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to obtain a flexib le regression estimator which behaves well in many problems even when the dimension dis rather large. The fact that the resulting partition depends on the full data (including the response) is however problematic for the theory since in this case, the local averaging estima tor is not a sum over indep... | https://arxiv.org/abs/2501.18204v1 |
In addition, the class of balls in Rdhas dimension equal to d+1. Definition 3. A local map Vis said to be VC whenever {V(x) :x∈SX} ⊂ A , a fixed VC collection of sets in Rd. Let us further define some quantities that will be instrument al in our analysis. For any set V, its diameter is given by the formula diam(V) = sup (... | https://arxiv.org/abs/2501.18204v1 |
now required to ensure that enoug h points are lying within each element of the local map. Definition 5. A VC local map x/ma√sto→ V(x)with dimension v >0is called (δ,n)-large whenever, for allx∈SX, almost surely, nmax(Pn(V(x)),P(V(x)))≥8log/parenleftbigg4(2n+1)v δ/parenrightbigg . 7 Note in particular that the latter in... | https://arxiv.org/abs/2501.18204v1 |
way to obtain aβ-SR (and therefore γ-SR) tree is to allow only for β-SR splits when growing the tree, i.e., valid splits in light of Definition 8. This is easily imposed as it only requires to restrict the optimization domain when finding the optimal split. We furth er develop this aspect in Section 5.2below. Note that, ... | https://arxiv.org/abs/2501.18204v1 |
3σ2/(κb)+Lγ1/d. 5 Data-dependent local regression maps In this section, we show that shape regularity is useful to an alyse local regression maps that are data-dependent. The first example is the nearest neighbors r egression estimator and the second one is a modified version of the CART-like algorithm. 5.1 Nearest neigh... | https://arxiv.org/abs/2501.18204v1 |
a general class of recursive data dependent trees. For a given cell V, a split is characterized by two parameters (p,u)∈S:={1,...,d}×(0,1). The resulting left and right child cells, V(l)andV(r), are such that for any k/ne}ationslash=p,hk(V(l)) =hk(V(r)) =hk(V), and fork=p,hk(V(l)) =hk(V)uandhk(V(r)) =hk(V)(1−u). We als... | https://arxiv.org/abs/2501.18204v1 |
Let Vbe the local regression map obtained from a CART-like tree wi th input parameters β,mand cost function Mn, then we have, with probability 1−3δ, for allx∈SX, |ˆgV(x)−g(x)| ≤/radicalbigg 2σ2log((n+1)2d/δ) m+L(V(x))β√ d/parenleftbigg5m nfX(x)κ/parenrightbigg1/d . 12 Note that the conditions on the value of mare satis... | https://arxiv.org/abs/2501.18204v1 |
mis of ordern2/(d+2), Corollary 16shows that our modified version of CART is rate optimal in sup- norm. 6 Purely random trees We consider now purely random trees (PRT), that are built by s uccessively refining a partition of the space, in a way that is independent from the initial sam pleDn. Before considering 13 uniform... | https://arxiv.org/abs/2501.18204v1 |
P(λ(V(x))≥e−αN)≤/parenleftbig αe1−α/parenrightbigN. Corollary 20. When the number of splits goes to infinity, it holds that, almo st surely, there existsn0≥1such that for all N≥n0, √ de−N/d−4√ Nlog(N)/d≤diam(V(x))≤√ de−N/d+2√ 2Nlog(N)/d and e−N−2√ Nlog(N)≤λ(V(x)))≤e−N+2√ Nlog(N). Moreover, if we denote the normalized di... | https://arxiv.org/abs/2501.18204v1 |
estimator. Corollary 25. Letn≥1,d≥1,x∈SXandN=dlog(n)/{(d+ 2)log(2) }. Suppose that V(x) =V(x,(Di,Si)N i=1)is obtained from a centered random tree as described in Propo sition 23. Under (E)and(X), suppose that gis Lipschitz on SX, there exists C >0, that only depends on the parameters of the problem but not on n, such t... | https://arxiv.org/abs/2501.18204v1 |
j=1gjis sub-Gaussian with parameter σ2 underPXn 1. Therefore, we obtain PXn 1 sup x∈Rd/summationtextn i=1εi1V(x)(Xi)/radicalig/summationtextn j=11V(x)(Xj)>t ≤/summationdisplay (g1,...,gn)∈Gexp/parenleftbigg−t2 2σ2/parenrightbigg ≤SV(n)exp/parenleftbigg−t2 2σ2/parenrightbigg . If we setδ=SA(n)exp/parenleftbig −t2/(... | https://arxiv.org/abs/2501.18204v1 |
we find E[(ˆgV(0)−g(0))2]≥ψ(λ(V)1/d)≥/parenleftbiggσ2d(cγ)d 2(2√ 2)dn/parenrightbigg2/(d+2)/parenleftbigg 1+2 d/parenrightbigg =C2 d/parenleftbigg¯γσ2 n/parenrightbigg2/(d+2) where Cd=/radicalbigg 1+2 d/parenleftbiggd 2/parenrightbigg1/(d+2)/parenleftbigg1 2d√ 72/parenrightbiggd/(d+2) . Proof of Theorem 12 By assumption... | https://arxiv.org/abs/2501.18204v1 |
fX(x)κhd −≤P(V)≤4 nlog/parenleftbigg4(2n+1)2d δ/parenrightbigg +2Pn(V)≤m n+4m n=5m n. In addition, diam(V)≤√ dh+≤√ dβh−≤√ dβ/parenleftbigg5m nfX(x)κ/parenrightbigg1/d . It remains to apply Theorem 4and to use that nPn(V)≥mfor the variance term to get the stated result. Proof of Corollary 16 It is sufficient to reason as ... | https://arxiv.org/abs/2501.18204v1 |
P(λ(V(x))≥e−Nα)≤(αe1−α)N. Proof of Corollary 20 We will use the Borel Cantelli lemma together with the inequal ities obtained in theorems 18and 19. To prove the upper bound on the diameter, we provide values βNleading to small enough probabilities. More precisely, by taking βN= 2/radicalbig 2log(N)/(dN), we gete−Ndβ2 N... | https://arxiv.org/abs/2501.18204v1 |
d+2Cd= max/parenleftig/radicalig d d+2,/radicalig 8 d+2/parenrightig ≤2to obtain the desired inequality. Proof of Proposition 22 At each stage, for each terminal leaf, draw uniformly Diin{1,...,d}as well as a uniform random variable Ui. Then we divide the cell according to coordinate k=Di. The corresponding lengthh... | https://arxiv.org/abs/2501.18204v1 |
β−αN. We proceed in the same way as before for the diameter upper bou nd. By a union bound and symmetry in the directions, we have P(diam(V(x))≤t)≤dP/parenleftbigg h1≤t√ d/parenrightbigg . Then, for any r∈(0,1)andλ>0, P(h1≤rN) =P/parenleftiggN/productdisplay i=12B(1) i≥r−N/parenrightigg ≤E /parenleftigg/producttex... | https://arxiv.org/abs/2501.18204v1 |
get that nP(V(x))/log(n)→ ∞. Fourth, by putting together the second and third point from a bove, we have the following inequality, with probability 1, fornlarge enough, |ˆgV(x)−g(x)| ≤/radicaligg 3σ2log((n+1)v+2) nκfX(x)λ(V(x))+L(V(x))diam(V(x)). Now, from Proposition 24, for a sufficiently large, we have diam(V(x))≤√ d... | https://arxiv.org/abs/2501.18204v1 |
first point, with probability at least 1−δ, we have for all A∈ A nPn(A)−nP(A)≥ −/radicalbig 4nP(A)log(4SA(2n)/δ), equivalently, nP(A)−/radicalbig 4nP(A)log(4SA(2n)/δ)−nPn(A)≤0. Settingx=/radicalbig nP(A),α=/radicalbig 4log(4SA(2n)/δ)andβ=nPn(A), we have that x2−αx−β≤0. Solving the inequality, we find (α−/radicalbig α2+4β... | https://arxiv.org/abs/2501.18204v1 |
45:5–32, 2001. [BS16] Gérard Biau and Erwan Scornet. A random forest guided to ur.Test, 25:197–227, 2016. [CKT22] Matias D Cattaneo, Jason M Klusowski, and Peter M Tia n. On the pointwise behavior of recursive partitioning and its implications fo r heterogeneous causal effect estimation. arXiv preprint arXiv:2211.10805 ... | https://arxiv.org/abs/2501.18204v1 |
Wang. On the convergence of CART under sufficient impurity decrease condition. Advances in Neural Information Processing Systems , 36, 2024. [Nad64] Elizbar A Nadaraya. On estimating regression. Theory Probab. Appl. , 9(1):141–142, 1964. [Nob96] Andrew Nobel. Histogram regression estimation usi ng data-dependent partitio... | https://arxiv.org/abs/2501.18204v1 |
arXiv:2501.18374v2 [cs.IT] 24 Apr 2025Proofs for Folklore Theorems on the Radon-Nikodym Derivative Yaiza Bermudez∗, Gaetan Bisson†, I˜ naki Esnaola‡§, and Samir M. Perlaza∗†§ ∗INRIA, Centre Inria d’Universit´ e Cˆ ote d’Azur, Sophia Ant ipolis, France. †Laboratoire GAATI, Universit´ e de la Polyn´ esie franc ¸ais e, Fa... | https://arxiv.org/abs/2501.18374v2 |
re. Finally, these novel theorems are used to prove a key identit y involving the sum of mutual and lautum information, which is often observed in statistical learning [42], [46]. Hopef ully, these contributions would provide a valuable reference for researchers and students in information theory. II. P RELIMINARIES Th... | https://arxiv.org/abs/2501.18374v2 |
observing that: (a)simple functions are dense in the space of Borel measurable functions [5, Theorem 1.5.5(b)], and (b)the integral is a continuous map on that space [5, Theorem 1.6.2]. From (a)and(b), it follows that (2) also holds for any Borel measurable function /u1D453. This completes the proof. Another reputed fo... | https://arxiv.org/abs/2501.18374v2 |
A ∈ℱ, it holds that /u1D444(A)=/uni222B.dsp Ad/u1D444 d/u1D443(/u1D465)d/u1D443(/u1D465). (16) Therefore, the equality in (16) and Theorem 1 yields (14). The ensuing folklore theorem relates the Radon–Nikodym derivative of a product measure to those of its component measures. This result also appears in [12, Exercise 3... | https://arxiv.org/abs/2501.18374v2 |
for all sets B ∈ℱY, /u1D443/u1D44C(B)/defines/u1D443/u1D44C/u1D44B(B ×X)=/u1D443/u1D44B/u1D44C(X ×B). (30) From the total probability theorem [5, Theorem 4.5.2], it follows that for all B ∈ℱY, /u1D443/u1D44C(B)=/uni222C.dsp Bd/u1D443/u1D44C|/u1D44B=/u1D465(/u1D466)d/u1D443/u1D44B(/u1D465); (31) and for all A ∈ℱX, /u1D4... | https://arxiv.org/abs/2501.18374v2 |
the set /hatwideAis defined in (27). The equality in (54) is obtained by performing a change of measure using Theorem 2 under assumption (/u1D44E); and the equality in (57) follows by exchanging the order of integration [5, Theorem 2.6.6]. The proof is completed from Theorem 1 and by combining equations (42), (47), (53)... | https://arxiv.org/abs/2501.18374v2 |
zation theory and hypothesis testing [42]. REFERENCES [1] D. Fudenberg and E. Maskin, “The Folk theorem in repeated games with discounting or with incomplete information,” Econometrica , vol. 54, no. 3, pp. 533–554, 1986. [2] D. Fudenberg, Game Theory , 1st ed. Cambridge, MA, USA: MIT Press, 1991. [3] J. Radon, Theorie... | https://arxiv.org/abs/2501.18374v2 |
New York, NY , USA: John Wiley & Sons, 1989. [27] R. J. McEliece, The Theory of Information and Coding , 1st ed. Cam- bridge, UK: Cambridge University Press, 2002. [28] M. Mezard and A. Montanari, Information, Physics, and Computation , 1st ed. Oxford, UK: Oxford University Press, 2009. [29] M. Pinsker, Information and... | https://arxiv.org/abs/2501.18374v2 |
1 One-Bit Distributed Mean Estimation with Unknown Variance Ritesh Kumar and Shashank Vatedka Department of Electrical Engineering, Indian Institute of Technology Hyderabad, India Abstract In this work, we study the problem of distributed mean estimation with 1-bit communication constraints when the variance is unknown... | https://arxiv.org/abs/2501.18502v2 |
have been studied. Practical algorithms were proposed in [31], [32]. Distributed settings with one-bit information sharing is challenging, and these have been extensively studied [33], [34]. In this paper, we consider distributed mean estimation in a setting where each client has a single i.i.d. sample and is allowed t... | https://arxiv.org/abs/2501.18502v2 |
n and outputs an estimate ˆµof the scale parameter µ. We say that an estimator is non-adaptive if each Yidepends only on the corresponding Xi; it is adaptive if Yi depends on XiandY1, . . . , Y i´1. The performance of a protocol is measured using the asymptotic mean squared error (MSE): lim nÑ8n¨MSEpˆµqfilim nÑ8n¨E“ |µ... | https://arxiv.org/abs/2501.18502v2 |
introduces dependencies that are tricky in the course of deriving the MSE. The works [21], [35] studied DME of Gaussian distributions with more general multi-bit communication constraints, and they also derive bounds on the minimax MSE for finite (but large) n. However, both works assume that the range ofµis bounded (a... | https://arxiv.org/abs/2501.18502v2 |
for non-adaptive protocols is derived by first obtaining an upper bound on the squared Hellinger distance between the distribution of the transcript under two different realizations of the mean, and then using Le Cam’s method [43], [44]. D. A Consistent Estimator for Location Families To provide some intuition for the ... | https://arxiv.org/abs/2501.18502v2 |
remaining n3users (if the protocol is sequential and the i’th user has access to Y1, . . . , Y i´1, then ˆµccan be computed by the last n3users using their observations). Fig. 2: Our adaptive protocol. We run the non-adaptive protocol using the first n1`n2users, and the server constructs coarse estimates ˆµc,ˆσc. The ˆ... | https://arxiv.org/abs/2501.18502v2 |
main challenge is to derive a suitable upper bound on the squared Hellinger distance, and for this, we use an approach similar to [35] which makes use of the special structure of the Gaussian density. Extending this result to more general log-concave distributions is part of future work. III. P ROOFS OF THEOREMS In thi... | https://arxiv.org/abs/2501.18502v2 |
p1´An3qAn3. (16) where FTn3|ˆµcis the conditional cdf of Tn3given ˆµc, and Φpxqis the standard normal cumulative distribution function. Now we can rewrite (15) as: Tn3“?n31a An3p1´An3qrpFXpα3q´An3qs Rearranging the terms, we have α3“F´1 X¨ ˝Tn3´a An3p1´An3q¯ ?n3˛ ‚`An3. Since FXis an increasing function, F´1 Xwill also... | https://arxiv.org/abs/2501.18502v2 |
and from Lemma III.1,?n3ppµ´ˆµcqpσ´ˆσcqqpÝ Ñ0. Consequently, ζÑ0in probability, and fX´ ˆµ´µ σ¯ ´ ˆσca An3p1´An3q¯ÑfXp0qm σ{2 almost surely. Thus, the R.H.S. converges to zero as nÑ8 , and we can write: Mf“?npµf´µqdÝ ÑNˆ 0,σ2 4fXp0q2˙ . Finally, this gives us lim nÑ8n¨MSEpˆµfq“σ2 4fXp0q2. This concludes the proof C. Pr... | https://arxiv.org/abs/2501.18502v2 |
independent of each other (but potentially nonidentically distributed). Using subadditivity of the squared Hellinger distance, H2pP´ Yn;P` Ynqďnÿ i“1H2pP´ Yi;P` Yiq. (34) The main technical challenge is to bound each squared Hellinger distance. Lemma III.4. Suppose that fXis the standard normal. Then, H2pP´ Yi;P` Yiqďϵ... | https://arxiv.org/abs/2501.18502v2 |
. PMLR, 2018, pp. 560–569. [7] T. Chen et al. , “LAG: Lazily Aggregated Gradient for Communication-Efficient Distributed Learning,” Advances in neural information processing systems , vol. 31, 2018. [8] H. Wang et al. , “Atomo: Communication-Efficient Learning via Atomic Aparsification,” Advances in neural information ... | https://arxiv.org/abs/2501.18502v2 |
, “Inference under Information Constraints II: Communication Constraints and Shared Sandomness,” IEEE Transactions on Information Theory , vol. 66, no. 12, pp. 7856–7877, 2020. [27] J. Acharya et al. , “Inference under Information Constraints III: Local Privacy Constraints,” IEEE Journal on Selected Areas in Informatio... | https://arxiv.org/abs/2501.18502v2 |
Press, 2002. [47] R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science . Cambridge University Press, 2018, vol. 47. [48] R. Zamir, “A proof of the fisher information inequality via a data processing argument,” IEEE Transactions on Information Theory , vol. 44, no. 3, pp. 1246–12... | https://arxiv.org/abs/2501.18502v2 |
1´řk j“1∆Fjpµq`g1 kpϵ1q. Substituting these into Iµ,i, we obtain: Iµ,i“´ 1 σřk j“1∆fjpµq¯2 ´řk j“1∆Fjpµq¯´ 1´řk j“1∆Fjpµq¯`hkpϵ1q. (36) We now make use of the following result, which is a corollary of [22, Lemma 11]: Lemma A.3. Assume the hypotheses of Theorem II.3. For every δą0, ´ 1 σřk j“1∆fjpµq¯2`δ ´řk j“1∆Fjpµq¯1`... | https://arxiv.org/abs/2501.18502v2 |
write P«ˇˇˇˇαi´θi´µ σˇˇˇˇěϵiˆθi´µ σ˙ff “P„ˇˇF´1 XpEFi`Giq´F´1 XpEFiqˇˇěϵiˆθi´µ σ˙ȷ . Using the Taylor series expansion F´1 XpEpFiq`Giq“F´1 XpEpFiqq`1 fXpF´1 XpEpFiqqq.Gi`OpG2 iq. (44) Thus (44) becomes P«ˇˇˇˇαi´θi´µ σˇˇˇˇěϵiˆθi´µ σ˙ff “P„ˇˇˇˇGi fXpF´1 XpEFiqq`OpG2 iqˇˇˇˇěϵiˆθi´µ σ˙ȷ . (45) For sufficiently large ni,Gii... | https://arxiv.org/abs/2501.18502v2 |
as: że´p1 ϵ´ϵq2{2 0x2 adaďże´p1 ϵ´ϵq2{2 0´ ϵ`a ´2 lnpaq¯2 da ďe´p1 ϵ´ϵq2{2¨ ˝ϵ`d 2p1 ϵ´ϵq2 2˛ ‚2 (53) “e´p1 ϵ´ϵq2{2ˆ1 ϵ“opϵ3q, (54) because e´p1 ϵ´ϵq2{2decays faster than any poly pϵqasϵÑ0. Now let us bound the second part of (52). For e´p1 ϵ´ϵq2{2ăaďa˚, we have xaą1 ϵ. This implies that a“fpxa´ϵq´fpxa`ϵqď3ϵxae´x2 a{2 ... | https://arxiv.org/abs/2501.18502v2 |
THE NO-UNDERRUN SAMPLER: A LOCALLY ADAPTIVE, GRADIENT-FREE MCMC METHOD BYNAWAF BOU-RABEE1,a, BOBCARPENTER2,b, SIFAN LIU3,c,AND STEFAN OBERDÖRSTER4,d 1Department of Mathematical Sciences, Rutgers University,anawaf.bourabee@rutgers.edu 2Center for Computational Mathematics, Flatiron Institute,bbcarpenter@flatironinstitut... | https://arxiv.org/abs/2501.18548v2 |
form of the Hit-and-Run sampler. Slice sampling. The slice sampler reduces the problem of sampling from an absolutely continuous target distribution µonRdto alternately sampling from two uniform distributions. Denote the target’s density also by µand for y≥0 define slice(y) = θ∈Rd:µ(θ)>y . The slice sampler generates ... | https://arxiv.org/abs/2501.18548v2 |
y. Recent analyses based on Dirichlet forms highlight the trade-offs between practicality and theoretical efficiency of hybrid slice samplers [ 32]. Extensions such as polar and elliptical slice samplers address challenges in specific high-dimensional settings [35,25]. Understanding the convergence properties of these ... | https://arxiv.org/abs/2501.18548v2 |
slice-free variant of NUTS is also known as multinomial NUTS [4]. Given a countable set Sand a discrete density f:S→R≥0satisfying ∑x∈Sf(x)<∞, we define the categorical distribution on Sweighted by fas: (1) X∼categorical (S,f)ifP(X=x)∝f(x)for all x∈S. In the context of NUTS, this means that each state in Ois weighted ac... | https://arxiv.org/abs/2501.18548v2 |
tion, NURS selects an update direction from the unit sphere, thus avoiding artifacts introduced by fixed coordinate directions. The algorithm employs orbit-based explo- ration, and incorporates a locally adaptive orbit length, where a stopping criterion adjusts the orbit length based on the local scale of the target di... | https://arxiv.org/abs/2501.18548v2 |
suggest that NURS is a promising alternative to existing methods, combining strong theoretical foundations with practical applicability in a way that has been largely unexplored. Organization of paper. Section 2 provides a detailed description of NURS. In Section 3, we present quantitative tuning guidelines and connect... | https://arxiv.org/abs/2501.18548v2 |
hybrid slice sampler and NUTS, orbits are constructed by repeated doubling. To double an orbit O, we first select a direction forward or backward uniformly. If we are moving forward, the extension is Oext=O+h|O|ρ, and if backward, Oext=O − h|O|ρ. Here, arithmetic is performed componentwise and |O|denotes the number of ... | https://arxiv.org/abs/2501.18548v2 |
of storing all states, the log sum of probabilities is accumulated along with a chosen state. These are initialized at the initial state and initial log density. Then as each state is visited, the log sum of probabilities is updated and a probabilistic choice is made as to whether to update the state to the new state. ... | https://arxiv.org/abs/2501.18548v2 |
NURS. To turn Hit-and-Run into an implementable algorithm, the exact sampling from the scalar displacement measure needs to be replaced by computation- ally feasible steps. By discretizing the line L(θ,ρ)into an infinite lattice and selecting the subsequent state from a categorical approximation of the scalar displacem... | https://arxiv.org/abs/2501.18548v2 |
the line L(θ,ρ). The No-Underrun condition for this continuous orbit can be expressed in terms of µθ,ρ, the restriction of the target measure µto the line. Let a,b∈Rrepresent the scalar displacements corresponding to the endpoints of the continuous orbit, i.e., ∆θ,ρ(a) =θleftand∆θ,ρ(b) =θright. Then condition (3)approx... | https://arxiv.org/abs/2501.18548v2 |
Connection to Random Walk Metropolis. When the lattice spacing is small relative to the scale of the target µat the current state θin direction ρ, NURS operates in the Hit-and-Run regime behaving similarly to Hit-and-Run (see Figure 7 (a)). However, when the lattice spacing his large compared to this local scale, the w... | https://arxiv.org/abs/2501.18548v2 |
However, by considering only the portion of the neck where a significant fraction of the probability mass lies, hcan be appropriately scaled to maintain efficient exploration (see Section 6). Inspired by the connection between NURS and RWM, similar control over the Metropolis step of NURS can be established by followin... | https://arxiv.org/abs/2501.18548v2 |
such that θ∼ηand ˜θ∼ν. Here, for any symmetric positive definite matrix A, the matrix A1/2denotes its principal square root. The following theorem extends the contraction for generalized Hit-and-Run, recently established in [8], to the setting of uniform-shifted infinite-orbit NURS. THEOREM 2.Forµ=γC, uniform-shifted i... | https://arxiv.org/abs/2501.18548v2 |
generalized Hit-and-Run and uniform-shifted infinite-orbit NURS coincide. PROOF OF THEOREM 2.Based on the above considerations on generalized Hit-and- Run, we now construct a coupling for uniform-shifted infinite-orbit NURS (Algorithm 4). For given θ∈Rd,ρ∈Sd−1and s∈[−h/2,h/2), in correspondence to (14), the scalar disp... | https://arxiv.org/abs/2501.18548v2 |
measures ηandνonRd, it can be expressed in terms of couplings: (25) TV η,ν=infP[θ̸=˜θ] where the infimum is taken over all couplings of ηandν. A coupling consists of a pair of random variables (θ,˜θ)defined on a common probability space such that θ∼η and ˜θ∼ν. Additionally, if ηandνare absolutely continuous with resp... | https://arxiv.org/abs/2501.18548v2 |
µθ,ρ(t) Zcat θ,ρ−µθ+sρ,ρ(t) Zcat θ+sρ,ρ . By the triangle inequality, (26) together with (28), and Lemma 3, we further obtain 1 2E" ∑ t∈hZ µθ,ρ(t) Zcat θ,ρ−µθ+sρ,ρ(t) Zcat θ+sρ,ρ # ≤1 2E∑ t∈hZ µθ,ρ(t) Zcat θ,ρ−Zt+h/2 t−h/2µθ,ρ(r)dr+Zt+h/2 t−h/2µθ,ρ(r)dr−µθ+sρ,ρ(t) Zcat θ+sρ,ρ 24 ≤1 2E∑ t∈hZ" Zt+h/2 t−h/2 µθ,ρ(t) hZcat ... | https://arxiv.org/abs/2501.18548v2 |
(d+1)-dimensional Gaussian distri- butions: N(0,Cneck)andN(0,Cmouth )with covariance matrices defined as: (31)Cneck=diag(1,L−1, . . .)andCmouth =diag(1,m−1, . . .)for 0<m≪L. By studying the behavior of NURS and RWM in these Gaussian distributions, we can gain theoretical insight into their performance in the more compl... | https://arxiv.org/abs/2501.18548v2 |
make larger moves in the flat mouth region, whereas RWM is limited to the same local moves it employs in the neck region (see Figure 12). Since NURS operates in the Hit-and-Run regime in the mouth region (see Figure 7 (a)), it inherits this advantage, leading to a quantitative speed-up compared to RWM in this region. 6... | https://arxiv.org/abs/2501.18548v2 |
comparable performance. LOCALLY ADAPTIVE, GRADIENT-FREE MCMC 29 2.In the flat mouth region, NURS considerably outperforms RWM both in terms of total complexity and the number of sequential computations required. These results suggest that NURS has the potential to outperform RWM in multi-scale target distributions like... | https://arxiv.org/abs/2501.18548v2 |
3 6 9 0.800.850.900.951.00Acceptance rate h 0.000625 0.00125 0.0025 0.005 0.01 0.02 (b) 10 5 0 5 100.000.020.040.060.080.100.12(c) Fig 13: (a) Squared jump distances (on a log scale) between consecutive states versus ω; (b) Metropolis acceptance rates versus ω; (c) Histogram of the funnel axis variable ω. Fig 14: Scatt... | https://arxiv.org/abs/2501.18548v2 |
. . . ,θright k) where the indices of the endpoints are defined as θleft k=θ−akhρ where ak=k ∑ i=1(Bi−1)2i−1, and θright k=θ+bkhρ where bk=k ∑ i=1Bi2i−1. In other words, the rightmost point of Okis obtained by taking bksteps from θ in the direction ρ, and the leftmost point of Okis obtained by taking aksteps in the opp... | https://arxiv.org/abs/2501.18548v2 |
by NSF grant No. DMS- 2111224. S. Oberdörster was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC- 2047/1 – 390685813. REFERENCES [1]ANDERSEN , H. and DIACONIS , P.(2007). Hit and run as a unifying device. J. Soc. Fr. Stat. & Rev. Stat. Appl. ... | https://arxiv.org/abs/2501.18548v2 |
GELMAN , A. (2014). The No-U-Turn Sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research 151593–1623. [22] JOULIN , A. and OLLIVIER , Y.(2010). Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 382418–2442. [23] MIRA, A.and ROBERTS... | https://arxiv.org/abs/2501.18548v2 |
ifBk=1: Oext=extend-orbit (θright,ρ,h, 2k−1) O=Oold⊙ Oext else: Oext=extend-orbit (θleft,ρ,−h, 2k−1) O=Oext⊙ Oold ifsub-stop (Oext,ϵ,h): break u∼Unif((0, 1)) ifu≤∑˘θ∈Oextµ(˘θ) ∑˘θ∈Oµ(˘θ): ˜θ∼categorical (Oext,µ) ifstop (O,ϵ,h): break return ˜θ Listing 1: The No-Underrun Sampler. The extend-orbit() function is given in ... | https://arxiv.org/abs/2501.18548v2 |
1 A New Statistical Approach to the Performance Analysis of Vision-based Localization Haozhou Hu, Student Member, IEEE , Harpreet S. Dhillon, Fellow, IEEE , R. Michael Buehrer, Fellow, IEEE Abstract —Many modern wireless devices with accurate po- sitioning needs also have access to vision sensors, such as a camera, rad... | https://arxiv.org/abs/2501.18758v1 |
The mathematical foundation for this mapping remains largely unexplored because of a key challenge: landmarks in vision-based systems can appear visually similar, making it difficult to distinguish between them. This similarity can lead to ambiguities when localizing the target, as different locations may produce simil... | https://arxiv.org/abs/2501.18758v1 |
improved with the use of more detailed maps [14], [15], including topography labels and 3-dimensional models. In [16], the landmarks are modeled as various types of nodes, and the geometric placements of these landmarks are used to localize the target. The graph of landmarks, which represents their adjacency relationsh... | https://arxiv.org/abs/2501.18758v1 |
contrast, [39] considers a stochastic environment where the landmark locations form a Poisson point process (PPP). They assume that at least three landmarks are visible and use Kalman filtering to estimate the target’s location and orientation. From this stochastic framework, an upper bound on the uncertainty in locali... | https://arxiv.org/abs/2501.18758v1 |
work provides a comprehensive mathematical framework and analytical results for vision-based positioning, offering insights into identifying landmarks in scenarios where the landmarks are not uniquely identifiable. II. M ODEL AND PROBLEM FORMULATION This section presents the system model based on stochastic geometry, d... | https://arxiv.org/abs/2501.18758v1 |
noise variance is assumed to be independent of noise-free distance diand is determined by the size and shape of the landmark. Thus, the variance depends on the mark of landmarks, denoted as σ2 i=σ2 mi. Since landmarks with the same mark are visually indistinguishable, we do not know the exact landmark with which the ra... | https://arxiv.org/abs/2501.18758v1 |
. . , r N]were obtained. Once the landmark combination is correctly iden- tified, the target’s location can be estimated using the range measurements to these landmarks. We represent all possible matching between range measurements and landmarks as the combination set , defined as C={[x1, . . . ,xN] :x1∈ Bm1, . . . ,xN... | https://arxiv.org/abs/2501.18758v1 |
the circle centered at the origin, represented as ∂b(0, d1). Since the distance from the target to the second landmark is d2, the landmark with mark m2must be located on ∂b(0, d1)⊕∂b(0, d2) =a(0,|d1−d2|, d1+d2),(7) where a(0,|d1−d2|, d1+d2)represents the annulus with radii|d1−d2|andd1+d2. This result shows that the dis... | https://arxiv.org/abs/2501.18758v1 |
as X3={xp,xq} ⊕b(0, d3). (9) Since the Lebesgue measure of X3isλ(X3) =|X3|= 0, when the landmark locations form a PPP, the probability that the third landmark will lie exactly on X3is almost surely zero. Consequently, in this scenario, the probability that any other combination of three landmarks, excluding the observe... | https://arxiv.org/abs/2501.18758v1 |
interested in the following two probabilities: A. True Positive Rate (TPR) The true positive rate ptrepresents the probability that the correct combination c∗is included in the solution set, defined as pt=P[ID∗= 1|r,m] =ED∗|r,m{ 1(P[A∗]≥T)},(15)where the random variable D∗represents the distance between landmarks in c∗... | https://arxiv.org/abs/2501.18758v1 |
and the corresponding distributions can be derived accordingly. The random policy refers to obtaining measurements from arbitrary landmarks within the visibility distance. In this setting, the PDF of D∗is identical to that of Dsince the probability of selecting any pair of the observed landmarks is the same. In contras... | https://arxiv.org/abs/2501.18758v1 |
combination c∗satisfies the geometric constraints and is contained in the solution set. When I∗= 0, meaning c∗is not included in the solution set, the above equation simplifies to P[|S|=k|r,m,|C|=m, I∗= 0] =P X cij∈C\{ c∗}Iij=k|r,m,|C|=m (21) (a)=m−1 k P[Iij= 1|r,m]k · P[Iij= 0|r,m]m−k−1(22) (b)=m−1 k pk f... | https://arxiv.org/abs/2501.18758v1 |
successfully localized. As a result, it serves as a fundamental benchmark for evaluating localization performance. The result in Theorem 1 is expressed in an expectation form and relies on the joint distribution of R,M, and|C|. To proceed further, we need to make assumptions about the observation policy, which will pri... | https://arxiv.org/abs/2501.18758v1 |
the mark of the selected points is independent of their locations, given by P[M1=p] =ΛpPM m=1Λm, (36) where Λp=λpπd2 p. The distribution of mark M2is identical. This completes the proof. Next, we give the conditional PDF of the range measure- ments in the following lemma. Lemma 5. The conditional joint distribution of ... | https://arxiv.org/abs/2501.18758v1 |
the scaled unmarked PPP defined in the proof of Lemma 4, ˜Φ[m]is the PPP of landmarks with mark m. Now, the joint PMF of random variables M1,Vpand Vcan be written as P[M1=p, Vp=v, W =w] =P[M1=p|Vp=v, W =w]P[Vp=v, W =w],(51) where the first component P[M1=p|Vp=v, W =w]repre- sents the probability that the selected landm... | https://arxiv.org/abs/2501.18758v1 |
select two arbitrary landmarks and associate them with measurements r1andr2, the possible locations for the target are the two intersection points of the circles with r1andr2, centered at the respective landmarks. The six inequalities ensure that all three circles 11 1r 2r3r 1x 2x3x Fig. 4. An illustration of the geome... | https://arxiv.org/abs/2501.18758v1 |
AOI, represented by |Cn|. Thus, using the result in Theorem 1, the probability of correctly identifying ci,ncan be written as ER,M,|Cn| P ˆci,n=ci,n|r,m,|Cn|,ˆc−n=c∗ −n =ER,M,|Cn|( pt·1−(1−pf)|Cn| |Cn| ·pf) . (64) This completes the proof of Theorem 3. Theorem 3 provides an analytical upper bound for the localizabil... | https://arxiv.org/abs/2501.18758v1 |
of landmark combinations re- moved by the proposed geometric constraint are presented in Fig. 5(b). When comparing the result differences in the two observation policies, the nearest policy removed more landmark combinations than the random policy , which means thenearest policy is more efficient in identifying the cor... | https://arxiv.org/abs/2501.18758v1 |
derive the probability of correctly identifying the landmark combination to evaluate the localiza- tion accuracy. This work is the first to tackle the challenge of landmark non-uniqueness in vision-based localization and to analyze their properties within this framework. Given the novel framework presented in this pape... | https://arxiv.org/abs/2501.18758v1 |
3D point clouds,” Large-Scale Visual Geo-Localization , pp. 147–163, 2016. [15] O. Saurer, G. Baatz, K. K ¨oser, L. Ladick `y, and M. Pollefeys, “Image- based large-scale geo-localization in mountainous regions,” in Large- Scale Visual Geo-Localization . Springer, 2016, pp. 205–223. [16] S. Verde, T. Resek, S. Milani, ... | https://arxiv.org/abs/2501.18758v1 |
Systems (IROS) , Sep. 2011. [30] R. F. Salas-Moreno, R. A. Newcombe, H. Strasdat, P. H. Kelly, and A. J. Davison, “SLAM++: Simultaneous localisation and mapping at the level of objects,” in Proc., IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) , Jun. 2013, pp. 1352–1359. [31] J. Engel, T. Sch ¨ops, and D.... | https://arxiv.org/abs/2501.18758v1 |
Targeted Data Fusion for Causal Survival Analysis Under Distribution Shift Yi Liu1,2, Alexander W. Levis3, Ke Zhu1,4, Shu Yang1, Peter B. Gilbert5, Larry Han∗6 1Department of Statistics, North Carolina State University, Raleigh, NC, USA 2Duke Clinical Research Institute, Durham, NC, USA 3Carnegie Mellon University, Dep... | https://arxiv.org/abs/2501.18798v2 |
the data fu- sion literature—may fail in the presence of covariate shift, heterogeneous censoring mechanisms, or other types of distribution shift. Traditionally, time-to-eventoutcomeshavebeenanalyzedusingdatafromsingle-siterandomized clinical trials, with survival curves estimated via the unadjusted Kaplan–Meier estim... | https://arxiv.org/abs/2501.18798v2 |
causal effect estimation in complex multi-source survival settings, even under distributional shifts. 3 2 Multi-Source Inference on Treatment-specific Survival Functions 2.1 Notation, estimand, and assumptions Consider Kstudies, each of which could be randomized or observational. For each participant in a given study, ... | https://arxiv.org/abs/2501.18798v2 |
(Westling et al. [48]) .The nonparametric EIF for θ0(t, a)given t∈[0, τ]and a∈ {0,1}isφ∗0 t,a(O;S0, G0, π0) =φ∗0 t,a(O;P) =φ0 t,a(O;P)−θ0(t, a), where φ0 t,a(O;P) =I(R= 0) P(R= 0) 1−I(A=a) π0(a|X)Ht,a(O;S0, G0) S0(t|a,X). Throughout, let Pn[f(O)] = n−1Pn i=1f(Oi)be the empirical average and bPdenote that the nuisance... | https://arxiv.org/abs/2501.18798v2 |
to a normal random variable with mean zero and variance σ2=P[(φ∗CCOD t,a )2]. If Conditions D.4–D.5 (which strengthen the consistency and product error conditions to hold uniformly) also hold, then sup u∈[0,t] bθCCOD n (u, a)−θ0(u, a)−Pn(φ∗CCOD u,a ) =op(n−1/2). In particular, {n1/2(bθCCOD n (u, a)−θ0(u, a)) :u∈[0, t]}... | https://arxiv.org/abs/2501.18798v2 |
sharing constraints; and (iii) for Skin the augmentation term, we train an S0model on the target site and apply its predictions to site k, since S0andSkare exchangeable under partial CCOD. Crucially, if partial CCOD is violated, we can detect site heterogeneity by the difference between bθk,0 n(t, a)andbθ0 n(t, a). Fur... | https://arxiv.org/abs/2501.18798v2 |
the federated estimator bθfed n(t, a)at each (t, a)∈[0, τ]× {0,1}has asymptotic distribution q n/bVt,an bθfed n(t, a)−θ0(t, a)o →dN(0,1), 10 wherebVt,ais a consistent estimator for the asymptotic variance of bθfed n(t, a)(see Appendix D.3 for its form), which is no greater than the asymptotic variance of the target-onl... | https://arxiv.org/abs/2501.18798v2 |
outcome, or conditional censoring mechanism. Case (v) combines all three shifts. Figure 5 in Appendix A.1 illustrates the true underlying survival curves under covariate and outcome shifts, providing a direct comparison of these two types of site heterogeneity. Simulation results are evaluated using: (A) Estimation Bia... | https://arxiv.org/abs/2501.18798v2 |
assumption. 13 0.920.940.960.981.00 0200400600Time (day)Survival ProbabilityGroupControlTreatedFED 0.920.940.960.981.00 0200400600Time (day)Survival ProbabilityGroupControlTreatedFED (BOOT) 0.920.940.960.981.00 0200400600Time (day)Survival ProbabilityGroupControlTreatedCCOD 0.920.940.960.981.00 0200400600Time (day)Surv... | https://arxiv.org/abs/2501.18798v2 |
curves. However, TGT occasionally has wider confidence intervals—reflecting lower effi- ciency—and fails to produce interval estimates at some early time points when its variance estimates are unavailable or too small. In contrast, FED and FED (BOOT) still provide interval estimates in those cases. The CCOD method gene... | https://arxiv.org/abs/2501.18798v2 |
Brantner, Ting-Hsuan Chang, Trang Quynh Nguyen, Hwanhee Hong, Leon Di Stefano, and Elizabeth A Stuart. Methods for integrating trials and non-experimental data to examine treatment effect heterogeneity. Statistical Science , 38(4):640–654, 2023. [6] Kate Bull and David J Spiegelhalter. Tutorial in biostatistics surviva... | https://arxiv.org/abs/2501.18798v2 |
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