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(173) This substitution is motivated by equation 20 of [FGV25] which proves that ncopies of a classical teleportation protocol (on the right) are equivalent to one classical teleportation protocol which is broadcast to nparties (on the left). For each split vertexv∈Vs(Gc,tp)and each outgoing edge e∈Out(v), let us denot... | https://arxiv.org/abs/2502.04168v2 |
For allv∈Vs(Gc,tp)ande∈Out(v), associate to Te vand outcome te v= 1the map acting on ρ∈L/parenleftig HIn(Tev)/parenrightig∼=L/parenleftig H(v,Tev)⊗H (Rev,Tev)/parenrightig MTe v 1(ρ) = Tr/bracketleftbigg EfTev t=1ρ/bracketrightbigg = Tr /summationdisplay x∈Xv/parenleftig |x⟩⟨x|H(v,Tev)⊗|x⟩⟨x|H(Rev,Tev)/parenrigh... | https://arxiv.org/abs/2502.04168v2 |
are equiv- alent to equation (181). Let us show that the maps associated to each vertex are equivalent in the two causal models, CmG′c,tpandCmGtp. We have: 1. For all vertices which are preserved from GtoG′ c,tp(andGtp),v∈Vthe maps associated in CmGtpandCmG′c,tpare equal (see equations (175) and (179)), i.e., for allρ∈... | https://arxiv.org/abs/2502.04168v2 |
acyclic, we can represent it on a page through such a diagram. We now add the pre- and post-selection vertices to this diagram by drawing all the Riat the bottom of the page, below all other vertices, and all the Tiat the top of the page, above all the other vertices. Since all pre-selection vertices Rionly have outgoi... | https://arxiv.org/abs/2502.04168v2 |
5 — if we establish proposition 10 in this case, the general result follows from corollary 16). We can now relate the success probability p(i) ✓of the causal model CmGtp,ion the teleportation graph Gtp,ifori∈{1,2}. By definition 9, we have p(1) ✓=/summationdisplay xPr(x,t=✓)Gtp,1 (199) =1 ptp/summationdisplay xPr(x,t=✓... | https://arxiv.org/abs/2502.04168v2 |
directed acyclic graph, lemma 7). Pr(x,{ti=✓}Ti∈Vpost)Gtp= Tr/bracketleftiggk/circlemultiplydisplay i=1E(i)Cx/parenleftiggk/circlemultiplydisplay i=1φ(i)/parenrightigg/bracketrightigg , (203) where the state/circlemultiplytext iφ(i)acts on/circlemultiplytext iH(Ri,Ti)⊗H e′ i, and consists of the tensor product of t... | https://arxiv.org/abs/2502.04168v2 |
Cyclic functional causal models beyond unique solvability with a graph separation theorem Carla Ferradini1, Victor Gitton1, and V. Vilasini1,2 1Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland 2Université Grenoble Alpes, Inria, 38000 Grenoble, France Functional causal models (fCMs) specify functi... | https://arxiv.org/abs/2502.04171v2 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Solvability and related properties of cyclic functional models 26 5.1 Number of solutions and consistency . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Examples on solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 Linking... | https://arxiv.org/abs/2502.04171v2 |
of the variables involved. However, to our knowledge, a general graph-separation property that applies to all (including non-uniquely solvable) consistent causal models is lacking, even in the case of finite-cardinality variables. This is complicated by the fact that existing methods do not uniquely fix the probability... | https://arxiv.org/abs/2502.04171v2 |
to witness the gap, and indeed there exist such examples of continuous variable models on this graph [FM17, FM18]. More generally, this motivates exploration of correlation gaps between finite vs infinite cardinality fCMs on a given cyclic causal structure, analogous to the highly studied problem of certifying correlat... | https://arxiv.org/abs/2502.04171v2 |
function fv:XPa(v)×Uv∝⇕⊣√∫⊔≀→Xv. One can think of functional models as assigning to each vertex v∈Va value from the setXv. The value assignment on a vertex, v∈V, depends stochastically on the value assignments of its parents through the functional dependency associated to it, fv. The stochastic character of such depend... | https://arxiv.org/abs/2502.04171v2 |
(y,z)∈XY×XZover the remaining observed vertices by marginalising equation (10) as Pracyc(y,z)G=/summationdisplay x∈XXPracyc(x,y,z )G=/summationdisplay x∈XXpX(x)δy,fY(x)δz,fZ(x,y).(11) Cyclicfunctionalmodels. Ifthegraphis cyclic, theliteratureconsiderstheprobability rule to be well-defined only for a subset of models [F... | https://arxiv.org/abs/2502.04171v2 |
the variables AandCthat “teleports” A’s distribution to C. Let us consider the following: take Cto be uniformly distributed and measure whether AandCare equal, if not apply a bit flip correction to C. For clarity let us denote with C′the variable Cafter the correction (which is trivial if AandCare equal) is applied. We... | https://arxiv.org/abs/2502.04171v2 |
the same classical post-selected teleportation protocol. Indeed, given ncopies of the same protocol we have the following result A... C1 T1 Cn Tneq. (16)≡ A... eq. (19)≡ A C T... . (20) The equivalence has to be understood at the level of conditional probabilities: explicitly, it holds that Pracyc({ck}k=1,...,n|{tk= 1}... | https://arxiv.org/abs/2502.04171v2 |
defining functional models on classical teleportation graphs. 11 Definition 9 (Family of functional models on acyclic classical teleportation graphs) . Given a functional model on a directed graph G= (V,E),fCmG, we can construct a family of functional models by defining a functional causal model, fCmGtp, on each acycli... | https://arxiv.org/abs/2502.04171v2 |
have constructed a family of acyclic functional models associated with a given, potentially cyclic, functional model (definition 9). We have proven that the acyclic distribution conditioned on successful post-selection is independent of which functional model we choose in this family. Here, we use these results to defi... | https://arxiv.org/abs/2502.04171v2 |
probability rule does not depend on which classical teleportation protocol one chooses (corollary 14). In principle, other classical teleportation protocols which are associatedwithanon-uniformprioronthepre-selectionverticescouldbeconsidered, which would yield the same probability. 3.4 Examples of cyclic functional cau... | https://arxiv.org/abs/2502.04171v2 |
methods would suggest that any distributions with x1=x2and arbitrary weights on their value being 0or1are allowed. Contrary to that, our method uniquely fixes the probability distribution to Pr(x1,x2|0,0)G=p3(0)p4(0)δx1,x2 p3(0)p4(0)/summationtext y1,y2δy1,y2=δx1,x2 2. (46) This is not a consequence of the specific tel... | https://arxiv.org/abs/2502.04171v2 |
causal models, which 17 embed finite-cardinality fCMs. Here, we present an alternative formulation of p-separation that is independent of the quantum formalism, and defined in terms of the classical causal modeling framework we have developed in this paper (section 3). The key difference lies in the mapping of cyclic t... | https://arxiv.org/abs/2502.04171v2 |
in V2isblockedbyV3. Otherwise, V1is said to be d-connected withV2givenV3, denoted as (V1̸⊥dV2|V3)G. The concept of d-separation only involves graph properties and is independent on whether a functional model is defined on the graph. However, the d-separation theorem relates the graph property to conditional independenc... | https://arxiv.org/abs/2502.04171v2 |
conditioned upon in the original cyclic graph, however, the logical consistency of the model imposes an effective post-selection on the values in the loop variables. In the next section, we introduce the notion of p-separation which generalizes d-separation through making the effective conditioning explicit. In this ex... | https://arxiv.org/abs/2502.04171v2 |
completeness of p-separation. These results prove that p-separation correctly captures conditional independences in finite-cardinality cyclic functional models. Theorem 20 (p-separation theorem) .Consider a directed graph Gand letV1,V2andV3 be any three disjoint sets of the vertices of GwithV1andV2being non-empty. Then... | https://arxiv.org/abs/2502.04171v2 |
a functional model onG, the outcomes aandbof the vertices AandBwill be conditionally independent even when conditioned on the collider Vpost:={T}in equation (64). This is because the post-selection on the unblocking collider Tis fine-tuned in our definition of the induced model(ithastocorrespondtoapost-selectedteleport... | https://arxiv.org/abs/2502.04171v2 |
the property) no longer applies, and cannot account for the correlations there. In contrast, the p-separation theorem proven here applies to all consistent functional causal models, but with the restriction to finite-cardinality variables. More generally, as shown in the com- panion paper [FGV25], the theorem also exte... | https://arxiv.org/abs/2502.04171v2 |
6, we further discuss possible directions for future 9In the quantum case, this discussion is related to the difficulty in defining an analogue of a maximally entangled state between infinite-dimensional Hilbert spaces while ensuring a bounded norm for that state. 25 research based on such comparisons between graph-sep... | https://arxiv.org/abs/2502.04171v2 |
previous proposition allows us to characterize inconsistent functional models, i.e., modelswhere ps= 0, intermsoftheaveragenumberofsolutions. Thisfollowsimmediately from proposition 23 and from the fact that ptp>0by definition of teleportation protocol. Corollary 24 (Characterization of inconsistent models) .A function... | https://arxiv.org/abs/2502.04171v2 |
functional causal model fCmGover a directed graph G= (V,E). If the joint distribution over observed eventsx={xv∈Xv}v∈Vfactorizes as in equation (6)i.e., Pr(x)G=/productdisplay v∈VPr(v)(xv|Pa(xv))G, (82) where Pr(v)is the conditional probability distribution specified by fCmGfor allv∈V, then we say that the model satisf... | https://arxiv.org/abs/2502.04171v2 |
the vertex vi. How- ever, it holds that (v2̸⊥dv3|v1,v4)G, since the path v2→v1←v3is a collider, hence conditioning on v1d-connectsv2andv3allowing the associated variables to be condition- ally dependent in a general functional model on this graph. This highlights that in cyclic graphs,d-separation, Markovianity (as in ... | https://arxiv.org/abs/2502.04171v2 |
structure. Bell’s seminal theorem [Bel64] provides the first instance of an acyclic graph with such a gap, which is verifiable experimentally and has lead to several quantum information processing applications (see [BCP+14] for a review on Bell nonlocality, including relevant experiments and applications). Our contribu... | https://arxiv.org/abs/2502.04171v2 |
cyclic graphs. It would be interesting to characterize the class of cyclic models in our framework for which the equivalence is recovered. In particular, al- though we found that the Markov factorization is recovered for all averagely uniquely solvable models, the example of [Nea00] (a uniquely solvable model) demonstr... | https://arxiv.org/abs/2502.04171v2 |
ble to functional causal models associated with finite-cardinality random variables. The generalization to infinite-cardinality models (including both discrete and contin- uous random variables) is not immediate because it would require post-selecting on a measure-zero event. Therefore, the question on whether and how ... | https://arxiv.org/abs/2502.04171v2 |
volume 216 ofProceedings of Machine Learning Research , 433–442, PMLR, URL https://proceedings.mlr.press/v216/claassen23a.html , 2023. [CS16] F. Costa, S. Shrapnel, “Quantum causal modelling”, New Journal of Physics , 18(6):063032, doi:10.1088/1367-2630/18/6/063032, 2016. [Deu91] D.Deutsch,“Quantummechanicsnearclosedti... | https://arxiv.org/abs/2502.04171v2 |
Conference on Uncertainty in Artificial Intelligence , UAI’96, 420–426, Morgan Kaufmann Publishers Inc., doi:10.48550/arXiv.1302.3595, 1996. [Pea09] J. Pearl, Causality , Cambridge University Press, 2nd edition, doi:10.1017/CBO9780511803161, 2009. [PL14] M. Petersen, M. Laan, “Causal models and learning from data integ... | https://arxiv.org/abs/2502.04171v2 |
implies that that the term before λ2should vanish for all c∈XC. This contradicts our assumption that pA̸=p′Aandptp(pA)̸=ptp(p′A). Lemma 5. Consider a classical teleportation protocol defined by the pair (f,PB,PC). Then, it holds:/summationdisplay b∈XBδf(a,b,c),1PB(b)PC(c) =ptpδa,c. (18) 37 Proof.Consider a deterministi... | https://arxiv.org/abs/2502.04171v2 |
t∪t0:={tT∈{0,1}}T∈V2 postas outcomes associated to the post-selection vertices of G2. Denoting with (x={xv∈ Xv}v∈V,t)an event on G1, consider the probability Pracyc(x,t)G1associated to the causal model fCmG1. As this is an acyclic causal model by construction, the probability is immediately given by applying the acycli... | https://arxiv.org/abs/2502.04171v2 |
u/producttext v∈Vpv(uv)δyv,fv(yPa(v),uv)(33) where we have defined the global event x={xv∈Xv}v∈V, the sum/summationtext yruns over all y={yv∈Xv}v∈Vand/summationtext uoveru={uv∈Uv}v∈V. Proof.We have shown that the probability rule is independent on the set of split vertices Vs(Gtp), hence we can consider the element G0=... | https://arxiv.org/abs/2502.04171v2 |
immediate that the probabilities for x′are identical. Now suppose we have a p-connection (V1̸⊥pV2|V3)G, and consider two cases based on whether or not the corresponding d-connection holds. We will establish the existence of a causal model with the corresponding conditional dependence separately in each case. Proof when... | https://arxiv.org/abs/2502.04171v2 |
(T̸⊥dT′|V3)G′tp.(109) We now construct a functional model fCmG′tponG′ tp(an acyclic graph) and use the d-connections of equation (109) to argue that the random variable ¯X1ofv1∈V1, with values ¯x1=xv1, and the random variable ¯X2ofv2∈V2, with values ¯x2=xv2, will become correlated given the outcomes of V3andT,T′∈Vposti... | https://arxiv.org/abs/2502.04171v2 |
˜ul T′ (112) for some vertices {˜uj}l j=1, none of which belong to V3, C R T R′ T′ (113) for someC∈V3. LetG′ tpbe defined to be formed by these three unblocked paths (i.e. all vertices and edges featuring in equation (111), equation (112), equation (113) included) and the remaining vertices of the original graph Gtp in... | https://arxiv.org/abs/2502.04171v2 |
appearing in any functional dependence with the outcome xvfor the corresponding vertex v(which has an edge to the same TasR, as in equation (110)). This way all functional dependences are expressed only in terms of the vertices of G, and fCmGis a fully specified functional model on G. This establishes that the construc... | https://arxiv.org/abs/2502.04171v2 |
equation (109), we must have C̸∈V3. The causal model of definition 31 for this case entails xC=xR, i.e., this is again equivalent to TandT′sharing the same common cause Ras in equation (114). Another case is where Cis a common cause of TandT′(thenC̸∈V3), here we already have the structure of equation (114) (it does not... | https://arxiv.org/abs/2502.04171v2 |
arXiv:2502.04179v1 [math.ST] 6 Feb 2025THE MAXIMUM LIKELIHOOD DEGREE OF GUMBEL’S TYPE-I BIVARIATE EXPONENTIAL DISTRIBUTION POOJA YADAV1. TANUJA SRIVASTAVA2 Abstract. In algebraic statistics, the maximum likelihood degree of a statis- tical model refers to the number of solutions (counted with m ultiplicity) of the scor... | https://arxiv.org/abs/2502.04179v1 |
organized as follows, in section 2, the maximum l ikelihood esti- mation problem of the parameter of GBED-I is introduced and s hown that the score equation is a rational function of the parameter and th e ML-degree of the parameter of GBED-I is defined. In section 3, the geometry of t he score equation is explained. In... | https://arxiv.org/abs/2502.04179v1 |
points where the score equation is not defined due to the cleared denominator. Therefore, the solutions of the sc ore equation are the zeros off(θ), which are not the zeros of g(θ). So, for the ML-degree, the common zeros off(θ)andg(θ)should be removed from the zeros of f(θ). To determine the common zeros of f(θ)andg(θ)... | https://arxiv.org/abs/2502.04179v1 |
is given that all pairs (gi,gj)i/\e}atio\slash=jhave common zeros, then all zeros of gk(θ)andgl(θ)are same, which is not true for generic data. Hence, by contradiction, the statement is true. /square Note 1. For the generic data, V(gi,gj)is either empty or a singleton set for any i/\e}atio\slash=j∈ {1,2,...,n}. Letg1(θ... | https://arxiv.org/abs/2502.04179v1 |
of exactly n1gi(θ)’s(2≤n1≤n). Proof. Suppose αis a common zero of exactly n1gi(θ), say g1(θ),g2(θ),...,g n1(θ). Sincen1≥2, at least one pair (gi,gj)have a common zero, then by theorem 3.1, V(f,g)/\e}atio\slash=∅andα∈V(f,g). To count the multiplicity of αinf(θ), suppose αi (α/\e}atio\slash=αi) are the other zeros of gi(... | https://arxiv.org/abs/2502.04179v1 |
ML-degree of the association parameter θis6−2 = 4. (2) IfV(g1,g2,g3) =∅, then the varieties are of the form : V(g1) ={α,α1}, V(g2) ={α,α2},V(g3) ={α1,α2},α/\e}atio\slash=α1/\e}atio\slash=α2. Then, by theorem 3.1, V(f,g) ={α,α1,α2}, and it is given that V(fi,gi) =∅for every i, then by lemma 4.2, the multiplicity of each... | https://arxiv.org/abs/2502.04179v1 |
double zero, so the remaining zeros ofg(θ)are2(n−l). Since each double zero is also a zero of at least one moregj(θ), which makes l≤2n 3. 6.Conclusion In this paper, the ML-degree of the association parameter in GBED(θ) is in- vestigated. The ML-degree of θis the number of solutions of the score equation (2.2) counted ... | https://arxiv.org/abs/2502.04179v1 |
in the interval [0,1], but the score equation has many complex solutions (more than or equal to 2depending on n). Still, the number of real solutions among them remains unknown and requires further i nvestigation. Acknowledgements The first author would like to thank the University Grants Com mission, India, for providi... | https://arxiv.org/abs/2502.04179v1 |
arXiv:2502.04208v2 [math.ST] 7 Feb 2025Supermartingales for One-Sided Tests: Sufficient Monotone Likelihood Ratios are Sufficient Peter Grünwalda,b, Wouter M. Koolena,c aCentrum Wiskunde & Informatica, Amsterdam, The Netherlands bLeiden University, Leiden, The Netherlands cTwente University, Enschede, The Netherlands Abstr... | https://arxiv.org/abs/2502.04208v2 |
(2) One-Sided Tests. Now fixδ+> δ0and consider the one-sided null hypothesis Hδ≤δ0:={Pδ:δ∈∆,δ≤δ0}. As alternative we may take either the Bayesian point alterna tiveH1={PW}withWa prior on{δ∈∆ :δ≥δ+}or a composite alternative H1⊆ {Pδ:δ∈∆,δ≥δ+}. For simplicity we concentrate on the case with H1={Pδ+}for now, returning to t... | https://arxiv.org/abs/2502.04208v2 |
invariant t-likelihood ratio pUn δ+is actually an e-value for the larger one- sided null Hδ≤0ifδ+≥0 [...]meaning that one can non-sequentially test that null using this statistic. However, it is unclear if this t-l ikelihood ratio process is an e-process for Hδ≤0.We leave this question open for future investigation . .... | https://arxiv.org/abs/2502.04208v2 |
increasing in t. Importantly, the likelihood ratio pUn δ+(u|un−1)neednotbe monotone in uand indeed it is not in the t-test setting; this caused the perceived difficu lty of the problem. Combining the lemma with Proposition 2 gives: Theorem 4. Let(Tn)n∈Nbe a sequence of sufficient statistics satisfying the monoton e like- l... | https://arxiv.org/abs/2502.04208v2 |
have for each factor in the product: pTn δ+(Tn|un−1) =pUn δ+(Un|un−1). Since also for each n, it holds that pUn δ(Un) =/producttextn i=1pUi δ(Ui| Ui−1), the processes (/producttextn i=1pTi δ(· |Ui−1))n∈Nand(pUn δ)n∈Nmust coincide. 5 3. Further Discussion and Future Work Priors on Alternative and Growth-Rate Optimality ... | https://arxiv.org/abs/2502.04208v2 |
Johnstone, and K. B. MacGibbon. Variation d iminishing transforma- tions: a direct approach to total positivity and its statist ical applications. Journal of the American Statistical Association , 76(376):824–832, 1981. P. D. Grünwald, R. de Heide, and W. M. Koolen. Safe testing. Journal of the Royal Statistical Societ... | https://arxiv.org/abs/2502.04208v2 |
Qnis a sufficient statistic. We also see that the likelihood ratio is increasing in Qnwhenever σ+≥σ0, thus establishing the MLR property. We may also observe thatQn σ2has a (central) chi squared distribution, and write the above as pUn σ+(un) =fχ2(n−1,σ2 +)(Qn) fχ2(n−1,σ2 0)(Qn), wherefχ2(ν,s)is theχ2density with νdegree... | https://arxiv.org/abs/2502.04208v2 |
directions a re dropped. By construction, the coarsening Unhas distribution solely dependent on δ, and is independent of the nuisance parameters βandσ. Let us characterise that distribution according to Pδ,β,σ. First, before the normalisation to unit length, we have AnYn∼N/parenleftbig δσbn,σ2Ik/parenrightbig where bn:... | https://arxiv.org/abs/2502.04208v2 |
is clearly positive, as 2Tn≥nandφ≥1. All in all, for all θ0,θ+∈[1/2,1]with θ0≤θ+we have, by Theorem 4, that (B.4) is a supermartingale under a llθ∈[1/2,θ0]. Symmetrically, for θ+≤θ0, (B.4) is also a supermartingale under θ∈[θ0,1]. Appendix C. No Free For All The examples in Appendix B might suggest that, more generall ... | https://arxiv.org/abs/2502.04208v2 |
Dimension estimation in PCA model using high-dimensional data augmentation U. Radojiˇ ci´ c Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Austria J. Virta Department of Matematics and Statistics, University of Turku, Finland Abstract We propose a modified, high-dimensio... | https://arxiv.org/abs/2502.04220v1 |
previous issues is that when either the data dimension pnor the augmentation dimension rnis comparable in magnitude to n, we en- ter the domain of high-dimensional asymptotics and the classical arguments in Luo and Li (2021); Radojiˇ ci´ c et al. (2022) stop working. As such, the purpose of the current work is three-fo... | https://arxiv.org/abs/2502.04220v1 |
of its leading pneigenvectors. Decomposing the eigenvectors as uj,n= (u′ j,n,A, u′ j,n,B, u′ j,n,C)′where the dimensionalities of the three parts are d, pn− d, rn, respectively, the predictor augmentation estimator of dis then based on the sequence of squared norms ∥u1,n,C∥2, . . . ,∥up,n,C∥2. Heuristically, a jump is ... | https://arxiv.org/abs/2502.04220v1 |
following result gives the probability limits of the eigenvalues τj,nillustrating further the inconsistency of the traditional esti- mators in the high-dimensional regime. Here F−1 γdenotes the quantile function of the Marchenko-Pastur distribution with the concentration parameter γ >0. Theorem 2. Under Assumption 1, w... | https://arxiv.org/abs/2502.04220v1 |
signal remains distinguishable, but the augmentation still yields an inconsistent estimate. Figure 1 illustrates these inconsistency regions. In the left plot, for γp= 0.75,σ2= 1, and d= 1, the light gray region bounded by the dashed line 5 0.000.250.500.75 0.75 1.00 1.25 1.50 λ12γr 0.000.250.500.751.00 0.00 0.25 0.50 ... | https://arxiv.org/abs/2502.04220v1 |
signal of the data is captured by a relatively small amount of factors. However, the search interval could also be widened and allowed to grow withn, in the same sense as discussed in Remark 1. Theorem 4 quantifies the limiting ”jump” in the successive differences of the adjusted augmented norms hj,n,j= 1, . . . , K > ... | https://arxiv.org/abs/2502.04220v1 |
corrected median eigenvalue estimator ˆ σ2 ⌊n/2⌋defined in Lemma 1. For each of two models, we replicate m= 1000 data sets of size n= 100,200,500,1000, and dimensionality pn=γpn, forγp= 0.05,0.2,0.5,1,1.5. To study the effect of the augmentation dimension rn, we estimate the latent dimension in all settings using rn=γr... | https://arxiv.org/abs/2502.04220v1 |
should attain its minimum at k= 1. The solid lines in the plot show the corresponding pointwise limiting values of the objective function ϕn(when n→ ∞ ), computed based on the asymptotic results in our Section 3. rn=0.01n rn=0.5n 0123456789100123456789100.31.03.010.0 kAverage of φ(k) pn pn=0.25n pn=0.025n Figure 3: The... | https://arxiv.org/abs/2502.04220v1 |
normal distribution, the form of the covari- ance matrix (4) means that the distribution of mj,nis orthogonally invariant. Hence, mj,n/∥mj,n∥is uniformly distributed in the unit sphere in Rp−d+rnand 11 ∥uj,n,C∥2=∥mj,n∥2Yj,nwhere Yj,n∼Beta{rn/2,(p−d)/2} →pγr/(γr+γp) as n→ ∞ . Consequently, ∥uj,n,C∥2→pγrσ2(λj+σ2) λj{λj+ ... | https://arxiv.org/abs/2502.04220v1 |
that g(ε)<0 for every 0 < ε < ε 0. To conclude, for every γr∈(0, ε0(ε0+ 2√γp)), we have both γr+γp< ε 0(ε0+2√γp)+γp= (ε0+√γp)2=λ2 1andϕ(1)> ϕ(2). Statement (iii): For simplicity, we show the proof of statement (iii) again ford= 1,σ2= 1. As in (ii), the general case is proven in an analogous way. Fix λ1>0. We need to sh... | https://arxiv.org/abs/2502.04220v1 |
arXiv:2502.04297v1 [cs.LG] 6 Feb 2025Statistical guarantees for continuous-time policy evaluation: blessing of ellipticity and new tradeoffs Wenlong Mou⋄ Department of Statistical Sciences, University of Toronto⋄ Abstract We study the estimation of the value function for continuous-time M arkov diffusion processes using ... | https://arxiv.org/abs/2502.04297v1 |
th et→0 limit βf⋆=/an}⌊∇a⌋k⌉tl⌉{tb,∇f⋆/an}⌊∇a⌋k⌉t∇i}ht+1 2Tr/parenleftbig Λ∇2f⋆/parenrightbig +r. (3b) We aim to estimate the value function f⋆from discrete-time observations of the diffusion process ( 1), with a snapshot observation of the state Xtand random reward Rt=Rt(Xt) everyη>0 time interval. A popular and practi... | https://arxiv.org/abs/2502.04297v1 |
proof of the results in Section 4, and conclude with discussions in Section 5. Notations: Here we summarize some notation used throughout the paper. W e use (Bt: t≥0) to denote d-dimensional standard Brownian motion. For a positive inte germ, we define the set [ m] :={1,2,···,m}. We use /⌊a∇⌈⌊l·/⌊a∇⌈⌊l2to denote the sta... | https://arxiv.org/abs/2502.04297v1 |
controlled SDEs. The techniques developed in this paper are potentially applicable to these settings. Statistical estimation of value functions: For discrete-time RL problems, theoretical analysis of policy evaluation algorithms is well-studied i n literature. [ TVR97] established approximation error guarantees for TD ... | https://arxiv.org/abs/2502.04297v1 |
processes. Additionally, the paper [ MZ24] also imposes the following regularity assumption on the fu nc- tion class K. 5 (Reg(c1,ω))For any function f∈Kwe have /⌊a∇⌈⌊l∇f/⌊a∇⌈⌊lL2(ξ)≤cmω/⌊a∇⌈⌊lf/⌊a∇⌈⌊lL2(ξ),and/⌊a∇⌈⌊l∇2f/⌊a∇⌈⌊lL2(ξ)≤cmω/⌊a∇⌈⌊l∇f/⌊a∇⌈⌊lL2(ξ). See the paper [ MZ24] for a detailed discussion on examples t... | https://arxiv.org/abs/2502.04297v1 |
solve the linear equation /hatwideθT=/braceleftBigT/η−ν/summationdisplay k=0ψ(Xkη)·/parenleftBig ψ(Xkη)−e−β(ν−1)ηψ(X(k+ν−1)η)/parenrightBig⊤/bracerightBig−1 ·/braceleftBig ηT/η−ν/summationdisplay k=0ν−1/summationdisplay i=0κiR(k+i)ηψ(Xkη)/bracerightBig , (14a) where the coefficients κiare defined as κi:=1 η/integraldispla... | https://arxiv.org/abs/2502.04297v1 |
statistical error in Theor em1to follow, the constant τcan be replaced by the hypercontractivity constant for the basis functions ( e⊤ jH−1/2ψ)m j=1, which can be easily verified for most bases. 8 Based on these assumptions, we can derive a non-asymptotic u pper bound for the error achieved by the LSTD estimator. Playin... | https://arxiv.org/abs/2502.04297v1 |
projection estimators [ Tsy08], this term grows at a rate Tr/parenleftbig H−1 1H0/parenrightbig , which can be potentially slower than m/T. As we will see in Section 3.3, various bounds on this term could lead to non-classical tradeoffs in the choice of m. See Section 3.3for a concrete example of such trade-offs. •Theν-t... | https://arxiv.org/abs/2502.04297v1 |
ρ−1 ∗/⌊a∇⌈⌊l∆∗/⌊a∇⌈⌊lW2,2p(ξ)/⌊a∇⌈⌊l∆∗/⌊a∇⌈⌊lW1,2p(ξ)+η/⌊a∇⌈⌊l∆∗/⌊a∇⌈⌊l2 W2,2p(ξ)/bracerightBig ,(17a) under the sample size requirement T≥Cτ4 ρ∗m1+2ωlog3/parenleftbigm δη/parenrightbig +cD2 m ρ∗log3/2/parenleftbigm δη/parenrightbig . (17b) Compared to Equation ( 16a), the Markovian part of the error bound in Equation ... | https://arxiv.org/abs/2502.04297v1 |
of/hatwideθTtakes the form √ T(/hatwideθT−¯θ)d− →N/parenleftbig 0,ηA−1(Σ∗ MG+Σ∗ Mkv)A−⊤/parenrightbig , whereA=E/bracketleftbig ψ(Xkη)·/parenleftbig ψ(Xkη)−e−β(ν−1)ηψ(X(k+ν−1)η)/bracketrightbig , and the matrices Σ∗ MG, Σ∗ Mkvcor- respond to the martingale and Markovian parts of the noise va riance, respectively. When ... | https://arxiv.org/abs/2502.04297v1 |
W1,4p p−1(ξ)/⌊a∇⌈⌊lg/⌊a∇⌈⌊l2 W1,2p(ξ)+c′η2/⌊a∇⌈⌊lf/⌊a∇⌈⌊l2 W2,2p p−1(ξ)/⌊a∇⌈⌊lg/⌊a∇⌈⌊l2 W2,2p(ξ) See Section 4.1.2for the proof of this lemma. Note that Lemma 2and3are applicable only for t=kη∈(0,T0]. When the time index is outside this range, we use alternative estimates given by the following lemma. Lemma 4. Under th... | https://arxiv.org/abs/2502.04297v1 |
2/bracketrightBigp−1 4p/bracerightBig /⌊a∇⌈⌊lg1/⌊a∇⌈⌊lW1,2p(ξ)·/⌊a∇⌈⌊lg2/⌊a∇⌈⌊lW1,2p(ξ)·/⌊a∇⌈⌊lf1/⌊a∇⌈⌊l W1,4p p−1(ξ)·/⌊a∇⌈⌊lf2/⌊a∇⌈⌊l W1,4p p−1(ξ), for constant c1depending on the constants ( β,d,Lb 0,LΛ 0,Lb 1,LΛ 1,Lξ). For the term E2, we apply H¨ older’s inequality to the three terms to obtain t hat /vextendsingle/... | https://arxiv.org/abs/2502.04297v1 |
them and obtain /integraldisplay f2(y)(β−A)g2(y)pt(y|x)dy =/integraldisplay f2(y)pt(y|x)/braceleftBig βg2−/an}⌊∇a⌋k⌉tl⌉{t∇g2, b/an}⌊∇a⌋k⌉t∇i}ht−1 2Tr/parenleftbig Λ∇2g2/parenrightbig (y)/bracerightBig dy =/integraldisplay/bracketleftBigg/braceleftBig ∇f2(y)+f2(y)∇ylogpt(y|x)/bracerightBig⊤/braceleftBig g2(y)b(y)+1 2Λ(y... | https://arxiv.org/abs/2502.04297v1 |
Proposition 3 ([Wan05], Proposition 1.2) .Under above setup, for any bounded Lipschitz functionuonRdandt>0, we have /⌊a∇⌈⌊l∇Ptu(x)/⌊a∇⌈⌊l2 2≤e2ctPt/parenleftBig /⌊a∇⌈⌊l∇u/⌊a∇⌈⌊l2 2/parenrightBig (x),for anyx∈X, for any constant csatisfying c≥1 λminsup x∈Rd/braceleftBig/summationdisplay i,j∈[d]/⌊a∇⌈⌊l∇Λi,j(x)/⌊a∇⌈⌊l2 2+... | https://arxiv.org/abs/2502.04297v1 |
k≤var/parenleftBig f(X0)·1 η/integraldisplayη 0e−βt(β−A)g(Xt)dt/parenrightBig ·var/parenleftBig E/bracketleftBig1 η/integraldisplayη 0P1/braceleftBig f·(β−A)Ptg/bracerightBig (Xkη−1)dt|Xη/bracketrightBig/parenrightBig ≤e−λminρ∗(kη−1−η)E/bracketleftBig/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(X0)·1 η/int... | https://arxiv.org/abs/2502.04297v1 |
+cD2 m ρ∗Tlog3/2/parenleftbigTm ηδ/parenrightbig . See Section 4.2.2for the proof of this lemma. As for the main error term in Eq ( 33), we have Lemma 9. Under the setup of Theorem 1, we have E/bracketleftBig /⌊a∇⌈⌊lH−1/2 1/parenleftbig/hatwidebN−b+/hatwideεN−(/hatwideAN−A)¯θ/parenrightbig /⌊a∇⌈⌊l2 2/bracketrightBig ≤τ... | https://arxiv.org/abs/2502.04297v1 |
have |||1 NN/summationdisplay k=0Yk|||op≤8/parenleftBig+∞/summationdisplay i=0/parenleftbig |||E/bracketleftbig Y0Y⊤ i/bracketrightbig |||op+|||E/bracketleftbig Y⊤ 0Yi/bracketrightbig |||op/parenrightbig/parenrightBig1/2/radicalBigg logm1+m2 δ N+320B λminρ∗ηNlogm1+m2 δ. See Appendix A.1for the proof of this lemma. Let ... | https://arxiv.org/abs/2502.04297v1 |
.Let(Yk)k≥0be a discrete-time matrix martingale inRm1×m2adapted to the filtration (Fj)j≥0, andXk=Yk−Yk−1fork≥1. Assume that |||Xk|||op≤Rfork= 1,2,···, and define the predictable quadratic variation processes Wk:=k/summationdisplay j=1E/bracketleftbig XjX⊤ j|Fj−1/bracketrightbig ,and/tildewiderWk:=k/summationdisplay j=1E/... | https://arxiv.org/abs/2502.04297v1 |
Ak¯θ−bk+εk/parenrightbig =1 NηN−1/summationdisplay k=0H−1/2 1ψ(Xkη)/integraldisplay(ν−1)η 0e−βt/parenleftbig βf(Xkη+t)−Af(Xkη+t)−¯rt/parenrightbig dt +1 NηN−1/summationdisplay k=0H−1/2 1ψ(Xkη)/integraldisplay(ν−1)η 0e−βt¯θ⊤∇ψ(Xkη+t)Λ1/2(Xkη+t)dBkη+t −1 NN−1/summationdisplay k=0H−1/2 1εk =:J1(N)+J2(N)+J3(N). Upper bound... | https://arxiv.org/abs/2502.04297v1 |
0≤k≤N−1 kmod (ν−1)=ℓE/bracketleftBig /⌊a∇⌈⌊lH−1/2 1ψ(Xkη)/⌊a∇⌈⌊l2 2/integraldisplay(ν−1)η 0e−2βt∇¯f(Xkη+t)⊤Λ(Xkη+t)∇¯f(Xkη+t)dt/bracketrightBig ≤λmax (Nη)2/summationdisplay 0≤k≤N−1 kmod (ν−1)=ℓ/integraldisplay(ν−1)η 0E/bracketleftbig /⌊a∇⌈⌊lH−1/2 1ψ(Xkη)/⌊a∇⌈⌊l2 2·/⌊a∇⌈⌊l∇¯f(Xkη+t)/⌊a∇⌈⌊l2 2/bracketrightbig dt ≤m/summa... | https://arxiv.org/abs/2502.04297v1 |
the elliptic structures play a key role in our statistical analysis. In pa rticular, since the effective horizon diverges to infinity as stepsize decreases, most cla ssical discrete-time RL theory becomes inapplicable. Second, through a new characterizat ion of the asymptotic covari- ance structure for functionals of Mar... | https://arxiv.org/abs/2502.04297v1 |
Kakade. A natural policy gradient. Advances in neural information pro- cessing systems , 14, 2001. (Cited on page 6.) [KB23] Z. Kobeissi and F. Bach. Temporal difference learning w ith continuous time and state in the stochastic setting. 2023. (Cited on pages 2,3, and4.) [KT99] V. Konda and J. N. Tsitsiklis. Actor-criti... | https://arxiv.org/abs/2502.04297v1 |
l inear stochastic approxima- tion andtd learning. In Conference on Learning Theory , pages 2803–2830. PMLR, 2019.(Cited on page 4.) [Sze22] Cs. Szepesv´ ari. Algorithms for reinforcement learning . Springer Nature, 2022. (Cited on page 2.) [Tan24] W. Tang. Fine-tuning of diffusion models via stochast ic control: entrop... | https://arxiv.org/abs/2502.04297v1 |
matrices. IfFsatisfies |||F(Ω)|||op≤Malmost surely and |||Eξ[F2(Ω)]|||op≤V, we have P/braceleftBig λmax/parenleftBign/summationdisplay k=1F(Ωk)/parenrightBig ≥t/bracerightBig ≤d2−π 4exp/parenleftBigg −π2(1−λ)t2 32(1+λ)nV+256 πMt/parenrightBigg , for anyt>0. In order to apply this result, we construct two discrete-tim e ... | https://arxiv.org/abs/2502.04297v1 |
. Substituting to the expression for u⊤E/bracketleftbig M0Mk/bracketrightbig uand invoking Cauchy–Schwarz inequality for each term, we obtain that /vextendsingle/vextendsingle/vextendsingleu⊤E/bracketleftbig (M0−E[M0])(Mk−E[Mk])/bracketrightbig u/vextendsingle/vextendsingle/vextendsingle ≤e−ρ∗λmin 2ℓ(k−1)ην2η2/summatio... | https://arxiv.org/abs/2502.04297v1 |
N+1 N+∞/summationdisplay k=1e−ρ∗λmin 4(ν−1)(k−1)η/radicalbig τ2η2·η2ν/bracerightBig ≤CTr/parenleftbig H−1 1H0/parenrightbig/braceleftBigτ2η2 T+τη1+ν Tρ∗λmin/bracerightBig . Combining with Equation ( 46), we conclude that E/bracketleftbig /⌊a∇⌈⌊l1 NN/(ν−1)/summationdisplay k=0H−1/2 1ψ(Xℓ(k)η)/parenleftbig/tildewideζℓ(k)... | https://arxiv.org/abs/2502.04297v1 |
α:/ba∇dblα/ba∇dbl1≤n1 1+/⌊a∇⌈⌊lα/⌊a∇⌈⌊l2 2≤/summationdisplay α:/ba∇dblα/ba∇dbl1≤n1 1+/⌊a∇⌈⌊lα/⌊a∇⌈⌊l2 1/d=n/summationdisplay j=01 1+j2/d/vextendsingle/vextendsingle/vextendsingle{α∈Zd:/⌊a∇⌈⌊lα/⌊a∇⌈⌊l1=j}/vextendsingle/vextendsingle/vextendsingle. 53 So we have the upper bound Tr/parenleftbig H−1 1H0/parenrightbig ≤cdn/... | https://arxiv.org/abs/2502.04297v1 |
Analysis of Diffusion Models for Manifold Data Anand Jerry George, Rodrigo Veiga, Nicolas Macris EPFL, School of Computer and Communication Sciences. CH-1015 Lausanne, Switzerland. Abstract —We analyze the time reversed dynamics of generative diffusion models. If the exact empirical score function is used in a regime o... | https://arxiv.org/abs/2502.04339v1 |
p-dimensional hyperplane which is then warped by applying a point-wise non-linear function (e.g., a sigmoid activation). Such manifold models have already been used in the learning theory and inference context where they provide a tractable setting (see, e.g., [13]–[15]). Closer to this work, Refs. [16], [17] have inve... | https://arxiv.org/abs/2502.04339v1 |
look at a regime of large dimensions and exponentially large number of samples. More precisely d, p→+∞,p/d= β,n=eαd,α >0and0< β < 1fixed. In Section III we analyze the specialization phenomenon. In this short note, we carry out the details for the simplest case of opposite centers µ++µ−= 0and odd activation functions. ... | https://arxiv.org/abs/2502.04339v1 |
row of F. The quantities λ± j≡f⊤ jµ±√penclose the information about the centers of the Gaussian clouds. By the central limit theorem, when p→ ∞ : ζ± j=Eu∼N(0,1) ϕ √ρ u+λ± j , (8) ζ± jl=Eu,v∼N(0,Θjl) ϕ √ρ u+λ± j ϕ √ρv+λ± j ,(9) where Θjl∈R2×2with matrix elements θjl=f⊤ jfl/p. In order to simplify the crossed-term... | https://arxiv.org/abs/2502.04339v1 |
columns, the sum over the functions Γ0can be rewritten as: dX j=1Γ0(λj)2=1 dtr" Γ0FM√p Γ0FM√p⊤# . (17) The matrix Fis assumed to be a random matrix with i.i.d standard Gaussian entries. We make use of the Gaus- sian Equivalence Principle [22]–[24] and write the following equivalence for Γ0 FM√p : U=ϱ01d1⊤ d+ϱ1(F/... | https://arxiv.org/abs/2502.04339v1 |
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