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Bwithout loss of generality. Therefore, the variance of ∆(w1,w2)is bounded by E (x,y)∼D[∆2(w1,w2)] = E (x,y)∼D[(σ(w1·x)−σ(w2·x))2(σ(w1·x) +σ(w2·x)−2y)2] ≤16B2E x∼Dx[(σ(w1·x)−σ(w2·x))2]. On the other hand, suppose without loss of generality that L(w1)≥ L(w2); then, the expectation of ∆(w1,w2)can be bounded below by E (x... | https://arxiv.org/abs/2502.08611v1 |
E.3 In this subsection, we provide technical lemmas that determine the number of samples required for each iteration. We start with Lemma E.6 that bounds the population gradient ∥g(w)∥2. Then in Lemma E.3 we provide the sufficient batch size of samples per iteration, utilizing the bounds on ∥g(w)∥2and the truncated upp... | https://arxiv.org/abs/2502.08611v1 |
shown in Claim C.6. Therefore, plugging the upper bound on the variance back into Equation (25), we get Pr[∥bg(w)−g(w)∥2≥t]≤dB2∥Tρσ′∥2 L2 nt2. Now choosing t=1 6∥T√ρcosθσ′∥2 L2sinθand setting n= ΘdB2∥Tρσ′∥2 L2 sin2θ∥T√ρcosθσ′∥4 L2δ , we obtain that with probability at least 1−δ, it holds ∥bg(w)−g(w)∥2≤1 6∥T√ρcosθσ′∥2... | https://arxiv.org/abs/2502.08611v1 |
that θ2∥Tcos(θ)σ′∥2 L2≤COPT +ϵ. Since θ(bw,w∗)≤θ0, combining with Proposition F.2, i.e., ∥P>1/θ2σ∥2 L2≲sin2θ∥Tcosθσ′∥2 L2, we further have ∥P>1/θ2σ∥2 L2≲ OPT +ϵ. Finally, usingthe errorboundon L(bw)developed inProposition D.7, we get L(bw)≤COPT +ϵ. As displayed in Theorem E.1, the main algorithm uses N1=˜Θ(dB2/ϵ+B2/ϵ)s... | https://arxiv.org/abs/2502.08611v1 |
0. The monotonic staircase functions (of M-bounded support) are defined by FM:=mX i=1Aiϕ(z;ti) +A0:A0∈R;Ai>0,|ti| ≤M,∀i∈[m];m <∞ . These staircase functions constitute a dense subset of the monotone function class and have a simple and easy-to-analyze form, therefore they serve well for our purpose. However, though t... | https://arxiv.org/abs/2502.08611v1 |
particular, let Φkbe a sequence of staircase monotonic functions (see Definition F.5) that converges toσuniformly; then, for ρ2≥1−C/M2where Mis the upper bound on the support of σ′(which is also the upper bound on the support of all Φ′ k’s) and Cis an absolute constant, from Proposition F.8, we conclude that ∥Φk−TρΦk∥2... | https://arxiv.org/abs/2502.08611v1 |
i=11 kϕ(z;ti) +σ(−M), where( m=⌈σ(M)−σ(−M)/k⌉+ 1, ti= min t∈[−M,M ]{σ(t)≥(i−1)(1/k) +σ(−M)}, i= 1, . . . , m −1;tm=M. By construction, we have |Φk(z)−σ(z)| ≤1/kfor all z∈R, therefore Φkconverges to σuniformly. To prove Proposition F.8, we decompose Ez∼N[(TρΦ(z)−Φ(z))2]into the following terms and provide upper bounds o... | https://arxiv.org/abs/2502.08611v1 |
integrable), TρΦ′(z)is well-defined and is continuous and smooth. Consequently, all the facts presented in Fact B.2 apply to TρΦ′(z)as well. Proceeding to the analysis of Ez∼N[(TρΦ(z)−Φ(z))2], however, technical difficulties arise when we try to relate Ez∼N[(TρΦ(z)−Φ(z))2]withEz∼N[(TρΦ′(z))2]. The main obstacle is that... | https://arxiv.org/abs/2502.08611v1 |
much from the uncentered augmentation TρΦ(z), as stated below. Lemma F.13. LetΦ∈ FM. Suppose 1> ρ2≥1−C/M2for an absolute constant C∈(0, M2/2]. Then: E z∼N[(TρΦ(z)−TρΦ(z/ρ))2]≤C′(1−ρ2)(∥TρΦ′(z/ρ)∥2 L2+∥TρΦ′(z)∥2 L2), where C′is an absolute constant. Proof.We first observe that since Tρis a linear operator on functionals... | https://arxiv.org/abs/2502.08611v1 |
2(1−ρ4)+ρ4titj 1−ρ4 ,for0< ti≤tj< ti/ρ≤tj/ρ. (37) We show: Claim F.14. Letti, tj>0satisfy ti≤tj≤ti/ρ. Then, for any ρ∈(0,1), it holds −t2 j 2≤ −ρ2(t2 i+t2 j) 2(1−ρ4)+ρ4titj 1−ρ4. The proof of Claim F.14 is deferred to Appendix F.2.3. Therefore, for each ti, tj,i, j∈[m], the expectation in Equation (36) is bounded abov... | https://arxiv.org/abs/2502.08611v1 |
z∼N[(TρΦ(z)−Φ(z))2]≲(1−ρ2)E z∼N[(TρΦ′(z))2]. Proof of Proposition F.8. Observe that Ez∼N[(TρΦ(z)−Φ(z))2]can be bounded as E z∼N[(TρΦ(z)−Φ(z))2]≤2E z∼N[(TρΦ(z)−Tρ1Φ(z/ρ1))2] + 2 E z∼N[(Tρ1Φ(z/ρ1)−Φ(z))2].(39) We first bound the second term in Equation (39). Since we have assumed that ρ2≥1−C/M2, where C is an absolute co... | https://arxiv.org/abs/2502.08611v1 |
Claim F.17. IfM2≥2Cthen ρ4 1/(1−ρ4 1)≤4/(1−ρ4). Proof of Claim F.17. For any fixed ρ∈(0,1), let us define h(M) =(ρ2+C(1−ρ2)/M2)(1−ρ4) 1−(ρ2+C(1−ρ2)/M2)2. It is easy to see that h(M)is a decreasing function with respect to M > 0, therefore, for any fixed ρ∈(0,1)and any M2≥2C, we have h(M) =(ρ2+C(1−ρ2)/M2)(1−ρ4) 1−(ρ2+C(... | https://arxiv.org/abs/2502.08611v1 |
(Diakonikolas et al. (2022c), Corollary of Lemma C.3 and Theorem C.1) .There is an algorithm that given a halfspace ϕ(w∗·x;t)and a distribution (x,˜y)∼ Dof labeled examples with OPT′-adversarial noise—meaning that Pr[ϕ(w∗·x;t)̸=˜y]≤OPT′, ifexp(−t2/2)/t≥C1OPT′, where C1>0is a universal constant—returns wsuch that θ(w,w∗... | https://arxiv.org/abs/2502.08611v1 |
Statistical tests based on R´ enyi entropy estimation Mehmet Siddik C ¸adirci1, Dafydd Evans2, Nikolai Leonenko2, Vitali Makogin3 and Oleg Seleznjev4 1Faculty of Science, Department of Statistics, Cumhuriyet University, Sivas, Turkey. 2School of Mathematics, Cardiff University, Cardiff, Wales, UK. 3Institute for Stocha... | https://arxiv.org/abs/2502.08654v1 |
positive definite m×mmatrix. •Themultivariate Gaussian distribution Nm(a,Σ) on Rmhas density function fG a,Σ(x) = (2 π)−m/2|Σ|−1/2exp −1 2(x−a)′Σ−1(x−a) . ForX∼Nm(a,Σ), we have a=E(X) and Σ = Cov( X), where Cov( X) =E[(X−a)(X−a)′] is the covariance matrix of the distribution. •Forν >0, the multivariate Student distri... | https://arxiv.org/abs/2502.08654v1 |
Pm(a,Σ, η) with Σ = (2 η+m+ 2)Candη= 1/(q−1). Applying (6) and (7) yields the following expressions for the maximum entropy. Corollary 4. (1) For m/(m+ 2)< q < 1 the maximum value of Hqis Hmax q=1 2log|Σ|+c2(m, q, ν ) with Σ = (1 −2/ν)Candν= 2/(1−q)−m. (2) For q >1 the maximum value of Hqis Hmax q=1 2log|Σ|+c∗ 2(m, η, ... | https://arxiv.org/abs/2502.08654v1 |
the Lpconvergence of ˆGk,N,q toIqasN→ ∞ . Because we only need to obtain a bound on left-hand side of (12), we can use results on the subadditivity of Euclidean functionals defined on the nearest-neighbors graph (Yukich 1998). We use the following result (Lemma 3.3) from Penrose and Yukich (2011), see also (Yukich 1998... | https://arxiv.org/abs/2502.08654v1 |
(17) is finite, so the first term on the right hand side of (15) is bounded by a constant which is independent of N. For a non-negative random variable Z >0, we know that E(Z) =Z∞ 0P(Z > z )dz, so the second term in (15) is bounded by WkZ∞ 0P max 1≤i≤N∥Xi∥b> u·N1−b/m du ≤Wk 1 +NZ∞ 1P ∥X1∥b> um/(m−b)N1−b/m du (1... | https://arxiv.org/abs/2502.08654v1 |
and ˜WN,k(m, η) using Monte Carlo methods. 7 5 Numerical experiments 5.1 Random samples Following Johnson 1987, random samples from the Tm(a,Σ, ν) and Pm(a,Σ, η) distributions can be generated according to the stochastic representation X=RBU +a, where Rrepresents the radial distance (X−a)′Σ−1(X−a)1/2,Bis an m×mmatrix... | https://arxiv.org/abs/2502.08654v1 |
ν 0). 8 5.2.1 Consistency Figure 1 shows the asymptotic behaviour of the test statistic WN,k(m, ν) for m∈ {1,2,3}andk∈ {1,2,3,4} on samples drawn from the Tm(ν) distribution with ν∈ {3,4,5,10,∞}, as the sample size N→ ∞ . In each plot, the lines represent the sample mean ¯ wN,k(m, ν 0) of our observations. Figure 1: Co... | https://arxiv.org/abs/2502.08654v1 |
N0.00.20.40.60.8WN,k(m,0) 0=3 0=4 0=5 0=10 0= (f)m= 3,ν= 10 10 0 1000 2000 3000 4000 5000 N0.00.10.20.30.4WN,k(m,0) 0=3 0=4 0=5 0=10 0= (g)m= 1,ν= 5 0 1000 2000 3000 4000 5000 N0.00.10.20.30.40.50.60.70.8WN,k(m,0) 0=3 0=4 0=5 0=10 0= (h)m= 2,ν= 5 0 1000 2000 3000 4000 5000 N0.00.20.40.60.81.0WN,k(m,0) 0=3 0=4 0=5 0=10 ... | https://arxiv.org/abs/2502.08654v1 |
0.3 0.4 w0.02.55.07.510.012.515.017.520.0 0=3 0=4 0=5 0=10 0= (b)m= 2,ν=∞ 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 w02468101214160=3 0=4 0=5 0=10 0= (c)m= 3,ν=∞ 0.05 0.00 0.05 0.10 0.15 0.20 0.25 w051015200=3 0=4 0=5 0=10 0= (d)m= 1,ν= 10 0.0 0.1 0.2 0.3 0.4 w0.02.55.07.510.012.515.017.50=3 0=4 0=5 0=10 0= (e)m= 2,ν= 10 0.1 0.0... | https://arxiv.org/abs/2502.08654v1 |
(e)m= 2,ν= 10 3 4 5 10 inf 0 0.1 0.00.10.20.30.40.5 w (f)m= 3,ν= 10 13 3 4 5 10 inf 0 0.05 0.000.050.100.150.200.25 w (g)m= 1,ν= 5 3 4 5 10 inf 0 0.1 0.00.10.20.3 w (h)m= 2,ν= 5 3 4 5 10 inf 0 0.00.20.40.60.8 w (i)m= 3,ν= 5 3 4 5 10 inf 0 0.1 0.00.10.20.3 w (j)m= 1,ν= 4 3 4 5 10 inf 0 0.00.20.40.60.81.0 w (k)m= 2,ν= 4 ... | https://arxiv.org/abs/2502.08654v1 |
2.5 2.0 1.5 WN,k(m,) 0=3 0=4 0=5 0=10 0= (g)m= 1,ν= 5 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 logN3.0 2.5 2.0 1.5 1.0 WN,k(m,) 0=3 0=4 0=5 0=10 0= (h)m= 2,ν= 5 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 logN2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 WN,k(m,) 0=3 0=4 0=5 0=10 0= (i)m= 3,ν= 5 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 l... | https://arxiv.org/abs/2502.08654v1 |
-0.38 -0.36 -0.32 -0.19 10 -0.13 -0.21 -0.27 -0.32 -0.38 -0.40 -0.40 -0.40 -0.36 -0.23 12 -0.12 -0.19 -0.24 -0.29 -0.36 -0.39 -0.40 -0.40 -0.38 -0.26 15 -0.10 -0.17 -0.22 -0.26 -0.33 -0.36 -0.39 -0.41 -0.41 -0.30 20 -0.09 -0.15 -0.20 -0.24 -0.30 -0.34 -0.37 -0.39 -0.40 -0.34 ∞ -0.06 -0.10 -0.13 -0.15 -0.19 -0.22 -0.25 ... | https://arxiv.org/abs/2502.08654v1 |
0.268 0.444 0.455 0.488 0.658 0.703 0.734 300 0.212 0.209 0.209 0.370 0.395 0.412 0.558 0.593 0.629 400 0.199 0.191 0.189 0.334 0.351 0.381 0.456 0.490 0.529 500 0.164 0.168 0.167 0.294 0.328 0.332 0.457 0.500 0.515 600 0.151 0.148 0.152 0.259 0.272 0.291 0.416 0.464 0.493 700 0.148 0.144 0.144 0.260 0.264 0.271 0.379 ... | https://arxiv.org/abs/2502.08654v1 |
the estimator ˆγ(ν) =1 MMX j=1I(wj>ˆwα) where Iis the indicator function. This is the proportion of observations that exceed the estimated critical value ˆwα, and serves as an estimate for the probability that the test correctly rejects the null hypothesis ν=ν0in favour of the alternative ν̸=ν0. Table 3 shows the estim... | https://arxiv.org/abs/2502.08654v1 |
the power of the test to detect ν̸=ν0asνdecreases. By contrast, when νis large, for example when samples are drawn from the multivariate Gaussian distribution Tm(∞), there are relatively few outlying values of WN,k(m, ν 0), and the resulting separation between the empirical distributions indicates that power of the tes... | https://arxiv.org/abs/2502.08654v1 |
0.015 0.010 0.005 0.0000.0050.010W* N,k(m,) k=1 k=2 k=3 k=4 (l)m= 3,η= 4 0 1000 2000 3000 4000 5000 0.030 0.025 0.020 0.015 0.010 0.005 0.0000.005W* N,k(m,) k=1 k=2 k=3 k=4 (m)m= 1,η= 2 0 1000 2000 3000 4000 5000 0.06 0.05 0.04 0.03 0.02 0.01 0.00W* N,k(m,) k=1 k=2 k=3 k=4 (n)m= 2,η= 2 0 1000 2000 3000 4000 5000 0.05 0... | https://arxiv.org/abs/2502.08654v1 |
Tm(ν), Figure 6 shows thatW∗ N,k(m, η 0) on samples from Pm(η) converges rapidly, and the rate of convergence increases as ηdecreases andPm(η) becomes increasingly light-tailed. Unfortunately, the differences between these limiting values are small compared to the variance of W∗ N,k(m, η), so the power of the test to d... | https://arxiv.org/abs/2502.08654v1 |
4 6 12 inf 0 0.05 0.000.050.10 w* (f)m= 3,η= 12 23 2 4 6 12 inf 0 0.075 0.050 0.025 0.0000.0250.0500.075 w* (g)m= 1,η= 6 2 4 6 12 inf 0 0.06 0.04 0.02 0.000.020.040.060.08 w* (h)m= 2,η= 6 2 4 6 12 inf 0 0.05 0.000.050.10 w* (i)m= 3,η= 6 2 4 6 12 inf 0 0.06 0.04 0.02 0.000.020.040.060.08 w* (j)m= 1,η= 4 2 4 6 12 inf 0 0... | https://arxiv.org/abs/2502.08654v1 |
6 0.63 0.32 0.11 -0.15 -0.21 -0.17 -0.16 -0.16 -0.16 8 0.23 0.20 -0.05 -0.18 -0.23 -0.18 -0.28 -0.22 -0.23 11 0.10 0.07 -0.07 -0.14 -0.23 -0.27 -0.28 -0.28 -0.26 15 0.16 0.15 -0.01 -0.13 -0.17 -0.21 -0.24 -0.23 -0.26 20 0.16 0.08 -0.02 -0.15 -0.20 -0.21 -0.25 -0.25 -0.26 ∞ 0.09 0.01 -0.05 -0.14 -0.18 -0.24 -0.25 -0.28 ... | https://arxiv.org/abs/2502.08654v1 |
100 0.318 0.248 0.214 0.389 0.306 0.301 0.388 0.321 0.298 200 0.215 0.154 0.140 0.247 0.210 0.197 0.292 0.243 0.229 300 0.174 0.134 0.115 0.218 0.171 0.170 0.251 0.204 0.198 400 0.170 0.118 0.107 0.188 0.152 0.148 0.208 0.180 0.170 500 0.144 0.107 0.099 0.161 0.134 0.129 0.184 0.171 0.170 600 0.133 0.096 0.088 0.147 0.... | https://arxiv.org/abs/2502.08654v1 |
. , w∗ M} independent observations of the test statistic W∗ N,k(m, ν 0) on independent samples from the Pm(η) distribution. As in section 5.2.6 we estimate the power function by the estimator ˆγ(η) =1 MMX j=1I(w∗ j>ˆw∗ α) where Iis the indicator function. Table 6 shows the estimated power for the case N= 5000 and k= 3 ... | https://arxiv.org/abs/2502.08654v1 |
entropy principle and a class of estimators for R´ enyi entropy based on nearest 27 neighbour distances. We have proved the L2-consistency of these statistics using results on the subadditivity of Euclidean functionals on nearest neighbour graphs, and investigated their distributions and rates of convergence using Mont... | https://arxiv.org/abs/2502.08654v1 |
E. (1987). Multivariate statistical simulation . John Wiley and Sons. Johnson, O. and Vignat, C. (2007). “Some results concerning maximum R´ enyi entropy distributions”. Annales de l’IHP Probabilit´ es et Statistiques 43, 339–351. Kotz, S. and Nadarajah, S. (2004). Multivariate t-distributions and their applications . ... | https://arxiv.org/abs/2502.08654v1 |
Statistical inference for L´ evy-driven graph supOU processes: From short- to long-memory in high-dimensional time series Shreya Mehta Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK s.mehta20@imperial.ac.uk Almut E. D. Veraart Department of Mathematics, Imperial College London... | https://arxiv.org/abs/2502.08838v1 |
on drift estimation for high-dimensional OU processes and general diffusions, see Cio lek et al. (2020); Ciolek et al. (2024). A powerful framework for more flexible serial dependence structures, including the potential for long-range dependence, was first advocated by Barndorff-Nielsen and Shep- hard (2001); Barndorff... | https://arxiv.org/abs/2502.08838v1 |
d×kmatrices. When k=d, we denote this set as Md(R). The linear subspace of d×dsymmetric matrices is denoted by Sd, the closed positive cone of symmetric matrices with non-negative real parts of their eigenvalues is denoted as S+ d, and the open positive definite cone of symmetric matrices with strictly positive real pa... | https://arxiv.org/abs/2502.08838v1 |
Theorem 3.1) states the conditions which ensure that such a process is well-defined. Theorem 2.2. (Theorem 3.1, Barndorff-Nielsen and Stelzer (2011)) Let Λbe anRd- valued L´ evy basis on M− d×Rwith generating quadruple (γ,Σ, ν, π)satisfyingR ∥x∥>1ln(∥x∥)ν(dx)< ∞and assume there exist measurable functions ρ:M− d→R+\{0}a... | https://arxiv.org/abs/2502.08838v1 |
of graph supOU type – Bridging between long and short memory In this section, we introduce new parametric models within the class of graph supOU processes. Specifically, we focus on model specifications that are parsimonious, analyti- cally tractable, and capable of capturing both short- and long-term memory. In the ca... | https://arxiv.org/abs/2502.08838v1 |
Adapting (Barndorff-Nielsen and Stelzer, 2011, Example 3.1), we assume that θ2∼ Γ(α,1), with π(dθ2) =1 Γ(α)θα−1 2e−θ21(0,∞)(θ2)dθ2,where α >1. (7) We note that setting the second parameter in the Gamma distribution to a constant (here chosen as 1) is needed to ensure the model identifiability. The moments of the graph ... | https://arxiv.org/abs/2502.08838v1 |
i= 1, . . . , d . We observe that the maximum eigenvalue of the scaled autocovariance function can be written as a function of πandcas follows: ρ(h∆;π, c) := max {ξ∈σ(R(h∆;π, c))} =Z∞ 01 θ2e−θ2(1+ca∗)hπ(dθ2)Z∞ 01 θ2π(dθ2)−1 .(9) Let us consider the two examples introduced earlier. Example 3.1. In the case when πis ... | https://arxiv.org/abs/2502.08838v1 |
sum of exponentials considered above. 3.2 Generalised Method of Moments (GMM) Estimator In addition to the previous discussion, we will also present the generalised method of moments framework for the parameter estimation of graph supOU processes, not necessarily restricted to the setting considered in Section 2.4. We ... | https://arxiv.org/abs/2502.08838v1 |
N(ξ), respectively. Assumption 3.5. Suppose that supξ∈Ξ|f(i) N(ξ)−g(i) N(ξ)|P−→0for all i= 1, . . . , q , as N→ ∞ . Moreover, the following assumption regarding the weighting matrix is also required. 11 Assumption 3.6. There is a sequence of deterministic, positive definite matrices de- noted by VNsuch that VN−VNP−→0, ... | https://arxiv.org/abs/2502.08838v1 |
supOU processes. Weak dependence in a process refers to a situation where the values of the process are not independent, but exhibit diminishing correlation or association as the time lag between observations increases. We give the definition of ζ-weakly dependent processes in the following, which were called θ-weakly ... | https://arxiv.org/abs/2502.08838v1 |
moment function, the partial derivative matrix∂f(Xt:t+m,ξ) ∂ξT is independent of Xt:t+m. Hence, we have FN(ξ) = E(∂f(Xt:t+m,ξ) ∂ξT ) =∂f(Xt:t+m,ξ) ∂ξ⊤ and can set F:=FN:=∂f(Xt:t+m, ξ) ∂ξ⊤|ξ=ξ0. An application of the continuous mapping theorem leads the property stated in Assump- tion 3.15. Assumption 3.16. We have a ce... | https://arxiv.org/abs/2502.08838v1 |
term involving the com- pensated small jumps Stand the big jumps Jt. Since there is extensive literature on the simulation of Gaussian processes, we will focus on the –possibly more interesting– jump case going forward. More precisely, we consider the scenario where the L´ evy basis Λ is given by a compound Poisson ran... | https://arxiv.org/abs/2502.08838v1 |
Figures 1a and 1b and depict the box- plots for the estimates of αandc, respectively, for different lags hused in the scaled autocovariance. The outliers are not depicted to improve readability. The solid red line presents the true parameter value in both cases. Figure 1c displays the corresponding error measures. Figu... | https://arxiv.org/abs/2502.08838v1 |
24 48 LagACF (b) 0.000.250.500.751.00 2012 2013 2014 2015 TimeLisbon time series data (c) (d) 0.000.250.500.751.00 0 24 48 LagACF (e) −0.6−0.30.00.30.6 2012 2013 2014 2015 TimeLisbon time series data (des.) (f) Figure 3: Plots of the 24-dimensional time series of wind capacity factors in Portugal: Heatmap of the 24 hou... | https://arxiv.org/abs/2502.08838v1 |
0.000.010.020.03 (b) 4812162024 4 8 12 16 20 24 ComponentComponentCov 0.000.010.020.03 (c) Figure 5: Estimated mean and variance of the L´ evy basis: Figure 5a Estimated mean cµL, Figure 5b estimated variancecσ2 L, and Figure 5c empirical covariance \Var(X). Finally, we estimate the mean and variance of the driving L´ ... | https://arxiv.org/abs/2502.08838v1 |
L. Using vectorisation, we get vec(Y) = vec( Q(θ)P+PQ(θ)⊤) = (Id×d⊗Q(θ) +Q(θ)⊤⊗Id×d)vec(P). We then get vec(P) = (Id×d⊗Q(θ) +Q(θ)⊤⊗Id×d)−1vec(Y). Hence, A(Q(θ))−1(Y) = vec−1((Id×d⊗Q(θ) +Q(θ)⊤⊗Id×d)−1vec(Y)). We note that P=A(Q(θ))−1(−σ2 L) = vec−1((Id×d⊗Q(θ) +Q(θ)⊤⊗Id×d)−1vec(−σ2 L)) =−A(Q(θ))−1(σ2 L). Hence, altogethe... | https://arxiv.org/abs/2502.08838v1 |
of the graph 23 supOU process to deduce that, for a constant C >0, using the notation Q=Q(θ), ζ2(r) =Z M− dZ−r −∞tr(e−Qsσ2 Le−Q⊤s)dsπ(dQ) +∥Z M− dZ−r −∞e−QsµLdsπ(dQ)∥2 ≤Z M− dZ−r −∞|tr(e−Qsσ2 Le−Q⊤s)|dsπ(dQ) +∥Z M− dZ−r −∞e−QsµLdsπ(dQ)∥2 ≤CZ M− dZ−r −∞∥e−Qs∥∥σ2 L∥∥e−Q⊤s∥dsπ(dQ) +∥Z M− dZ−r −∞e−QsµLdsπ(dQ)∥ ≤CZ M− dZ∞ r... | https://arxiv.org/abs/2502.08838v1 |
http://www.jstor.org/stable/29779366 2, 4, 7, 8, 21, 24 Bennedsen, M., Lunde, A., Shephard, N. and Veraart, A. E. (2023), ‘Inference and fore- casting for continuous-time integer-valued trawl processes’, Journal of Econometrics 236(2), 105476. URL: https://www.sciencedirect.com/science/article/pii/S0304407623001926 11 ... | https://arxiv.org/abs/2502.08838v1 |
arXiv:2502.08840v1 [math.ST] 12 Feb 2025Thresholds for Reconstruction of Random Hypergraphs From Graph Projections Guy Bresler∗Chenghao Guo†Yury Polyanskiy‡ Abstract Thegraph projection of a hypergraph is a simple graph with the same vertex set and w ith an edge between each pair of vertices that appear in a hyperedge.... | https://arxiv.org/abs/2502.08840v1 |
‡Supported in part by the NSF grant CCF-2131115 and the MIT-IB M Watson AI Lab. 1 Beyond serving as a justifying principle for employing hypergraph-t o-graph projections in algorithm creation, the task ofrecoveringhypergraphsfrom their graphp rojectionsarisesnaturallyin networkanalysis. The phenomenon of intrinsic hype... | https://arxiv.org/abs/2502.08840v1 |
2This definition of random hypergraph is equivalent to the defi nition of random d-complex [ TOG17] except here we use the language of hypergraphs instead of simplicial complexes. T he model considered in [ YPP21] is an inhomogeneous generalization of our model where each hyperedge has a distinct probability of appearing... | https://arxiv.org/abs/2502.08840v1 |
(or efficiently achievable) for δ=δ1. The lemma is proved via a simple reduction: given G= Proj(H) whereHhas density p(δ1,n,d), we independently sample a random hypergraph H′so thatH∪H′has density p(δ2,n,d) and give algorithm A (presumed to achieve exact recovery at δ1) the graph Proj( H)∪Proj(H′) = Proj(H∪H′). We then r... | https://arxiv.org/abs/2502.08840v1 |
nvertices, parameterized by q1andq2. A sampleHis generated as follows. First an assignment of labels σ∈{±1}nfor the vertices is sampled uniformly at random from all assignments with equal number of +1 and−1 (nis assumed to be even). Conditional on σ, for each h∈/parenleftbig[n] d/parenrightbig , the hyperedge h={i1,···... | https://arxiv.org/abs/2502.08840v1 |
simple algorithm that produces a hypergraph from a graph by including every possible hyperedge. T his can result in a hypergraph that has many more hyperedges than the maximum a posteriori (MAP) hypergraph, and therefore has far lower posterior probability. Correspondingly, this simple algorithm succee ds in a smaller ... | https://arxiv.org/abs/2502.08840v1 |
probability mass function of the random hypergra phgiven the projected graph. Therefore, if we do not worry about time complexity, the informatio n theoretically optimal algorithm should simply output a hypergraph with maximum posterior likelihood, i.e., follow ing the maximum a posteriori (MAP) rule. As discussed next... | https://arxiv.org/abs/2502.08840v1 |
different from th e usual definition of 2-connectivity in a simple graph. 7 (a) An example of hypergraph H. Triangles filled with colors represent hyperedgesin H. Two hyperedgesare 2- connected if they share two nodes. Different 2-connected components are marked in different colors.(b) The corresponding graph GH. Each node ... | https://arxiv.org/abs/2502.08840v1 |
try to understand why the probability of having the graph in Hcdecreases with the growth. Suppose Kis a set of hyperedges, and Cli(Proj( K)) is 2-connected. For Kto get larger, it must include one of its 2-neighbors h∈Hc. How did happear inHc? The somewhat delicate aspect of this is that hmay not be inH:hmight exist in... | https://arxiv.org/abs/2502.08840v1 |
78 Figure 2: A graph with non-unique minimum preimage in the case d= 3. The green hyperedges are the two possible minimum preimages. It follows that to prove impossibility of (exact) recovery, we only ne ed to find an ambiguous graph that is a 2-connected component with probability Ω n(1). Let the ambiguity threshold ,δ... | https://arxiv.org/abs/2502.08840v1 |
If for all finite ambiguous graphs Ga, IP/parenleftbig Cli(Ga)is a 2-connected component of Hc/parenrightbig =on(1), then we have IP(A∗(Gp) =H)≥1−on(1). The lemma is proved in Appendix C.4. Here we provide a sketch of the proof. If there is no ambiguous graph in Gp, the projections of every 2-connected components have a... | https://arxiv.org/abs/2502.08840v1 |
hyperedge hinH where every pair of nodes in his also included in other hyperedges in H. In this case the graph H\{h}has higher probability and has the same graph projection. Because the optimal algorithm outputs a minimum preimage, it does not output the original hypergraph H: deleting hforms a smaller preimage. We for... | https://arxiv.org/abs/2502.08840v1 |
valid preimages, so both are minimum preimages for Ga,d. Property 2: The Graph Ga,dAppears with Probability Ωn(1).The next lemma shows that Ga,d appears in Gpwith non-negligible probability using Lemma 2.6. Lemma 3.2. LetGa,dbe as in Defn. 2. For any δ≥2d−4 2d−1, IP(Ga,d⊂Gp) = Ωn(1). Proof.Let us focus on one possible ... | https://arxiv.org/abs/2502.08840v1 |
SK,h 2={2,4}respectively in two hyperedges. Possible 2-neighbors of a sub-hypergraph and Definition of Grow.In Section 2.4.3, we discussed that the size of 2-connected components can be bounded by exam ining how 2-connected sub-hypergraphs can grow. Specifically, we will look at a sub-hypergraph H⊂Hand the possible ways ... | https://arxiv.org/abs/2502.08840v1 |
and the set of nodes that are in hibut not in Kwhich has size d−|SK,h i|. Therefore, we have vK′−vK≤d−k+/summationdisplay i∈I(d−|SK,h i|). We have inequality instead of equality because some vertices may be d ouble-counted. Therefore, IEXK′ IEXK=On(nd−k+/summationtext i∈I(d−|SK,h i|)p|I|). Usingp=n−d+1+δ, we have that ... | https://arxiv.org/abs/2502.08840v1 |
i=1xi(xi−1) 2≥(d 2)−(k 2)/braceleftBigM/summationdisplay i=1(xi−1−δ)+k−d/bracerightBig = min M∈Z+, M≤(d 2)−1min x0,x1,···,xM≥2:/summationtextM i=0xi(xi−1) 2≥(d 2)/braceleftBigM/summationdisplay i=0xi−M(1+δ)−d/bracerightBig . Here in the equality we substituted kwithx0. By setting yi=xi(xi−1) 2, the above can be written... | https://arxiv.org/abs/2502.08840v1 |
[BCI+20] Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Lator a, Maxime Lucas, Alice Patania, Jean-Gabriel Young, and Giovanni Petri. Networks beyond pairwis e interactions: Structure and dynamics. Physics Reports , 874:1–92, 2020. [Bol81] B´ ela Bollob´ as. Threshold functions for small subgraphs. I nMathe... | https://arxiv.org/abs/2502.08840v1 |
First Search (DFS) over Hypergraphs. First, instead of searching over graphs Ga, we can search over preimages of the graphs, i.e., hypergraphs. The claim in Lemma2.10is equivalent to For any sub-hypergraph Kwhere Cli(Proj( K)) is 2-connected and IP( K⊂H) = Ωn(1), Khas unique minimum preimage. We will prove a sufficient c... | https://arxiv.org/abs/2502.08840v1 |
edges in Proj(E1) between v and other nodes in h. Alld−1 edges are included in some hyperedges in E1. But they cannot be in a single 21 hyperedge inE1, otherwise that hyperedge would be h, contradicting with E1∩E2=∅. Sovhas degree at least 2 inE1. Therefore, vE1≤dk minimum degree≤dk/2. So by Lemma 2.3, IP(E1⊂H)≤IEXE1= ... | https://arxiv.org/abs/2502.08840v1 |
is that h /∈ EHand all of the following hyperedges are in EH: •{u1,u2,···,ud}and •{v,ui,w(1) i,···,w(d−2) i}for all 1≤i≤d−1. LetKfdenote the hypergraph, seeFigure 7foran illustration(weabuse notationand let Kfinclude the non- hyperedge h). There are vKf=d2−2d+3nodes and eKf=dedgesinKf.p=n−d+1+δ= Ωn(n−d+2+3/d) = Ωn(n−vK... | https://arxiv.org/abs/2502.08840v1 |
first case. We will prove f(δ) = min S′⊂S,|S′|≥d: Proj([d])⊂Proj(S′)/summationdisplay S∈S′(|S|−1−δ) =d. This part of the proof is similar to what we did in Lemma 4.3. We can get a lower bound on f(δ) by relaxing the set of possible S′to be the set of cliques with at least/parenleftbigd 2/parenrightbig number of edges. A... | https://arxiv.org/abs/2502.08840v1 |
([ d]\{v})∈S′, denote this set of nodes U. Then there are at least d−ssets inS′with sized−1. By counting degree, there are at least 2 s/2 =ssize-2 set inS′. Since|S′|=d, there must be d−ssets with size d−1 andssets with size 2 in S′. Thessize-2 sets must be between nodes in T. So nodes in Uare not in any size-2 sets. I... | https://arxiv.org/abs/2502.08840v1 |
method LetK1,K2,···,Ktbe all copies of such sub-hypergraph on the complete graph of [ n], we have t=/parenleftbiggn vK/parenrightbigg(vK)! aut(K)= Θn(nvK). Here aut( K) is the number of automorphisms of K. LetIibe the indicator that Kiis inH. AndXK=/summationtextt i=1Iibe the number of such event happening. We have IE[... | https://arxiv.org/abs/2502.08840v1 |
that this is reached by the whole hypergraph, i.e., when K=S1∪S2∪{h1}. LetLbe the set of hyperedges in S1that is a subset of K,Rbe the set of hyperedges in S2that is a subset of K. Case 1:h1∝\e}atio\slash∈K,R=∅. eK vK=|L| (d−1)|L|+1≤d−1 (d−1)2+1. The maximum is achieved when L=S1. The case where R∝\e}atio\slash=∅andL=∅... | https://arxiv.org/abs/2502.08840v1 |
a size-ksubset of h. Note that any clique in Shhas size at least 2, gk(δ) is always non-negative. Therefore, by union bound over all hyperedges in Akfor any 2≤k≤d, IP/parenleftbig NHc(Cli(E1))∝\e}atio\slash=∅|E1⊂EH/parenrightbig =d/summationdisplay k=2|Ak|On(n−gk(δ)) =On(nmink{gk(δ)+k−d}). Given the bound for min k{gk(... | https://arxiv.org/abs/2502.08840v1 |
On (in)consistency of M-estimators under contamination∗ Jens Klooster†& Bent Nielsen‡ February 13, 2025 Abstract We consider robust location-scale estimators under contamination. We show that commonly used robust estimators such as the median and the Huber estimator are inconsistent under asymmetric contamination, whil... | https://arxiv.org/abs/2502.09145v1 |
on the location depends on the contamination, which is usually unknown. Thus, inference for M-estimators is fraught under contamination. Another type of robustness analysis uses the breakdown point (Hampel, 1971). The finite sample breakdown point asks how many arbitrary observations can be added to a sample without di... | https://arxiv.org/abs/2502.09145v1 |
in practice. 2.2 M-estimators with unknown scale In practice the scale σwill be unknown. Suppose a preliminary estimator ˆ σis available. Inserting this estimator in the above objective function for the location case gives a new objective function Rn(µ) =nX i=1ρyi−µ ˆσ . (2.1) Note, the median computed from ρ(x) =|x|... | https://arxiv.org/abs/2502.09145v1 |
The finite sample breakdown point with ϵ-contamination is defined as follows (Donoho and Huber, 1983). Let Xbe a sample of size hof ‘good’ observations. Adjoin n−h arbitrary values Yto get a sample X∪Yof size n. The sample X∪Ycontains a fraction ϵ= (n−h)/nof arbitrary values. The breakdown point for an estimator Tis th... | https://arxiv.org/abs/2502.09145v1 |
yi, with i= 1, . . . , n , has h≤n‘good’ observations and n−h‘outliers’. The models are indexed by n, so that h→ ∞ asn→ ∞ and the proportion of ‘good’ observations satisfies h/n→λwith 1 /2< λ≤1. Let ζnbe a deterministic sequence of h-sets from (1 , . . . , n ) of indices of ‘good’ observations. The observations satisfy... | https://arxiv.org/abs/2502.09145v1 |
µ◦ with large probability. In Assumption 3.2, part ( i) requires that the smallest ‘outlier’ errors diverges. This holds, for example, if the ‘good’ errors are N(0,1), while ‘outliers’ are outside the range of the ‘good’ errors. Part ( ii) requires that ρis Lipschitz, which holds for the Tukey estimator. Part ( iii) re... | https://arxiv.org/abs/2502.09145v1 |
are bounded in probability. Assumption 3.5. Thepquantiles of εifori∈ζnare bounded in probability for 0< p <1. Theorem 3.5. Suppose Assumption 3.5. (a): Let λ >3/4. Then, ˆσIQRis bounded in probability. (b): Let λ >1/2. Then ˆσMAD is bounded in probability. Theorem 3.5 shows that the IQR is bounded when the asymptotic p... | https://arxiv.org/abs/2502.09145v1 |
. . , n ) with indices of the ‘good’ observations. This is estimated by ˆζ= argmin ζX i∈ζ(yi−¯yζ)2, while the estimators of location and scale are ˆµLTS= ¯yˆζ, ˆσ2 LTS=1 hX i∈ˆζ(yi−ˆµLTS)2. 9 The LTS estimator is maximum likelihood in a contamination model satisfying Assumption 3.1, where ‘good’ observations are i.i.d.... | https://arxiv.org/abs/2502.09145v1 |
This only affects the Huber, Tukey and LTS estimators. As 105 10 Table 1: Bias for known scale and trimming. DGP1 DGP2 DGP3 DGP4 DGP5 DGP6 n= 25 Mean 0 .000 0 .002 0 .573 0 .973 1 .282 0 .198 Median 0 .001 0 .001 0 .315 0 .315 0 .353 0 .052 Huber 0 .001 0 .001 0 .412 0 .412 0 .468 0 .067 Tukey 0 .001 0 .000 0 .394 0 .0... | https://arxiv.org/abs/2502.09145v1 |
0 .001 0 .486 0 .493 0 .624 0 .068 Tukey 0 .001 0 .000 0 .480 0 .306 0 .357 0 .010 LTS 0 .002 0 .001 0 .018 0 .001 0 .036 0 .000 n= 100 Huber 0 .001 0 .000 0 .499 0 .499 0 .626 0 .016 Tukey 0 .001 0 .000 0 .516 0 .196 0 .101 0 .000 LTS 0 .001 0 .000 0 .002 0 .000 0 .000 0 .001 n= 400 Huber 0 .000 0 .000 0 .501 0 .501 0... | https://arxiv.org/abs/2502.09145v1 |
>1, the bias is zero up until ς= 1.2 (n= 100) or ς= 1.3 (n= 1000) and converges towards the bias of the average. When ςis large, the ‘outliers’ are not down-weighted so that the Tukey estimator behaves as the average. Nevertheless, it is consistent since the ‘outliers’ diverge and therefore for any fixed ς >1 the ‘outl... | https://arxiv.org/abs/2502.09145v1 |
easier problem to solve theoretically. Indeed, an estimator proposed in Berenguer-Rico et al. (2023) appeared successful in the simulations. A Proofs We consider estimators for µthat are location-scale invariant and estimators for σthat location invariant and scale equivariant. Thus, throughout we set µ◦= 0 and σ◦= 1, ... | https://arxiv.org/abs/2502.09145v1 |
+ 3 ϵ, for small ϵ. Thus, onSn, we get (2 −n/h−Cn/A2 0)> ϵ > 0. We can then bound Ln(µ) from below using |µ|>ˆςB0and get, on Sn, that Ln(µ)> Lwhere L=ϵ(ρ∗+ψ∗B0)−[˜ρς+ϵ+ψ∗{x∗+A0/(ς−ϵ)}]. Given A0, we can now choose B0so large that L >0. For measurability, apply the argument of Jennrich (1969), see also Johansen and Niel... | https://arxiv.org/abs/2502.09145v1 |
Proof of Theorem 3.4. We have that ˆ σ= 1 by Assumption 3.4( i), so that ( yi−µ)/ˆσ= εi−µ. Since ˆ µis bounded in probability by Theorem 3.1 using Assumptions 2.1, 3.1, it suffices to consider |µ| ≤B. Note that Assumption 3.4( i)−(iii) implies Assumption 3.1. ‘Outliers’ satisfy εj=ε(h)+ξwith ξ >0 for j̸∈ζn, while ε(h)=... | https://arxiv.org/abs/2502.09145v1 |
< 1and any ϱ. Given c, we choose dso that Ac,d= Φ(c+d)−Φ(c−d) = 1 /(2λ). We show d > qΦ 3/4. Now, Ac,d≤Φ(d)−Φ(−d), which increases in d. Thus, dis least for c= 0. As λ <1 then 1/(2λ)>1/2, and d > qΦ 3/4, which solves Φ(d)−Φ(d) = 1 /2. References Avella Medina, M. and Ronchetti, E. (2015). Robust statistics: A selective... | https://arxiv.org/abs/2502.09145v1 |
Regularization can make diffusion models more efficient Mahsa Taheri1Johannes Lederer1 Abstract Diffusion models are one of the key architectures of generative AI. Their main drawback, however, is the computational costs. This study indicates that the concept of sparsity, well known espe- cially in statistics, can prov... | https://arxiv.org/abs/2502.09151v1 |
distribution Q0and the approximated counterpart P0. For example, one tries to ensure TV(Q0, P0)≤τ for a fixed error level τ∈(0,∞), where TV(Q0, P0)··= supA⊂Rd|Q0(A)−P0(A)|is called the total variation (van de Geer, 2000). The many very recent papers on this topic highlight the large interest in this topic (Block et al.... | https://arxiv.org/abs/2502.09151v1 |
samples xi∈Rdwith an unknown target distribution q0 (xi∼q0fori∈ {1, . . . , n }). The goal of probabilistic generative modeling is to use the dataset Dnto learn a model that can sample from q0. The score of a probability density q(x), the gradient of the log-density with respect toxdenoted as ∇xlogq(x), are the key com... | https://arxiv.org/abs/2502.09151v1 |
which can generate samples from a probability density using the true score function, as follows: xt−1=1√αt xt+(1−αt)∇xtlogqt(xt) +r 1−αt αtzt =ut(xt) +σtzt forzt∼ N (0d,Id)andσt··=p 1−αt/αt. Let’s Pt be the marginal distribution of xtin the true reverse pro- cess, which is the reverse process by employing the true sc... | https://arxiv.org/abs/2502.09151v1 |
networks are sparse. Motivated by the denoising score matching objective, a scalable alternative to the objective function in (1)(Vincent, 2011), we define a regularized denoising score-matching estimator as (bΘℓ1,ˆκ)∈arg min Θ∈B1 κ∈(0,∞)Et∼U[0,T] Xt∼Qt||κsΘ(Xt, t)− ∇ Xtlogqt(Xt)||2 +rκ2, (3) where κ∈(0,∞)represents th... | https://arxiv.org/abs/2502.09151v1 |
and also is utilized in previous works like Huang et al. (2024). As discussed extensively in Liang et al. (2024, Section 5), this assumption is relatively mild, for example for distributions with finite variance or Gaussian mixtures. Theorem 3.5 (Non-asymptotic rates of convergence for regularized diffusion models) .Un... | https://arxiv.org/abs/2502.09151v1 |
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