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is the computational complexity of our adaptation of the No-U-Turn criterion, which currently scales as O(n2) per iteration with nthe number of events —significantly worse than the complexity of the corresponding criterion in HMC. Developing more efficient stopping rules remains an important direction for future resear... | https://arxiv.org/abs/2503.11479v1 |
the no-u-turn sampler using Gibbs self tuning,” arXiv preprint arXiv:2408.08259 , 2024. [15] T. Liu, N. Surjanovic, M. Biron-Lattes, A. Bouchard-Cˆ ot´ e, and T. Campbell, “Autostep: Locally adaptive involutive MCMC,” arXiv preprint arXiv:2410.18929 , 2024. [16] C. Modi, A. Barnett, and B. Carpenter, “Delayed rejection... | https://arxiv.org/abs/2503.11479v1 |
THE KOLMOGOROV-SMIRNOV STATISTIC REVISITED BYELVIS HANCUI*3,a, YIHAO LI†2,b, ZHUANG LIU‡1,c 1Department of Biostatistics, University of California, Los Angeles,aelviscuihan@g.ucla.edu 2Department of Biostatistics, University of California, Los Angeles,byihaoli1992@gmail.com 3College of Environmental and Resource Scienc... | https://arxiv.org/abs/2503.11673v1 |
=t, i.e., Xis uniform on [0,1], then P τn(λ) =ϵ+j n =ϵn j ϵ+j nj−1 1−ϵ−j nn−j (2) for all j= 0,1,···,⌊n−λ√n⌋. Remark 1 : The assumption that F(t) =tcan always be true in practice since the distribu- tions of Zn(t)andD+ ndo not depend on F. If we do not wish to make such an assumption, then we simply make an inv... | https://arxiv.org/abs/2503.11673v1 |
the walk arrives at position n−m, as there are nsteps in the positive direction and msteps in the negative direction. The total number of such paths is n+m n . The event {D+ n,m≥δ}is equivalent to the event where the path hits or exceeds the level k−lat some step, where δ=k n−l m>0for integers k= 1,...,n andl= 0,1,..... | https://arxiv.org/abs/2503.11673v1 |
is based on concentration bounds for Bernoulli variables and laborious calculus. • The one-sided DKWM is proven by bounding a Riemann summation using mean value interpolation. That is, P D− n> ϵ ≤e−2ϵ2. • Put two parts together, P(Dn> ϵ)≤2e−2ϵ2. (6) 5. Two-sided Two Sample Kolmogorov-Smirnov Statistic. The two-sided ... | https://arxiv.org/abs/2503.11673v1 |
and T INGEY , F. H. (1951). One-sided confidence contours for probability distribution func- tions. The Annals of Mathematical Statistics 592–596. [4] D ELBARRIO , E., D EHEUVELS , P. and V ANDEGEER, S. (2007). Lectures on empirical processes . Euro- pean Mathematical Society Zurich. [5] D URBIN , J. (1973). Weak conve... | https://arxiv.org/abs/2503.11673v1 |
Proposal for the Application of Fractional Operators in Polynomial Regression Models to Enhance the Determination Coe fficientR2on Unseen Data Anthony Torres-Hernandez∗,a,b aDepartment of Information and Communication Technologies, Music and Machine Learning Lab, Universitat Pompeu Fabra, Barcelona bDepartment of Physi... | https://arxiv.org/abs/2503.11749v2 |
that f,f(n)∈L1 loc(a,b). Then, the Riemann–Liouville fractional integral also allows constructing the operator Caputo fractional derivative, which is defined as follows [2, 4]: C aDα xf(x) :=( aI−α xf(x), ifα<0 aIn−α xf(n)(x),ifα≥0, (3) wheren=⌈α⌉andaI0xf(n)(x) :=f(n)(x). Furthermore, if the function ffulfills thatf(k)... | https://arxiv.org/abs/2503.11749v2 |
fractional operator Dαf(x) = lim h→0(1 +hβ)f(x+hα)−f(x) h, α +β= 1 Table 1: Di fferent fractional operators which is non-empty since it includes the following set of fractional operators: On 0,x,α(h) := oα x:∃oα kh(x) = ∂n k+µ(α)∂α k h(x),lim α→nµ(α)∂α kh(x) = 0∀k≥1 . (8) Consequently, the following result holds: I... | https://arxiv.org/abs/2503.11749v2 |
the classical Hadamard product: o0 x◦h(x) :=h(x)∀oα x∈mMO∞,u x,α(h). (27) For each operator oαx∈MO∞,u x,α(h), we can define the fractional matrix operator [44]: Aα(oα x) = [Aα(oα x)]jk := (oα k). (28) Next, we define a modified Hadamard product [42]: opα i,x◦oqα j,x:=opα i,x◦oqα j,x,ifi,j(horizontal Hadamard p... | https://arxiv.org/abs/2503.11749v2 |
under the operation (41): mG∗(Aα(oα x),M14)={A◦1 α,A◦3 α,A◦5 α,A◦9 α,A◦11 α,A◦13 α}. (43) Corollary 3.4. LetZ+prepresent the set of positive residual classes less than a prime p. For each fractional operator oαx∈ mMO∞,u x,α(h), we can define the Abelian group of fractional matrix operators under the operation (41) as: ... | https://arxiv.org/abs/2503.11749v2 |
the data, whereas a high-degree polynomial can lead to excessive fluctuations, resulting in an overly complex model. The selection of an appropriate degree often involves techniques such as cross-validation and domain knowledge. Polynomial regression has numerous applications across various fields: •Economics: Used to ... | https://arxiv.org/abs/2503.11749v2 |
if the model performs poorly on unseen data, it may be an indication that it has failed to generalize properly, potentially due to overfitting or underfitting. •Overfitting: Occurs when the model learns the intricacies and noise of the training data to such an extent that it negatively impacts its performance on unseen... | https://arxiv.org/abs/2503.11749v2 |
variable. A negative R2suggests that the model has a poor fit and is performing worse than random guessing. This often occurs in cases of overfitting or model mis-specification. In general, a higher R2indicates better model performance and the ability to explain the variance in the target variable. However,R2alone shou... | https://arxiv.org/abs/2503.11749v2 |
following fractional regression model for any operator oαx∈MO1 x,α(y): σ(α,x) :=β0+mX i=1βioα xxiwithα∈(−1,1). (49) 9 UNAM Faculty of Science It is important to note that the intercept value must be kept intact to respect the value of the polynomial regression model in the case of x= 0. On the other hand, if a multidim... | https://arxiv.org/abs/2503.11749v2 |
Science (a) Box plots of average prices grouped by month (b) Logarithmic scale graph of positive R2values for interpolation and extrapolation parts simultaneously (c) Average price distribution using the monthly median - Frac- tional regression model with α= 0 (d) Average price distribution using the monthly median - F... | https://arxiv.org/abs/2503.11749v2 |
positive R2values for interpolation and extrapolation parts simultaneously (c) Average price distribution using the monthly median - Frac- tional regression model with α= 0 (d) Average price distribution using the monthly median - Frac- tional regression model with α,0 Figure 9: Comparison of metrics and fractional reg... | https://arxiv.org/abs/2503.11749v2 |
Statistics , 9(3):309–318, 2021. DOI: 10.13189/ms.2021.090312. [14] Eduardo De-la Vega, Anthony Torres-Hernandez, Pedro M Rodrigo, and Fernando Brambila-Paz. Fractional derivative-based performance analysis of hybrid thermoelectric generator-concentrator photovoltaic system. Applied Thermal Engineering , 193:116984, 20... | https://arxiv.org/abs/2503.11749v2 |
Mohamad Rafi Segi Rahmat. A new definition of conformable fractional derivative on arbitrary time scales. Advances in Di fference Equations , 2019(1):1–16, 2019. [33] J Vanterler da C Sousa and E Capelas De Oliveira. On the ψ-hilfer fractional derivative. Communications in Nonlinear Science and Numerical Simulation , 6... | https://arxiv.org/abs/2503.11749v2 |
1 Excess Mean Squared Error of Empirical Bayes Estimators Yue Ju, Bo Wahlberg, Fellow, IEEE , and Håkan Hjalmarsson, Fellow, IEEE Abstract — Empirical Bayes estimators are based on min- imizing the average risk with the hyper-parameters in the weighting function being estimated from observed data. The performance of an... | https://arxiv.org/abs/2503.11863v1 |
[9], [26], [40], [41], [45]. Note that various regularized estimators can be interpreted as EB estimators with different forms of weighting functions. These regularized estimation methods have been increasingly recognized as a complement to classical system identification [34], [36]. In particular, for the kernel-based... | https://arxiv.org/abs/2503.11863v1 |
one for large sample sizes. The main contributions of this work are: 1) We derive an explicit expression for the XMSE of an EB estimator equipped with a general data-dependent hyper-parameter estimator. 2) As specific instances, we present XMSE expressions for generalized Bayes estimators, and kernel-based regular- ize... | https://arxiv.org/abs/2503.11863v1 |
[Φ]:,1= [u(0),···, u(N−1)]⊤and known as the regression matrix, andθ∈Rnis the unknown parameter vector. We also need the following assumption. Assumption 1: 1) The FIR model order nis fixed, and larger or equal to the true order. 2) The inputs {u(t)}N−1 t=1−nare deterministic and known withu(t) = 0 fort≤0. Moreover, the... | https://arxiv.org/abs/2503.11863v1 |
derive explicit results for other estimators using Theorem 2 below.•Scaled EB: The scaled EB hyper-parameter estimator is given by ˆηEB,α= arg min η∈DηFEB,α(η), (8a) FEB,α(η) =Y⊤Q(η)−1Y+αlog det( Q(η)),(8b) where Q(η) =ΦP(η)Φ⊤+σ2IN. For convenience, ˆηEB,αandFEB,α(η)withα= 1 will be denoted ˆηEB andFEB(η), respectively... | https://arxiv.org/abs/2503.11863v1 |
of the regularized estimator in (6), or more generally, the MSE of the EB estimator in (5). Remark 1: Note that the performance comparison between the ML and kernel-based regularized estimators has been par- tially discussed in [31]. It concludes that when the regression matrix Φand the kernel matrix P(ˆη)are well-cond... | https://arxiv.org/abs/2503.11863v1 |
are functions of the ML estimator ˆθML, denoted as ˆη(ˆθML). To understand why this is the case, we notice that the Rao-Blackwell theorem [23, 5 Theorem 1.7.8] gives that if there exists a minimal sufficient statistic (MSS) S, then MSE(E(ˆθ|S))≤MSE( ˆθ), with equality if and only if ˆθ=E(ˆθ|S), for any estimator ˆθ. Th... | https://arxiv.org/abs/2503.11863v1 |
implicit condition that θπ(θ|ˆη(ˆθML))p(Y|θ)andπ(θ|ˆη(ˆθML))p(Y|θ)(21) 6 are both well-defined and integrable. We can see from Section A.4 that the derivation of the XMSE of ˆθEB(ˆη(ˆθML))only involves the Taylor expansions of (21) and their integrals. Hence, Theorem 2 remains true for improper π(θ|ˆη(ˆθML)). The expre... | https://arxiv.org/abs/2503.11863v1 |
η=ˆη(ˆθML)=0. (26) 3) There exists fFN(ˆθML,η) =aF(ˆθML)FN(ˆθML,η) + bF(ˆθML)with aF(ˆθML)>0andbF(ˆθML)being independent of η, such that fFN(θ0,η)converges to a nonzero deterministic function W(θ0,η)uniformly for allη∈ DηasN→ ∞ , i.e., sup η∈Dη|fFN(θ0,η)−W(θ0,η)| →0,asN→ ∞ .(27) Moreover, when η= arg minη∈DηW(θ0,η), we... | https://arxiv.org/abs/2503.11863v1 |
However, surprisingly at least to the authors, there are settings where estimating ηreduces the XMSE. We have the following example. Example 1: Consider the SS kernel (7) with γ= 0.5,Σ= diag{[10,500]}andθ0= [2.53,1]⊤. By utilizing (A.12), we have Tr[XVarHPE( ·)]≈ −1.45α(σ2)2<0forˆθR(ˆηSy,α) andˆθR(ˆηGCV ,α). IV. A F IN... | https://arxiv.org/abs/2503.11863v1 |
accuracy of XMSE and apx. XMSE, we generate more systems and display their corresponding apx. XMSE in the following example. Example 2: We randomly generate 20collections of θ0 and{u(t), y(t)}N t=1using the method in Section II-B with n= 20 andN= 50 ,100,150,200. Then we demonstrate the sample ∆MSE(·)ofˆθR(ˆηEB)andˆθR(... | https://arxiv.org/abs/2503.11863v1 |
ˆθR(ˆηEB)). ˆθML ˆθR(ˆηEB) ˆθR(ˆηSy) ˆθR(ˆηGCV) average FIT 68.23 60 .94 69 .36 69 .17 sample MSE 9.01×10−11.37 8 .64×10−18.76×10−1 In Fig. 4, we illustrate the components of sample ∆MSE(·) and apx. XMSE( ·)/N2of three regularized estimators. Their correspondence has been shown in Theorem 1 and summarized in Table III.... | https://arxiv.org/abs/2503.11863v1 |
than ˆθR(ˆηSy), as shown in Tables II and V. Remark 9: Notice that in Theorem 2, we have proved that the limit of N2Tr[ΥR HOT]is zero. However, Tr[ΥR HOT]in Fig. 4-6 seems not always negligible compared with the other three components. We can approximate this term by deriving its higher order limit, i.e., limN→∞N3Tr[ΥR... | https://arxiv.org/abs/2503.11863v1 |
larized estimators, and could be used to quantify the alignment between the true model parameter θ0and 11 the kernel matrix P(η). Then, we look to further construct measures of such alignment and analyze their influences on the performance of regularized estimators. This will provide a more detailed expla- nation for t... | https://arxiv.org/abs/2503.11863v1 |
sequences ξN∈CandaN∈C, iflimN→∞ξN/aN= 0, we denote it as ξN=o(aN). 12 +o(∥(Φ⊤Φ)−1∥F), (A.3) Eθ|ˆθML[θπ(θ|η)] =ˆθMLπ(ˆθML|η) +σ2(Φ⊤Φ)−1∂π(θ|η) ∂θ θ=ˆθML +σ2 2ˆθMLTr∂2π(θ|η) ∂θ∂θ⊤ θ=ˆθML(Φ⊤Φ)−1 +o(∥(Φ⊤Φ)−1∥F). (A.4) It follows that ˆθEB(η) =Eθ|ˆθML[θπ(θ|η)]/π(ˆθML|η) 1 + [Eθ|ˆθML[π(θ|η)]/π(ˆθML|η)−1] +o(∥(Φ⊤Φ)−1∥F) =Eθ... | https://arxiv.org/abs/2503.11863v1 |
can be similarly proved like [18, Proof of Theorem 1]. To derive the expression of Bb, we first calculate Bb= ∂2Wb,α(η)/∂η∂θ⊤ 0in (32b) and then prove the uniform convergence of ∂2Fb,α(η)/∂η∂(ˆθML)⊤|ˆθML=θ0. We have sup η∈Dη∥∂2Fb,α(η)/∂η∂(ˆθML)⊤|ˆθML=θ0−Bb∥F ≤X ksup η∈Dη∥∂2Fb,α(η)/∂ηk∂(ˆθML)⊤|ˆθML=θ0−[Bb]k,:∥2 ≤X k2∥ˆθ... | https://arxiv.org/abs/2503.11863v1 |
, 41(1/2):56–61, 1954. [6] T. Chen. On kernel design for regularized LTI system identification. Automatica , 90:109–122, 2018. [7] T. Chen, H. Ohlsson, and L. Ljung. On the estimation of transfer func- tions, regularizations and Gaussian processes - Revisited. Automatica , 48:1525–1535, 2012. [8] H. Daniels. The asympt... | https://arxiv.org/abs/2503.11863v1 |
Transactions on Automatic Control , 2023. [29] B. Mu, T. Chen, and L. Ljung. Asymptotic properties of generalized cross validation estimators for regularized system identification. IFAC- PapersOnLine , 51(15):203–208, 2018.[30] B. Mu, T. Chen, and L. Ljung. On asymptotic properties of hyperpa- rameter estimators for ke... | https://arxiv.org/abs/2503.11863v1 |
Science and Technology, Nanjing, China, in 2017 and the Ph.D. degree in computer and information en- gineering from the Chinese University of Hong Kong, Shenzhen, China, in 2022. She is currently a postdoc at the KTH Royal Institute of Technology. She has been mainly working in the area of system identification. 15 Bo ... | https://arxiv.org/abs/2503.11863v1 |
TrainingDiagonalLinearNetworkswithStochastic Sharpness-AwareMinimization GabrielClara *‡1,SophieLanger† ‡2,andJohannesSchmidt-Hieber‡1 1FacultyofElectricalEngineering,Mathematics,andComputerScience UniversityofTwente 2FacultyofMathematics,RuhrUniversityBochum March18,2025 Abstract We analyze the landscape and training ... | https://arxiv.org/abs/2503.11891v1 |
Thisnotationalconvenienceisstandardinthestudyoflinear neuralnetworks[ACH18;Aro+19a;Bah+21]andcorrespondstotheFréchetderivative,seeAppendix Eformoredetails. Wemostlyconsiderdiagonalmatrices,inwhichcasethismatrix-valuedgradient equals ∇𝐴𝑓(𝐴)=⎡ ⎢⎢⎢⎢ ⎣𝜕 𝜕𝐴11𝑓(𝐴) ⋱ 𝜕 𝜕𝐴𝑑𝑑𝑓(𝐴)⎤ ⎥⎥⎥⎥ ⎦. 4 𝑥1 𝑥2 𝑥𝑑∑𝑑 𝑖=1𝑤... | https://arxiv.org/abs/2503.11891v1 |
computed with respect to the empirical measure,aswasdonein (3). Incase𝐿= 2,theregularizerreduces(uptoanadditiveconstant)to 𝑅(𝑊1,𝑊2)=𝜂2⋅(‖𝑊1‖2+‖𝑊2‖2),whichmatchesthecommonlyusedweight-decaymethod[KH91; DAn+23]. Thisconnectioninthetwo-layercasewasalreadynotedin[Orv+23]. Fordeepnetworks, 𝑅insteadpenalizesthesquare... | https://arxiv.org/abs/2503.11891v1 |
𝑚forall𝓁≠𝑚. (b) Everycriticalpoint (𝑊1,…,𝑊𝐿)ofℒ𝑅hastheform 𝑊𝓁,ℎℎ=𝑠𝓁,ℎ𝜆ℎ⋅|𝑊∗|1∕𝐿 ℎℎ, (13) wherethesigns 𝑠1,ℎ,…,𝑠𝐿,ℎ∈ {−1,0,1}eithersatisfy 𝑠1,ℎ⋯𝑠𝐿,ℎ= sign(𝑊∗,ℎℎ), or𝑠1,ℎ=⋯= 𝑠𝐿,ℎ=0. For𝑠1,ℎ⋯𝑠𝐿,ℎ≠0theshrinkagefactor 𝜆ℎ∈(0,1)isanypositiverootof 0=𝜆2 ℎ−𝜆1−1∕(𝐿−1) ℎ+𝜂2 |𝑊∗|2∕𝐿 ℎℎ. Moreover,�... | https://arxiv.org/abs/2503.11891v1 |
Analogoustotheresultforgradientflows,thetheoremstatesthatthegradientdescentiterates ofanalyticfunctionseitherdivergetoinfinityorconvergetoacriticalpointofthefunction,given regularityconditions. Thesolerequirementforthisresultistheso-called strongdescentcondition . In ournotation,theconditionmaybestatedastheexistenceof ... | https://arxiv.org/abs/2503.11891v1 |
⎠. (22) Bydefinition, theresultingtrajectoriesalwaysstayconfinedtotheballofradius 𝑟andeverylimit pointoftheiteratesmustthenbeacriticalpointof ℒ𝑅. Theorem4.4. Supposethestep-sizes 𝛼𝑘satisfy∑∞ 𝑗=0𝛼𝑗=∞and∑∞ 𝑗=0𝛼2 𝑗<∞. Pickaradius 𝑟such that 𝑟≥max{1,√ 𝐿 2𝜂𝐿−1}⋅‖𝑊∗‖ 18 Then,everylimitpointoftheprojectedS-SAM... | https://arxiv.org/abs/2503.11891v1 |
issn:0378-3758. [Cha04] DjalilChafaï.“Entropies,Convexity,andFunctionalInequalities:On Φ-Entropiesand Φ- SobolevInequalities”.In: JournalofMathematicsofKyotoUniversity 44.2(2004),pp.325– 363.issn:0023-608X. [Cha22] S.Chatterjee. ConvergenceofGradientDescentforDeepNeuralNetworks .2022.arXiv: 2203.16462 [cs.LG] . [Che+23... | https://arxiv.org/abs/2503.11891v1 |
Curran Associates Inc., 2020, pp. 1–12. isbn: 9781713829546. [MK23] T. Möllenhoff and M. E. Khan. “SAM as an Optimal Relaxation of Bayes”. In: 11th InternationalConferenceonLearningRepresentations .2023. [Neu+21] G.Neu,G.K.Dziugaite,M.Haghifam,andD.M.Roy.“Information-TheoreticGeneral- izationBoundsforStochasticGradient... | https://arxiv.org/abs/2503.11891v1 |
𝑟ℎ =( 𝑊∗−˜𝑊𝐿⋯˜𝑊1) ℎℎ foreachℎ=1,…,𝑑. Togetherwiththeexpressionfor ∇𝑊𝓁(𝑌−𝟏𝗍˜𝑊𝐿⋯˜𝑊1𝐗𝑖)2computedinLemma 2.1,thisresultsin 1 𝑛⋅𝑛∑ 𝑖=1∇𝑊𝓁( 𝑌−𝟏𝗍˜𝑊𝐿⋯˜𝑊1𝐗𝑖)2 =−2⋅( 𝑊∗−˜𝑊𝐿⋯˜𝑊1) ⋅𝐿∏ 𝑚=1 𝑚≠𝓁˜𝑊𝑚 (25) Bydefinition, 𝜉𝓁,ℎℎ𝑖.𝑖.𝑑.∼𝒩(0,𝜂2)andso 𝔼[𝜉2 𝓁,ℎℎ]=𝜂2forevery𝓁andℎ. Whenever 𝑟 > ... | https://arxiv.org/abs/2503.11891v1 |
sign matrices 𝑆𝓁∈ {−1,0,1}𝑑×𝑑may be chosen arbitrarily, as long as𝑆𝐿⋯𝑆1= sign(𝑊). In particular, we enforce the convention that 𝑆𝓁,ℎℎ= 0for every𝓁=1,…,𝐿,whenever sign(𝑊)ℎℎ=0,whichispossibleduetothebalancingconstraint implying𝑊𝓁,ℎℎ=0forevery𝓁whenever𝑊ℎℎ=0. Underthisconvention,forany 𝐼 ⊊{1,…,𝑑} and𝑚∉�... | https://arxiv.org/abs/2503.11891v1 |
the difference of squares 𝑊2 𝓁(𝑡)−𝑊2 𝓁+1(𝑡)features entirely non- negativeentries,beingaproductofpositive-definitefactors. Incase 𝑊𝑟(𝑡)=0forevery𝑟≠𝓁,𝓁+1, itsdiagonalentriestakeontheirminimalachievablevalue min ℎ=1,…,𝑑⎛ ⎜ ⎜ ⎝𝜂2⋅𝐿∏ 𝑟=1 𝑟≠𝓁,𝓁+1(𝑊2 𝑟(𝑡)+𝜂2⋅𝐼𝑑)⎞ ⎟ ⎟ ⎠ℎℎ≥𝜂2𝐿−2>0. (36) Since𝑊𝑟,ℎℎ(... | https://arxiv.org/abs/2503.11891v1 |
2‖‖‖‖‖‖‖‖‖‖‖‖‖‖( 𝑊2 𝓁+𝑊2 𝑚)⎛ ⎜ ⎜ ⎝𝐿∏ 𝑟=1 𝑟≠𝓁,𝑚(𝑊2 𝑟+𝜂2⋅𝐼𝑑)−𝐿∏ 𝑠=1 𝑠≠𝓁,𝑚𝑊2 𝑠⎞ ⎟ ⎟ ⎠‖‖‖‖‖‖‖‖‖‖‖‖‖‖ ≤max 𝓁=1,…,𝐿‖‖‖‖‖‖‖‖‖‖‖‖‖‖⎛ ⎜ ⎜ ⎝𝐿∏ 𝑟=1 𝑟≠𝓁(𝑊2 𝑟+𝜂2⋅𝐼𝑑)−𝐿∏ 𝑠=1 𝑠≠𝓁𝑊2 𝑠⎞ ⎟ ⎟ ⎠‖‖‖‖‖‖‖‖‖‖‖‖‖‖(44) wherethesecondboundfollowsbyapplyingthetriangleinequality,notingthat 𝑊2 𝑟,ℎℎ≤𝑊2 𝑟,ℎℎ+... | https://arxiv.org/abs/2503.11891v1 |
⎜ ⎜ ⎝𝑊2 𝓁+1(𝑘)⎛ ⎜ ⎜ ⎝𝐿∏ 𝑟=1 𝑟≠𝓁,𝓁+1(𝑊2 𝑟(𝑘)+𝜂2⋅𝐼𝑑)−𝐿∏ 𝑠=1 𝑠≠𝓁,𝓁+1𝑊2 𝑠(𝑘)⎞ ⎟ ⎟ ⎠+𝜂2𝐿∏ 𝑟=1 𝑟≠𝓁,𝓁+1(𝑊2 𝑟(𝑘)+𝜂2⋅𝐼𝑑)⎞ ⎟ ⎟ ⎠2 . Switchingtherolesof 𝓁and𝓁+1yieldsananalogousformulaforthesquaredgradientwithrespect 44 to𝑊𝓁+1(𝑘). Subtractingthesquaredgradientsandcancelingliketermsresultsin ... | https://arxiv.org/abs/2503.11891v1 |
≥𝐿⋅‖𝑊𝐿⋯𝑊1‖2−𝐿‖𝑊∗‖⋅‖𝑊𝐿⋯𝑊1‖+𝜂2𝐿−2⋅𝐿∑ 𝓁=1‖𝑊𝓁‖2,(57) wherethesecondestimatefollowsfromtheCauchy-Schwarzinequality. Wemustnowpickaradius 𝑟2=∑𝐿 𝓁=1‖𝑊𝓁‖2fortheprojectionmap,suchthatAssumption(b)ofLemmaD.6holds. Asafunction of𝑧=‖𝑊𝐿⋯𝑊1‖,(57)takesthequadraticform 𝑝(𝑧)=𝐿⋅𝑧2−𝐿‖𝑊∗‖⋅𝑧+𝜂2𝐿−2𝑟2. Overt... | https://arxiv.org/abs/2503.11891v1 |
𝑓∶ℳ(𝑑1,𝑑0)×⋯×ℳ(𝑑𝐿,𝑑𝐿−1)→ℝ. Choosing the inner product2⟨𝐴,𝐵⟩=∑𝐿 𝓁=1⟨𝐴𝓁,𝐵𝓁⟩onthisspaceyieldstheadditivedecomposition ⟨∇𝑓(𝐴),𝐵⟩=𝐿∑ 𝓁=1⟨∇𝐴𝓁𝑓(𝐴),𝐵𝓁⟩ ofthegradient ∇𝑓(𝐴). Fréchetdifferentiationcommuteswithboundedlinearmaps(CorollaryXIII.3.2 of[Lan93]),sothesecondderivativeof 𝑓maybecomputedvia 𝐷2... | https://arxiv.org/abs/2503.11891v1 |
TWO STATISTICAL PROBLEMS FOR MULTIVARIATE MIXTURE DISTRIBUTIONS RICARDO FRAIMAN, LEONARDO MORENO, AND THOMAS RANSFORD Abstract. In an earlier work [12], it was shown that mixtures of mul- tivariate Gaussian or t-distributions can be distinguished by projecting them onto a certain predetermined finite set of lines, the ... | https://arxiv.org/abs/2503.12147v1 |
the problem. Two algorithms implementing these ideas are described, and then illus- trated with some simulations. The simulations indicate a high degree of agreement between several well-known distance-measures and the one that we propose. Finally, in Section 4, we present a new procedure for estimating mix- tures of m... | https://arxiv.org/abs/2503.12147v1 |
that ν >1) and a variance of σ2ν/(ν−2) (if ν >2). In the multivariate case, where d >1, a Gaussian measure PonRdis defined by a density of the form (2)1 (2πdet(Σ))d/2exp −1 2(x−µ)TΣ−1(x−µ) (x∈Rd), where µ∈Rd, and where Σ is a real d×dpositive-definite matrix. 4 RICARDO FRAIMAN, LEONARDO MORENO, AND THOMAS RANSFORD AG... | https://arxiv.org/abs/2503.12147v1 |
Scontaining ( d2+d)/2 vectors is also an sm-uniqueness set. We now recall the Cram´ er–Wold theorem for Gaussian mixtures [12, The- orem 4.1]. Theorem 2.1. LetPandQbe convex combinations of ℓandmGaussian measures on Rdrespectively. Let Sbe a strong sm-uniqueness set for Rd containing at least (1/2)(ℓ+m−1)(d2+d−2) + 1 v... | https://arxiv.org/abs/2503.12147v1 |
quote: “Through its basis in a statistical modeling framework, model-based clustering provides a principled and repro- ducible approach to clustering. In contrast to heuristic approaches, model- based clustering allows for robust approaches to parameter estimation and objective inference on the number of clusters, whil... | https://arxiv.org/abs/2503.12147v1 |
for mixtures of Gaussian or t-distributions. Indeed, given the two samples ℵ1 andℵ2we consider the statistical test problem: H0 :P=QvsHA:P̸=Q. Next we perform the algorithm and calculated MAk. Once MAkhas been calculated, the critical value can be found using bootstrap as in [11]. 3.2.2. One-sample and different cluste... | https://arxiv.org/abs/2503.12147v1 |
and 1. If the distributions F1i:=ℓX j=1αjN(µ1ij, σ2 1ij) and F2i:=mX j=1βjN(µ2ij, σ2 2ij), are equal, then MAk= 0, and varies according to the difference between the distribution of the mixtures fitted by both partitions, increasing as long as they are further away. 3.3.Simulations. 3.3.1. A simple example of distances... | https://arxiv.org/abs/2503.12147v1 |
sample of size n= 500 from the mixture of two bivariate Gaussians distributions, with parameters λ:= (0 .3,0.7), µ1:= (1 ,−1) and µ2= (−2,2), Σ1:= 1 0 .5 0.5 2 and Σ 2:= 3 1 1 4 . The clustering methods implemented are k-means, see [19], k-medoids (PAM), [27], Spectral clustering, see [34], Hierarchical Ward, see [... | https://arxiv.org/abs/2503.12147v1 |
shown for the location case, where the covariances are all equal and known. In contrast, we just prove strong con- sistency, but this is done for the general case where the covariances are unknown and may be different, in an almost universal framework assum- ing only a differentiability condition on the characteristic ... | https://arxiv.org/abs/2503.12147v1 |
. . , Rebψu(tm),Imbψu(t1), . . . , Imbψu(tm)t , where Re zand Im zdenote the real and imaginary parts of the complex number zrespectively. We define the estimator ( bυu,(1),bσ2 u,(1),bΛu,(1)) as the values of the parameters that minimize the product Zu−Fu(υ, σ2)tW Zu−Fu(υ, σ2) , where Wis some weighting matrix de... | https://arxiv.org/abs/2503.12147v1 |
Γ = diag( γ1, . . . , γ d), then ΠS+,d(S) =UΓ+Ut, where Γ+= diag( γ+ 1, . . . , γ+ d) and γ+ j= max( γj,0) (j= 1, . . . , d ). If we want a strictly positive-definite matrix, then we can add a term δId, where Idis the d×didentity matrix and δis a very small positive constant. (iv) Lastly, we note that the algorithm can... | https://arxiv.org/abs/2503.12147v1 |
adequate fitting of our estimators (at least within this brief example). In each replicate, we can calculate the L2-distance (error) between the estimated mean and mean vectors and their respective true values, see Fig- ure 6. The estimation of the mixture parameter is similar, giving on average cλ1EM−st= 0.312 and cλ1... | https://arxiv.org/abs/2503.12147v1 |
financing). In the sample, 4852 and 1585 students attend public and private schools, respectively. By means of a multiple-choice test, using Item Response Theory, a score in Mathematics is assigned to the item. In addition, an index of each student’s socioeconomic and cultural level is constructed from data collected i... | https://arxiv.org/abs/2503.12147v1 |
are not addressed in the present manuscript. The number of directions kinto which we project is not a smoothing pa- rameter that should be chosen in an optimal way. In practice we suggest to use a larger value for k(without increasing too much the computational time) than the one provided by Theorem 2.1 which correspon... | https://arxiv.org/abs/2503.12147v1 |
Rev. Stat. Appl. , 10:573–595, 2023. [15] L. P. Hansen. Large sample properties of generalized method of moments estimators. Econometrica , 50(4):1029–1054, 1982. [16] C. R. Heathcote. The integrated squared error estimation of parameters. Biometrika , 64(2):255–264, 1977. [17] N. J. Higham. Computing a nearest symmetr... | https://arxiv.org/abs/2503.12147v1 |
Bulletin , 91(3):461–481, 1982. [37] D. Xu and J. Knight. Continuous empirical characteristic function estimation of mix- tures of normal parameters. Econometric Rev. , 30(1):25–50, 2011. [38] G. Youness and G. Saporta. Comparing partitions of two sets of units based on the same variables. Adv. Data Anal. Classif. , 4(... | https://arxiv.org/abs/2503.12147v1 |
arXiv:2503.12151v1 [math.ST] 15 Mar 2025Optimal ANOVA-based emulators of models with(out) derivatives Matieyendou Lamboni1a,b aUniversity of Guyane, Department DFR-ST, 97346 Cayenne, Fren ch Guiana, France b228-UMR Espace-Dev, University of Guyane, University of Réun ion, IRD, University of Montpellier, France. Abstrac... | https://arxiv.org/abs/2503.12151v1 |
stabilit y and accuracy of polynomial chaos expansions, the number of m odel runs needed is firstly estimated at the square of the dimension of the basis u sed ([36]), and then reduce at that dimension up to a logarithm factor ([ 37,38]). Note that for dinputs, such a dimension is about (w+ 1)dfor the tensor-product bas... | https://arxiv.org/abs/2503.12151v1 |
that are easy to fit and compute, and can cope with every distribution of conti nuous input variables; • examine the convergence analysis of our emulators with a pa rticular focus on i) dimension-free upper-bounds of the biases and MSEs; ii) the parametric rates of convergence (i.e., O(N−1)); and iii) the number of mode... | https://arxiv.org/abs/2503.12151v1 |
vector of independent and continuous variables, supported on an open domain Ω. Assumption 2 (A2).f(·)is a deterministic function with f(·)∈Wd,2. 2.1. Full derivative-based emulators Under (A2), every sufficiently smooth unction f(·)admits the derivative-based ANOVA (Db-ANOVA) expansion (see [ 22,13]), that is, ∀x∈Ω, f(x)... | https://arxiv.org/abs/2503.12151v1 |
([ 42,43,44,45])) of orderd0is given by fT,d0(X) :=E[f(X′)]+/summationdisplay v⊆{1,...,d} 0<|v|≤d0fv(Xv). WhilefT,d0is an approximation of fin general, the equality holds for some functions. Given an integer α≥0, consider the space of functions Lα,0:= f:Rd→Rn:/vextendsingle/vextendsingle/vextendsingle/vextendsi... | https://arxiv.org/abs/2503.12151v1 |
emulators of models, even wh en all the inputs are important according to screening measures. 3.1. Stochastic surrogates of functions using Db-ANOVA Consider integers L >0, q >0;βℓ∈Rwithℓ= 1,...,L ;h:= (h1,...,h d)∈Rd +, and denote with V:= (V1,...,V d)ad-dimensional random vector of independent variables satisfying: ∀... | https://arxiv.org/abs/2503.12151v1 |
moments allows for deriving the emulator of any simulator or the estimator of any function. To that end , we are given two independent samples of size N, that is, {X′ i}N i=1:=/braceleftbig/parenleftbig X′ i,1,...,X′ i,d/parenrightbig/bracerightbigN i=1fromX′and {Vi}N i=1:={(Vi,1,...,V i,d)}N i=1fromV. The full and con... | https://arxiv.org/abs/2503.12151v1 |
(see Corollary 1). To provide such results, consider Kmax 1,r∗,d0:= max w⊆{1,...,d} r∗<|w|≤d0/braceleftbig Kw,(L−r∗−1)M|w|+2(L−r∗−1)/bracerightbig ; Kmax 2,r∗,d0:= max w⊆{1,...,d} r∗<|w|≤d0/braceleftbig M|w|+2(L−r∗−1)Γ|w|+2(L−r∗−1)/bracerightbig ; K1,ρmin,r∗:= 2ρmin/parenleftbiggd 2ρmin/parenrightbiggr∗+1/parenleftB... | https://arxiv.org/abs/2503.12151v1 |
Corollary 3. Given ( 5),r∗=d0−1, assume f∈ Hαwithα∈ {0,max(d,d0+2(L−d0))}; hk=h→0and (A1)-(A3) hold. Then, the MSE and IMSE share the same uppe r- bound given as follows: E/bracketleftbigg/parenleftBig /hatwidefc N,d0−fc/parenrightBig2/bracketrightbigg ≤2Dd0,ρmin/vextendsingle/vextendsingle/vextendsingle/vextendsingleh... | https://arxiv.org/abs/2503.12151v1 |
2ρ∗ min/parenrightbigg . However, different emulators are going to be built in order to estimate f(x)for any valuexofX. Constructions of balls of given nodes and the radius 1/ρ∗ minare an interesting perspective. Remark 2. Whenf /∈ Ld0,0, in-depth structural assumptions on fthat should allow to enjoy the above MSEs conc... | https://arxiv.org/abs/2503.12151v1 |
N/bracketleftBig 2(κ2d(d+0.8))d01 Id0≤d∗ 0+2d(κ2(d+0.8))d01 Id0>d∗ 0/bracketrightBig , 18 provided that κ2(d+0.8)≥1. Corollary 9. Letr∗=d−1andL=d+1. Assume f∈ Hαwithα∈ {0,d+1}; ξ≤/bracketleftbig dMd+1Γd+1κd 1/bracketrightbig−1;hk=h∝N−ηwithη∈]1 2,1[; and (A1)-(A2) hold. Then, E/bracketleftbigg/parenleftBig /hatwiderfc N... | https://arxiv.org/abs/2503.12151v1 |
all the inputs are important. Thus, we have to include a lot of ANOVA compone nts in our emulator with small effective effects since the variance of th at function is fixed. More information is needed to better design the structure of this function. Figures 2-3depict the predictions versus the simulated outputs (i.e., obs... | https://arxiv.org/abs/2503.12151v1 |
Figure 4(right-bottom panel) show the observations versus predictions for derivative-free e mulators using only the com- ponents for which UBj>0.01. It turns out that our emulators provide reliable estimations. As expected (see MSEs), the derivative-based emulator using exact values of derivatives performs better. 6. C... | https://arxiv.org/abs/2503.12151v1 |
and /vectorw:= (1 Iw(1),...,1 Iw(d))lead toD(/vectorw)f=D|w|f. Also, using Ek:= Gk(X′ k)−1 IX′ k≥xk gk(X′ k)implies that Rk=Vk hkσ2Ek,k= 1,...,d . Firstly, by evaluating the above expansion at X′and taking the expectation w.r.t. V,A:=/summationtextL ℓ=1C(|u|) ℓEV/bracketleftBig f(X′+βℓhV)e(1:d) u(R(x,X′,V))/bracketrigh... | https://arxiv.org/abs/2503.12151v1 |
we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleD|w|f(x)−L/summationdisplay ℓ=1C(|w|) ℓE/bracketleftBigg f(x+βℓhV)/productdisplay k∈wVk hkσ2/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤σ2L′ wM|w|+2L′wK1,L′w/vextendsingle/vextendsingle/vextendsingl... | https://arxiv.org/abs/2503.12151v1 |
V/bracketleftBig /hatwidefc N,d0(x)/bracketrightBig ≤/parenleftbig ̥max d0/parenrightbig2 Nd0/summationdisplay p=1/parenleftbiggd p/parenrightbigg/parenleftBigg E/bracketleftbig Z2 1||Z||2 2/bracketrightbig 3ρ2 min/parenrightBiggp ≤/parenleftbig ̥max d0/parenrightbig2 N/parenleftBigg E/bracketleftbig Z2 1||Z||2 2/brack... | https://arxiv.org/abs/2503.12151v1 |
7 (3) (2024) 1–22. [14] A. Chkifa, A. Cohen, R. DeVore, C. Schwab, Sparse adapti ve Taylor approxi- mation algorithms for parametric and stochastic elliptic P DEs, ESAIM: Math- ematical Modelling and Numerical Analysis 47 (1) (2013) 253 –280. [15] P. Patil, H. Babaee, Reduced-order modeling with time- dependent bases f... | https://arxiv.org/abs/2503.12151v1 |
Mathematics, 1990. doi:10.1137/1.9781611970128 . [33] R. K. W. Wong, C. B. Storlie, T. C. M. Lee, A frequentist ap proach to computer model calibration, Journal of the Royal Statistical Societ y Series B: Statistical Methodology 79 (2) (2016) 635–648. [34] J. H. Friedman, B. E. Popescu, Predictive learning via r ule en... | https://arxiv.org/abs/2503.12151v1 |
On self-training of summary data with genetic applications Buxin Su * Jiaoyang Huang†Jin Jin‡Bingxin Zhao§ March 18, 2025 Abstract Prediction model training is often hindered by limited access to individual-level data due to privacy concerns and logistical challenges, particularly in biomedical research. Resampling- ba... | https://arxiv.org/abs/2503.12155v1 |
to assess genetic risk for various complex traits and diseases (Purcell et al., 2009). Millions of genetic variants in GWAS are used as predictors, each contributing only a small amount of information (Boyle et al., 2017). Over the last two decades, a wide variety of statistical methods have been developed to improve P... | https://arxiv.org/abs/2503.12155v1 |
as their counterparts trained using individual-level data. We provide detailed an- alytical evaluations of ridge-type estimators and marginal thresholding estimators, both of which, along with their variants, are widely used in PRS applications (Ma and Zhou, 2021). These anal- yses provide deep insights for practical a... | https://arxiv.org/abs/2503.12155v1 |
aspect ratio p/n→γ>0. Condition 2 outlines the regularity conditions on Xrequired by technical lemmas in random matrix theory, such as bounded moments and eigenvalues. Notably, these conditions do not assume a Gaussian distribution for the elements of Xand have been shown to be robust to minor potential violations, suc... | https://arxiv.org/abs/2503.12155v1 |
is defined as h2= limn,p→∞var(Xβ)/var(y)=limn,p→∞βTXTXβ/(βTXTXβ+ϵTϵ). Thus, we have h2∈[0,1]. As n,p→∞ with p/n→γ>0,h2can be asymptotically represented as h2=lim n,p→∞∥β∥2 Σ ∥β∥2 Σ+σ2ϵ=lim n,p→∞κσ2 β·trace(Σ)/p κσ2 β·trace(Σ)/p+σ2ϵ. (2.2) 2.2 Model training and performance measures In this section, we model the process... | https://arxiv.org/abs/2503.12155v1 |
detailed in later sections, this key insight motivates the summary data-based model training approach. 5 2.2.2 Summary data-based model training Now we consider the practical scenario where only summary statistics, XTyandWTW, are available, rather than having access to X(tr)Ty(tr),WTW, and individual-level data ( X(v),... | https://arxiv.org/abs/2503.12155v1 |
a p-dimensional Gaussian random variable with mean ( n(tr)/n)XTyand covariance [n(tr)(n−n(tr))]/n2·Cov( XTy), where Cov( XTy) is defined to be Cov( XTy)= XTy−nΣβ XTy−nΣβT. This sampling distribution is chosen based on the fixed-dimension scenario where pis fixed, while the sample size n→∞ . In such a setting, this sa... | https://arxiv.org/abs/2503.12155v1 |
the class of linear estimators defined in Equation (2.3). 3.1 Resampling-based ridge estimator We quantify the out-of-sample performance of reference panel-based ridge estimators bβR(θ) in Equa- tion (2.5) and bβR(θ)∗in Equation (2.6) for θ∈R+, denoted as R2 ind,R(θ) and R2 sum,R(θ), respectively. Our analysis mainly r... | https://arxiv.org/abs/2503.12155v1 |
datasets does not lead to overfitting or reduced out-of- sample prediction performance in high dimensions. Below, we provide a proof sketch to provide insights into this counterintuitive result. 9 Proof sketch of Theorem 3.3. By the continuous mapping theorem, it su ffices to show that the nu- merator and denominator o... | https://arxiv.org/abs/2503.12155v1 |
the matched second moment Cov( XTy) ins(tr). The limiting behavior of I(2) sumprecisely cancels out this inflation, ensuring that the resulting R2 sum,R(θ) aligns exactly with the right-hand side of Equation (3.6). Thus, the dependence between s(tr)ands(v)does not negatively impact prediction accuracy. Our proof sketch... | https://arxiv.org/abs/2503.12155v1 |
selected by Algorithm 1 and Algorithm S.1, respectively. We compare the prediction accuracy across varying levels of heritability, dimensionality, and sparsity. The param- eters are set as follows: h2∈{2/5,1/2,2/3,4/5},p∈{5000,10000},n=5000,κ∈{0.05,0.5,0.9}, nblock=20, and nw=1000. Corollary 3.4. Under the same conditi... | https://arxiv.org/abs/2503.12155v1 |
for the marginal thresholding es- timator. The left panel of Figure 2 supports Theorem 3.5, demonstrating that the out-of-sample R2pattern obtained from resampling-based self-training closely aligns with that of individual-level training. Additionally, the right panel of Figure 2 illustrates that models selected by res... | https://arxiv.org/abs/2503.12155v1 |
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