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routing tasks consisting of approximately 30k prompts and responses from eleven ( M= 11 ) different LLMs. The data includes prompts from 8 benchmarks covering commonsense reasoning, knowledge-based understanding, conversation, math, and coding. Open LLM leaderboard: The Open LLM leaderboard v2 *[Fourrier et al., 2024] ... | https://arxiv.org/abs/2502.03261v2 |
vs. Binary Routers on Routerbench. 0.000 0.001 0.002 0.003 Cost Per Query, $0.500.550.600.650.700.750.800.850.90Accuracygpt-3.5-turbo-1106gpt-4-1106-preview mistralai/mixtral-8x7b-chatzero-one-ai/Yi-34B-Chat CARROT (Roberta) CARROT (KNN) Routerbench (Roberta) Routerbench (KNN) (b) CARROT vs. Routerbench on Routerbench.... | https://arxiv.org/abs/2502.03261v2 |
process the test set at a fraction of the cost of o3-mini, but at its best CARROT cannot exceed the o3-mini’s accuracy. On the other hand, we showed that CARROT can outperform the best model (Qwen2-72B) by a large margin in Open 8 LLM leaderboard v2 (see Figure 3b). The difference is likely due to the existence of a si... | https://arxiv.org/abs/2502.03261v2 |
, 24:49–64, 1996b. T. T. Cai and H. Wei. Transfer Learning for Nonparametric Classification: Minimax Rate and Adaptive Classifier. arXiv:1906.02903 [cs, math, stat] , June 2019. V . Castelli, R. Chakravarti, S. Dana, A. Ferritto, R. Florian, M. Franz, D. Garg, D. Khandelwal, S. McCarley, M. McCawley, M. Nasr, L. Pan, C... | https://arxiv.org/abs/2502.03261v2 |
Cohen, and X. Lu. PubMedQA: A dataset for biomedical research question answering. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP) , pages 2567–2577. Association for Computational Linguistic... | https://arxiv.org/abs/2502.03261v2 |
Araki, A. Gundroo, B. Wang, R. Menon, M. Parvez, and Z. Feng. Delucionqa: Detecting hallucinations in domain-specific question answering. pages 822–835, 01 2023. doi: 10.18653/v1/2023.findings-emnlp.59. 11 T. Shnitzer, A. Ou, M. Silva, K. Soule, Y . Sun, J. Solomon, N. Thompson, and M. Yurochkin. Large language model r... | https://arxiv.org/abs/2502.03261v2 |
-3 -1 -70b- instruct ’, 7 ’llama -3 -1 -8b- instruct ’, ’llama -3 -2 -1b- instruct ’, 8 ’llama -3 -2 -3b- instruct ’, ’llama -3 -3 -70b- instruct ’, 9 ’llama -3 -405b- instruct ’, ’mixtral -8 x7b - instruct -v01 ’] Each key corresponds to another list. "prompt" contains the model queries, the "dataset" list indicates w... | https://arxiv.org/abs/2502.03261v2 |
each LLM are collected using a corresponding chat template with a generic prompt andzero shot prompting . Given the use of chat templates and zero-shot prompting, evaluation is challenging because model responses will not necessarily follow a specific format. To alleviate this, we adopt the evaluation protocol from Mix... | https://arxiv.org/abs/2502.03261v2 |
$0.20.30.40.50.60.70.8Accuracy WizardLM/WizardLM-13B-V1.2gpt-3.5-turbo-1106gpt-4-1106-preview mistralai/mixtral-8x7b-chatzero-one-ai/Yi-34B-Chat0.544 0.7720.99 ERM router CARROT (Roberta) CARROT (Known Cost) Figure 5: Router Bench Supplementary. B.2 Open LLM Leaderboard V2 LLMs and costs: Table 3 gives all models used ... | https://arxiv.org/abs/2502.03261v2 |
local polynomial regression in eq. (C.1) and (C.1). These conditions are taken directly from Audibert and Tsybakov [2007, Section 3]. Definition C.2 (Kernel regularity) .For some β >0we say that a kernel K:Rd→[0,∞)satisfies the regularity condition with parameter β, or simply β-regular if the following are true: for so... | https://arxiv.org/abs/2502.03261v2 |
bg(X) =mifbf(X)has the minimum value at the m-th coordinate. This further implies bfm(X)≤bfg(X)(X). The only way this could happen if |bfm(X)−fm(X)| ≥δm(X)/2or|bfg(X)(X)−fg(X)(X)| ≥δm(X)/2. Otherwise, if both are|bfm(X)−fm(X)|<δm(X)/2and|bfg(X)(X)−fg(X)(X)|<δm(X)/2this necessarily implies bfg(X)(X)< fg(X)(X) +δm(X) 2 =... | https://arxiv.org/abs/2502.03261v2 |
establish (C.10) for each k∈[K1]. To obtain this, we construct a finite family of probability measures Mr⊂ P (indexed by [r]) and study max P∈Mr. The technical tool which allows this to be fruitful is a generalized version of Fano’s lemma. 20 Lemma C.5 (Generalized Fano’s lemma) .Letr≥2be an integer and let Mr⊂ P conta... | https://arxiv.org/abs/2502.03261v2 |
3)β ≤Kγ,k∥x−x′∥β ∞≤Kγ,k∥x−x′∥β 2. In order transfer the inequality in Fano’s lemma to a statement on rate of convergence, we need an upper bound on KL(Pσ1, Pσ2)and a lower bound on the semi-metric EPσ0(µ, g⋆ µ,σ1). These are established in the next two lemmas. Lemma C.9. Consider the probability distribution Pσfor the ... | https://arxiv.org/abs/2502.03261v2 |
k0n−2γk0+d 2γk0+d(because his defined as L×µ1 γk0 k0n−1 2γk0+d) ≤CL2γk0+dµd γk0 k0logr logM(because r≥Mm0 4) ≤CL2γk0+dlogr logM=βr In the Lemma C.5 we would likeβr+log 2 logr≤3 4so that we have 1−βr+log 2 logr≥1 4. Note that, βr+log 2 logr−3 4=βr logr+log 2 logr−3 4 =CL2γk0+d logM+log 2 log 4−3 4(because r≥4, βr=CL2γk0... | https://arxiv.org/abs/2502.03261v2 |
that Sandy will have 375 ,000 more tokens than any of her siblings , which is a precise numerical value . The model ’s answer translates this scenario into a fraction of the total , saying Sandy will have more tokens than any sibling by 3/8 million . 1 million tokens * 3/8 =375 ,000 tokens . So the model provided an an... | https://arxiv.org/abs/2502.03261v2 |
Europe s monetary union in coordination with Merkel , hoped the issues to be raised in his speech would be taken into account in Germany s coalition negotiations . One Elysee official said a eurozone budget , one of Macron s most contentious ideas , would be necessary in due course and that the president would therefor... | https://arxiv.org/abs/2502.03261v2 |
had a strong interest in the strength of France . 51 52Golden Answer (s) --- After German election , Macron to set out his vision for Europe 53 54Model ’s Answer --- French President Emmanuel Macron to introduce plans for reforming the European Union amid the uncertain aftermath of German elections 55<dmf > assistant 5... | https://arxiv.org/abs/2502.03261v2 |
and fairness. •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in... | https://arxiv.org/abs/2502.03261v2 |
contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. (b)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. (c)If the contribution is a new model (e.g., a large language model), then there should e... | https://arxiv.org/abs/2502.03261v2 |
defined or other appropriate information about the statistical significance of the experiments? Answer: [No] Justification: We present results for the version of CARROT we release to the public. 29 Guidelines: • The answer NA means that the paper does not include experiments. •The authors should answer "Yes" if the res... | https://arxiv.org/abs/2502.03261v2 |
impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? 30 Answer: [NA] Justification: We feel that there are no specific societial impacts of this work that must be highlighted. Guidelines: • The answer NA means that there is no societal im... | https://arxiv.org/abs/2502.03261v2 |
used. Guidelines: • The answer NA means that the paper does not use existing assets. • The authors should cite the original paper that produced the code package or dataset. • The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should b... | https://arxiv.org/abs/2502.03261v2 |
32 •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. •For initial submissions, do not include any information that would break anonymity (if applica- ble), such as ... | https://arxiv.org/abs/2502.03261v2 |
On the limits of some Bayesian model evaluation statistics Hien Duy Nguyen1,2, Mayetri Gupta3, Jacob Westerhout4, and TrungTin Nguyen5 1Department of Mathematics and Physical Science, La Trobe University, Melbourne, Australia. 2Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan. 3School of Mathema... | https://arxiv.org/abs/2502.03846v1 |
widely applicable Bayesian information criterion (WBIC; Watanabe, 2013). In this work, we seek to obtain general regularity conditions under which the almost sure limits of the DIC, BPIC, and WBIC can be obtained. Our technique relies on the concept of almost sure weak convergence, otherwise known as almost sure condit... | https://arxiv.org/abs/2502.03846v1 |
function with respect to the n-fold product of some dominating measure non(X,BX). Then, we may identify the sequence of posterior measures (Πn(ω,·))n∈Nas a sequence of random measures, where Πn(ω,·) = Π ( ·|X1(ω), . . . , X n(ω)), (3) is defined by its density function θ7→π(θ|X1, . . . , X n) =p(X1, . . . , X n|θ)π(θ)R... | https://arxiv.org/abs/2502.03846v1 |
Prop. 6.2). For another example, if the sequence (θn)n∈Nconverges P-a.s. toθ0, then since Tis separable, we can determine weak convergence by checking condition (1) using only a countable number of bounded continuous functions (cf. Bain & Crisan, 2009, Thm. 2.18), and determine that (δθn)n∈Nconverges to δθ0, P-a.s.w., ... | https://arxiv.org/abs/2502.03846v1 |
1 nnX i=1logp(Xi|θ)−E[logp(X|θ)] −→ n→∞0. (8) A3There exists a unique maximizer θ0∈Tof E[logp(X|θ)], in the sense that, for each θ∈T\{θ0}, E[logp(X|θ)]<E[logp(X|θ0)]. A4There exists a compact set S ⊊ Tcontaining θ0, such that there is a P-a.s. set on which lim sup n→∞sup θ∈Sc1 nnX i=1logp(Xi|θ)<E[logp(X|θ0)]. Propositi... | https://arxiv.org/abs/2502.03846v1 |
say that fis AUI with respect to (Qn)n∈N. Here, χA:T→Ris the usual characteristic function for the set A⊂T. A direct modification of Feinberg et al. (2020a, Cor. 2.8) for the stochastic setting yields the following general Lebesgue dominated convergence theorem. Theorem 1. Assume that (Qn)n∈Nconverge to Q,P-a.s.w., and... | https://arxiv.org/abs/2502.03846v1 |
for βn= 1, then BPIC nP-a.s.−→ n→∞−2E [log p ( X|θ0)]. If A1–A5 hold with nβn→ ∞, then WBIC nP-a.s.−→ n→∞−2E [log p ( X|θ0)]. 2.4 Convergence of the DIC Finally, to obtain the limit of the DIC, we require the convergence of the limit of the second right-hand term of (5). To this end, we require the following assumption... | https://arxiv.org/abs/2502.03846v1 |
conclude by considering a nonconjugate prior pair, taking a Laplace model for X∈X=R, with PDF p(x|θ) =1 2γexp −|x−µ| γ , where θ= (µ, γ)∈T= [−m, m ]×[1/s, s], with m > 0ands >1. We will endow θwith a prior measure Πwhose density πis strictly positive on T. Write Med (X)as the median of Xand assume the regularity assu... | https://arxiv.org/abs/2502.03846v1 |
sequence βn= 1/lognof Watanabe (2013) along with alternative choices βn= 1/log log n,βn= 1, and βn= 1/√n. We will also consider choices of βn= 1/nandβn= 1/{nlogn}in order to investigate the necessity of the condition nβn→ ∞. Figure 2 displays our results for 10replicates of each sample sizes nranging from 101to105, wit... | https://arxiv.org/abs/2502.03846v1 |
that uis continuous on every compact set K, asuis the uniform limit of continuous functions un(ω,·)(cf. Remmert 1991, Thm. 3.1.5). This then implies that uis continuous on T⊆Rp. For any A,B⊂Rpandϵ >0, let ¯Bϵ(A) = x∈Rp: inf y∈A∥x−y∥ ≤ϵ , and define the distance d (A,B) = inf x∈Ainfy∈B{∥x−y∥}. When Tis open, d(S,Tc)>0... | https://arxiv.org/abs/2502.03846v1 |
we have lim sup n→∞an(ω)≤lim n→∞Π Bc ϵ1(θ0)∩¯Bϵ0(S) Π ¯Bϵ2(θ0) exp (γn{u∗(ϵ1)−u∗(ϵ2) + 2δ}) =Π Bc ϵ1(θ0)∩¯Bϵ0(S) Π ¯Bϵ2(θ0) exp lim n→∞γn{u∗(ϵ1)−u∗(ϵ2) + 2δ} = 0, and via (22) we have lim sup n→∞bn(ω)≤lim n→∞Π ¯Bc ϵ0(S) Π ¯Bϵ2(θ0)exp γn sup θ∈¯Bcϵ0(S)un(ω, θ)−u∗(ϵ2) +δ =Π ¯Bc ϵ0(S) Π ¯Bϵ2(θ0)e... | https://arxiv.org/abs/2502.03846v1 |
be continuous as a consequence. (ii) For any ϵ >0, such that define the sphere of radius ϵas Sϵ={θ∈T:∥θ−θ0∥=ϵ}, and write αn=un(θ0)−sup θ∈Sϵun(θ), and α=u(θ0)−sup θ∈Sϵu(θ). SinceSϵis compact, unconverges uniformly to u, onSϵ, and since uandunare continuous, for each n, we have limn→∞αn=α. Additionally, the extreme valu... | https://arxiv.org/abs/2502.03846v1 |
together with A1 and A2 imply that the conditions for Theorem 1 are met, and we therefore have lim n→∞Z T1 nnX i=1logp(Xi(ω)|·) Πn(ω,dθ) =Z TE[logp(X|θ)] Π0(dθ), (26) for almost every ω∈Ω. Thus, given expression (6), we have lim n→∞BPIC n=−2 lim n→∞Z T1 nnX i=1logp(Xi(ω)|·) Πn(ω,dθ) + 2 lim n→∞p n =−2Z TE[logp(X|θ)] Π0... | https://arxiv.org/abs/2502.03846v1 |
criteria based on minima of risk functions can achieve model selection consistency. This consistency is understood as the asymptotic selection of the minimal complexity model that minimizes risk as n→ ∞. Conversely, finite sample properties of information criteria-like, penalty-based methods have been studied in works ... | https://arxiv.org/abs/2502.03846v1 |
E [∆ (X)]<∞whenever E X <∞, which implies A2b and thus A2. To show A3, we simply observe that E[logp(X|θ)] =EXlog (1−θ) + log θ (28) is maximized at θ0= 1/{1 +EX}, by Fermat’s first order condition and the fact that (28) is strictly concave. Finally, the concavity of θ7→logp(x|θ), for each x∈X, implies quasiconcavity, ... | https://arxiv.org/abs/2502.03846v1 |
X⊤θand∥θ∥2are continuous in θ. Thus, the Weierstrass extreme value theorem implies that logp(X|θ)≤p 2log (2 π) +∥X∥2+ 2|X|⊤sup θ∈¯Bρ(0)|θ|+ sup θ∈¯Bρ(0)∥θ∥2= ∆ ( X), where |·|, when applied to a vector, denotes the elementwise absolute value function. Observe that E[∆ (X)]<∞whenever Eh ∥X∥2i <∞, and thus A2b is verifie... | https://arxiv.org/abs/2502.03846v1 |
where m > 0ands >1. Since Tis compact, the analysis is the same for any prior distribution Πwhose density πis strictly positive on T. Writing the log-likelihood as logp(X|θ) =−log (2 γ)−1 γ|X−µ|, we observe that θ7→logp(X|θ)is is continuous for each fixed X∈X, thus it is Caratheodory and satisfies 29 A1. We verify A2a ... | https://arxiv.org/abs/2502.03846v1 |
We then apply the Poincare-type approximation ψ(a)≈loga−1 2a 31 to get the approximation ¯Xn logbn an+bn −an 2bn(an+bn) + logan an+bn −bn 2an(an+bn) . By substitution of ¯θn=EY, we also have 1 nnX i=1logp Xi|¯θn =¯Xnlogbn an+bn + logan an+bn . Our approximation is then obtained via the definition of DIC n... | https://arxiv.org/abs/2502.03846v1 |
Forbes, F., Nguyen, H. D., Nguyen, T., & Arbel, J. (2022). Summary statistics and discrepancy measures for approximate Bayesian computation via surrogate posteriors. Statistics and Computing , 32(5), 85. (Cited on page 4.) Ghosal, S. & Van der Vaart, A. (2017). Fundamentals of nonparametric Bayesian inference , volume ... | https://arxiv.org/abs/2502.03846v1 |
on page 4.) Serfozo, R. (1982). Convergence of Lebesgue integrals with varying measures. Sankhya A , 44, 380–402. (Cited on pages 2 and 7.) Shapiro, A., Dentcheva, D., & Ruszczynski, A. (2021). Lectures on Stochastic Programming . SIAM. (Cited on page 21.) 34 Sin, C.-Y. & White, H. (1996). Information criteria for sele... | https://arxiv.org/abs/2502.03846v1 |
Consistent model selection in a collection of stochastic block models Lucie ARTS1 1Sorbonne Universit´ e and Universit´ e Paris Cit´ e, CNRS, Laboratoire de Probabilit´ es, Statistique et Mod´ elisation, F-75005 Paris, France February 5, 2025 Abstract We introduce the penalized Krichevsky-Trofimov (KT) estimator as a c... | https://arxiv.org/abs/2502.03848v1 |
consistency without such bounds is a significant challenge, as it requires addressing the complexities of unbounded parameter spaces while ensuring the robustness of the estimator. Another important aspect of model selection is the pursuit of minimal penalties. In the literature on order estimation, there is a focus on... | https://arxiv.org/abs/2502.03848v1 |
2 relevant for networks that evolve across multiple dimensions or layers. On the other hand, in the DynSBM [Matias and Miele, 2017] a sequence of graphs is observed and node groups evolve over time via independent Markov chains. While in MLSBM, index Tcorresponds to the level (i.e. number of layers), it represents time... | https://arxiv.org/abs/2502.03848v1 |
both models, we assume that, for each t, the nnodes are split into klatent groups, as encoded by the random variables Z1:T n= (Zt i)1≤t≤T,1≤i≤nwith Zt i∈ {1, . . . , k }denoting the label of the i-th vertex at level or time t. We also assume that, conditionally on the collection of latent groups {Zt i}1≤t≤T,1≤i≤n, the ... | https://arxiv.org/abs/2502.03848v1 |
iT−1Y t=1πZt iZt+1 i,∀i∈J1, nK. Since, with discrete latent random variables, identifiability (i.e. it is possible to uniquely determine the model parameters based on the distribution of the observed data) can only be obtained up to label switching on the node groups for a permutation σwhich acts globally (meaning it i... | https://arxiv.org/abs/2502.03848v1 |
If a multi-layer or dynamic SBM has order k0then it cannot be reduced to a model with less communities than k0. This implies that there exists 1≤t≤Tsuch that the matrix P0,t does not have two identical columns (or rows). Before defining the penalized Krichevsky–Trofimov estimator, we define a prior distribution on the ... | https://arxiv.org/abs/2502.03848v1 |
1 (in practice 1 + ϵ) times log n, is the penalty term that ensures the estimator does not overestimate the right number of clusters. The structure of the penalty also results from the proof, and it is certainly too large. We can now define the estimator: first in a multi-layer SBM, then in a dynamic SBM. The only diff... | https://arxiv.org/abs/2502.03848v1 |
≤logKTT k A1:T n×n|z1 n +k 2log(n2T) +k(k−1) 2log(nT) +Tk(k−1) 2logn+ck,T, (10) where ck,T=k 3T[k(k−1) + 2] + Tk(k−1) + 2 k. The proof can be found in Appendix A.1. 8 4.2 Non-overestimation in the consistency proof We now establish that the estimator ˆkKTdoes not overestimate the true number of communities k0. We sta... | https://arxiv.org/abs/2502.03848v1 |
entries equal to 1. Moreover, the matrix Qn(¯ z,z)satisfies ∥Qn(¯ z,z)∥1=kX a=1k0X a′=1[Qn(¯ z,z)]aa′= 1, for all (¯ z,z)and Eθ0 mod[˜oab(¯ z, At n×n)|Z=z] =n2[Qn(¯ z,z)P0,tQn(¯ z,z)⊺]ab, where θ0 mod= (π0,P0)if mod =ML and θ0 mod= (Π0,P0)if mod =dyn, and where ˜oab(¯ z, At n×n) =X 1≤i,j≤n1{¯zi=a,¯zj=b}at ij, (12) for ... | https://arxiv.org/abs/2502.03848v1 |
ρtnna(z⋆ n,k)nb(z⋆ n,k)! ≤1 2X 1≤a,b≤kπ⋆ aπ⋆ bτ S⋆,t ab , almost surely. Moreover, for all 1≤t≤T, the pair (π⋆, S⋆,t)is given by π⋆ a= [R⋆1k0]a, a∈ {1, . . . , k } S⋆,t ab=[R⋆S0,t(R⋆)⊺]ab [R⋆1k01⊺ k0(R⋆)⊺]ab, a, b ∈ {1, . . . , k }, (13) for a matrix R⋆∈[0,1]k×k0satisfying ∥R⋆∥1= 1and having one and only one non-zero... | https://arxiv.org/abs/2502.03848v1 |
, ρT n) (resp. ρnnot depending on t) with ρt n≥Clogn/n andρt n→0for all t∈ {1, . . . , T }. Then, the ˆkKT A1:T n×n order estimator does not underestimate k0, eventually almost surely when n→ ∞ . Proof. We start with case of the multi-layer SBM. Following the approach of Cerqueira and Leonardi [2020] in the proof of ... | https://arxiv.org/abs/2502.03848v1 |
by (13). Finally, by Lemma 6 we have that the difference of (17) and (16) is lower bounded by 1 2 X 1≤a,b≤k0πaπbτ S0,t ab −X 1≤a,b≤kπ⋆ aπ⋆ bτ S⋆,t ab ≥0 14 with a strict inequality if S0,tdoesn’t have two identical columns (there exists such a tby definition of the model). Then we have proven the proposition fo... | https://arxiv.org/abs/2502.03848v1 |
and k0= 6 communities with uniform proportions. For each layer the connection probability matrix P0,tis defined as shown in Figure 15 1, where u1,u2,u3andu4are independently and identically distributed according to a uniform distribution U(0.6,1). This represents a mix of communities and disassortative behaviour. We ha... | https://arxiv.org/abs/2502.03848v1 |
0.00 0.83 1.00 1.00 1.00 1.00 1.00 1.00 1.00 BHMC accuracy 0.00 0.83 1.00 1.00 1.00 1.00 1.00 1.00 1.00 NCV accuracy 0.01 0.78 0.98 0.99 0.99 0.99 0.98 0.96 0.92 This table indicates that the accuracy of the penalized KT estimator, as well as that of the other estimators, improves significantly as ρincreases. We also o... | https://arxiv.org/abs/2502.03848v1 |
collection. Thus, the penalized KT estimator is of course slower than a spectral method but it is very competitive with the other two. By interpreting these results in parallel with those of the first experiment shown in Figure 2, the penalized KT estimator emerges as the fastest among the estimators that successfully ... | https://arxiv.org/abs/2502.03848v1 |
a model to fit data from an exponential family. The Annals of Statistics , 16(1), 1988. URL http://www.jstor.org/stable/2241441 . Paul W. Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmodels: First steps. Social Networks , 5(2), 1983. doi: 10.1016/0378-8733(83)90021-7. URL https: //www.scienc... | https://arxiv.org/abs/2502.03848v1 |
ab/nab. Then with the same calculations as in Cerqueira and Leonardi [2020], we have that Pπ,P zn,a1:T n×n KTT k zn,a1:T n×n≤eC(zn,a1:T n×n), where C zn,a1:T n×n = log Γ 1 2 Γ n+k 2 Γ k 2 Γ n+1 2! +TX t=1X 1≤a≤b≤klog Γ 1 2 Γ (nab+ 1) Γ nab+1 2! . Then, by using Stirling’s formula for the Γ function, we w... | https://arxiv.org/abs/2502.03848v1 |
η >0 such that dH(B(R⋆, ϵ), B)> η, where dHis the Hausdorff distance and B(R⋆, ϵ) is the ball for the ∥ · ∥ 1norm over matrices centered at R⋆with radius ϵ. Since ( Qmax nj)j→R⋆there exists n0such that for all j > n 0we have ( Qmax nj)j∈ B(R⋆, ϵ). So, for all j > n 0 dH(Bnj(c), B) = max {sup M∈Bnj(c)d(M, B ),sup N∈Bd(B... | https://arxiv.org/abs/2502.03848v1 |
by using this in (24) to bound (23) that ∞X n=1Pπ0,P0 ˆkKT a1:T n×n ∈(k0,logn] ≤eck0,T∞X n=1logn n1+ϵ<∞. Now the result follows by the first Borel Cantelli lemma. The proof for a dynamic SBM is the same using the part of the dynamic SBM in Lemma 1. B.3 Proof of Lemma 3 We write the proof in the case of a multi-laye... | https://arxiv.org/abs/2502.03848v1 |
by 2, given by 2 At ij1{¯zi= ¯zj=a}, with expected value given by 2 P0,t zizj1{¯zi= ¯zj=a}, and the sum of na(¯ z) independent Bernoulli random variables, given by At ii1{¯zi=a}, with expected value given by Pzizi1{¯zi=a}. Using Hoeffding’s inequality and the fact that 4 naa(¯ z) +na(¯ z)≤2n2, we have that for any δ >0... | https://arxiv.org/abs/2502.03848v1 |
aπ⋆ bτ S⋆,t ab . This concludes the proof of the lemma. C.3 Proof of Lemma 6 AsR⋆has one and only one non-zero entry in each column, we have that there is a surjective function h: [k0]→[k] connecting each community in [ k0] (columns of R⋆) with its corresponding community in [ k] (line with non-zero entry). Then for ... | https://arxiv.org/abs/2502.03848v1 |
be attained at one of the vertices of the polyhedron; that is, on those matrices Rsuch that at most one entry per column is greater than zero. Since αa>0 for all a∈ {1, . . . , k 0}, it follows that each column must have at least one strictly positive entry. Thus, the maximum is achieved on matrices where one and only ... | https://arxiv.org/abs/2502.03848v1 |
submitted ISSN 2824-7795A fast algorithm to compute a curve of confidence upper bounds for the False Discovery Proportion using a refer- ence family with a forest structure Guillermo Durand1Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay Date published: 2025-01-31 Last modified: 2025-01-31 Abstract This p... | https://arxiv.org/abs/2502.03849v1 |
analysis, like Genome-Wide Association Studies, where multiple features are tested to find promising ones. Classical multiple testing theory like Family-Wise Error Rate (FWER) control or False Discovery Rate (FDR) control (Benjamini and Hochberg, 1995) can be used, but a more recent trend consists in the computation of... | https://arxiv.org/abs/2502.03849v1 |
path of selection sets (St)t∈N∗m, for example the hypotheses attached to the tsmallest p-values. Whereas the algorithm provided in the aforementioned work (Durand et al., 2020, Algorithm 1), which is reproduced here, see Algorithm 1 , is fast for a single evaluation, it is slow and inefficient to repeatedly call it to ... | https://arxiv.org/abs/2502.03849v1 |
that will be re-used in the remainder of the paper. Example 2.1 (Gaussian one-sided) .In this case we assume that X= (X1, . . . , X m)is a Gaussian vector and the null hypotheses refer to the nullity of the means in contrast to their positivity. That is, formally, (X,X) = (Rm,B(Rm)),P={N(µ,Σ) :∀j∈N∗ m, µj≥0,Σpositive s... | https://arxiv.org/abs/2502.03849v1 |
1. We also need to introduce the notion of depth with the following function: ϕ:K → N∗ k7→1 +|{k′∈ K:Rk⊊Rk′}|.(10) Example 2.2. Letm= 25 ,R1={1, . . . , 20},R2={1,2},R3={3, . . . , 10},R4={11, . . . , 20}, R5={5, . . . , 10},R6={11, . . . , 16},R7={17, . . . , 20},R8={21,22},R9={22}. This is the same example as Exampl... | https://arxiv.org/abs/2502.03849v1 |
The family is assumed complete, otherwise the first step would be to complete it. In the original paper, Khused to designate the elements of Kat depth hplus the atoms at depth ≤h. Actually one can realize that the last assumption is not needed for this algorithm to perform exactly the same, with the added benefit of no... | https://arxiv.org/abs/2502.03849v1 |
that, when setting i0=iandip=i′+ 1, the sequence (i0, . . . , i p)is strictly increasing, (ij−1, ij−1)∈ K for all 1≤j≤pand finally ζ(i,i′)=ζ(i0,ip−1)≥Pp j=1ζ(ij−1,ij−1). An important note is that for a removed (i, i′)∈ K\Kpr, we can always choose the indices i1, . . . , i p−1 such that actually (ij, ij+1−1)∈ Kprand not... | https://arxiv.org/abs/2502.03849v1 |
a by-product. The following proposition states that Algorithm 2 indeed produces the pruned region as in Definition 3.1. Proposition 3.2. The final Lreturned by Algorithm 2 is equal to Kpr:L=Kpr. Proof. First,K \ L ⊆ K \ Kpris trivial: a ksuch that ζk≥P k′∈Succ kV ec k′obviously satisfies the condition of Definition 3.1... | https://arxiv.org/abs/2502.03849v1 |
mand1≤h≤H, we denote by k(t,h)the element of Khsuch that it∈Rk(t,h)if it exists, and we denote by hmax(t)the highest hsuch that k(t,h)exists. Example 3.1 (Continuation of Example 2.2 and Example 2.3) .Assume that the reference family of Example 2.2 has been labeled as in Example 2.3 and completed. Let (i1, . . . , i 25... | https://arxiv.org/abs/2502.03849v1 |
regions of the current partition-realizing Pt. In particular, we always have, for any t∈Nm,K1⊆ K tandPt⊆ K t. We can also remark that the sequence (Kt)0≤t≤mis non-increasing for the inclusion relation, and that K0=K. Theorem 3.1 (Fast curve computation) .Let any t∈Nm. Then, Pt∈P, and for all k∈ K t, we have V∗ R(St∩Rk)... | https://arxiv.org/abs/2502.03849v1 |
. . , h max(t)do 10: findk(t,h)∈ Khsuch that it∈Rk(t,h) 11: ηk(t,h)←ηk(t,h)+ 1 12: ifηk(t,h)< ζkthen 13: pass 14: else 15: K−← K−∪ {k(t,h)} 16: break the loop 17: end if 18: end for 19: Vt←Vt−1+ 1 20: end if 21: end for 22: return (Vt)1≤t≤m 23:end procedure First, we apply Algorithm 2 to the family. This results in pru... | https://arxiv.org/abs/2502.03849v1 |
of Example 2.2 at t= 0in Algorithm 3 . P1:5 ζ(1,5)= 6 η1 (1,5)= 1P8 ζ(8,8)= 3 η1 (8,8)= 0 P1 ζ(1,1)= 2 η1 (1,1)= 0P2:3 ζ(2,3)= 1 η1 (2,3)= 0P4:5 ζ(4,5)= 4 η1 (4,5)= 1P6 ζ(6,6)= 1 η1 (6,6)= 0P7 ζ(7,7)= 0 η1 (7,7)= 0 P2 ζ(2,2)= 2 η1 (2,2)= 0P3 ζ(3,3)= 4 η1 (3,3)= 0P4 ζ(4,4)= 2 η1 (4,4)= 1P5 ζ(5,5)= 3 η1 (5,5)= 0 Figure 6... | https://arxiv.org/abs/2502.03849v1 |
of (18) We first derive (18) from (16) and (17). First note that for all Q∈P, Q=[ k∈K1{k′∈Q:Rk′⊆Rk} (19) and the union is disjoint. From (12), letQ∗∈Psuch that V∗ R(St) =P k′∈Q∗ζk′∧ |St∩Rk′|. Then by (19), V∗ R(St) =X k′∈Q∗ζk′∧ |St∩Rk′| =X k∈K1X k′∈Q∗ Rk′⊆Rkζk′∧ |St∩Rk′| =X k∈K1X k′∈Q∗ Rk′⊆Rkζk′∧ |St∩(Rk∩Rk′)| =X k∈K1X... | https://arxiv.org/abs/2502.03849v1 |
have both ¯k∈ Ptandk′ max∈ Ptif they are distinct. So k′ max=¯k, so¯k∈ K− t, but it cannot have been added to K− tduring a previous step of the algorithm, otherwise it would have been added to Pt, too. Hence ¯k∈ K− 0which means that ζ¯k= 0andζ¯k= 0≤ |St∩R¯k|. 3.4.2.1.2 Second subcase: ¯khas been added to Ptat a previou... | https://arxiv.org/abs/2502.03849v1 |
=V∗ R(St∩Rk) + 1 by (17) . (24) Note that by the joint construction of K− tandPton lines 17 and 18, the fact that it+1̸∈S k∈K− tRk implies that ¯kis the index of an atom, so actually hmax(t+ 1) = ϕ(¯k),¯k=k(t+1,ϕ(¯k))and the Rk, k∈ K t, such that R¯k⊆Rkare nested and are exactly indexed by the k(t+1,h),1≤h≤ϕ(¯k). We no... | https://arxiv.org/abs/2502.03849v1 |
K tsuch that it+1∈Rkand so for all such kthat are in Kt+1. So every inequality in (24)becomes an equality and we have proven that V∗ R(St+1∩Rk) =V∗ R(St∩Rk) + 1 = ηt k+ 1 = ηt+1 k, that is, (16) is true at t+ 1. Looking at the first line of (24) , we also proved that V∗ R(St+1∩Rk) =X k′∈Pt Rk′⊆Rkζk′∧ |St+1∩Rk′|. (26) T... | https://arxiv.org/abs/2502.03849v1 |
specific structure that is used to store the information of K. Speaking of the data structure, we briefly describe it, with an example. We represent (Rk)k∈Kby two lists, Candleaf_list .leaf_list is a list of vectors, where leaf_list[[i]] is the vector listing the hypotheses in the atom Pi.Cis a list of lists. For 1≤h≤H... | https://arxiv.org/abs/2502.03849v1 |
(2020) running macOS 15.1.1. The package microbenchmark allows to run code snippets a given number n_repl of times, and to compute summary statistics on the computation time. The script executing the computation can be found in the same repository as this manuscript. Four scenarios are studied, all based on a common se... | https://arxiv.org/abs/2502.03849v1 |
3.4614076 3.5061541 3.8822089 100 fast.not.pruned 0.1332744 0.1367000 0.1383806 0.1385039 0.1392707 0.1768691 100 fast.pruned 0.0921422 0.0945472 0.0974025 0.0954231 0.0978687 0.1908498 100 Table 3: Scenario 2 expr min lq mean median uq max neval naive.not.pruned 3.7280744 3.8025695 3.8514710 3.8451367 3.8831009 4.1891... | https://arxiv.org/abs/2502.03849v1 |
0.078% of the curve had been computed. Now, simulation studies with an adequate number of replications and 100% of the curve become feasible. A lot of work remains to be done on the sanssouci package. For example, to make the data format of a forest structure (Rk)k∈Kless convoluted and more user-friendly is an interest... | https://arxiv.org/abs/2502.03849v1 |
J. Goeman. A region-based multiple testing method for hypotheses ordered in space or time. Stat. Appl. Genet. Mol. Biol. , 14(1):1–19, 2015. ISSN 2194-6302. doi: 10.1515/sagmb-2013-0075. URL https://doi.org/10.1515/sagmb-2013-0075. Nicolai Meinshausen. False discovery control for multiple tests of association under gen... | https://arxiv.org/abs/2502.03849v1 |
A method for sparse and robust independent component analysis Lauri Heinonena,∗, Joni Virtaa aDepartment of Mathematics and Statistics, University of Turku, 20014 Turku, Finland Abstract This work presents sparse invariant coordinate analysis, SICS, a new method for sparse and robust independent component analysis. SIC... | https://arxiv.org/abs/2502.04046v1 |
Email address: lauri.k.heinonen@utu.fiarXiv:2502.04046v1 [stat.ME] 6 Feb 2025 the scatter matrices are the covariance matrix and the FOBI-matrix. Also ICS with other scatter matrices can lead to the solution of the ICA problem and we discuss this relation closer in Section 2. The purpose of this work is to develop, usi... | https://arxiv.org/abs/2502.04046v1 |
parameters. Of earlier approaches to robust ICA, our work is most similar with [35] who likewise used robust scatter matrices to achieve robust estimation of independent components. Besides this, the previous literature on robust ICA includes maximizing robust measures of shape [3] or divergence [8], and using rank [28... | https://arxiv.org/abs/2502.04046v1 |
vectors b∈Rp. Definition 2. Letx∈Rpbe a random vector. A matrix S∈Rp×p, calculated from x, is a scatter matrix if it is positive definite and a ffine equivariant, in the sense that S(Ax+b)=AS(x)A′ for all full rank matrices A∈Rp×pand vectors b∈Rp. Scatter matrices are a generalization of the regular covariance matrix Σ... | https://arxiv.org/abs/2502.04046v1 |
different. Later we focus on a particular class of scatters that allows us to solve the problem in an outlier-resistant way. 2.2. Regression formulation for scatter matrix ICA Fix next two scatter matrices, S1,S2, both having the independence property. Computing the respective IC solution is simple to do via eigendecom... | https://arxiv.org/abs/2502.04046v1 |
be found using two scatter matrices, let us now take a look at one class of robust scatter matrices, robust M-estimators. By robust we mean that small deviations from assumptions do not impair the model’s performance too much and large deviations from model do not cause a catastrophe [23]. For us, the interesting devia... | https://arxiv.org/abs/2502.04046v1 |
symmetrization can be expected to lower the breakdown point of a scatter matrix, see [16], but, as our simulations later in Section 4 show that robust scatters, even when symmetrized, tolerate outliers particularly well. Another measure of robustness is the influence function for a statistic T. Let Qbe a probability di... | https://arxiv.org/abs/2502.04046v1 |
from representing (2) (for fixed B, so without needing the penalty term) as pX j=1∥S1(x)−1/2rj−AB′rj∥2 2=∥S1(x)−1/2S1/2 2−AB′S1/2 2∥2 2=∥S1/2 2S1(x)−1/2−S1/2 2S1(x)−1/2S1(x)1/2BA′∥2 2 and using the reduced rank Procrustes rotation [50, Theorem 4] to minimize this expression with respect to A. As this estimate is only c... | https://arxiv.org/abs/2502.04046v1 |
sequence of lower bounds for the di fference of the two objective functions and showing that, as soon as λn1 vanihses at a suitable rate, the corresponding minimizers are within an ε-neighbourhood of each other with increasing probability. The combining of the two results then yields Theorem 4. 4. Simulations 4.1. Simu... | https://arxiv.org/abs/2502.04046v1 |
forced Robustness non−robust robustFig. 1: Median absolute error by sample size for di fferent methods. The error ribbon has width 0 .2×MAD Then, we chose the number of variables to be p=15 and the number of observations to be n=1500. We varied the number of non-zero coe fficients q=2,3,5,8,11,15 for again N=1000 repet... | https://arxiv.org/abs/2502.04046v1 |
are the variables and whose edges represent causal relations between the variables [39]. The acyclicity and directedness of the graph then ensures that “e ffects cannot precede causes”. [40] showed the remarkable fact that ICA can be used for non-Gaussian linear causal discovery. Essentially, this is because a linear n... | https://arxiv.org/abs/2502.04046v1 |
is that most of the 10 explanatory variables are estimated to be causes of the disease progression. Additionally, a relation between the covariates LDL and TC was discovered. We note that these plots should not be interpreted as the full causal graphs between the 11 variables, but rather as estimates of the set of stro... | https://arxiv.org/abs/2502.04046v1 |
are easier interpretability, as there are less coe fficients to interpret, and trying to avoid overfitting. It can also be beneficial to try di fferent values of r. The most typical way of choosing the number of non-zero coe fficients (or the amount of regularization), cross-validation, is not possible in our unsupervi... | https://arxiv.org/abs/2502.04046v1 |
the solutions of Versions I and II are close and then doing the same for Versions II and III. For a vector m∈Rp, we define a∗ n,λn,mto be the minimizer of the objective function a7→fn,λn(a,bn,0+m), over a∈Rp,∥a∥2=1. As the objective function is symmetric in the sense that fn,λn(a,b)=fn,λn(−a,−b), it is su fficient to r... | https://arxiv.org/abs/2502.04046v1 |
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