialized modes will evolve independently. 33 Spectral Bias in Diffusion Learning Dynamics Note this assumption can also b... | https://arxiv.org/abs/2503.03206v1 |
INITIALIZATION :QUALITATIVE ANALYSIS OF OFF DIAGONAL DYNAMICS Next we can write down the dynamics of the non diagonal part of the weight, i.e. overlaps between modes d dt(qT kqm) =qT kd dtqm+qT md dtqk =−η[−4(λk+λm)(qT mqk) + 2X n 4σ2+ 2λn+λm+λk (qT kqn)(qT mqn)] = 4η[(λk+λm)(qT mqk)−X n 2σ2+λn+λm+λk 2 (qT kqn)(qT ... | https://arxiv.org/abs/2503.03206v1 |
Per our previous convention the relevant factor is ψk(σ;τ) =λk σ2+λk1 (1 Qkλk σ2+λk−1)e−8ητλk+ 1 and the integration Φ(σ) = exp −Zσψk(σ′;τ)−1 σ′dσ′ d dσuT kx=−1 σ ψk(σ;τ)−1 uT kx =−1 σλk σ2+λk1 (1 ∥qk(0)∥2λk σ2+λk−1)e−8ητλk+ 1−1 uT kx Note that the integrand is just a fraction function of σ2which can be integrate... | https://arxiv.org/abs/2503.03206v1 |
the eigenbasis of Σ,[u1, ...u d] W∗=X ktλk−(1−t) t2λk+ (1−t)2ukuT k Asymptotics, Consider the limit , t→0 W∗ t→0=−I W∗ t→1=I b∗ t→0=µ b∗ t→1=µ−W∗ t→1µ= 0 E.1. Solution to the flow matching sampling ODE with optimal solution Solving the sampling ODE of flow matching integrating from 0 to 1, with the linear vector field ... | https://arxiv.org/abs/2503.03206v1 |
t2λk+ (1−t)2ukuT k+X k Qk−tλk−(1−t) t2λk+ (1−t)2 ukuT kexp −2ητ t2λk+ (1−t)2 Ignoring the bias part, consider the weight integration along ck(t) =uT kx(t) d dtck(t) = [tλk−(1−t) t2λk+ (1−t)2+ Qk−tλk−(1−t) t2λk+ (1−t)2 exp −2ητ t2λk+ (1−t)2 ]ck(t) Integration of the coefficient yields I=Z1 0dthtλk−(1−t) t2λk... | https://arxiv.org/abs/2503.03206v1 |
solution is non positive definite. So different from the diffusion case, there are two scenarios: • When t >1 λk+1,A >0,Q∗>0. The dynamics is normal, converging to Q∗.limτ→∞∥qk∥2(τ) =Q∗ k. •When t <1 λk+1,A < 0, Q∗<0. In this case, the ideal solution is “non-achievable” by a two layer network. limτ→∞e−Aτ→ ∞ , solimτ→∞∥... | https://arxiv.org/abs/2503.03206v1 |
X (which may have 2-4 dim) sigma_vec = sigma.view([-1, ] + [1, ] *(X.ndim - 1)) c_skip = self .sigma_data **2 / (sigma_vec **2 + self .sigma_data **2) c_out = sigma_vec *self .sigma_data / (sigma_vec **2 + self .sigma_data ** 2).sqrt() c_in = 1 / ( self .sigma_data **2 + sigma_vec **2).sqrt() c_noise = sigma.log() / 4 ... | https://arxiv.org/abs/2503.03206v1 |
covariance Σ=R D RT. This rotation matrix Ris the eigenbasis of the true covariance matrix. To obtain training samples {xi} ⊂Rd, we draw xi fromN 0,Σ . In practice, we generate a total of 10,000samples and stack them as pnts . We compute the empirical covariance of the training set, Σemp= Cov( pnts ),and verify that ... | https://arxiv.org/abs/2503.03206v1 |
to GPU memory for training, and we estimate its empirical covariance Σemp= Cov( pnts )for reference. F.4.2. N ETWORK ARCHITECTURE AND TRAINING SETUP Since the MNIST dataset is higher dimensional than the synthetic data in the previous experiment, we use a deeper MLP network: model =UNetBlockStyleMLP backbone (ndim = 78... | https://arxiv.org/abs/2503.03206v1 |
Estimating weak Markov-switching AR (1)models Yacouba Boubacar Maïnassaraa,∗, Armel Brab, Landy Rabehasainab aUniv. Polytechnique Hauts-de-France, INSA Hauts-de-France, CERAMATHS - Laboratoire de Matériaux Céramiques et de Mathématiques, F-59313 Valenciennes, France bUniversité Marie et Louis Pasteur, Laboratoire de ma... | https://arxiv.org/abs/2503.03316v1 |
et al. (2008) who studied a general AR model with Markov regime-switching, allowing for AR with infinite order. Under some regular assumptions they demonstrated the consistency of the maximum likelihood estimators. We can also cite Douc et al. (2004) who studied the asymptotic properties of the maximum likelihood estim... | https://arxiv.org/abs/2503.03316v1 |
Markovian regime changes involving such noise. A notable exception Francq and Zakoïan (2001) who investigated the stationarity conditions of such models in a multivariate framework. The authors demonstrated that local stationarity of these processes is neither sufficient nor necessary to ensure global stationarity. And... | https://arxiv.org/abs/2503.03316v1 |
loss of generality, we will assume that σ2= 1. An example of weak white noise is the GARCH model (see Francq and Zakoïan (2010)). It is customary to say that (Xt)t∈Zis a strong ARHMC (1)representation and we will do this henceforth if in (2.1) (ηt)t∈Zis a strong white noise, namely an i.i.d. sequence of random variable... | https://arxiv.org/abs/2503.03316v1 |
i-th component of the stationary distribution π. 4 3. Estimation of the ARHMC (1)model parameters We state by the following theorem which provides an explicit expression of the autocovariance function of order kof the centered process (Xt)t∈Z. Theorem 1. Under Assumptions (A1),(A2),(A3)and(A6), the joint moments of the... | https://arxiv.org/abs/2503.03316v1 |
is defined as ˆck,0:= (n−k)−1n−kX t=1Xt+kXt,∀1≤k≤N < n. (3.6) Note that ˆck,0converges a.s. to ck,0(θ0)asn→ ∞for all 1≤k≤Nby the ergodic theorem and the fact that the process (Xt)t∈Zis stationary. Let ˆθnbe the estimator of θ0obtained by the Newton method through the estimation function FN,n. Formally, for large n, we ... | https://arxiv.org/abs/2503.03316v1 |
by r(θ)and is stationary from a certain rank onward. 4. Asymptotic properties 4.1. Consistency and asymptotic normality of the moments estimator The asymptotic properties of the estimator ˆθnobtained via the Newton algorithm 1 are stated in the following two theorems. Theorem 2. Let us assume that the limiting rank of ... | https://arxiv.org/abs/2503.03316v1 |
IK−A2 θ0P′ θ0−1) Vθ0π′ θ0, Qii 2:=1′( P′ θ0JiiP′ θ0A2 θ0P′ θ0 IK−A2 θ0P′ θ0−1) Vθ0π′ θ0, Qii 3:=1′( 2a0P′ θ0JiiP′ θ0 IK−A2 θ0P′ θ0−1+ 2a0P′ θ0A2 θ0P′ θ0 IK−A2 θ0P′ θ0−1JiiP′ θ0 IK−A2 θ0P′ θ0−1) Vθ0π′ θ0. 9 Followingasimilarlineofreasoningto(4.3), for i, j= 1, . . . , Kthereexistmatrixcoefficients (Rij s)1≤s≤3 ... | https://arxiv.org/abs/2503.03316v1 |
satisfied, so that λ3is well defined and different from 0. 4.3. Expression of the matrix Iwhen (ηt)t∈Zis assumed i.i.d. The aim of this subsection is to show that the covariance matrix I=IN(θ0)given in Theorem 3 has an explicit, albeit not simple, expression in the particular case when the noise sequence is i.i.d. This... | https://arxiv.org/abs/2503.03316v1 |
i= 1, . . . , s −1}. We then define Psas follows Ps:={κi, ιi, i= 1, . . . , sand(κi, ιi)i=1,···,s∈ Ps} representing all distinct individual elements extracted from each pair in Ps. This construction ensures thatPscontains only unique values from both components of the pairs. LetL(Ps)be the set of sorted elements of Pso... | https://arxiv.org/abs/2503.03316v1 |
i1−1)}), I2:=I({(0, i1−1),(−m1, i1−1)}), I3:=I({(k, i1+k−1),(−m2+k, i1+k−1)}). Notice that, under Assumption (A2), the respective spectral radii of the matrices Qaφ(ζ), for ζ∈S3 i=1Ii∪ Ji, are strictly less than 1. Note also that, in practice the infinite sums involved in I(m1, m2)are truncated. 2 5. Estimation of the ... | https://arxiv.org/abs/2503.03316v1 |
the noise (ηt)t∈Zis independent (particularly when ηtD=N(0,1)), in view of Section 4.3, we have ˆΩS:=ˆM−1 nˆJ′ nˆISˆJ′ nˆM−1 nwhere ˆISis a consistent estimator of the matrix ISdefined for a fixed integers r,r2>0as: 16 ˆIS(m1, m2) =r1X k=−r1 3r2X i1=01′ Y ζ∈I1ˆQlength( ζ) ˆaφ(ζ) ˆπˆf4 +r2X i1,i2=0 i1̸=i2−m11′ Y... | https://arxiv.org/abs/2503.03316v1 |
a range of scenarios. For each experiment, Rindependent realizations were generated and we estimated the coefficient vector θ0:= (α11, α22, β11, β21, γ11, γ22)′= (−0.4,0.3,0.3,0.2,1.0,0.5)′. The parameter space Θassociated is chosen to satisfy the assumptions of Theorem 3. The simulation procedure was as follows: start... | https://arxiv.org/abs/2503.03316v1 |
5). The failure of the standard estimator ofΩin the weak ARHMC setting may have important consequences in terms of hypothesis testing for instance. Figure 2: Simulation of length 400 of model (2.1) with θ0:= (α11, α22, β11, β21, γ11, γ22)′= (−0.4,0.3,0.3,0.2,1.0,0.5)′ and(ω0, a0, β0) = (0 .2,0.1,0.5). 19 α11 α22 β11 β2... | https://arxiv.org/abs/2503.03316v1 |
model (2.1) with noise ηt=ut(|ut−1|+ 1)−1, based on 1000 replications for sequence sizes of 300 and 2000, respectively. α11 α22 β11 β21 γ11 γ22 θ0 -0.4 0.3 0.3 0.2 1.0 0.5 Min -1.51651 -0.52734 0.02252 0.01694 -0.67843 -0.18852 Q1-0.62838 0.26199 0.22981 0.19064 0.95115 0.47577 Mean -0.39454 0.36688 0.33641 0.27990 1.0... | https://arxiv.org/abs/2503.03316v1 |
(f). 7. Application to real data In this section, we consider the hourly meteorological data from the Los Angeles region from Jan- uary 1st to January 31, 2022, denoted by (Ht)t=1,...,744. The data were obtained from the website Open-Meteo.com . In this dataset, we are specifically interested in the variable wind-speed... | https://arxiv.org/abs/2503.03316v1 |
plosive regime 1 with autoregressive parameter ˆα11. At the 5% significance level, the autoregressive parameters ˆα11andˆα33, as well as the volatility parameters ˆγ11,ˆγ22andˆγ33are all significant (see Table 12). The fact that in regime 2, the autoregressive parameter α22is not significant can be explained by the nat... | https://arxiv.org/abs/2503.03316v1 |
β32 β13 β23 β33 γ11 γ22 γ33 Estimate: 0.263 0.086 0.170 0.670 0.637 -1.275 -0.988 Standard error: – – – – 0.263 0.255 0.276 p-value: – – – – 0.015 0.004 0.000 Table 12: Re-estimation of the parameters of model (2.1) based on (Xt)t=1,...,743with K= 3. 8. Conclusion In this paper, we studied a first-order autoregressive ... | https://arxiv.org/abs/2503.03316v1 |
=∞X n1=k1 E n1−k1−1Y j=−2k1−1a(∆−j)f(∆k1−n1)n1−k1−1Y j=0a(∆−j)f(∆−n1+k1) =∞X n1=k1 E −1Y j=−2k1−1a(∆−j)n1−k1−1Y j=0a2(∆−j)f2(∆−n1+k1) . 29 Using Lemma 1, we obtain for odd kthat: c2k1+1,0=∞X n1=k1 1′ 2k1+1Y l=1Qan1−k1Y l=1Qa2! πf2! =1′∞X n1=k1 Q2k1+1 a Qn1−k1 a2 πf2. For any k∈Nwe draw the conclu... | https://arxiv.org/abs/2503.03316v1 |
obtain that√n ˆθn−θ0 has a limiting normal distribution with mean 0 and covariance matrix M−1J′IJM−1. 2 Lemma 2 (Davydov (1968)). LetXandYbe two random variables, and let σ(X)andσ(Y)be the σ-fields generated by XandYrespectively. Consider three strictly positive numbers p,q, and rsuch thatp−1+q−1+r−1= 1. Then, |Cov(X... | https://arxiv.org/abs/2503.03316v1 |
2∞X i3=0∞X i4=0Cρi1+i2+i3+i4|Cov(ηt−i1ηt+l−i2, ηt−k−i3ηt−k+r−i4)|, u3:=∞X i1=0∞X i2=0X i3>k 2∞X i4=0Cρi1+i2+i3+i4|Cov(ηt−i1ηt+l−i2, ηt−k−i3ηt−k+r−i4)|, u4:=∞X i1=0∞X i2=0∞X i3=0X i4>k 2Cρi1+i2+i3+i4|Cov(ηt−i1ηt+l−i2, ηt−k−i3ηt−k+r−i4)|, u5:=X i1≤k 2X i2≤k 2X i3≤k 2X i4≤k 2Cρi1+i2+i3+i4|Cov(ηt−i1ηt+l−i2, ηt−k−i3ηt−k+r−i... | https://arxiv.org/abs/2503.03316v1 |
∆t−k)is defined in Notation 1 (See Equation (9.9)). We then have √nFN,n(θ0) =1√nnX t=1(Yt(θ0)−Eθ0(Yt(θ0)) =1√nnX t=1(Yt,s−Eθ0(Yt,s)) +1√nnX t=1(Zt,s−Eθ0(Zt,s)). The process (Yt,s)t∈Zdepends on ηkand∆kforkin finite set. Furthermore, as the processes (∆t)t∈Z and(ηt)t∈Zare strongly mixing, according to the assumption (A4)... | https://arxiv.org/abs/2503.03316v1 |
and (9.19), we deduce that sup nVar 1√nnX t=1(Zt,s(p)−Eθ0(Zt,s(p)))! − − − → s→∞0. And the proof is complete using Anderson (1971, Corollary 7.7.1, p. 426). 2 9.4. Proof of the convergence of the variance matrix estimator We proceed to demonstrate the proof of Theorem 4 by employing a series of Lemmas. We consider the ... | https://arxiv.org/abs/2503.03316v1 |
have for all X∈CN, X′f(ω)X=X′ UωDωUω′ X= U′ ωX′ Dω U′ ωX ≤sup ω∥f(ω)∥∥X∥2. (9.24) Furthermore, 1 2πZ [−π,π] rX m=1γ(r) meiωm!′ rX n=1γ(r) neiωn! dω=1 2πZ [−π,π] rX m=1rX n=1γ(r) mγ(r) neiω(m−n)! dω =1 2πrX m=1rX n=1γ(r) mγ(r) nZ [−π,π]eiω(m−n)dω =rX m=1rX n=1γ(r) mγ(r) nδmn =∥γ(r)∥2. (9.25) By combining Equations... | https://arxiv.org/abs/2503.03316v1 |
v1(s, h, ℓ, e, m 1, m2) :=X i1>⌊h/2⌋X 0≤i2,...,i8≤∞ Cov(d1 i1d1+m1 i2d1+s i3d1+s+m2 i4, dℓ+h i5dℓ+h+m1 i6d1+e+h i7d1+e+h+m2 i8) , v2(s, h, ℓ, em 1, m2) :=X i2>⌊h/2⌋X 0≤i1,i3,...,i8≤∞ Cov(d1 i1d1+m1 i2d1+s i3d1+s+m2 i4, dℓ+h i5dℓ+h+m1 i6d1+e+h i7d1+e+h+m2 i8) , v3(s, h, ℓ, e, m 1, m2) :=X i3>⌊h/2⌋X 0≤i1,i2,i4,...,i8≤∞ C... | https://arxiv.org/abs/2503.03316v1 |
×αν 2+ν ∆{1 +e−ℓ−i7−m1}, (9.33) and for ⌊h/2⌋ ≥1 +e−ℓ−m1„ Cov dℓ+h i5dℓ+h+m1 i6, d1+e+h i7d1+e+h+m2 i8 = Cov d1+e+h i7d1+e+h+m2 i8, dℓ+h i5dℓ+h+m1 i6 ≤C0∥dℓ+h i5dℓ+h+m1 i6∥2+ν∥d1+e+h i7d1+e+h+m2 i8∥2+ν ×αν 2+ν ∆{ℓ−1−e−i5−m2}. (9.34) 44 Thus, from Equations (9.33), (9.34) and the fact that 0≤i5, i7≤ ⌊h/2⌋, it follow... | https://arxiv.org/abs/2503.03316v1 |
d1+s+m2 i4d1+e+h i7d1+e+h+m2 i8 + Cov d1 i1d1+m1 i2dℓ+h i5, d1+s i3 E dℓ+h+m1 i6 E d1+s+m2 i4d1+e+h i7d1+e+h+m2 i8 + Cov d1+s+m2 i4d1+e+h i7d1+e+h+m2 i8, dℓ+h+m1 i6 E d1 i1d1+m1 i2dℓ+h i5 E d1+s i3 −Cov d1 i1d1+m1 i2dℓ+h i5, dℓ+h+m1 i6 Cov d1+s i3, d1+s+m2 i4d1+e+h i7d1+e+h+m2 i8 −Cov d1+s i3, d1+s+m2... | https://arxiv.org/abs/2503.03316v1 |
We now focus on the term u9(s, h, ℓ, e, m 1, m2). Assume ⌊h/2⌋ ≥s+m0. By applying Lemma 2, we have ∞X ⌊h/2⌋=s+m0⌊h/2⌋X i1,···,i8=0ρP8 k=1ik Cov ϵ(2) i1,1,m1,i2ϵ(2) i3,1+s,m2,i4, ϵ(2) i5,ℓ+h,m1,i6ϵ(2) i7,1+e+h,m2,i8 ≤K0∞X ⌊h/2⌋=s+m0⌊h/2⌋X i1,···,i8=0ρP8 k=1ik∥ηt∥8 8+4ναν 2+νη{h−s−m2−i5+i4} ≤K0 ∞X i1,···,i8=0ρP8 k=1i... | https://arxiv.org/abs/2503.03316v1 |
m 1, m2)|<∞,and sup s,ℓ,e∈N∞X h=0|w3(s, h, ℓ, e, m 1, m2)|<∞. The same bounds clearly hold for h≤0. Thus, we have demonstrated that sup s,ℓ,e∈N∞X h=−∞|Cov(Y1(m1)Y1+s(m2),Yℓ+h(m1)Y1+e+h(m2))|<∞. This completes the proof. 2 Lemma 9. We assume that the condition E|ηt|8+4ν<∞holds and that Assumption (A4)is satisfied for so... | https://arxiv.org/abs/2503.03316v1 |
the covariance and appropriately bounding each term, as previously established and it is omitted. 2 Lemma 11. Under the assumptions of Theorem 4, the terms√r∥ˆΣYr−ΣYr∥,√r∥ˆΣY−ΣY∥and√r∥ˆΣY,Yr−ΣY,Yr∥tend towards 0 in probability as n→ ∞when r= o(n1/3). 53 Proof. To demonstrate this lemma, we will focus solely on the proo... | https://arxiv.org/abs/2503.03316v1 |
(E(Z1))2+n−2n−1X k=−n+1(n− |k|)Cov(Zt,Zt−k)! ≤NX m1,m2=1rX r1,r2=1r (E(Z1))2+n−1X k=−n+1n− |k| n2Cov(Zt,Zt−k)! ≤NX m1,m2=1rX r1,r2=1r (E(Z1))2+n−1∞X k=−∞Cov(Zt,Zt−k)! ≤N2r3 (E(Z1))2+n−1C13r= o(n1/3)− − − − − − − → n→∞0, where C13is a strictly positive constant and independent of r1, r2, m1, m2, r, and n. When r= o(n1... | https://arxiv.org/abs/2503.03316v1 |
i=1φiLi. Using Lemmas 14, 15 and Equation (9.21), we obtain ∥ˆφ(1)−φ(1)∥ ≤ rX i=1( ˆφi,r−φr,i) + rX i=1(φr,i−φi) + ∞X i=r+1φi ≤ ˆφr−φrrX i=1Ki + φ⋆ r−φrrX i=1Ki +∞X i=r+1∥φi∥ ≤ rX i=1Ki n ˆφr−φr + φ⋆ r−φr o +∞X i=r+1∥φi∥ ≤√ N√rn ˆφr−φr + φ⋆ r−φr o +∞X i=r+1∥φi∥= oP(1). We also have ˆΣˆur=ˆΣˆY−ˆφrˆΣ′ ˆY,ˆYr,... | https://arxiv.org/abs/2503.03316v1 |
Teor. Verojatnost. i Primenen. , 13:730–737. den Haan, W. J. and Levin, A. T. (1997). A practitioner’s guide to robust covariance matrix estimation. Douc, R., Moulines, É., and Rydén, T. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. , 32(5):... | https://arxiv.org/abs/2503.03316v1 |
Safety Verification of Nonlinear Stochastic Systems via Probabilistic Tube Zishun Liu1, Saber Jafarpour2and Yongxin Chen1 Abstract —We address the problem of safety verification for nonlinear stochastic systems, specifically the task of certifying that system trajectories remain within a safe set with high probability.... | https://arxiv.org/abs/2503.03328v1 |
and evaluate safety. Besides brutal force Monte Carlo, there are two other commonly used strategies for stochastic safety verification, one based on reachability analysis and one based on barrier functions. 1Zishun Liu and Yongxin Chen are with Georgia Institute of Technology, Atlanta, GA 30332 {zliu910}{yongchen}@gate... | https://arxiv.org/abs/2503.03328v1 |
probabilistic tube (PT), the tube in which stochastic trajectories stay with a high probability. Equipped with this PT, the safety verification problem for stochastic systems reduces to one for deterministic systems with an eroded safe set. The effectiveness of this set-erosion strategy is captured by the depth of eros... | https://arxiv.org/abs/2503.03328v1 |
to denote the unit sphere: {x∈Rn:∥x∥= 1}. Given two sets A, B⊆Rn, the Minkowski sum of them is defined by A⊕B= {x+y:x∈A, y∈B}, and the Minkowski difference is defined byA⊖B= (Ac⊕(−B))c, where Ac, Bcare the complements of A, B and−B={−y:y∈B}. For a random variable X,X∼ G means Xis independent and identically drawn from ... | https://arxiv.org/abs/2503.03328v1 |
Stochastic Systems Consider the deterministic dynamics ˙xt=f(xt, dt, t)(CT), (3) xt+1=f(xt, dt, t)(DT), (4) which can be viewed as the noise-free version of their associated stochastic systems (1)and (2)respectively. Given a time horizon (a) Set Erosion Strategy (b) Probabilistic Tube Fig. 1: An illustration of set-ero... | https://arxiv.org/abs/2503.03328v1 |
.Consider a stochastic system (1)(respectively (2)) and its associated deterministic system (3) (respectively (4)). Given a finite time horizon [0, T]and a prob- ability level δ∈(0,1), a curve rδ,t: [0, T]→R≥0, the set T={(t, y)|0≤t≤T,∥y∥ ≤rδ,t}is said to be a probabilistic tube (PT) of the stochastic system if for any... | https://arxiv.org/abs/2503.03328v1 |
quite the same as the trajectory level bound in PT. C. Union Bound Approach For DT systems, it is possible to combine state level bound at one time step using the union bound inequality to establish trajectory level probabilistic bound. Proposition 3. [14, Theorem 2] Consider the stochastic system (2) and its associate... | https://arxiv.org/abs/2503.03328v1 |
(1)under Assumption 1. Recall the definition of AMGF and its energy function. Definition IV .2 (AMGF & Energy Function) .Given a constant λ∈R, the Averaged Moment Generating Function (AMGF) EX(Φn,λ) : Rn→Ris defined as EX(Φn,λ(X)) :=EXEℓ∼Sn−1 eλ⟨ℓ,X⟩ , (15) where Φn,λ(X) =Eℓ∼Sn−1 eλ⟨ℓ,X⟩ (16) is called the Energy F... | https://arxiv.org/abs/2503.03328v1 |
Φn,λ(rℓ),∀t≤T ≥P eλ2σ2(T−t) 2 Φn,λ(St)≤Φn,λ(rℓ),∀t≤T ≥1−eλ2σ2T 2 Φn,λ(rℓ),∀ℓ∈ Sn−1(Lemma IV .1) ≥1−(1−ε2)−n 2exp λ2σ2T 2−ελr (Lemma IV .2-3)(26) Minimizing the last line of (26) overλ, we get λ∗=εr σ2T. Plugging λ=λ∗into (26) yields P(∥St∥ ≤r,∀t≤T)≥1−(1−ε2)−n 2e−ε2r2 2σ2T, (27) for every r≥0. For a given δ∈(0,1), ... | https://arxiv.org/abs/2503.03328v1 |
larger than coefficient σq e2ct−1 2c, especially when t≪T, meaning the trajectory level probabilistic bound in Theorem 2 is much worse than the state level probabilistic bound in Proposition 1. To illustrate the bound (17), consider the linear system dXt=cXtdt+σdWt, X t∈R, X0= 0, σ=√ 0.1,(36) whose associated determini... | https://arxiv.org/abs/2503.03328v1 |
the second “ ≥” directly follows Theorem 2 by setting the time period as ∆tand the initial time as k∆t. Next, we combine the probabilistic bound (38) with (42) to complete the proof. Define the following sequences of events: E(1) k:∥Xk∆t−xk∆t∥ ≤rk∆t E(2) k:∥Xt−y(k) t∥ ≤r∆,∀t∈(k∆t,(k+ 1)∆ t).(43) Then by (38),(42) and u... | https://arxiv.org/abs/2503.03328v1 |
in Section IV, in this section, we derive the PT of DT systems. We first introduce the DT affine martingale and then prove an AM-based PT for general DT systems. We also improve its tightness for contractive systems by leveraging the union-bound inequality. A. AM Based Probabilistic Tube To begin with, we introduce the... | https://arxiv.org/abs/2503.03328v1 |
σ2(L2k∆t−1) L2−1(ε1n+ε2logN δ), Fig. 6: The bound rδ,tderived from Theorem 5 with respect to ∆tat different time t.Left:L= 0.9999 with∆t∈ {1, . . . , 450}. The opti- mal choice is ∆t= 19 .Right :L= 0.99with∆t∈ {1, . . . , 40}.The optimal choice is ∆t= 1. and (79) is relaxed to P ∥Xt−xt∥ ≤r σ2(L2t−1) L2−1(ε1n+ε2logN δ)!... | https://arxiv.org/abs/2503.03328v1 |
that ∀x0∈ X0⇒xt∈ C ⊖ Bn(rδ,t,0), (55) which is sufficient to conclude the safety of the stochastic system with1−δguarantee by Theorem 1. Proposition 4 converts the stochastic safety verification problem into a deterministic safety verification on a time-varying set. This conversion offers tremendous flexibility as one ... | https://arxiv.org/abs/2503.03328v1 |
3. Given a safe set C ⊆Rn, let X0∈ X0⊆ C. For a stochastic trajectory Xtstarting from X0and its associated deterministic trajectory xt, if there exists a control law ut∈ U such that f(xt, ut)∈ C ⊖ Bn(rδ,t,0),∀t≤T, (58) where rδ,tis as Table I, then by Proposition 4, P(Xt+1∈ C)≥ 1−δif the same utis applied to the stocha... | https://arxiv.org/abs/2503.03328v1 |
0.722} ∪ {(px−6.2)2+ (py−0.5)2≤0.752}is the union of the circumcircle of red obstacles shown in Figure 8, and the safe region is C=Rn\ Cu. To accomplish this task, vtandωtare designed as the feedback controllers proposed in [ 45]. The details of the controller design can be found in [23, Section VIII]. Our goal is to v... | https://arxiv.org/abs/2503.03328v1 |
are plotted in Figure 9. For each trajectory, it satisfies the Fig. 9: Visualization of stochastic trajectories of the Mass-Spring- Damper system (60). In each figure, the stochastic curves in different colors are 3000 independent stochastic trajectories. The black dashed lines represent the boundary of the safe constr... | https://arxiv.org/abs/2503.03328v1 |
IEEE, 2018, pp. 5182– 5188. [9]R. K. Cosner, P. Culbertson, A. J. Taylor, and A. D. Ames, “Robust safety under stochastic uncertainty with discrete-time control barrier functions,” arXiv preprint arXiv:2302.07469 , 2023. [10] P. Mohajerin Esfahani, D. Chatterjee, and J. Lygeros, “The stochastic reach-avoid problem and ... | https://arxiv.org/abs/2503.03328v1 |
[26] M. Ono, M. Pavone, Y . Kuwata, and J. Balaram, “Chance-constrained dynamic programming with application to risk-aware robotic space exploration,” Autonomous Robots , vol. 39, pp. 555–571, 2015. [27] K. M. Frey, T. J. Steiner, and J. P. How, “Collision probabilities for continuous-time systems without sampling,” in... | https://arxiv.org/abs/2503.03328v1 |
dt≤atE(M(vt, t)) +bt. (61) We define ψt=eRT taτdτandfMas in (14) and observe thatdψt dt= −atψt. Thus, we get dE fM(vt, t) dt =dE(M(vt, t)) dtψt+E(M(vt, t))dψt dt+d dtZT tbτψτdτ =dE(M(vt, t)) dtψt−atE(M(vt, t))ψt−btψt ≤(atE(M(t, vt)) +bt)ψt−(atE(M(t, vt)) +bt)ψt= 0,which implies that fM(vt, t)is a super-martingale. Us... | https://arxiv.org/abs/2503.03328v1 |
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