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adequate reducing the need for large -scale demand response capacity . Scenario B1 resembles the traditional situation where power generation primarily from large controllable synchronous generators adjust in real -time to match demand and the electricity flow towards the customers . Scenario B4, the Highly distributed... | https://arxiv.org/abs/2503.08232v1 |
two ways. First, it estimates the probabilities of different grid management scenarios based on the power generation, power storage , and demand response capacities. Second, it employ s a Bayesian Networks approach to model these dependencies. This innovation enables the utili sation of the constructed Bayesian network... | https://arxiv.org/abs/2503.08232v1 |
, are interconnected by arcs that delineate the conditional dependencies between these nodes as depicted in Fig. 2 [12], [13], [14]. An a rc represents either a statistical or causal dependency between the nodes . For the purposes of this study, a pre-defined four-layer hierarchical structure (cf. Fig. 2a) has bee n ch... | https://arxiv.org/abs/2503.08232v1 |
true , i.e., the study applies the Noisy -OR approach . This implies , for instance , that a specific component of power generation capacity mix can theoretically trigger the bulk capacity on its own, without the need for other components, allowing E q. 1 to be applied. However, the computation al effort required for l... | https://arxiv.org/abs/2503.08232v1 |
domestic bulk generation power capacity, assuming its value will be greater or equal to 13 GW in 2035. Both capacities are situated at layer L3. The trigger values, i.e., the discreti sation levels of 5 GW and 13 GW, were selected based on the findings from the literature review (Table 1) to approximately represent the... | https://arxiv.org/abs/2503.08232v1 |
model and presents the findings from the target optimi sation analysis conducted with this model . 4.1 Review of existing 2035 scenarios for the Finnish electricity system As mentioned already , multiple electrification scenarios have been devised in Finland for 2035, 2045 and 2050 to assess es alternatives for the pow... | https://arxiv.org/abs/2503.08232v1 |
coding since it is not domestic and might not be an option even though there would high demand for electricity in Finland. The t wo columns on the right provide a summary of the controllable domestic bulk generation (green background colour) and the new cost-effective large -scale domestic balancing power (blue colour)... | https://arxiv.org/abs/2503.08232v1 |
which started regular elect ricity production in 2023) and the balancing power capacity 1,3 GW [25],[26],[27]. By 2035, the mean capacities are projected to be 13,8 GW for controllable domestic bulk generation and 13,2 GW for new large scale balancing power (the second row from the bottom). However, these figures exhib... | https://arxiv.org/abs/2503.08232v1 |
capacity mix, given the multitude of other external factors that can impact the power generation capaci ty mix components. Policy and incentives have a substantial impact on the development of small -scale nuclear capacity, as it was listed 12 out of 15 panellists . This is expected, as both regulatory framework and pu... | https://arxiv.org/abs/2503.08232v1 |
more. Notably the Leak has the second highest impact after DSR. This suggests that, according to the experts, there may be other balancing power solutions or reasons not included in the analysis that could affect the total balancing power capacity. Based on the Question Set 3b, examples given by experts for missing sol... | https://arxiv.org/abs/2503.08232v1 |
13; Balance< 5 22,9 16,5 29,4 31,2 Bulk < 13; Balance≥ 5 24,9 20,8 39,5 14,8 Bulk≥ 13; Balance < 5 31,0 26,2 30,3 12,5 Bulk ≥ 13; Balance ≥ 5 53,2 11,9 26,7 8,2 Mean Probability 33,0 18,9 31,5 16,7 4.3. Bayesian Network construction and analysis Fig. 8 illustrates the structure of the constructed hierarchical Bayesian ... | https://arxiv.org/abs/2503.08232v1 |
power generation capacity mix components, as well as for total domestic 13 controllable bulk generation and new cost -effective domestic large -large balancing power capacities , are outcomes of the ICI process described in Chapters 2 and 3 and are indicated in the nodes of Fig. 8. In Bayesian statistics a posteriori d... | https://arxiv.org/abs/2503.08232v1 |
capacity mix components through the ICI approach and constructed Bayesian Network (a posteriori estimations in Fig. 8). As seen from the Table 4, Wind, Large -Scale Nuclear, Solar, and DSR capacities increase by 2 ,4 GW, 1 ,5 GW, 1 GW, and 0 ,7 GW, respectively , while the capacities of pumped hydro and batteries sligh... | https://arxiv.org/abs/2503.08232v1 |
Top-down grid is maximi sed by applying the following equation for each power generation capacity mix componen t: Score = w1 x Ii x w2 x Ei x w3 x 1 /Ci Eq. 2 where : • Score is the value which will be maximi sed • Ii is the impact (effect) size (%) of an power generation capacity mix component Xi in the Bayesian Netwo... | https://arxiv.org/abs/2503.08232v1 |
be adjusted down according to power grid needs, the implementation of DSR capacity could be fairly inexpensive; otherwise, its cost could be prohibi tively high. Finland having currently peak power of about 15 GW [33] and committed DSR capacity of 1,3 GW (c.f. Table 1), additional DSR capacity of several GWs is not pos... | https://arxiv.org/abs/2503.08232v1 |
in 2023 [35]. For peak season, the availability assumptions are largely similar to those for peak hour but exclude battery, pumped hydro, and P2X -X2P storage solutions, which might lack the capacity for sustained output over several weeks. Battery limitations relate to energy density and physical size constraints, pum... | https://arxiv.org/abs/2503.08232v1 |
Nuclear 0,3 0,3 0,3 Fossil 0,6 0,5 0,5 Wind 21,4 1,3 1,3 Solar 6,9 0,0 0,0 Battery 1,0 0,9 0,0 Pumped Hydro 0,5 0,5 0,0 Gas 1,9 1,7 1,7 Bio 3,0 2,7 2,7 P2X-X2P 0,7 0,6 0,0 Total Capacity 46,7 17,2 15,2 The results of the target optimisation (Table 5) reveal that the policy makers and industry stakeholders should priori... | https://arxiv.org/abs/2503.08232v1 |
). [2] The Finnish Innovation Fund Sitra , Enabling cost -efficient electrification in Finland , 2021 . https://www.sitra.fi/en/publications/enabling -cost-efficient -electrification -in-finland/ (accessed 9 April 2024 ). [3] Valtioneuvosto (Finnish government) Hiilineutraalisuustavoitteen vaikutukset sähköjärjestelmää... | https://arxiv.org/abs/2503.08232v1 |
, P. Lucas, Th. van der Weide , A generic qualitative characterization of independence of causal influence , International Journal of Approximate Reasoning, vol. 48, no.1 (2008). https://doi.org/ 10.1016/j.ijar.2007.08.012 . [21] L. Zhang , Q. Pan, X. Wu, X. Shi, Building Bayesian networks from GWAS statistics based on... | https://arxiv.org/abs/2503.08232v1 |
(2022 ). https://doi.org/ 10.1016/j.est.2022.105861 . 19 APPENDIX A The Question Sets The questions regarding peak power capacities at hierarchy layer L2, as well as the impact of various variables at hierarchy layer L1 on these capacities, comprise Question Set 1 . It consists of three distinct sub-questions: Question... | https://arxiv.org/abs/2503.08232v1 |
parent variables and the four potential grid management scenarios , which act s as a child node. • Question Se t 4: What are the probabilities for the following scenarios, where their sum is 100%: Scenario B1Top -Down grid, Scenario B2 Decentrally balanced grid, Scenario B3 Centrally balanced grid and Scenario B4 Highl... | https://arxiv.org/abs/2503.08232v1 |
arXiv:2503.08355v3 [math.ST] 20 Mar 2025Pointwise Minimax Vector Field Reconstruction from Noisy O DE Hugo Henneuse1,2 hugo.henneuse@universite-paris-saclay.fr March 21, 2025 Abstract This work addresses the problem of estimating a vector field f rom a noisy Ordinary Differential Equation (ODE) in a non-parametric regres... | https://arxiv.org/abs/2503.08355v3 |
the authors propose regr ession-basedestimation strategies and provide a detailed analysis of their convergence rates, depending on the reg ularity of f. To our knowledge, these works represent the most advanced contributions to the statistical un derstanding of the vector field reconstruction problem. In this paper, we... | https://arxiv.org/abs/2503.08355v3 |
by our main convergence results. Framework Assumption on the vector field. We consider the autonomous Ordinary Differential Equation (ODE) defined by ( 1) and assume that fbelongs to Lip( L,M), the set of L-Lipschitz functions from RDtoRDwith norm bounded by M. LetX ⊂RDbe the space of initial values. By the Cauchy-Lipschi... | https://arxiv.org/abs/2503.08355v3 |
as: wl=6l2 k(k+1)(2k+1). This choice of weights minimizes the variance of the derivative estimat or under the constraint that their sum equals 1. Further details can be found in the proof of Theo rem 1 in De Brabanter et al. (2013). We omit derivative estimation near the temporal boundaries t= 0 and t=Tas the loss of i... | https://arxiv.org/abs/2503.08355v3 |
n/parenrightbigg1 b ,1 m/parenrightigg , where the infimum is taken over all possible estimators of f, andK2is a constant depending only on L,M, a,b,D,T, andσ. By examining the proof of Theorem 1, one can observe that the Gaussian assumption on the noise can be relaxed. Indeed, the result remains valid if the family ( ... | https://arxiv.org/abs/2503.08355v3 |
(2013), which bounds the expectation of the variance and the bias of our derivative estimators conditionally to Sr(x). Lemma 1. We have, for all l∈ {1,...,r}ands∈ {1,...,D} E/bracketleftig/vextendsingle/vextendsingle/vextendsingleE/bracketleftig ˆfs(φ(Xil(x),tjl(x)))|Sr(x)/bracketrightig −fs(φ(Xil(x),tjl(x)))/vexten... | https://arxiv.org/abs/2503.08355v3 |
m+Lexp(LT)C(b)/parenleftigr na/parenrightig1 b /lessorsimilar/parenleftbigg1 nm/parenrightbigg1 b+5 Ifn5/(5+b) mb/(5+b)<1 thenm≥n5/b,r= 1 and k≃m(4+b)/(5+b) n1/(5+b), thus : (5)+(6)+(7)≤C1/2 4m k3 2+C3k m+2MLk m+Lexp(LT)C(b)/parenleftbigg1 na/parenrightbigg1 b /lessorsimilarn3 2(5+b) m3(4+b) 2(5+b)−1+1 n1 5+bm1 5+b+1... | https://arxiv.org/abs/2503.08355v3 |
makes use of the Total Variation distance (TV) between prob ability distributions, which can be stated in our context as the following inequality (adapted from Yu,1997): inf ˆfsup f∈Lip(L,M),µ∈P(a,b)sup x∈Envf(X,T)E/bracketleftig ||ˆf(x)−f(x)||/bracketrightig ≥||f0(x)−f1(x)|| 4(1−TV(Qn 0,Qn 1)) ≥h 4(1−nTV(Q0,Q1)). It... | https://arxiv.org/abs/2503.08355v3 |
phenomenon is also observed in the models studied by Sch¨ otz and Siebel (2024);Sch¨ otz(2025), where classical nonparametric regression methods are adapted t o the setting of noisy ODEs. A key advan- tage of the model they consider is that it mitigates the challenge pos ed by the dependence of observation locations on... | https://arxiv.org/abs/2503.08355v3 |
Harish Bhat, Majerle Reeves, and Ramin Raziperchikolaei. Estimating vector fields from noisy time series. 12 2020. doi: 10.48550/arXiv.2012.03199. Fr´ ed´ eric Chazal, Marc Glisse, Catherine Labru` ere Chazal, and B ertrand Michel. Convergence rates for persistence diagram estimation in topological data analysis. 31st I... | https://arxiv.org/abs/2503.08355v3 |
TransPCA for Large-dimensional Factor Analysis with Weak Factors: Power Enhancement via Knowledge Transfer Yong He∗, Dong Liu†, Yunjing Sun‡, Yalin Wang‡ Early work established convergence of the principal component estimators of the factors and loadings up to a rotation for large dimensional approximate factor models ... | https://arxiv.org/abs/2503.08397v1 |
and He et al. (2025a). 1.1 Closely Related Work Among the extensive studies on the estimation of factor models, few pay attention to the availability of large number of auxiliary source datasets which could be informative for the target factor analysis. Actually, it is growing important to extract information and trans... | https://arxiv.org/abs/2503.08397v1 |
the convergence rates for the weak factors? In this article, we give an affirmative answer and address this challenging problem elegantly by proposing a Transfer Principal Component Analysis (TransPCA) method, which leverages useful information from a large body of panel source datasets to enhance the estimation accura... | https://arxiv.org/abs/2503.08397v1 |
when weak factors exist, the TransPCA estimators outperform the traditional PCA estimators derived solely based on the target dataset under certain assumptions, in terms of achieving remarkably superior convergence rates. Our TransPCA method essentially improves the estimation accuracy of the target factor model by inc... | https://arxiv.org/abs/2503.08397v1 |
Let A⊤, tr(A) and rank( A) be the transpose, the trace and the rank of A, respectively. For any vector a, denote its ℓ2norm as ∥a∥2. We use ⌊x⌋and⌈x⌉to denote the largest previous and smallest following integers of x. For two scalars sequences {an}n≥1and{bn}n≥1, we say an≲bn(an≳bn) if there exists a universal constant ... | https://arxiv.org/abs/2503.08397v1 |
X(0)−F(0)Λ(0)⊤ 2 F, 6 subject to Λ(0)⊤Λ(0)=D, where Dis a diagonal matrix with diagonal elements ( D)ii≍Nαi,i∈[r0]. The solution to the above least squares problem is bΛ(0)=bQ(0)D1/2, where bQ(0)is composed of the top r0eigenvectors of the sample covariance matrix bΣ(0)=1 T0X(0)⊤X(0), bF(0)can be further obtained by di... | https://arxiv.org/abs/2503.08397v1 |
known in advance. To distinguish it from the procedure equipped with useful dataset selection capability in Section 6, we refer to the TransPCA method in this section as Oracle TransPCA. We utilize a weighted average projection matrix to aggregate useful information from auxiliary datasets, which is the primary idea an... | https://arxiv.org/abs/2503.08397v1 |
Output: The estimators of loadings and factors for the target model, bΛ(0) w,bΛ(0) s,bF(0) wandbF(0) s. 9 3 Theoretical Results In this section, we investigate the theoretical properties of the proposed TransPCA estimators. We first give some assumptions. Assumption 1.(Factors) For any k∈ {0} ∪[K], assume that (1)E∥f(k... | https://arxiv.org/abs/2503.08397v1 |
known in advance. In Section 4 and Section 5, we will provide consistent estimators for factors numbers and factor strengths, respectively. Theorem 3.1 shows the error rates in terms of estimating the loading and factor spaces corresponding to weak factors. Theorem 3.1. (General stochastic order for estimation errors) ... | https://arxiv.org/abs/2503.08397v1 |
Proposition 1 yields that 1√ N bΛ(0) w−Λ(0) wHw =√ Nα1 √ N·Op √T0√ N1+α1 TNαs+T0 TNαs+1√ T+1 N+ε! . (3.3) By comparing (3.2) and (3.3), when T0≪NorT0≍N,ε=o maxn√T0√ N1+α1 TNαs,T0 TNαs,1√ T,1 No and Assumption 6 holds, our TransPCA estimator would achieve a faster convergence rate than that of the estimator in Bai and... | https://arxiv.org/abs/2503.08397v1 |
the number of factors There is a large amount of literature investigating eigenvalue-based methods for selecting the number of factors in approximate factor model. For auxiliary dataset X(k), k∈ A , the number of factors rkcan be estimated by IC or PC criteria in Bai and Ng (2002), Eigenvalue Ratio (ER) method in Lam ... | https://arxiv.org/abs/2503.08397v1 |
borrowing idea from Chen and Lam (2024). As mentioned earlier, the factor loading Λ(0)can be written as Λ(0)=Q(0)D1/2, andD= diag( Ds,Dw) is a diagonal matrix, where ( Ds)ii≍N,i∈[r0−s], (Dw)ii≍Nαi,i∈[s]. Define cM=bQ(0)⊤ 1 T0X(0)⊤X(0) bQ(0), where bQ(0)is a column orthogonal matrix composed of the leading r0eigenvect... | https://arxiv.org/abs/2503.08397v1 |
the estimation accuracy by increasing the effective sample size. Theorem 5.1 indicates that the estimation accuracy of the factor strengths does not depend on the sample size of the 17 target dataset T0, and therefore the TransPCA method would not be able to improve estimation accuracy of the factor strengths. 6 Useful... | https://arxiv.org/abs/2503.08397v1 |
s−c2hτ], where c1andc2are constants satisfying 0< c2< c1<1, the weak loading matrix estimator bQ(0) wis a local maximum of (6.2) with probability tending to 1. In Theorem 6.1, we do not require hτto be a constant; instead, we allow it to tend to zero, provided that √ N1−α1δ+ max k∈[K]\AT−1/2 k=o(hτ). As N, min k∈{0}∪[K... | https://arxiv.org/abs/2503.08397v1 |
diag N0.7, N0.6 . We consider all combinations of dimension N∈ {50,100}, T0∈ {50,100},Tk∈ {200,300,400}fork∈[K] and the number of auxiliary datasets K∈ {4,8}. K N T 0TkD(Oracle) D(Target-only) MSE(Oracle) MSE(Target-only) 45050200 0.087(0.011) 0.249(0.024) 0.082(0.009) 0.128(0.011) 300 0.085(0.011) 0.249(0.025) 0.081... | https://arxiv.org/abs/2503.08397v1 |
estimating loading matrices. This further validates that our method inherently improves estimation accuracy by increasing the total effective sample size. 22 Next, we consider the scenario where some of the panel datasets are non-informative, necessitating the useful dataset selection step to avoid negative transfer. F... | https://arxiv.org/abs/2503.08397v1 |
0.095(0.009) 0.278(0.023) 0.382(0.052) 300 0.090(0.009) 0.091(0.024) 0.276(0.024) 0.429(0.061) 400 0.088(0.009) 0.088(0.009) 0.277(0.023) 0.451(0.063) 100200 0.077(0.005) 0.077(0.005) 0.196(0.013) 0.284(0.045) 300 0.071(0.005) 0.071(0.005) 0.197(0.014) 0.337(0.046) 400 0.067(0.005) 0.067(0.005) 0.197(0.013) 0.370(0.057... | https://arxiv.org/abs/2503.08397v1 |
oil crisis in 1979, the global financial crisis in 2008, and the COVID-19 pandemic in 2020. Figure 2 shows the time series of two macroeconomic variables, from which it can be seen that fluctuations occur at the identified change time points emphasized by gray dashed lines. The final segment, from March 2020 to August ... | https://arxiv.org/abs/2503.08397v1 |
0.73(0.047) 0.62(0.046) 100200 0.71(0.041) 0.63(0.043) 0.73(0.034) 0.62(0.034) 300 0.71(0.039) 0.63(0.042) 0.73(0.033) 0.62(0.032) 400 0.71(0.040) 0.63(0.043) 0.72(0.033) 0.62(0.033) 10050200 0.71(0.041) 0.61(0.050) 0.72(0.036) 0.62(0.040) 300 0.71(0.039) 0.61(0.046) 0.72(0.036) 0.62(0.037) 400 0.71(0.044) 0.60(0.046) ... | https://arxiv.org/abs/2503.08397v1 |
Dickey-Fuller test rejects the null hypotheses for all the three panels of series, indicating their stationarity. The Eigenvalue-Ratio (ER) method in Ahn and Horenstein (2013) indicates that ( r0, r1, r2) = (1 ,1,1) for the three preprocessed datasets, whereas our proposed factor strength estimation method suggests tha... | https://arxiv.org/abs/2503.08397v1 |
show that when weak factors exist, the TransPCA estimators outperform the traditional PCA estimators derived solely based on the target dataset under certain assumptions. TransPCA estimators would also achieve superior or at least comparable convergence rates with those of PCA estimators in Bai 30 (2003) when all facto... | https://arxiv.org/abs/2503.08397v1 |
arXiv preprint arXiv:2402.05789 . Connor, G., Hagmann, M., Linton, O., 2012. Efficient semiparametric estimation of the Fama-French model and extensions. Econometrica 80, 713–754. Duan, J., Pelger, M., Xiong, R., 2024. Target pca: Transfer learning large dimensional panel data. Journal of Econometrics 244, 105521. Fan,... | https://arxiv.org/abs/2503.08397v1 |
Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables ELISE COLIN ,1RAZVIGOR OSSIKOVSKI ,2,* 1DTIS-ONERA, University Paris-Saclay, 91123 Palaiseau, France 2LPICM, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France *elise.colin@onera.fr Abstract:... | https://arxiv.org/abs/2503.08808v1 |
section, we extend this result to derive the joint PDF of two correlated Gamma- distributedvariables,obtainedasthesumof 𝑘independentexponentialvariables. Theparameter 𝑘 canbeinterpretedasthenumberofindependentspecklesintegratedwithinasinglemeasurement, a quantity of interest in various imaging applications. Building ... | https://arxiv.org/abs/2503.08808v1 |
integrations by parts: ⟨𝑋1𝑋2⟩=𝐶∫2𝜋 0∫2𝜋 0𝑥1𝑥2𝑒−𝛼(𝑥1+𝑥2)𝐼0(𝛽√𝑥1𝑥2)𝑑𝑥1𝑑𝑥2=(2𝜎2 𝑧)2(1+𝜌2 𝑧).(17) From the moments thus obtained, we find the correlation coefficient to be: 𝜌𝑋1,𝑋2=⟨𝑋1𝑋2⟩−⟨𝑋1⟩⟨𝑋2⟩ 𝜎𝑋1𝜎𝑋2=𝜌2 𝑧. (18) 3. Derivation of the Joint PDF of Two Correlated Gamma Variables Having es... | https://arxiv.org/abs/2503.08808v1 |
As already mentioned, this parameter is widely used in dynamic speckle imaging, where it is commonly referred to as the Fujii index. To obtain its distribution, we apply a change of variable from the ratio 𝑍=𝑋1/𝑋2to𝐷. We introduce the auxiliary variable: 𝐷′=𝑍−1 𝑍+1, (34) which provides a one-to-one mapping betwe... | https://arxiv.org/abs/2503.08808v1 |
forms the basis of our approach: two exponentially distributed variables 𝑋and𝑌with correlation 𝜌2can be simply obtainedfromtwocorrelatedcircularcomplexGaussianvariables 𝑍𝑥and𝑍𝑦. ,usingtherelation: 𝑍𝑦=𝜌𝑍𝑥+√︃ 1−𝜌2𝑊, where𝑊isanindependentcomplexGaussianvariable. Thisensuresthattheresultingintensities 𝑋=|𝑍... | https://arxiv.org/abs/2503.08808v1 |
Fig. 5. Comparison between the empirical histogram and the theoretical density of the Normalized Dissimilarity Ratio 𝐷(𝑋,𝑌)for fixed values of 𝜌,𝜎, and𝑘. 7.4. Evolution of the Normalized Dissimilarity Ratio with 𝜌 Figure 6 presents the evolution of the Normalized Dissimilarity Ratio as a function of the correlat... | https://arxiv.org/abs/2503.08808v1 |
XavierOrlikfortheirinvaluableparticipationinallourdiscussionsandfortheirconstantsupport throughout this work. References 1.M.Nakagami,“Them-distribution—ageneralformulaofintensitydistributionofrapidfading,”in Statistical methods in radio wave propagation, (Elsevier, 1960), pp. 3–36. 2. A. Papoulis, Random variables and... | https://arxiv.org/abs/2503.08808v1 |
“All-Something-Nothing” Phase Transitions in Planted k-Factor Recovery Julia Gaudio* Colin Sandon†Jiaming Xu‡Dana Yang§ March 13, 2025 Abstract This paper studies the problem of inferring a k-factor, specifically a spanning k-regular graph, planted within an Erd ˝os–R ´enyi random graph G(n, λ/n ). We uncover an intere... | https://arxiv.org/abs/2503.08984v1 |
reduces to the planted clique problem, which has AoN phase transition at threshold m∗= 2 log2(n).As another example, when Hconsists of m-paths and p=λ/n for a constant λ,then we arrive at the planted path problem, which has AoN phase transition at threshold m∗= log( n)/log(1/λ)[11]. More interestingly, the critical thr... | https://arxiv.org/abs/2503.08984v1 |
intuitive argument above is straightforward, formalizing it rigorously is highly non-trivial and constitutes the main contribution of this paper. Theorem 1.1. Consider the planted k-factor model with nnodes and p=λ/n. The following hold with probabilities tending to 1asn→ ∞ : •(Exact recovery) If λ=o(1), then µGis a de... | https://arxiv.org/abs/2503.08984v1 |
2λ′ n,then we can still test the hypothesis based on the minimum degree or the existence of a k-factor. The test based on the existence of a k-factor succeeds as long as λ≤logn+ (k−1) log log n−ω(1), as the null model does not contain any k-factors with high probability. In summary, we see that detection is much easie... | https://arxiv.org/abs/2503.08984v1 |
It is known that finding a k-factor in general graphs can be done efficiently in total time O(n3k)[12]. Alternatively, for the planted k-factor model, we can show a linear-time iterative pruning algorithm [19] outputs a set of edges ˆH(which may not necessarily be a valid k-factor) that achieves the thresholds for the ... | https://arxiv.org/abs/2503.08984v1 |
3 Proof Overview In this section, we present the main proof ideas. The formal proofs are deferred to appendices. 3.1 Alternating Circuits For ease of visualization, we color the planted edges red and unplanted edges blue. Our starting point is the following key observation: for any k-factor H, the symmetric difference ... | https://arxiv.org/abs/2503.08984v1 |
number of k-factors in Gthat are close to H∗. To lower-bound the number of k-factors that are far away from H∗, recall that for any k-factor H,H∆H∗can be decomposed into a union of disjoint alternating circuits. Moreover, given any union of disjoint alternating circuits C, the XOR C⊕H∗is ak-factor. Therefore, it suffic... | https://arxiv.org/abs/2503.08984v1 |
edge. Similarly, we say that a tree Rjis blue-connected to a tree-facing endpoint vinE⋆ Rif some red vertex in Rjis connected to vby a blue edge. If the corresponding linking vertices are connected by a blue edge (as in the long blue edge in Figure 4), then we say that the ithandjthtrees are connected by a “five edge c... | https://arxiv.org/abs/2503.08984v1 |
tree-facing endpoint of an edge eLinE⋆ L, and similarly Rjis blue-connected to the tree-facing endpoint of an edge eRinE⋆ R. In turn, the linking endpoints of eLandeRare connected by a blue edge. 3.3 Proof Ideas for Theorem 2.5 Establishing the impossibility of partial recovery when λ=ω(1)reduces to showing the posteri... | https://arxiv.org/abs/2503.08984v1 |
at least (1−ρ)2−o(1). Interestingly, this cycle construction differs from the one used in the impossibility proof for almost exact recovery. Here, we build a single large two- sided tree of size√nlognand connect the two sides via a three-edge (blue-red-blue) sprinkling. In comparison, the impossibility proof for almost... | https://arxiv.org/abs/2503.08984v1 |
complexity of graph partitioning problems. Discrete Applied Math- ematics , 57(2-3):193–212, 1995. [10] Cheng Mao, Alexander S Wein, and Shenduo Zhang. Detection-recovery gap for planted dense cycles. In The Thirty Sixth Annual Conference on Learning Theory , pages 2440–2481. PMLR, 2023. [11] Laurent Massouli ´e, Ludov... | https://arxiv.org/abs/2503.08984v1 |
vj),(v′ i, v′ j)}is disjoint from {(vi′, vj′),(v′ i′, v′ j′)}for all (i′, j′)̸= (i, j), so the events Eijare mutually independent for all i < j . Hence, P{∪i<jEij}= 1−P ∩i<jEc ij = 1−Y i<jP Ec ij = 1−(1−λ2/n2)⌈n 4k⌉(⌈n 4k⌉−1)/2= Ω(1) , where the last equality holds by the assumption that λ= Ω(1) andkis a fixed consta... | https://arxiv.org/abs/2503.08984v1 |
Gsatisfy ℓ(H, H∗)≥2δ k. Proof of Lemma C.1. Applying Lemma 3.1, we get that E[|Hgood| |H∗] =X t<δn|{H∈ H:|H∆H∗|=t}| ·λ nt/2 ≤X t<δn(kn)t/2·λ nt/2 =X t<δn 2(kλ)t ≤(kλ)δn 2+1−1 kλ−1≤kλ kλ−1(kλ)δn 2. The conclusion follows by applying Markov’s inequality. In order to prove Lemma C.2, we will provide an algorithm for c... | https://arxiv.org/abs/2503.08984v1 |
Remove u0andu′ 0from both AandF. Remove all the planted neighbors of u0andu′ 0 fromF. 6 (Grow the left tree rooted at u0.) Initialize the leaf queue to be L ← { u0}, and the cumulative size to be s←1. 7 whileL ̸=∅ands <2ℓdo 8 Letu← L.pop. 9 (Find the children of u.) LetCu← {v∈ F: (u, v)is an unplanted edge }; i.e.,Cu i... | https://arxiv.org/abs/2503.08984v1 |
the “tree-facing” vertex and designate vas the “linking” vertex. 4Initialize an empty (bipartite) graph G. 5fori∈[K1]do 6 ifLiis blue-connected to at least dunmarked tree-facing endpoints among E⋆ Land the same is true for Riwith respect to E⋆ Rthen 7 Let the first dof these edges be denoted E(Li)⊂E⋆ LandE(Ri)⊂E⋆ R. 8 ... | https://arxiv.org/abs/2503.08984v1 |
on the realization of H∗. In order to characterize the size of the trees, we compare the trees to branching processes, where the offspring distribution is kindependent copies of a suitable binomial random variable. At a high level, the probability that a given tree reaches a prescribed depth can be related to the survi... | https://arxiv.org/abs/2503.08984v1 |
the size of 2ℓ nodes and the construction of the i-th tree is finished. In this case, we randomly add additional offspring to Bito ensure the offspring distribution of Biis exactly Binom( n−5γn, λ/n ). We can check that under this coupling, when the i-th tree has not reached the size of 2ℓnodes, it has at least as many... | https://arxiv.org/abs/2503.08984v1 |
on input (G,2γn/k ), so that |E⋆|= 2γn/k . Next, let Tbe the output of Algorithm 2 on input (G, E⋆, s). Let E1be the event thatT={Ti= (Li, Ri)}icontains at least K1two-sided trees with at least ℓred vertices in each subtree, where K1=Kϵ2 2(ϵ+ 2k)2=γnϵ2 4(2ℓ+k)k(ϵ+ 2k)2. By Theorem C.5, we have P(E1|H∗) = 1 −e−Ω(n). On ... | https://arxiv.org/abs/2503.08984v1 |
the graph Gcan be coupled to a bi-colored bipartite graph H with at least cnvertices on each side, a perfect (red) matching, and random blue edges which exist with probabilityλd2 2nindependently, due to (16) (and independently of E). To apply Lemma C.7, we need to verify λd2 2n·cn≥256 log(32 e). We simply let c=512 log... | https://arxiv.org/abs/2503.08984v1 |
that neither unorvhas degree k. Ifuis isolated in G0, then uwill have degree kinG. Letting Xbe the number of isolated vertices in G0, we see that |ˆH△H∗| |H∗|≤|H∗| −1 2kX |H∗|= 1−X n. Here the factor of 1/2accounts for the possibility that both endpoints of a given edge have degree kinG. Since each vertex is isolated i... | https://arxiv.org/abs/2503.08984v1 |
By definition, E[Zℓ(H∗, G)] =X H∈H:|H∗∩H|=ℓP{His ak-factor in G}=X H∈H:|H∗∩H|=ℓpkn/2−ℓ≤(nkp)kn/2−ℓ, where the last inequality follows from (9) in Lemma 3.1 that |{H∈ H:|H∗∩H|}| ≤ (nk)nk/2−ℓ. F Proof of Theorem 2.6 In this section, we prove Theorem 2.6. 30 F.1 Proof of Error Upper Bound In this subsection, we show that ... | https://arxiv.org/abs/2503.08984v1 |
connecting to vertices in [n]\V(G2t−1 e).Then Ru≤k. Thus, |∂G2t e| ≤k|∂G2t−1 e|.Hence, P C2t|G2(t−1) e, C2(t−1) =P |∂G2t−1 e| ≤2λ(2kλ+ 2)t−1logn|G2(t−1) e, C2(t−1) ≥P X≤2λ(2kλ+ 2)t−1logn ≥P{X≤2E[X]} ≥1−n−λ/3. Finally, conditional on C2t, V(G2t e) =|V(G0 e)|+tX s=1 ∂G2s−1 e + ∂G2s e ≤2 + (1 + k)2λtX s=1(2λk+ 2)s−1... | https://arxiv.org/abs/2503.08984v1 |
equality holds when we decompose V(G2t+1 e)intoV(G2t e) 33 and∂G2t+1 e; the last equality holds because w∈∂G2t+1 eif and only if w /∈V(G2t e)is connected to some w′∈∂G2t evia a blue edge. It follows from a union bound that P ∃u∈∂G2t+1 e, w∈V(G2t+1 e) :w∈NR u|H∗, G2t e, C2t ≤P ∃u′∈∂G2t e, u /∈V(G2t e), w∈V(G2t e) :u∈N... | https://arxiv.org/abs/2503.08984v1 |
edges connect to distinct vertices in [n]\V(G2t+1 e). Thus, on the event E2t+1∩E2t∩ {Bu=˜Bu,∀u∈∂G2t e}, there exists a one-to-one mapping from the vertices in ∂G2t+2 e to the vertices in ∂T2t+2 e, so that G2t+2 e=T2t+2 e.In conclusion, we get that P G2t+2 e=T2t+2|G2t e=T2t, C2t ≥Pn E2t+1∩E2t∩ {Bu=˜Bu,∀u∈∂G2t e} |G2t e... | https://arxiv.org/abs/2503.08984v1 |
we prove P{e∈Cn} ≥(1−ρ)2−o(1). Note that this is trivially true when kλ≤1asρ= 1. Thus it suffices to focus on kλ > 1. Lemma F.6. A planted edge eis in the core Cnif it belongs to an alternating circuit in the graph G. Remark F.1.We remark that the reverse direction of the above lemma is not true. A planted emay remain ... | https://arxiv.org/abs/2503.08984v1 |
e′(and connected on the correct side) with probability at least λ2logn/(4n).It follows that LandRare simultaneously connected to some reserved edge with probability at least 1− 1−λ2logn/(4n)γn≥1−exp γnλ2logn/(4n) = 1−exp(−Ω(log n)). In conclusion, we have shown that there exists an alternating cycle containing ewit... | https://arxiv.org/abs/2503.08984v1 |
of the vi’s, where vihas at most nvertex labels and vi+1has at most kvertex labels for all odd ifrom 1to2t. The last vertex v2t+1has at most klabels, as it is connected to v1via a red edge. Thus in total, we have at most ntkt+1different such “almost” alternating cycles. Thus, the probability that Gcontains an “almost” ... | https://arxiv.org/abs/2503.08984v1 |
that by replacing 1−xkwithx, it suffices to show sup x∈[0,1] (1−λx/n )m−e−αλx ≤dn Now, (1−λx/n )m−e−αλx = e−αλx emlog(1−λx/n )+αλx−1 ≤ emlog(1−λx/n )+αλx−1 . Note that h(x)≜mlog(1−λx/n ) +αλx is concave in x. Thus, for all x∈[0,1], h(x)≥min{h(0), h(1)}= min {0, mlog(1−λ/n) +αλ} Moreover, since log(1−λx/n )≤ −λx/n , i... | https://arxiv.org/abs/2503.08984v1 |
a cycle. Lemma I.1. LetHbe a random 2-factor on mvertices. With probability at least 1/m,His a cycle. Proof. There are m!possible directed cycles with starting points on mvertices, so there are (m− 1)!/2possible cycles on mvertices. Meanwhile, every possible 2-factor can be converted to the cycle decomposition of a per... | https://arxiv.org/abs/2503.08984v1 |
Addressing pitfalls in implicit unobserved confounding synthesis using explicit block hierarchical ancestral sampling Xudong Sun * Institute of AI for Health Helmholtz Munich GermanyAlex Markham Department of Mathematical Sciences University of Copenhagen Denmark Pratik Misra Department of Mathematics Technical Univers... | https://arxiv.org/abs/2503.09194v2 |
idiosyncratic covariance matrix. This construction restricts the spectrum of the partial correlation matrix and fails to cover the entire space of possible distributions. •Similarly, we point out a common restriction on the possible graphical structures that excludes valid causal models. •To address these limitations, ... | https://arxiv.org/abs/2503.09194v2 |
if 𝑊(𝑢,𝑣)≠0then there is an arrow 𝑣→𝑢. The random vector 𝑌is defined by 𝑌=𝑊𝑌+𝜖𝑌, (2) so that𝑌can equivalently be expressed in terms of the idiosyncratic noise vector 𝜖𝑌: 𝑌=(𝐼−𝑊)−1𝜖𝑌. (3) The covariance matrix of the idiosyncratic noise is given by Ω:=E(𝜖𝑌𝜖𝑇 𝑌), (4)where a diagonal Ωcorresponds t... | https://arxiv.org/abs/2503.09194v2 |
an ancestral ADMG, the protocol first uniformly samples directed edges from a lower triangular matrix with a specified probability. Then, bidirected edges are added only among pairs of observable vertices that do not share an ancestral relationship [ 33]. This constraint, which is critical to prevent the formation of a... | https://arxiv.org/abs/2503.09194v2 |
𝑂) spans the space of all symmetric positive definite matrices of size |𝐽𝑂|×|𝐽𝑂|. PROPOSITION 3.[Explicit Reformulation of Implicit Parameterization] Suppose the observable variables 𝑌𝑂are generated according to an inter-observable causal structure defined by the adjacency matrix 𝑊𝑜=𝑊where𝑊in Equation (5), w... | https://arxiv.org/abs/2503.09194v2 |
invariant Σ11in Σ′′:=E(𝑌′′𝑌′′𝑇) (29) =Σ11Σ′′ 12 Σ′′ 21Σ′′ 22 (30) The adjacency matrix 𝑊′′ 1results from the similarity transformation with matrix 𝑄𝑇(instead of𝑄) to the adjacency matrix 𝑊′of a DAG, where 𝑊′is the adjacency matrix in Equation (15), 𝑄=𝐼Σ−1 11Σ12 0𝐼 Similarly, the adjacency matrix 𝑊′′ 2 ... | https://arxiv.org/abs/2503.09194v2 |
FOR A DIAGONALLY DOMINANT MATRIX Ω).For a diagonally dominant matrix Ω, the spectral radius 𝜌(Ω)satisfies 0<𝜌(Ω)≤2 max 𝑖|Ω𝑖,𝑖|. (36) PROOF .Inequality (36) follows directly from the Gershgorin circle theorem [ 15, Theorem 6.1.1]. According to the definition of diagonal dominance (see Definition 7), the Gershgorin ... | https://arxiv.org/abs/2503.09194v2 |
E(ˆ𝜖2 𝑖|𝑖)E(ˆ𝜖2 𝑗|𝑗). (45) PROOF . See Appendix D. □ LEMMA 5.Let Σ−1=h (𝐼−𝑊)−1Ω(𝐼−𝑊)−𝑇i−1 be the precision matrix. Then, the partial correlation between variables𝑖and𝑗can be expressed as 𝜌𝑖,𝑗|· =−[Σ−1]𝑖,𝑗√︃ [Σ−1]𝑖,𝑖[Σ−1]𝑗,𝑗(46) =−h (𝐼−𝑊)−1Ω−1(𝐼−𝑊)−𝑇i 𝑖,𝑗√︂h (𝐼−𝑊)−1Ω−1(𝐼−𝑊)−𝑇i 𝑖,𝑖h (�... | https://arxiv.org/abs/2503.09194v2 |
Traditional methods (e.g., generating large DAGs via Erdo-Renyi graphs) tend to produce homogeneous graphs. To remedy this, we adopt a block-hierarchical approach (see Section 4.1). 4.1 Hierarchical data generation via ancestral sampling 4.1.1 An example DAG generated hierarchically. We first present the output and the... | https://arxiv.org/abs/2503.09194v2 |
set of hidden nodes 𝑈. The conditional independence (CI) relationships are preserved; for example, in the left-hand-side (l.h.s.) DAG, 𝐶⊥𝐵|𝐷holds, and this CI statement is maintained in the right-hand-side (r.h.s.) ancestral graph. Similarly, 𝐴⊥𝐷|𝐵,𝐶is valid in both representations. 𝑣, we examine if we can add... | https://arxiv.org/abs/2503.09194v2 |
examples presented in Figure 4 and Figure 5. In Figure 4, the standard setting is shown where the unobserved confounder is a root variable. Figure 5 presents an interesting scenario in which the unobserved confounders 𝑋2and𝑋5are themselves confounded by the observed variable𝑋1. In this case, the ground truth ancestr... | https://arxiv.org/abs/2503.09194v2 |
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