text string | source string |
|---|---|
(3) 2.4974 (2) 1.0 1.2287 (1) 1.1594 (1) 1.7519 (2) 1.5048 (2) 2.7231 (3) 2.0234 (2) 4.6926 (5) 3.0924 (3) 1.5 1.2331 (1) 1.1566 (1) 1.8106 (2) 1.5189 (2) 4.1842 (4) 2.4067 (2) 14.2539 (14) 5.6002 (6) Table 3 shows results analogous to those in Table 1 for λ∈ {0.1,0.001}. Table 3: Accuracy on GSM8K for LLaMA 3.1 8B usi... | https://arxiv.org/abs/2505.19371v1 |
ICS FOR COMPLEX DATA WITH APPLICATION TO OUTLIER DETECTION FOR DENSITY DATA A P REPRINT Camille Mondon Mathematics and Statistics Toulouse School of Economics Toulouse, 31000 camille.mondon@tse-fr.euHuong Thi Trinh Faculty of Mathematical Economics Thuongmai University Hanoi trinhthihuong@tmu.edu.vnAnne Ruiz-Gazen Math... | https://arxiv.org/abs/2505.19403v1 |
also used in the context of blind source separation and more precisely for Independent Component Analysis (ICA) which is a model-based approach as opposed to ICS (see Nordhausen and Ruiz-Gazen 2022, for more details). ICS has later been adapted to more complex data, namely compositional data (Ruiz-Gazen et al. 2023), f... | https://arxiv.org/abs/2505.19403v1 |
both compositional and functional data. The literature on outlier detection for density data is very sparse and recent with, as far as we know, the papers by Menafoglio (2021), Lei, Chen, and H. Li (2023) and Murph, Strait, et al. (2024) only. Two types of outliers have been identified for density data: the horizontal-... | https://arxiv.org/abs/2505.19403v1 |
intrinsic. In particular, it leads to more interpretable invariant components that are of the same nature as the considered complex random objects. In the case of functional or distributional data, the usual framework assumes that the data objects reside in an infinite-dimensional Hilbert space, which leads to non-orth... | https://arxiv.org/abs/2505.19403v1 |
the classical formulation of invariant coordinate selection by Tyler et al. (2009). In the ICS problem Equation 1, the scatter operators S1andS2do not play symmetrical roles. This is because the usual method of solving ICS(X,S 1,S2)is to use the associated inner product of S1[X], which requires S1[X]to be injective. In... | https://arxiv.org/abs/2505.19403v1 |
random object X∈Ew1∩Ew2(with the notations from Definition 2), the equality CovF wℓ[φ(X)] =φ◦CovE wℓ[X]◦φ−1(4) holds forℓ∈{1,2}, as well as the equivalence between the following assertions, for any basis H= (h1,...,hp)ofE, and any finite non-increasing real sequence Λ = (λ1≥...≥λp): (i)(H,Λ)solves ICS(X,CovE w1,CovE w2... | https://arxiv.org/abs/2505.19403v1 |
expressions (in Definition 2) for instance) of the form Ef(X)for any function fare discrete and equal to1 n/summationtextn i=1f(xi). Now, let us assume that we observe an i.i.d. sample Dn= (X1,...,Xn)following the distribution of an unknown E-valued random object X0. We can estimate solutions of the problem ICS(X0,S1,S... | https://arxiv.org/abs/2505.19403v1 |
grid (t1,...,tN), we need to reconstruct an E-valued random function ufrom its noisy observed values (u(t1) +ε1,...,u (tN) +εN). There are many well-documented approximation techniques to carry out this preprocessing step, such as interpolation, spline smoothing, or Fourier methods (for a detailed presentation, see Eub... | https://arxiv.org/abs/2505.19403v1 |
to smooth each histogram into a compositional spline in E. In the latter, we assume that a random density is observed through a finite random sample (X1,...,XN)drawn from it. The preprocessing step consists in estimating the density from the observed sample. To perform the estimation, we need a nonparametric estimation... | https://arxiv.org/abs/2505.19403v1 |
data for outlier detection We propose using ICS to detect outliers in complex data, specifically in scenarios with a small proportion of outliers (typically 1 to 2%). For this, we follow the three-step procedure defined by Archimbaud, Nordhausen, and Ruiz-Gazen (2018), modifying the first step based on the implementati... | https://arxiv.org/abs/2505.19403v1 |
Gaussian distribution. For a given sample size and number of variables, we generate 10,000 standard Gaussian samples and compute the empirical quantile of order 97.5% of the ICS-distances using the three steps previously described. An observation with an ICS distance larger than this quantile is flagged as an outlier. ... | https://arxiv.org/abs/2505.19403v1 |
scree plots of the ICS eigenvalues following the recommendations in Section 3.1. For each of the experimental scenarios detailed below, we compute the squared ICS distances of the 112 observations as defined in Section 3.1, using κ= 4. Observations are classified as outliers when their squared ICS distance exceeds the ... | https://arxiv.org/abs/2505.19403v1 |
a rather small impact on the ICS results for this data set. Regarding the impact of the λparameter, the outlier pattern remains relatively stable when the number of knots is small (less than or equal to 6), especially when looking at the densities from the south of Vietnam in red. For a large number of knots, the obser... | https://arxiv.org/abs/2505.19403v1 |
already existing in the literature, such as median-based approaches (Murph, Strait, et al. 2024), the modified band depth method (Sun and Genton 2011) and MUOD indices (Ojo, Fernández Anta, et al. 2022). Our simulation uses three density-generating processes with 2%of outliers. The scheme named GP_clr , based on model ... | https://arxiv.org/abs/2505.19403v1 |
in (Murph, Strait, et al. 2024) and implemented in the DeBoinR package from (Murph and Strait 2023) using the recommended default parameters. For each combination between a generating scheme and a method, we average the TPR (True Positive Rate, or sensitivity) and the FPR (False Positive Rate, one minus specificity) ov... | https://arxiv.org/abs/2505.19403v1 |
However, these regions cover areas with varied climates. To achieve more climatically homogeneous groupings, we use clusters of provinces based on climatic regions as defined by Stojanovic et al. (2020). Figure 6 displays the three climatic regions covering Northern Vietnam. We focus on region S3, composed of 13 provin... | https://arxiv.org/abs/2505.19403v1 |
characterised by a smaller mass of the temperature values on the interval [5; 20] , compared to the uniform distribution, a mass similar to the uniform on [20; 35] , and a much 15 A P REPRINT - M AY27, 2025 0.81.01.21.4 IC.1IC.2IC.3IC.4IC.5IC.6IC.7IC.8IC.9ICS eigenvaluesselected FALSE TRUE (a) 1987 19871987198719871987... | https://arxiv.org/abs/2505.19403v1 |
IC.2, and correspond to densities with very few days with maximum temperature less than 15 degrees Celsius compared to other densities. 16 A P REPRINT - M AY27, 2025 0.0000.0250.0500.0750.1000.125 10 20 30 40 temperature (deg. Celsius)densityoutlying years 1987 1994 1998 2007 2010 (a) −10010 10 20 30 40 temperature (de... | https://arxiv.org/abs/2505.19403v1 |
an outlying density is often characterised by a behaviour that differs from the other densities in the tails of the distribution. This is not surprising because the Bayes inner product defined by equation Equation 9 involves the ratio of densities which can be large when a density is small (at the tails of the distribu... | https://arxiv.org/abs/2505.19403v1 |
the two reviewers who gave us constructive comments that allowed us to improve our article. Appendix Scatter operators for random objects in a Hilbert space Let us first discuss some definitions relative to scatter operators in the framework of a Hilbert space (E,⟨·,·⟩). We consider an E-valued random object X: Ω→Ewher... | https://arxiv.org/abs/2505.19403v1 |
Bayes spaces approach can be found in (Van Den Boogaart, Juan José Egozcue, and Pawlowsky-Glahn 2014). For the present work, we will identify the elements of a Bayes space, as defined by Van Den Boogaart, Juan José Egozcue, and Pawlowsky-Glahn (2014), with their Radon–Nikodym derivative with respect to a reference meas... | https://arxiv.org/abs/2505.19403v1 |
propose a basis of zero-integral splines in L2 0(a,b)that are called ZB-splines. The corresponding inverse images of these basis functions by clr are called CB-splines. A ZB-spline basis, denoted by Z={Z1,...,Zk+d−1},is characterised by the spline of degree less than or equal to d (orderd+ 1), the number kand the posit... | https://arxiv.org/abs/2505.19403v1 |
B[y]B⟩Rp =⟨Covwℓ(GB[X]B)[x]B,[y]B⟩Rp =⟨Covwℓ([X]B)GB[x]B,GB[y]B⟩Rp,(14) where Equation 14 (1) comes from the equation Equation 4, and the equalities Equation 14 (2) and Equation 14 (3) come from the affine equivariance of Covwℓ. Proposition 3. Let us decompose S1[X]−1(X−EX)over the basis H, which is orthonormal in (E,⟨... | https://arxiv.org/abs/2505.19403v1 |
Lei, Xinyi, Zhicheng Chen, and Hui Li (July 2023). “Functional Outlier Detection for Density-Valued Data with Application to Robustify Distribution-to-Distribution Regression”. In: Technometrics 65.3, pp. 351–362. ISSN : 0040-1706. DOI: 10.1080/00401706.2022.2164063. (Visited on 03/27/2024) (cit. on p. 2). Li, Bing et ... | https://arxiv.org/abs/2505.19403v1 |
“Clustering in high dimension for multivariate and functional data using extreme kurtosis projections”. eng. PhD thesis. Universidad Carlos III de Madrid. DOI: 10016/25286. (Visited on 03/14/2024) (cit. on p. 2). Rousseeuw, Peter (1985). “Multivariate Estimation with High Breakdown Point”. en. In: Mathematical Statisti... | https://arxiv.org/abs/2505.19403v1 |
arXiv:2505.19438v1 [math.ST] 26 May 2025Sampling from Binary Quadratic Distributions via Stochastic Localization Chenguang Wang* 1 2Kaiyuan Cui* 3Weichen Zhao4Tianshu Yu1 Abstract Sampling from binary quadratic distributions (BQDs) is a fundamental but challenging problem in discrete optimization and probabilistic infe... | https://arxiv.org/abs/2505.19438v1 |
Tweedie, 1996). Intuitively, rather than di- rectly sampling from the continuous target distribution, SL achieves sampling by iteratively adding Gaussian noise to posterior estimates of the target distribution. This results in posterior distributions that are Gaussian convolutions of the target distribution which effec... | https://arxiv.org/abs/2505.19438v1 |
al., 2022). Our experimental results demonstrate that SL consistently im- proves the sampling performance of all three samplers across every dataset, providing strong empirical support for our theoretical guarantees.2. Preliminaries In this section, we introduce key concepts and notations that will be used throughout t... | https://arxiv.org/abs/2505.19438v1 |
3. Binary Quadratic Sampling via Stochastic Localization In this section, we present the SL-based sampling frame- work for binary quadratic distributions. Based on SL frame- work introduced in Section 2.3, estimating the posterior expectation uα t(y) =R xqα t(x|y)dxis crucial for sampling. The most straightforward appr... | https://arxiv.org/abs/2505.19438v1 |
case (Martinelli et al., 1994; Martinelli & Olivieri, 1994; Keri- mov, 2012). Intuitively, when this field becomes sufficiently large, the quadratic interaction term becomes negligible in comparison, and the posterior distribution approximates to: qα t(x|y)≈∝e⟨x,b+α(t)Yt σ2t⟩. This simplifies to a product of independen... | https://arxiv.org/abs/2505.19438v1 |
stochas- tic dynamics satisfying Poincar ´e inequality exhibit polynomial mixing time. 4 Sampling from Binary Quadratic Distributions via Stochastic Localization Glauber Dynamics Glauber dynamics, or Gibbs sam- pling, provides a fundamental approach for sampling from BQDs. The algorithm iteratively updates each spin ib... | https://arxiv.org/abs/2505.19438v1 |
for all x= ˆxoff j: P(xi|x)−P(ˆxi|ˆx) ≤CLip(β, h)|βWijxixj|, where CLip(β, h)decreases exponentially to 0 as the exter- nal field |h|tends to infinity. Then for the Gibbs measure (13) satisfying the large field Condition 4.1, the following Poincar ´e inequality holds: Varνβ,h(f)≤1 1−CLip(β, h)|h|EMH(f, f), (18) where E... | https://arxiv.org/abs/2505.19438v1 |
optimization problems with the form of binary quadratic distributions: maximum independent set, maximum cut, and maximum clique problems, each contain- ing multiple diverse datasets (14 datasets in total). Detailed information about these problems and datasets is provided in Appendix D.1. Experimental Settings We emplo... | https://arxiv.org/abs/2505.19438v1 |
+ MCMC sampler under identical MCMC steps. Results on MIS Table 1 shows the performance compar- ison on MIS problems across five datasets, including four Erd˝os-R ´enyi random graphs with different densities (0.05- 0.25) and instances from SATLIB dataset. For each MCMC sampler, our SL variant consistently achieves bett... | https://arxiv.org/abs/2505.19438v1 |
DMALA87.387.487.587.6Ratio MAXCLIQUE RB GWG PAS DMALA0.0160.0180.0200.0220.0240.026Ratio +1.019e2MAXCUT ER-256-300 GWG PAS DMALA0.0400.0450.0500.0550.0600.0650.0700.075Ratio +1.001e2MAXCUT ER-512-600 GWG PAS DMALA99.96299.96499.96699.96899.97099.97299.974Ratio MAXCUT BA-256-300 GWG PAS DMALA100.78100.80100.82100.84100.... | https://arxiv.org/abs/2505.19438v1 |
guarantees are particularly general, encom- passing both Glauber dynamics and Metropolis-Hastings based discrete MCMC samplers without restrictive assump- tions on the underlying distributions. Extensive experiments on QUBO problems demonstrate that SL consistently im- proves the sampling efficiency of various discrete... | https://arxiv.org/abs/2505.19438v1 |
Distributions via Stochastic Localization Chen, Y . and Eldan, R. Localization schemes: A frame- work for proving mixing bounds for markov chains. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , pp. 110–122. IEEE, 2022. Dai, H., Singh, R., Dai, B., Sutton, C., and Schuurmans, D. Learning ... | https://arxiv.org/abs/2505.19438v1 |
Advances in neural information process- ing systems , 33:6840–6851, 2020. Huang, B., Montanari, A., and Pham, H. T. Sampling from spherical spin glasses in total variation via algorithmic stochastic localization. arXiv preprint arXiv:2404.15651 , 2024. Huang, X., Dong, H., Yifan, H., Ma, Y ., and Zhang, T. Re- verse di... | https://arxiv.org/abs/2505.19438v1 |
Hochreiter, S., and Lehner, S. A diffusion model framework for unsupervised neural combinatorial optimization. In Forty-first International Conference on Machine Learning, ICML 2024, Vienna, Austria, July 21-27, 2024 , 2024. Sly, A. and Sun, N. The computational hardness of counting in two-spin models on d-regular grap... | https://arxiv.org/abs/2505.19438v1 |
the gradient information and updates all dimensions in parallel by factorizing the joint distribution. When second-order information of the target distribution is available, (Zhang et al., 2012; Rhodes & Gutmann, 2022; Sun et al., 2023a) leverage the Gaussian integral trick (Hubbard, 1959) to transform discrete samplin... | https://arxiv.org/abs/2505.19438v1 |
space equipped with the Borel field B, and the Gibbs measures νon spin space ET. Glauber Dynamics is a Markov process of pure jumps. If the configuration at present is x, then at each site i, it will change to xiaccording to the conditional distribution ν(xi|x∼i). The generator of Glauber dynamics is LGDf(x) =X i∈T(Eν[... | https://arxiv.org/abs/2505.19438v1 |
Dobrushin interdependence matrix C= (cij)i,j∈Tdefined in (26) (which is an eigenvalue of C by the Perron–Frobenius theorem). If rsp(C)<1, then (1−rsp(C))ν(f, f)≤EνX i∈Tνi(f, f)∀f∈L2(ET, ν), where ν(f, g)denotes the covariance of f,gunder ν, and νi(f, g) =νi(fg)−νi(f)νi(g)is the conditional covariance of f,gunder νiwith... | https://arxiv.org/abs/2505.19438v1 |
k̸=i,jWikxk−β 2Wijxj+h ≤2 e3h0 2+e−3h0 2=2 e3|h| 4+e−3|h| 4. Hence, for 2βsup i∈TX k̸=i|Wik| ≤ |h|, we get W1,d(νβ,h(xi|x∼i), νβ,h(xi|ˆx∼i)) = sup A∈B|νβ,h(A|x∼i)−νβ,h(A|ˆx∼i)| ≤2 e3|h| 4+e−3|h| 4 β 2Wij(xj−ˆxj) ≤2β|Wij| e3|h| 4+e−3|h| 4, which implies cij:= sup x=yoffjW1,d(νi(· |x), νi(· |y)) d(xj, yj)≤2β|Wij| e3|h|... | https://arxiv.org/abs/2505.19438v1 |
i=1Z Ei(f(xyi)−f(x))P(xyi|x) =TX i=1(f(xi)−f(x))Ψ(xi|x) min{1,π(xi)Ψ(x|xi) π(x)Ψ(xi|x)}, Define the local kernel as follows νi(x, xi) =Ψ( xi|x) min{1,νβ,h(xi)Ψ(x|xi) νβ,h(x)Ψ(xi|x)}. Obviously, νi(x, Ei) = 1 for any x, then η= 1. ForB={+1,−1}N,i̸=j ci(j) = sup x=ˆxoffj|νi(x,·)−νi(ˆx,·)|TV= sup x=ˆxoffjsup B|νi(x, B)−νi... | https://arxiv.org/abs/2505.19438v1 |
2h0 −3h0≤βX k̸=i,jWikxk±βWijxj−h≤ −h0, and for h=−2h0 h0≤βX k̸=i,jWikxk±βWijxj−h≤3h0. 24 Sampling from Binary Quadratic Distributions via Stochastic Localization Putting everything together we get for |h|= 2h0 βX k̸=i,jWikxk±βWijxj−h ≥h0. For|y| ≥h0 0≤d dy1 1 +ey ≤1 (eh0 2+e−h0 2)2. Because for any i∈[N],0<Ψ(xi|x)≤1,... | https://arxiv.org/abs/2505.19438v1 |
comprehensive ablation studies on various components of our framework, such as the α-schedule in SL and the influence of hyperparameters on sampling performance and convergence properties. D.1. QUBO Instances In this section, we describe the form of Binary Quadratic Distributions for the QUBO instances used in our expe... | https://arxiv.org/abs/2505.19438v1 |
number of steps Nminto each time step. Then, we distribute the remaining steps Ntot−KN minexponentially according to Nt=Nmin+c·rt/K, where ris the decay rate and cis a normalization constant ensuring the sum constraint. To handle discretization effects, we floor the continuous schedule to integers and carefully distrib... | https://arxiv.org/abs/2505.19438v1 |
and GEOM(2,1) demonstrate superior performance over the CLASSIC schedule (defined as α(t) =t) in MIS problems. However, this impact becomes negligible in MaxCut instances, where all schedules perform similarly with variations within 0.1%. Second, the number of SL iterations K shows a problem-dependent sensitivity patte... | https://arxiv.org/abs/2505.19438v1 |
Let Nbe the dimension of the problem (number of binary variables), Tbe the total number of SL iterations, and Mbe the total budget of DMCMC steps across all SL iterations. E.1. Baseline DMCMC Complexity For standard DMCMC methods used as baselines in our study, such as Gibbs With Gradients (GWG) (Grathwohl et al., 2021... | https://arxiv.org/abs/2505.19438v1 |
σ=10 100.16 ±0.12 100.17 ±0.12 100.14 ±0.13Ablation SL-GWG SL-PAS SL-DMALA CLASSIC 100.08 ±0.14 100.12 ±0.14 100.06 ±0.14 GEOM(1,1) 100.10 ±0.14 100.12 ±0.13 100.07 ±0.14 GEOM(2,1) 100.10 ±0.13 100.12 ±0.14 100.08 ±0.14 K=256 100.09 ±0.14 100.12 ±0.14 100.08 ±0.14 K=512 100.09 ±0.14 100.12 ±0.14 100.07 ±0.14 K=1024 100... | https://arxiv.org/abs/2505.19438v1 |
MaxClique Method ER-0.05 ER-0.10 ER-0.20 ER-0.25 SATLIB RB TWITTER DMALA 23.873 35.511 77.866 103.636 188.901 3025.712 120.634 SL-DMALA 24.997 37.621 78.494 104.367 200.743 3013.018 117.597 PAS 32.648 44.558 86.655 112.305 377.038 3049.668 125.222 SL-PAS 36.134 49.425 90.428 115.630 388.979 3041.619 125.502 GWG 27.249 ... | https://arxiv.org/abs/2505.19438v1 |
arXiv:2505.19824v1 [math.ST] 26 May 2025Weighted Tail Random Variable: A Novel Framework with Stochastic Properties and Applications Sarikul Islam∗and Nitin Gupta∗∗ ∗,∗∗Department of Mathematics, Indian Institute of Technology Kharagpur, 721302, West Bengal, India. Email address:∗sarikul phdmath@kgpian.iitkgp.ac.in,∗∗n... | https://arxiv.org/abs/2505.19824v1 |
probability models derived from the logit of the beta distribution, highlighting the beta-normal distribution as a particular case. Their research demonstrated flex- ibility in modeling symmetric, skewed, and bimodal data, su pported by maximum likelihood estimates and empirical analysis. Some researchers have pr oposed... | https://arxiv.org/abs/2505.19824v1 |
X; see Bon and Illayk [4]. We denote the corresponding random variable by /tildewideX. Hence, if w(x)=x,x≥0, then the framework constructs the equilibrium random variable /tildewideX. One may refer to Bon and Illayk [4], Gupta [9], Gupta and Sankaran [10], Li and Xu [14], Nanda et al. [18], and the references therein f... | https://arxiv.org/abs/2505.19824v1 |
cumulative distribution function (CDF) by FX(·); the SF by FX(·)=1−FX(·); the failure rate function by rX(·)=fX(·)/FX(·); the reversed failure rate function by /tildewiderX(·)= fX(·)/FX(·); the mean residual life (MRL) function by mX(·)=/integraltext∞ ·FX(x)dx/FX(·); the support of the PDF fX(·)of the variable XbySX=(l... | https://arxiv.org/abs/2505.19824v1 |
[decreasing] on SX. One may refer to Shaked and Shanthikumar [21] and the referen ces therein for details of stochastic orderings and stochastic aging/reliability no tions. The following facts on aging properties of a random variable are useful: (see Misra et al. [17]). (i) If fX(·)is log-concave, then Xhas the IFR pro... | https://arxiv.org/abs/2505.19824v1 |
function of Xis established in Liu [15]. The formula is given by the follow ing lemma: (see Theorem 1 of Liu [15]). Lemma 4. Let the continuous variable Xhas support (lX,uX)⊂R. Let a differentiable function w(·)be defined on (lX,uX)with its first derivative absolutely integrable with respec t toX. IflXis real-valued, then... | https://arxiv.org/abs/2505.19824v1 |
some random variables more straightforwardly, despite their survival or failure rate functions not admitting closed-form expressions. The following theorem assumes Xis IFR [DFR] and w′(·)is log-concave [log-convex] to ensure that Xwbelongs to the ILR [DLR] aging class. 8 Proposition 1. If (a)Xis IFR [DFR], and (b)w′(·)... | https://arxiv.org/abs/2505.19824v1 |
the conditions (i)-(iii) of Lemma 2 for Z1andZ2 to show that∆1≥0. Since lX−θ<x1<x2<uX−θ<uX, it follows that SZ1∩SZ2=(lX−θ,uX−θ)∩(lX,uX)=(lX,uX−θ)/nequal∅. Now define φ2(x,y)=w′(x+θ)w′(y) rX(x+θ)rX(y)I(x1≤x<uX−θ)I(x2≤y<uX), and φ1(x,y)=w′(x+θ)w′(y) rX(x+θ)rX(y)I(x2≤x<uX−θ)I(x1≤y<uX), where (x,y)∈SZ1×SZ2. Then, utilizing ... | https://arxiv.org/abs/2505.19824v1 |
MRL function mX(·)to ensure that Xwhas the DMRL property. Theorem 3. LetXis DMRL. If w′(·)/rX(·)is increasing and log-concave on SXandmX(·)is log-convex on SX, then Xwis IFR and hence DMRL. Proof. By applying Theorem 2.5(a) of Misra et al. [17], which assume s the DMRL property ofXand the log-convex property of mX(·)on... | https://arxiv.org/abs/2505.19824v1 |
Consequently, X≤f rYandw′ 2(·)/w′ 1(·)is increasing. Observe that, l1=l2=0andu1=u2=∞. Now, by applying Theorem 5, it follows that Xw1≤lrYw2. In Theorem 5, if we set w1(x)=w2(x)=x, then the WTRVs Xw1andYw2becomes /tildewideXand/tildewideY ofXandY, respectively. Then we have the following corollary. 14 Corollary 2. Suppo... | https://arxiv.org/abs/2505.19824v1 |
fact that w′ 1(·)/w′ 2(·)is increasing on SX∩SY, we obtain FY(t)≥FX(t)for all t∈S1∩S2=SX∩SY. Hence, X≤stY. 16 Example 6. LetXandYhave CDFs FX(x)=1−e−2x,x>0and FY(x)=1−e−x,x>0, respectively. Let w1(x)=x2andw2(x)=x,x>0. Then observe that w′ 1(x) w′ 2(x)=2x,x>0, is increasing. Furthermore, the function FYw2(x) FXw1(x)=2e2... | https://arxiv.org/abs/2505.19824v1 |
1 for the independ ent variable XandY. Since SX∩SY=(lY,uX)/nequal∅, it follows that∆φ21(x,y)≥0for every (x,y)∈SX×SYwith x≤y. Using the non-negative and decreasing properties of w′ 1(·)/rX(·)onSX, and non-negative and increasing properties of w′ 2(·)/rY(·)onSY, we have for each fixed y∈SY, the∆φ21(x,y)is decreasing in xo... | https://arxiv.org/abs/2505.19824v1 |
the condition of Theorem 5 is not satisfied. The PDFs of Xw1andYw2are given by fXw1(x)=ex(1−x) e−2,0<x<1,and fYw2(x)=e−x,x>0. Now, consider the function fYw2(x) fXw1(x)=e−2x(e−2) 1−x,0<x<1. We plot fYw2(·)/fXw1(·)to demonstrate that it is not increasing. Therefore, the lik elihood ratio order Xw1≤lrYw2does not hold. 0.0... | https://arxiv.org/abs/2505.19824v1 |
real-world data using the random variable sX,Xw, and the Beta distribution, and assess their goodness-of-fit metrics by employing the me thod of maximum likelihood estimation (MLE), followed by statistical tests to compare their fitting performance. Poondi Kumaraswamy introduced a family of double-bounded continu ous pro... | https://arxiv.org/abs/2505.19824v1 |
distributions employing the same o ptimisation technique. Table 2 presents the descriptive statistics of monsoon rainfall da ta (in millimeters) from Northwest (NW) and Northeast (NE) India, while Table 3 shows the estima ted parameters of the fitted probability models for these datasets. Table 2: Descriptive statistics... | https://arxiv.org/abs/2505.19824v1 |
met rics than the input random variable of the framework, as evidenced by the weighted Kumaraswamy d istribution, which exhibits a superior goodness-of-fit compared to the original Kumaras wamy probability model. Future research may comprehensively investigate the proposed fra mework, with particular emphasis on the tai... | https://arxiv.org/abs/2505.19824v1 |
arXiv:2505.19835v1 [math.ST] 26 May 2025On a retarded stochastic system with discrete diffusion modeling life tables Tom´ as Caraballo1, Francisco Morillas2, Jos´ e Valero3 1Universidad de Sevilla, Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico Apdo. de Correos 1160, 41080-Sevilla Spain E.mail: caraba... | https://arxiv.org/abs/2505.19835v1 |
that we are interested in studying variables like the probability of death, which take values in the interval [0 ,1]. Although the results given by the numerical simulations are fine from the qualitative point of view, it was necessary to make a correction of the estimates by using the average annual improvement rate. ... | https://arxiv.org/abs/2505.19835v1 |
of global positive solutions. Lemma 1 Assume that gr(t)≥0,for all randt,and that αi≤1, for all i. Then for any ϕ∈C([−h,0],Rm +) there exists a unique globally defined solution u(·)such that u(t)∈Rm +almost sure for t≥τ. Proof. The existence of a unique local solution to problem (5) follows from standard results for fun... | https://arxiv.org/abs/2505.19835v1 |
conditions are as well. Lemma 3 Assume that gr(t)∈[0,1],for all randt,and that αi≤1, for all i. Then, for any ϕ∈ C([−h,0],Rm +)such that ϕi(s)∈(0,1), for any i∈Dands∈[−h,0], the unique solution u(·)to (7) satisfies almost surely that ui(t)∈(0,1)for all i∈Dandt≥τ. Proof. The existence and uniqueness of local solution is... | https://arxiv.org/abs/2505.19835v1 |
small enough such that h <1 2(1−b2)log 1−b2 1−δ(h0)−b2 , the fact that δ(h)is non-increasing implies that h <1 2(1−b2)log 1−b2 1−δ(h)−b2 . Remark 6 Letf(λ) = 2(1 −δ(h)−b2)eλh−λ. Condition (11) implies that f(2(1−b2))<0. Then choosing λ∗close enough to 2(1−b2)we have that f(λ∗)<0, soλ∗> L(h, λ∗). Proof. We define th... | https://arxiv.org/abs/2505.19835v1 |
0eλsX i∈DIi(s)dwi(s) and, taking expectations, we infer eλtE∥u(t)−u∥2 Rm≤E∥u(0)−u∥2 Rm+2b2∥u∥2 Rm λeλt+ 2(1−δ(h)−b2)Zt 0sup s∈[−h,0]E∥u(l+s)−u∥2 Rmeλldl. Again, the last term in the previous equation is well defined thanks [31, Lemma 2.3, P. 150]. Notice that as the solutions belong to (0 ,1) almost surely, the term in... | https://arxiv.org/abs/2505.19835v1 |
[34] a non-parametric dynamic model, based on kernel smoothing techniques to estimate mortality rates, was proposed. This model avoids rigid functional assumptions and uses a system of non-local differential equations to approximate the evolution of qt xover time. Although it successfully reproduces qualitative feature... | https://arxiv.org/abs/2505.19835v1 |
( t):qt x,x∈D={0, . . . , 100}, t∈ {1908, . . . , 2018}. Also, we consider the values of gx(t), which is the rate of death either at ”negative ages” or after the actuarial infinite (we have chosen it equal to 100). 2. We estimate the improvement rates for each age and delay, that is: year delay1 2 3 4 ... 109 110 r1909... | https://arxiv.org/abs/2505.19835v1 |
consider only integer delays, we can discretize the interval [0 , h] by using a finite number of integer delays s={0,1, . . . , d max}, where dmax=h≤n−1, which is the maximum delay to be considered. Thus, instead of (14) we will use the discretized probability function f∗(s) =ˆfλ(s) Pdmax s=0ˆfλ(s), (15) which approxim... | https://arxiv.org/abs/2505.19835v1 |
defined as follows. We consider a finite set of ages A={−m,−m+ 1, ..., x, . . . , M −1, M}, where 0 ≤x≤100, m≥0, M≥100. We define the set of distances Dx={d−m, d−m+1, . . . , d 0, d1, . . . , d M}={x+m, x +m−1, ...,1,0,1, ..., M −x−1, M−x}. Then, we define the truncated gaussian kernel bKb(·) as bKb(k) =Kb(k)P ξ∈DKb(ξ)... | https://arxiv.org/abs/2505.19835v1 |
we put It c,1−α= # x:qt,obs x̸∈IC1−α . (19) For each year It c,1−αsummarizes the number of ages of the observed data at year tthat do not belong to theα−synthetic confidence interval, α∈[0,1]. We will use 1 −α= 0.98,0.90 and 0 .80. Central measures and variability. It is important to point out that a stochastic model ... | https://arxiv.org/abs/2505.19835v1 |
3a-3c show, for the 5-year horizon of forecasting, if the observed rates belong or not to the intervals IC0.98,IC0.90andIC0.80forb= 0.1 and b= 0.025. 16 0 20 40 60 80 100−10 −8 −6 −4 −2 0 AgeMortality rates Observed rates Mean of NLSD NLSD−IC0.98(a) Confidence Level: 1 −α= 0.98 0 20 40 60 80 100−10 −8 −6 −4 −2 0 AgeMor... | https://arxiv.org/abs/2505.19835v1 |
Complementarialy, we can use the quantitave measures to determine the goodness of each model and to compare them. We start with the count indicators. Table 1 shows the number of ages (for each year into the period of the validation) that do not belong to the confidence interval, IC0.98, and for each of the evaluated me... | https://arxiv.org/abs/2505.19835v1 |
is the most accurate method in four out of five years, while RH performs best in one (although the error value for NLSD is nearly identical to that of RH in that case). The LC model does not achieve the lowest error in any of the five years for either indicator. Central measures are also very useful to assess the accur... | https://arxiv.org/abs/2505.19835v1 |
−2 0 AgeMortality rates NLSD LC RHObserved Rates Mean of NLSD Ensemble of Realizations Figure 7: Mean trajectories 22 From a qualitative point of view, it is worth noting that the analysis of historical time series indicates a decreasing intensity of the social hump over time. In this regard, the NLSD model exhibits a ... | https://arxiv.org/abs/2505.19835v1 |
comparison across different regions and time periods. From a technical perspective, it would be valuable to incorporate optimization techniques for parameter estimation and to assess the applicability of cross-validation strategies. Acknowledgements. The research has been partially supported by the Spanish Ministerio d... | https://arxiv.org/abs/2505.19835v1 |
25 November 2009 on the taking-up and pursuit of the business of Insurance and Reinsurance. Available online: https://eur-lex.europa.eu/ [on-line: 2020/12/6]. [18] Eurostat ,Population projections in the EU. Statistics Explained, 2024, March 4, https://ec.europa.eu/eurostat/statistics-explained/index.php?oldid=596339 [... | https://arxiv.org/abs/2505.19835v1 |
arXiv:2505.20005v1 [math.ST] 26 May 2025Existence of the solution to the graphical lasso Jack Storror Carter Dept. of Economics and Business, Universitat Pompeu Fabra, Spain Data Science Center, Barcelona School of Economics, Spain Abstract The graphical lasso (glasso) is an l1penalised likelihood estimator for a Gauss... | https://arxiv.org/abs/2505.20005v1 |
for these existence results will be provided that do not utilise the du al optimisation problem, but instead show how the objective function acts when certain eigenva lues are allowed to tend to infinity. The idea of these proofs can be extended to any penalty function t hat is separable in the entries of the precision ... | https://arxiv.org/abs/2505.20005v1 |
the MLE for positive definite and positive semidefinite S. While these results hardly need proving, the existence of the glasso and odglasso for positive definite Seasily follow, they provide simple examples of the style of proofs that will be used for the glasso and odglasso, and understanding why the MLE does not exist ... | https://arxiv.org/abs/2505.20005v1 |
ij. All terms in the full penalty function are non-negative, so removing the off-diagonal penalty terms obtains the upper bound G(σ,W|λ,V)≤p/summationdisplay i=1log(σi)−σi/parenleftBiggp/summationdisplay j=1λj(wT ivj)2+ρw2 ij/parenrightBigg , Since the eigenvectors w1,...,w pare orthonormal, for each i= 1,...,p there ex... | https://arxiv.org/abs/2505.20005v1 |
already see n that both must have at least two non-zero entries. If w1,w2both have exactly two non-zero entries in the same position, then, since w1,w2are in the null space of S, all non-null space eigenvectors of Smust be equal to 0 in these two entries by orthogonality. This would r esult inShaving a diagonal entry e... | https://arxiv.org/abs/2505.20005v1 |
boundeded or sub-logarithmic penalty is preferred for t he off-diagonals, the solution will still exist for all positive semidefinite Sas long as it is paired with a suitably strong penalty on the diagonal. The diagonal penalty could be allowed to depend on the samp le size of the data in such a way that it disappears wh... | https://arxiv.org/abs/2505.20005v1 |
Kernel Ridge Regression with Predicted Feature Inputs and Applications to Factor-Based Nonparametric Regression Xin Bing∗Xin He†Chao Wang‡. May 27, 2025 Abstract Kernel methods, particularly kernel ridge regression (KRR), are time-proven, powerful non- parametric regression techniques known for their rich capacity, ana... | https://arxiv.org/abs/2505.20022v1 |
function space, as any continuous function can be approximated arbitrarily well by an intermediate function in the RKHS under the infinity norm (Micchelli et al., 2006; Steinwart, 2005). More importantly, the representer theorem (Kimeldorf and Wahba, 1971) ensures that ef in (1) admits a closed-form solution and can be... | https://arxiv.org/abs/2505.20022v1 |
the dependence between XandZ. Example 2 (Feature extraction via autoencoder) .Unlike PCA, an autoencoder is a powerful al- ternative in machine learning and data science for constructing a nonlinear low-dimensional repre- sentation of high-dimensional features (Goodfellow et al., 2016). Specifically, an autoencoder lea... | https://arxiv.org/abs/2505.20022v1 |
Xi)}n i=1∈(R,X)nwithX ⊆ Rpand the response Yigenerated according to the following non-parametric regression model Y=f∗(Z) +ϵ, (4) where Z∈ Z ⊆ Rris some random, unobservable, latent factor with some r∈N,f∗:Z → R is the true regression function, and ϵis some regression error with zero mean and finite second moment σ2<∞.... | https://arxiv.org/abs/2505.20022v1 |
α∈[0,1], which is related to but weaker than (6). Alternatively, Fischer and Steinwart (2020); Zhang et al. (2023) handle f∗/∈ H Kby relying on the smoothness condition f∗=Lr Kg∗for some g∗∈ L2(ρ) and some r∈(0,1/2] as well as the upper bound requirement in condition (6). The other proof strategy for analyzing KRR adop... | https://arxiv.org/abs/2505.20022v1 |
eigenvalues or eigenfunctions of the RKHS hold. As explained in the next section, our new framework of analyz- ing KRR removes such restrictions under the squared loss in (1), while remaining applicable to the analysis of general convex loss functions. More substantially, when the feature inputs Ziin (1) must be predic... | https://arxiv.org/abs/2505.20022v1 |
regularity or decaying conditions on µj’s, such as those in (6), in contrast to existing integral operator approaches, for instance, Caponnetto and De Vito (2007); Fischer and Steinwart (2020); Rudi and Rosasco (2017); Steinwart et al. (2009). In Section 3.2.1 we further derive more transparent expression of E∆bg. For ... | https://arxiv.org/abs/2505.20022v1 |
used in place of the squared loss, as demonstrated in Section 5. Another fundamental difficulty in our analysis arises from handling predicted feature inputs. As noted in the end of Section 1.1, the primary challenge lies in establishing a precise connection between the local Rademacher complexity and the kernel comple... | https://arxiv.org/abs/2505.20022v1 |
risk are later given in Bing et al. (2021), where a general linear predictor of Xi, including 8 Figure 1: Proof strategy of relating local Rademacher complexity to kernel complexity R(δ). PCA as a particular instance, is used to predict Zi. In a more recent work (Fan and Gu, 2024), the authors investigate the predictiv... | https://arxiv.org/abs/2505.20022v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.