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et al. / Quantitative Scaling Limits of Stochastic Iter ative Algorithms 42 |1 nn/summationdisplay i=1∇ℓi(θ(n) ⋆)⊗2−1 nn/summationdisplay i=1∇ℓi(θ⋆)⊗2| ≤1 nn/summationdisplay i=1|∇ℓi(θ(n) ⋆)⊗2−∇ℓi(θ⋆)⊗2| Now, ∇ℓi(θ(n) ⋆) =∇ℓi(θ⋆)+/integraldisplayθ(n) ⋆ θ⋆∇⊗2ℓi(θ)dθ, so ∇ℓi(θ(n) ⋆)⊗2−∇ℓi(θ⋆)⊗2 = 2∇ℓi(θ⋆)/integraldisplay... | https://arxiv.org/abs/2501.12212v1 |
log(1+It)/parenrightBig , and consequently, using Eq. ( E.5), we can write E/parenleftbigg sup 0≤s≤t|Xs|/parenrightbigg ≤3 sup 0≤s≤t/braceleftBigg e−as/radicalBigg γ+/integraldisplays 0q(u)2du/bracerightBigg E/parenleftBig/radicalbig log(1+It)/parenrightBig . Wang et al. / Quantitative Scaling Limits of Stochastic Iter... | https://arxiv.org/abs/2501.12212v1 |
log/parenleftbig e2B+2B/parenrightbig = 2B+log/parenleftbig 1+2Be−2B/parenrightbig ≤4B, and hence E/parenleftBigg sup 0≤t≤1/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−Bt/integraldisplayt 0eBs/parenleftBig√ A−/radicalbig ˜A/parenrightBig dWs/vextendsingle/vextendsingle/vextendsingle/vextendsinglep/parenrig... | https://arxiv.org/abs/2501.12212v1 |
βξj/parenrightBigg4 +E/bracketleftBigg/summationdisplay 0<r<s<k +1Q(r+1,k)3/parenleftBigg h bb/summationdisplay i=1ψI(r,i)+/radicalBigg 2h βξr/parenrightBigg3 ×Q(s+1,k)/parenleftBigg h bb/summationdisplay i=1ψI(s,i)+/radicalBigg 2h βξs/parenrightBigg/bracketrightBigg +E/bracketleftBigg/summationdisplay 0<r<s<k +1Q(r... | https://arxiv.org/abs/2501.12212v1 |
/lessorsimilar Eα−1/summationdisplay j=0η2 j/parenleftBigg/parenleftBigg hΣ−h bb/summationdisplay i=1σI(j,i)/parenrightBigg ηj+h bb/summationdisplay i=1ψI(j,i)+/radicalBigg 2h βξj/parenrightBigg2 1/2 /lessorsimilar Eα−1/summationdisplay j=0η2 j /parenleftBigg hΣ−h bb/summationdisplay i=1σI(j,i)/pare... | https://arxiv.org/abs/2501.12212v1 |
Thus, it follows from Eqs. ( F.4) to (F.6) that |Eg(Y)−Eg(Y)| /lessorsimilar∝⌊a∇d⌊lg∝⌊a∇d⌊lM/braceleftBigg wαK1CR/parenleftbiggh2Ω b+h β/parenrightbigg +w3α3C3 RK3 3/parenleftbiggh6 b3/parenleftBig Ω3+(Eψ4 I(1,1))3/2+Eψ4 I(1,1)Ω+Eψ6 I(1,1)/parenrightBig +h3 β3/parenrightbigg +w3/braceleftBigg α2h4 b2/bracketleftBigg αh... | https://arxiv.org/abs/2501.12212v1 |
−b/summationdisplay i=1σI(K,i)+b/summationdisplay i=1σI′(K,i)/parenrightBigg ηK+b/summationdisplay i=1ψI(K,i)−b/summationdisplay i=1ψI′(K,i)/parenrightBigg +/radicalBigg 2h β(ξK−ξ′ K). Wang et al. / Quantitative Scaling Limits of Stochastic Iter ative Algorithms 60 F.7. Proof of Lemma 10 F.7.1. Decomposition of the err... | https://arxiv.org/abs/2501.12212v1 |
αα−1/summationdisplay k=0αw2h2 b2/radicalbig E∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l2 ×/radicaltp/radicalvertex/radicalvertex/radicalbtE/parenleftBiggb/summationdisplay i=1σI(k,i)b/summationdisplay i=1ψI(k,i)−bΣ(b/summationdisplay i=1ψI(k,i))+bσΣΨ/parenrightBigg2 η2 k =1 αα−1/summationdisplay k=0αw2h2 b2/radicalbig E∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l... | https://arxiv.org/abs/2501.12212v1 |
α)(t), α/summationdisplay r=k+1/parenleftbig Q(k+1,r)+(1−hΣ)r−k−1/parenrightbig I[r α,r+1 α)(t)/bracketrightBigg ≤1 αα−1/summationdisplay k=0αw2h2Ω bEsup m∈{k+1,···,α−1}/vextendsingle/vextendsingle2/parenleftbig Q(k+1,m)−(1−hΣ)m−k−1/parenrightbig/vextendsingle/vextendsingle∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l ≤2 αα−1/summationdisplay ... | https://arxiv.org/abs/2501.12212v1 |
( F.24), ǫcov.ΨΨ−zG.2.3.3≤w2h3ΩΣ 2bE∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l(1+2α(1+2hαΣ)). (F.25) Thus, it follows from Eqs. ( F.22) to (F.25) that ǫcov.ΨΨ−zG.2.3 /lessorsimilarǫcov.ΨΨ−zG.2.3.1+ǫcov.ΨΨ−zG.2.3.2+ǫcov.ΨΨ−zG.2.3.3 /lessorsimilarα2w2h4ΩΣ2E∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l b+w2h3αΩΣ bE∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l +w2h3ΩΣ bE∝⌊a∇d⌊lD2f(Y)∝⌊a∇d⌊l(1+... | https://arxiv.org/abs/2501.12212v1 |
the second line, define Ykξ t:=Yt−wα/summationdisplay m=k+1Q(k+1,m)/radicalBigg 2h βξkI[m α,m+1 α)(t). (F.31) ThenYkξ tis independent with ξk. Thus using f’s properties, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleEw2h βα−1/summationdisplay k=0/parenleftbig ξ2 k−1/parenrightbig D2f(Y)/bracketle... | https://arxiv.org/abs/2501.12212v1 |
i=1σI(m−1,i)Tm−1+sm−1/parenleftBigg 1 bb/summationdisplay i=1σI(m−1,i)−Σ/parenrightBigg/bracketrightBigg . So Esup m=k+1,···,αT2 m =Esup m=k+1,···,α/bracketleftBigg T2 m−1−2hTm−1/parenleftBigg 1 bb/summationdisplay i=1σI(m−1,i)Tm−1+sm−1/parenleftBigg 1 bb/summationdisplay i=1σI(m−1,i)−Σ/parenrightBigg/parenrightBigg +h... | https://arxiv.org/abs/2501.12212v1 |
Wang et al. / Quantitative Scaling Limits of Stochastic Iter ative Algorithms 82 ≤αh2bL4α/summationdisplay j=0j−1/summationdisplay k=0Eη2 k. References Anastasiou, A. ,Balasubramanian, K. andErdogdu, M. A. (2019). Normal Approx- imation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT. InProce... | https://arxiv.org/abs/2501.12212v1 |
(2021). Stein’s method of exchangeable pairs in multi- variate functional approximations. Electronic Journal of Probability 261–50. Fischer, M. andNappo, G. (2008). On the Moments of the Modulus of Continuity of Itˆ o Processes. Stochastic Analysis and Applications - STOCHASTIC ANAL APP L28. Ge, R.,Huang, F. ,Jin, C.an... | https://arxiv.org/abs/2501.12212v1 |
the American Statistical Association 116433–450. Nemirovski, A. ,Juditsky, A. ,Lan, G. andShapiro, A. (2009). Robust Stochastic Approximation Approach to Stochastic Programming. SIAM Journal on Optimization 191574-1609. Pflug, G. C. (1986). Stochastic Minimization with Constant Step-Size: Asympto tic Laws. SIAM Journal... | https://arxiv.org/abs/2501.12212v1 |
Hardware-in-the-Loop Evaluation of Goodness of Fit (GoF) Testing for Dynamic Spectrum Sharing Mir Lodro, Simon Armour, Mark A. Beach Communication Systems and Networks Research Group School of Electrical, Electronic and Mechanical Engineering University of Bristol Bristol, United Kingdom mir.lodro@bristol.ac.uk Abstrac... | https://arxiv.org/abs/2501.12693v1 |
for the emulation and the data acquisition for spectrum sensing. The results and discussion are explained in Section V . In Section VI, we conclude our work. II. S PECTRUM SENSING SCENARIO In a dynamic spectrum sensing scenario, the secondary user (SU) must detect not only the presence of primary user devices but also ... | https://arxiv.org/abs/2501.12693v1 |
in [10]. The Cramer-von Mises test statistic W2is shown in eq.(2): W2≜nZ+∞ −∞(F1(y)−F0(y))2dF0(y) (2) It can be seen that eq. (2) does not give enough weight to the tails of the distribution. Anderson and Darling introduced a more flexible solution and introduced the weight function ϕ(F0(y))to the CM test statistic. He... | https://arxiv.org/abs/2501.12693v1 |
controller RFSoC 4x2Eigenvalue DetectionsFig. 2: Measurement setup. 1.3 1.4 1.5 1.6 1.7 T est Statistic0.00.20.40.60.81.0CDF ValueMME SU 1 MME SU 1 CDF Normal CDF 1.15 1.20 1.25 1.30 1.35 T est Statistic0.00.20.40.60.81.0CDF ValueME-AM SU 1 ME-AM SU 1 CDF Normal CDF 1.15 1.20 1.25 1.30 1.35 T est Statistic0.00.20.40.60... | https://arxiv.org/abs/2501.12693v1 |
can see when the spectrum is occupied the KL test is not satisfied i.e. DKL(P∥Q)>0 which means that the alternative hypothesis H1is true. Fig. 4 shows the noise and signal data CDFs in SU 1 using MME eigenvalue detection. We can see that the CDFs of the test statistics of the eigenvalue of the signal data do not follow... | https://arxiv.org/abs/2501.12693v1 |
the data samples for both cases where there was no primary user signal and when the primary user signal was present. The RF channel emulator was configured to emulate the 3GPP EPA channel model between PU and all SUs. The test statistics for different variants of eigenvalue detection algorithms are used for the perform... | https://arxiv.org/abs/2501.12693v1 |
arXiv:2501.12747v2 [stat.ML] 11 Feb 2025Singular leaning coefficients and efficiency in learning the ory Miki Aoyagi College of Science & Technology, Nihon University, 1–8–14, Surugadai, Kanda, Chiyoda-ku, Tokyo 101–8308 Japan. (aoyagi.miki@nihon-u.ac.jp) Abstract : Singular learning models with non-positive Fisher infor... | https://arxiv.org/abs/2501.12747v2 |
Assume that its log likelihood function has relatively finit e variance, Ex,y[logp(x,y|w0) p(x,y|w)]≥cEx,y[(logp(x,y|w0) p(x,y|w))2], w0∈W0,w∈W, for a constant c>0. Then, we have a unique probability density function p0(x,y) =p(x,y|w0)for allw0∈W0, meaning that the probability density function is the same fo r allw0∈W0.... | https://arxiv.org/abs/2501.12747v2 |
learning coefficients for Vandermonde matrix- type singularities, which are related to the three-layered neural networks and normal mixture models, among others [8, 1, 3, 5, 6]. We have also exact values for the restri cted Boltzmann machine [4]. Additionally, Rusakov and Geiger [20, 21] and Zwiernik [24], respectively,... | https://arxiv.org/abs/2501.12747v2 |
We haveh2(x,w) =vtC(x)v≥0,andK(w) =/integraltext Xh2(x,w)q(x)dx=vt/integraltext XC(x)q(x)dxv≥0.Also we haveK(w) = 0 if and only if v= 0, and therefore/integraltext XC(x)q(x)dxis positive definite. By setting α1andα2as the maximum and minimum eigenvalues of/integraltext XC(x)q(x)dx, respectively, we complete the proof. (... | https://arxiv.org/abs/2501.12747v2 |
(Rectified Linear Unit function). For a matrix A= (aij),aij∈R, define A+= (max{0,aij}). Denote the input value by x∈RH(L+1)with probability density function q(x)and output value y∈RH(1)for the multiple-layered neural network with ReLU units, which is given byh+(x,A,B) =F(1) +◦F(2) +◦···◦F(L) +(x). Consider the statistica... | https://arxiv.org/abs/2501.12747v2 |
Lare linearly independent. These conditions complete the proof of the theorem. (Q.E.D.) Using Theorem 5, we have the following theorem. Theorem 6 Let h+(x,A,B) =A(1)(A(2)x+B(2))+ The model has H(3)input units,H(1)output units, and H(2)hidden units in one hidden layer. Let us divide the hidden layer into k(2)groups: H(2... | https://arxiv.org/abs/2501.12747v2 |
contributed to the enhancement of our paper. References [1] M. Aoyagi. The zeta function of learning theory and gener alization error of three layered neural perceptron. RIMS Kokyuroku, Recent Topics on Real and Complex Singulari ties, 1501:153–167, 2006. [2] M. Aoyagi. A Bayesian learning coefficient of generaliza tion... | https://arxiv.org/abs/2501.12747v2 |
arXiv:2501.13037v1 [math.ST] 22 Jan 2025Causality for VARMA processes with instantaneous effects: The global Markov property, faithfulness and instrumental variables Ignacio González-Pérez Department of mathematics, École Polytechnique Fédérale d e Lausanne ignacio.gonzalezperez@epfl.ch Abstract. Causal reasoning has ga... | https://arxiv.org/abs/2501.13037v1 |
the field has branched in multiple directions. Causal discov ery methods rely on different assumptions to try and recover the true causal stru cture (or at least its Markov equivalence class). Classical regression metho ds are in some contexts used to estimate quantities which have well-defined causal i nterpretations. Th... | https://arxiv.org/abs/2501.13037v1 |
character- izations for directed acyclic graphs and acyclic directed m ixed graphs. Most of the proofs of the results presented in this chapter can be fou nd in the given ref- erences. However, we have stated some results which we formu lated to cater the specific needs of the theory developed in this work. The proof s ... | https://arxiv.org/abs/2501.13037v1 |
the DAG to an undirected graph calle d the moralized graph of a DAG. Definition 5 (Moralized graph of a DAG [11, Sec. 2.1.1]). Given a DAG G, we construct its moralized graph, and denote it by Gm, by drawing an edge between every pair of nodes which have a common child, and then removing the direction of all edges. Prop... | https://arxiv.org/abs/2501.13037v1 |
the structures in the path: →vi←,→vi↔,↔vi←,↔vi↔. Otherwise viis called a non-collider in the path. Definition 9 ( m-separation in an ADMG [17, Sec. 2.1]). LetGbe an ADMG over nodes V. Given two distinct nodes v,w∈VandB⊂V\{v,w}, we say that a path pbetweenvandwis blocked by Bif one of the following holds: 1. There is a n... | https://arxiv.org/abs/2501.13037v1 |
in Vare preserved by the latent projection operation. Therefore, the sets of parents, ance stors, children and de- scendants of nodes of VinVdo not change in the latent projection, as these concepts involve only directed edges. Causality for VARMA processes with instantaneous effects 7 As mentioned before, the purpose o... | https://arxiv.org/abs/2501.13037v1 |
series Sfollows a VAR( p) process with instantaneous effects if there exists p∈Nand coefficient matrices A0,A1,...,A p∈Rd×dsuch that diag(A0) = 0 and for all t∈Z: St=A0St+A1St−1+...+ApSt−p+εt, whereA0,A1,...,A pare such that det((Id−A0)λp−A1λp−1−...−Ap) = 0 implies|λ|<1, andεtis an i.i.d. process with finite second moments... | https://arxiv.org/abs/2501.13037v1 |
t, (2) withεti.i.d.∼ N(0,I2). Its associated full-time DAG GIcan be seen in Figure 1. The distribution-equivalent VAR(1) process without insta ntaneous effects is: /braceleftigg Xt=1 2Xt−1+δX t Yt=1 6Xt−1+1 2Yt−1+δY twithδti.i.d.∼ N/parenleftbigg/parenleftbigg0 0/parenrightbigg ,1 9/parenleftbigg9 3 3 10/parenrightbigg... | https://arxiv.org/abs/2501.13037v1 |
series Sfollows a VARMA( p,q) process with instantaneous effects if there exist p,q∈Nand coefficient matrices A0,A1,...,A p,B1,,...,B q∈Rd×d such that diag(A0) = 0and for all t∈Z: St=A0St+A1St−1+...+ApSt−p+εt+B1εt−1+...Bqεt−q, whereA0,A1,...,A pare such that det((Id−A0)λp−A1λp−1−...−Ap) = 0 implies|λ|<1, andεtis an i.i.d.... | https://arxiv.org/abs/2501.13037v1 |
these full-t ime DAGs, and therefore it also extends to the marginalized full-time ADM Gs. Also worth men- tioning is the fact that the ADMG in Figure 6 has some bi-direc ted edges which do not exist in the ADMG in Figure 4. This owes to the fact that w hen we rewrite the process without instantaneous effects we introd ... | https://arxiv.org/abs/2501.13037v1 |
em 3) is not a 14 I. González-Pérez Xt YtXt−1 Yt−1Xt−2 Yt−2Xt+1 Yt+1 Fig.4. Full-time marginalized ADMG GI,Sof the time series defined in Equation (4). consequence of [7, Thm. 1], even if the absolute-continuity assumption of the innovation distribution were added. Thus, the families of t ime series for which [7] and th... | https://arxiv.org/abs/2501.13037v1 |
this process as Gfull. Assume that the components of the process are or- dered according to the topological order of the instantaneo us effects, so that A0 16 I. González-Pérez Xt YtXt−1 Yt−1Xt−2 Yt−2Xt+1 Yt+1 Fig.6. Full-time marginalized ADMG GII,Sof the time series defined in Equation (5). X YZγ α β Fig.7. Associated ... | https://arxiv.org/abs/2501.13037v1 |
a VARMA( p,q) process with instantaneous effects as per Definition 14. Let Gfullbe its full-time DAG. Let Y:=Si0 tbe a node in Gfull, and letX= (Si1 t−l1,...,Sim t−lm)⊤be a finite collection of endogenous nodes in this DAG not containing Y. 1. A directed path from XtoYinGfullis calledX-causal (or simply causal) if it only... | https://arxiv.org/abs/2501.13037v1 |
(3)to be satisfied, as E[Cov(X,I|B)]has dimensions dim(X)×dim(I), a necessary condition is that dim(X)≤dim(I). This motivates the following definition. Definition 19. In the context of Theorem 6, if conditions (1) and (2) hold, but condition (3) is violated due to dim(X)>dim(I), we say the total causal effect is under-iden... | https://arxiv.org/abs/2501.13037v1 |
between VARM A models and linear Gaussian systems, a faithfulness statement is given in Theorem 5. At this point, with both the global Markov property and faith fulness at hand, we sink into the field of causal inference. Extending th e work of [24], in Section 5 we define the concept of total causal effect for VAR MA pro... | https://arxiv.org/abs/2501.13037v1 |
85. Sp ringer Science & Business Media (2012) 5. Francq, C., Zakoian, J.M.: GARCH models: structure, stat istical inference and financial applications. John Wiley & Sons (2019) 6. Hamilton, J.D.: Time Series Analysis. Princeton Univers ity Press, Princeton (1994). https://doi.org/doi:10.1515/9780691218632 , https://doi.... | https://arxiv.org/abs/2501.13037v1 |
C. Regarding all other nodes in (Gm)V\B(if there were any), as they are in other connected components of (Gm)V\B, they are separated from both A+andC+byBinGm, or equivalently they are d-separated from both A+andC+byBinG. Therefore they can be arbitrarily added to A+or toC+while keeping these sets disjoint andd-separate... | https://arxiv.org/abs/2501.13037v1 |
are non-colliders not in B. Furthermore, when taking this descendant path back to γand then following p,γbecomes a non-collider not in B. Now we will modify pto obtain a m-connecting path. (a) Ifphas no colliders, or all colliders in pare under i.), then pis an open path between AandCgivenBinG. (b) If there is a collid... | https://arxiv.org/abs/2501.13037v1 |
from SjtoSiof the product of the coefficients of the edges in those paths (this is the tota l causal effect of Sj tonSi tin the DAG of instantaneous effects as per Definition 17). The p roof of Equation (10) is done by induction of the dimension of the pro cessd. Ford= 2 it holds, as:/parenleftbigg1 0 −α2,11/parenrightbigg−... | https://arxiv.org/abs/2501.13037v1 |
I. González-Pérez process without instantaneous effects as defined in Equation (1). IfSi ttoSj thave no common instantaneous ancestors ( ANGI(Si t)[t]∩ANGI(Sj t)[t]=∅), thenδi tand δj tare independent. Proof. Recall that we can, w.l.o.g. take A0to be triangular inferior with zeroes on its diagonal, so that B= (Id−A0)−1is... | https://arxiv.org/abs/2501.13037v1 |
that all edges in G∗point “forward in time”, meaning for all k∈Nandi,j∈{1,...,d}there is no edgeSi t→Sj t−k,andG∗is a subgraph of the full-time graphGfullinduced byG0:=G∗ [s0−q,s0−1]and ANGfull(V+)[s0,t0]=V+,whereV+:=V∗ [s0,t0]and|V+|=n, for all VAR( p) processes with structure specified by G0, for allA∗,B,C∗⊆V∗such tha... | https://arxiv.org/abs/2501.13037v1 |
second independence statement. We consider two cases: Causality for VARMA processes with instantaneous effects 31 (a) Assume that: PAG+(λ)∩A∗/\e}atio\slash=∅ or ANG∗(PAG∗(λ)[s0−q,s0−1])∩ANG∗(PAG∗(A∗)[s0−q,s0−1])/\e}atio\slash=∅. (12) Then it holds that: PAG+(λ)∩C∗=∅ and ANG∗(PAG∗(λ)[s0−q,s0−1])∩ANG∗(PAG∗(C... | https://arxiv.org/abs/2501.13037v1 |
pro of, as any other step in the proof of Theorem 1 generalizes directly to the case wit h instantaneous effects. Theorem 3 (Global Markov property for VARMA processes with i n- stantaneous effects). Consider p,q∈N, andSa VARMA( p,q) process with instantaneous effects as per Definition 14. Consider finite pa irwise disjoint... | https://arxiv.org/abs/2501.13037v1 |
and due to the continuity of the solutions of an algebraic equation w .r.t. its coefficients, 3IfAis ann×mreal-valued matrix, we define vec( A) as the Rnmvector obtained from stacking the columns of Aone after another. If Bis an×nreal-valued matrix, we define vech( B) as theRn(n−1)/2vector obtained from stacking the elemen... | https://arxiv.org/abs/2501.13037v1 |
faithfulness violation has Lebesgue measure z ero, and therefore is aP-null set. This concludes the proof. Theorem 5 (Faithfulness for VARMA processes with instantan eous effects). Consider Sa VARMA( p,q) process with instantaneous effects as per Definition 14, such that A0is triangular inferior with a diagonal of zeroes ... | https://arxiv.org/abs/2501.13037v1 |
have independent and normally distributed inn ovations. Yet, the stationary distribution of the process is indeed Gaussian, and the innovations are jointly independent. This suffices to apply Theorem 3.5 in [23], meaning that ifX/\e}atio\slash⊥GfullY|BthenCov(X,Y|B)does not trivially vanish in PQ; but as PQ S=PS, we conc... | https://arxiv.org/abs/2501.13037v1 |
condition holds as well (3)E[Cov(X,I|B)]has rank dim(X), that is full row rank, then we can identify the total causal effect, that is: 38 I. González-Pérez (ii)βis the only solution to Equation (7). Additionally, to consistently estimate βwe have the following result: (iii) If X, I, B, Y are observations of X,I,B,YandWi... | https://arxiv.org/abs/2501.13037v1 |
B∪X ) andm is not a collider in q. Thus,qis not blocked by BinGS. Next we will show that this path is still not blocked by Bwhen we remove all outgoing edges from Xon a causal path from XtoY. This owes to the fact thatqcontains none of these edges. Assume that qcontains one of these edges. As the descendant path from m... | https://arxiv.org/abs/2501.13037v1 |
only intersects X∪B at the initial node (therefore this path is causal). Thus, Aiis a descendant ofX, and by (2)we conclude that SPGS(Ai)∩ANGS(B) =∅. –ForI, assume the opposite: there exists ω∈SPGS(Ai)∩ANGS(I). Then we can consider the path r:I ←···← ω↔Ai→···→ Y. We know that no node in the descendant path Ai→···→ Yis ... | https://arxiv.org/abs/2501.13037v1 |
Guaranteed Recovery of Unambiguous Clusters Kayvon Mazooji and Ilan Shomorony Electrical and Computer Engineering University of Illinois Urbana-Champaign Urbana IL, United States Email: mazooji2@illinois.edu, ilans@illinois.edu Abstract —Clustering is often a challenging problem because of the inherent ambiguity in wha... | https://arxiv.org/abs/2501.13093v3 |
We implement and test a version of the algorithm that is modified to handle overlapping clusters well, and observe that it requires little parameter selection, and delivers improved performance on artificial and benchmark datasets compared to widely used algorithms for non-convex cluster recovery. This implementation i... | https://arxiv.org/abs/2501.13093v3 |
union does not necessarily include all points in X. We say that a clustering Cextends (or is an extension of) a partial clustering C′if there exists a bijective function fthat maps the clusters inC′to the clusters in Csuch that for each cluster c′∈C′, c′is a non-empty subset of the cluster f(c′)∈C. We say a point x1isϵ... | https://arxiv.org/abs/2501.13093v3 |
by {c(v)forv∈S}. Any such clustering is called a dendrogram clustering. In fact, any (partial) clustering that consists of maximal clusters is a dendrogram (partial) clustering since every maximal cluster corresponds to a node in the dendro- gram. The ϵ-cut of a dendrogram is the (partial) clustering that includes all ... | https://arxiv.org/abs/2501.13093v3 |
is to find a weakly separable clustering. Because there usually does not exist a unique weakly separable K- clustering even when an intuitively correct K-clustering exists, weak separability alone as a sufficient condition for clustering recoverability is not adequate. B. Local Maximum Separability To define local maxi... | https://arxiv.org/abs/2501.13093v3 |
implying that recovery is guaranteed. The orange, red, and pink points correspond to local maxima. The red points correspond to points in X∗ cforc∈C. The pink point corresponds to the local maximum x∈Xsuch thatmax y∈c(x):ϵNp(y)≤ϵNp(x)A(x, y) =Aℓ(C)(i.e. the local maximum that looks the most separated from its cluster).... | https://arxiv.org/abs/2501.13093v3 |
time. Algorithm 2 is an auxiliary algorithm used to define Al- gorithm 1. Algorithm 2 works by outputting a partial K- clustering of Xthat can then be postprocessed to form a K-clustering. For Algorithm 2, Agoverns the maximum variation in density within a partial cluster, Mgoverns the minimum size of a partial cluster... | https://arxiv.org/abs/2501.13093v3 |
a new cluster c′′∈C andx∈X∗ c′′. The ith partial cluster c∗(x, Aℓ(C)·ϵNp(x)) must only include points from c′′by the definition of LM- separability and Aℓ(C). Because only a partial cluster of a new cluster in Ccan be output in each iteration of the algorithm, a partial K-clustering is output. By Lemma 3, Amin is no la... | https://arxiv.org/abs/2501.13093v3 |
found in O(|X|3log(|X|))time. Proof: We implement the greedy step of Algorithm 2 to have a deterministic rule for deciding whether to choose x∗ to be xoryifϵNp(x) =ϵNp(y)(such as taking the point that comes first in the dataset). For a given X, N p, M, D, the number of clusters in Cg(X, N p, A, M, D )monotonically decr... | https://arxiv.org/abs/2501.13093v3 |
Clearly, Smust include an unclustered point that contains a clustered point that it is ϵ′-connected to in its ϵ′-ball. Thus, at each greedy step of Algorithm 3, we can pick (x, c)that minimizes ϵ!(x, y). To initialize the algorithm, for each unclustered point x∈X\(∪c∈ˆCc), and each clustered point y, we compute max( d(... | https://arxiv.org/abs/2501.13093v3 |
refer to the version of HDBSCAN where NpandM+1are not tied as “HDBSCAN (2).” We use the OPTICS implementation from scikit-learn, where Np, the minimum cluster size M, and the cluster selection parameter Xiare set by the user. Neither HDBSCAN nor OPTICS uses the number of clusters K. SpectACl uses knowledge of Kand acce... | https://arxiv.org/abs/2501.13093v3 |
Iris, 30 for Wine, 50 for Seeds, 3 for Glass, 100 for Cancer and Digits, 70 for Letters, and 600 for MNIST. For MSE, Dwas set to 20 for Iris, Wine, Glass, and Cancer, 2 for Seeds, Letters and MNIST, and 1.5 for Digits. For spectral clustering, in order of dataset appearance in Table II, the chosen Knvalues are 4,6,18,3... | https://arxiv.org/abs/2501.13093v3 |
ARI NMI |ˆC| IrisMSE 0.886 0.871 3 MSE (auto) 0.835 0.833 3 WineMSE 0.439 0.430 3 MSE (auto) 0.359 0.420 3 SeedsMSE 0.725 0.682 3 MSE (auto) 0.725 0.682 3 GlassMSE 0.232 0.379 7 MSE (auto) 0.232 0.379 7 CancerMSE 0.743 0.628 2 MSE (auto) 0.694 0.595 2 DigitsMSE 0.864 0.898 10 MSE (auto) 0.819 0.879 10 LettersMSE 0.193 ... | https://arxiv.org/abs/2501.13093v3 |
HDBSCAN 1.000 1.000 2 HDBSCAN (2) 0.699 0.737 4 OPTICS 1.000 1.000 2 SpectACl 1.000 1.000 2 Fixed Variance BlobsMSE 0.964 0.948 3 Spectral 0.976 0.961 3 K-means 0.970 0.954 3 HDBSCAN 0.867 0.847 4 HDBSCAN (2) 0.568 0.729 3 OPTICS 0.726 0.734 3 SpectACl 0.964 0.942 3 AnisotropicMSE 1.000 1.000 3 Spectral 0.994 0.989 3 K... | https://arxiv.org/abs/2501.13093v3 |
2, pp. 2–11, 1993. [4] R. Ding, Q. Wang, Y . Dang, Q. Fu, H. Zhang, and D. Zhang, “Yading: Fast clustering of large-scale time se-ries data,” Proceedings of the VLDB Endowment , vol. 8, no. 5, pp. 473–484, 2015. [5] U. V on Luxburg, “A tutorial on spectral clustering,” Statistics and computing , vol. 17, pp. 395–416, 2... | https://arxiv.org/abs/2501.13093v3 |
D. Moulavi, P. A. Jaskowiak, R. J. G. B. Campello, A. Zimek, and J. Sander, “Density-based clustering val- idation,” in Proceedings of the 2014 SIAM International Conference on Data Mining, Philadelphia, Pennsylvania, USA, April 24-26, 2014 (M. J. Zaki, Z. Obradovic, P. Tan, A. Banerjee, C. Kamath, and S. Parthasarathy... | https://arxiv.org/abs/2501.13093v3 |
and weak separability hold precisely when a clustering is unambiguous. In fact, strong separability implies LM-separability. Lemma 10 For a given Np, ifCis strongly separable, then it is LM-separable. Proof: If there is only one local maximum per cluster, then this holds trivially. Suppose Cis strongly separable but th... | https://arxiv.org/abs/2501.13093v3 |
arXiv:2501.13187v1 [stat.AP] 22 Jan 20251 Sequential One-Sided Hypothesis Testing of Markov Chains Greg Fields, Tara Javidi, and Shubhanshu Shekhar Abstract —We study the problem of sequentially testing whether a given stochastic process is generated by a known Markov chain. Formally, given access to a stream of random... | https://arxiv.org/abs/2501.13187v1 |
a sequence of samples X1,X2,...,X t,... one-by-one and tasked with deciding if the samples are drawn from a given stochastic process–the null hypothesis–or if they are drawn from a diff er- ent, unknown process. At each time step, the tester must deci de if more samples are needed or, given the observed samples so far,... | https://arxiv.org/abs/2501.13187v1 |
or expectation taken when the null hypothesis is true, i.e. dat a is generated according to the null hypothesis’ distributio n. Finally, we will make use of symmetric Dirichlet distributi ons– Dirichlet distributions where all parameters are equal–de noted byD(γ)for the shared parameter γ >0. II. P ROBLEM SETUP In this... | https://arxiv.org/abs/2501.13187v1 |
detection, our work deviates from prior formulatio n and removes any assumption, including ǫ-distance to the null, on the true distribution under the alternative hypothesis. In fact, the only assumption made in this work is the existence of a stationary distribution for the data-generating chain . Fur- thermore, consid... | https://arxiv.org/abs/2501.13187v1 |
combining the following three building blocks. Data-driven Estimator: In the absence of a known alter- native distribution for the observed data, we will need to deploy a causal sequence of empirical estimators, {ˆq(·|X1:t−1) :t≥1}, of the data-generating distribution. We note that our performance will depend on the ch... | https://arxiv.org/abs/2501.13187v1 |
with regretrt=O(logt). We discuss in Section III-D the existence of such estimators and how to choose suitable estimators for different problem settings. To analyze the performance of this algorithm we define its stopping time, τ:= inf{t∈N:Lt≥1/α}. We then show that it satisfies our desired conditions; under H0it may end... | https://arxiv.org/abs/2501.13187v1 |
1/α=α. It then remains to bound the expected stopping time under the alternative, where PX=Q∝ne}ationslash=P. First define the oracle test statistic with full knowledge of the true Markov chain Q: L∗ t:=t/productdisplay i=1Q(Xi|Xi−1) P(Xi|Xi−1). Then we can consider the value of the test statistic, Lt, at the stopping t... | https://arxiv.org/abs/2501.13187v1 |
the i.i.d. case the Krichevsky-Trofimov (KT, “add- 1/2”) estimator 6 was shown to have regret O/parenleftbigm−1 2log/parenleftbigt 2π/parenrightbig/parenrightbig [16]. ˆqt+1(x) =/summationtextt i=1 /BD{Xi=x}+1/2 t+m/2. (6) This estimator also has a Bayesian interpretation: it resul ts from the choice of the Jeffreys pri... | https://arxiv.org/abs/2501.13187v1 |
Thi s leads to a variety of interesting directions for practical i m- provements and implementations in settings lacking a simpl e Markovian structure, which we discuss in Section V. IV. S IMULATIONS A. Sequential Adaptivity To illustrate the value of sequential testing in this settin g, we ran simulations comparing ou... | https://arxiv.org/abs/2501.13187v1 |
show that, as expected, both algorithms take significantly more samples to perform well on the harder problem instances. The important distinction here, though , is that the fixed length algorithm has to be given a trajectory of some set length without knowing the difficulty of the problem – meaning that to guarantee a po... | https://arxiv.org/abs/2501.13187v1 |
These observations suggest that the need for the more elaborate estimator is tied to our interest not to restrict the alternative hypothesis. An important question for futu re work is the interplay between side information/assumption on alternative hypothesis and the complexity of the estimator . This is of particular ... | https://arxiv.org/abs/2501.13187v1 |
gives that, if X0,X1,... is a sequence of i.i.d. realizations of a finite mean random variable and τis a finite expectation stopping time, and θis an appropriately bounded function, then E[/summationtextτ i=0θ(Xi)] =E[τ]E[θ(X0)]. The Markov version reproduces this result, plus a drift term tha t depends on how closely th... | https://arxiv.org/abs/2501.13187v1 |
and closeness of distributions,” ArXiv , vol. abs/2205.06069, 2022. [Online]. Available: https://api.semanticscholar.org/CorpusID:248721783 [9] J. Takeuchi, T. Kawabata, and A. R. Barron, “Properties o f Jeffreys mixture for Markov sources,” IEEE transactions on information theory , vol. 59, no. 1, pp. 438–457, 2012. [... | https://arxiv.org/abs/2501.13187v1 |
arXiv:2501.13265v1 [econ.EM] 22 Jan 2025Continuity of the Distribution Function of the argmax of a Gaussian Process∗ Matias D. Cattaneo†Gregory Fletcher Cox‡Michael Jansson§ Kenichi Nagasawa¶ January 24, 2025 Abstract An increasingly important class of estimators has members whose as ymptotic distribution is non-Gaussi... | https://arxiv.org/abs/2501.13265v1 |
that i s, we have a result of the form sup t∈Rd/vextendsingle/vextendsingle/vextendsingleP/bracketleftBig rn(ˆθn−θ0)≤t/bracketrightBig −Fˆs(t)/vextendsingle/vextendsingle/vextendsingle→0. (3) Whenµis a quadratic form and Cis a bilinear form, the distribution of ˆsis Gaussian. More generally, under mild conditions on µt... | https://arxiv.org/abs/2501.13265v1 |
by Samorodnitsky and Shen (2013b, p. 3494), “very little is known about the random location of the supremum” of a Gaussian proc ess. In addition to providing results of the form ( 1),Kim and Pollard (1990) gave sufficient con- ditions for almost sure uniqueness of the argmax of a multiva riate Gaussian process; for more ... | https://arxiv.org/abs/2501.13265v1 |
a random vari- able with a well-known continuous distribution, namely the Chernoff (1964) distribution. For d >1, on the other hand, it would appear to be an open question whe therFˆsis continu- ous. We provide an affirmative answer to that question below, h ereby buttressing the inference procedures for maximum score est... | https://arxiv.org/abs/2501.13265v1 |
squares estimator ( ˆβ′ n,ˆδ′ n,ˆθ′ n)′of (β′ 0,δ′ n,θ′ 0)′is a minimizer of n/summationdisplay i=1/parenleftbig yi−x′ iβ−x′ iδ /BD{qi>w′ iθ}/parenrightbig2 over (β′,δ′,θ′)′∈R2k+d. Assuming /bar⌈blδn/bar⌈bl →0,n/bar⌈blδn/bar⌈bl2→ ∞, and¯δn=δn//bar⌈blδn/bar⌈bl →¯δ(for some ¯δ∈Sd−1={¯δ∈Rd:/bar⌈bl¯δ/bar⌈bl= 1}),Yu and Fan... | https://arxiv.org/abs/2501.13265v1 |
Nt=⌈|t|⌉+1, letting eℓdenote the ℓth standard basis vector of Rd, and noting that {ˆsℓ=t}=/braceleftBigg e′ ℓargmax s∈RdG(s) =t/bracerightBigg ⊆∞/uniondisplay N=Nt/braceleftBigg e′ ℓargmax s∈[−N,N]dG(s) =t/bracerightBigg , a sufficient condition for ( 6) to hold is that P/bracketleftBigg e′ ℓargmax s∈[−N,N]dG(s) =t/brack... | https://arxiv.org/abs/2501.13265v1 |
in each of the examples of Section 2. To explain why this is no coincidence, suppose the followin g “local uniform” version of the characterization ( 11) is valid: for any ηn>0 withηn=O(r−1 n) and any θn=θ0+O(ηn), C(s,t) = lim n→∞E[{mn(z,θn+ηns)−mn(z,θn)}{mn(z,θn+ηnt)−mn(z,θn)}] ηn.(12) Then validity of the displayed p... | https://arxiv.org/abs/2501.13265v1 |
RandPwis the probability measure induced by w. Then (13)-(14) hold with ω= (ω1,w′)′, eN(ω;s) =/bracketleftbig/BD{w′s≤ω1<0}+ /BD{0≤ω1≤w′s}/bracketrightbig/radicalBig E·|q,w[(¯δ′xu)2|w′θ0,w]fq|w(w′θ0|w) and lN(ω) =−1 2 /BD{|ω1| ≤N√ d/bar⌈blw/bar⌈bl}E·|q,w[(¯δ′x)2|w′θ0,w]/radicalBigg fq|w(w′θ0|w) E·|q,w[(¯δ′xu)2|w′θ0,w]. ... | https://arxiv.org/abs/2501.13265v1 |
the leading special case where µ(s) =−c|s|γfor some c,γ >0, (15) Theorem 1therefore implies that Fˆsis continuous whenever γ >1/2. The same condition on γ is necessary and sufficient in order to deduce continuity of Fˆsby applying Cattaneo et al. (2024, Lemma A.2), which replaces Assumption 1(iii)with the Brownian motion... | https://arxiv.org/abs/2501.13265v1 |
(2005): “On the Bootstrap of the Maximum Score Estimator,” Econometrica , 73, 1175–1204. Aza¨ıs, J.-M. and M. Chassan (2020): “Discretization Error for the Maximum of a Gaussian Field,”Stochastic Processes and their Applications , 130, 545–559. Aza¨ıs, J.-M. and M. Wschebor (2005): “On the Distribution of the Maximum o... | https://arxiv.org/abs/2501.13265v1 |
J. P. Romano, and M. Wolf (1999):Subsampling , Springer. Samorodnitsky, G. and Y. Shen (2012): “Distribution of theSupremumLocation of Stationa ry Processes,” Electronic Journal of Probability , 17, 1–17. ——— (2013a): “Intrinsic Location Functionals of Stationar y Processes,” Stochastic Processes and their Applications... | https://arxiv.org/abs/2501.13265v1 |
Signal-to-noise ratio aware minimax analysis of sparse linear regression Shubhangi Ghosh1, Yilin Guo1, Haolei Weng2, and Arian Maleki1 1Columbia University 2Michigan State University Abstract We consider parameter estimation under sparse linear regression – an extensively studied problem in high-dimensional statistics ... | https://arxiv.org/abs/2501.13323v1 |
approach focuses on the rate-optimal minimaxity with the goal to derive the order of R(Θ(k), σ). A representative result implied by the work [30] states that as k/p→0 and ( klog(p/k))/n→0, R(Θ(k), σ)∼σ2klog(p/k), (4) where the notation “ an∼bn” means an/bnremains bounded away from zero and infinity. Further- more, it h... | https://arxiv.org/abs/2501.13323v1 |
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