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orthonormal sequence in Dform∈N. Then there exists a countable sequence (di)i⊂D such that (di)iis an ONB ofHanddi=eifori≤m. Lemma A.2 ( [6, Lemma A.4]) .LetHbe a Hilbert space and A∈B(H). ThenA >0if and only if A≥0andAis injective. Lemma A.3 ( [18, Theorem 4.3.1]) .LetH,Kbe Hilbert spaces, and A∈B(H,K)be compact. Then ... | https://arxiv.org/abs/2503.24209v1 |
It holds that VarX∼µpos(/an}bracketle{tX,C−1/2 prwi/an}bracketri}ht) VarX∼µpr(/an}bracketle{tX,C−1/2 prwi/an}bracketri}ht)= 1+λi=1 1+−λi 1+λi,∀i∈N, (10) and for any subspace Vr⊂ranC1/2 prof dimension r∈N, min z∈(C−1/2 prVr)⊥\{0}VarX∼µpos(/an}bracketle{tX,z/an}bracketri}ht) VarX∼µpr(/an}bracketle{tX,z/an}bracketri}ht)= ... | https://arxiv.org/abs/2503.24209v1 |
that (i)Cpos=SposS∗ posandCy=SyS∗ yandS−1 pos: ranC1/2 pr→HandS−1 y∈B(Rn)exist, (ii)/bardblh/bardbl2 C−1 pos=/bardblS−1 posh/bardbl2for allh∈ranC1/2 pr= ranC1/2 pos, (iii)Spos(ranC1/2 pr) = ranCpr= ranCpos. Proof of Lemma 5.3. We recall that λi= 0 for i > nby Proposition 3.4. Since C1/2 obshas a bounded inverse, Lemma ... | https://arxiv.org/abs/2503.24209v1 |
[3, Theorem 2.4.7] or [17, Proposition 4.45], the Cameron–Martin space of a Gaussian measure is contained in eve ry measurable linear subspace of full measure. Thus, since AY∈ranC1/2 poswith probability 1, the Cameron–Martin space of AY, which is ran( ACyA∗)1/2, is contained in ran C1/2 pos= ranC1/2 pr. Because Ahas fin... | https://arxiv.org/abs/2503.24209v1 |
(20). Then a solution of Problem 5.1 for i= 2is given by Aopt,(2) r=C1/2 pr(/summationtextr i=1/radicalbig −λi(1+λi)wi⊗ϕi)C−1/2 obs∈M(2) r. Furthermore, ranAopt,(2) r⊂ranCpos, the corresponding loss is/summationtext i>r−λi 1+λi, and the solution Aopt,(2) ris unique if and only if the following holds: λr+1= 0orλr< λr+1.... | https://arxiv.org/abs/2503.24209v1 |
posCprG∗C−1 obsSy−C1/2 prG∗C−1/2 obsto find a solution /tildewideBoptto the approximation problem (34). For the given choices of TandS, we have that T†=I, while for the finite-rank operator Swe have from (20) and (21) that S=C−1/2 pr/parenleftBig C1/2 prG∗C−1/2 obs/parenrightBig C−1/2 obsSy=C−1/2 pr/parenleftBiggn/summat... | https://arxiv.org/abs/2503.24209v1 |
(22b) with ϕi←wiandx←wfor arbitary w∈Hyields the desired equation for S−1 posAopt,(1) rSy. Since/radicalBig −λi (1+λi)3=/radicalBig −λi 1+λi/parenleftBig 1+−λi 1+λi/parenrightBig , /tildewideAopt,(1) rSy=S−1 posAopt,(1) rSy=/summationdisplay i>r/radicalbigg −λi 1+λi3 wi⊗ϕi+n/summationdisplay i=1/radicalbigg −λi 1+λiwi⊗... | https://arxiv.org/abs/2503.24209v1 |
(16), andAopt,(2) ris a solution to Problem 5.1 for i= 2. Proof.SincePC1/2 prwi=C1/2 prwifori≤rand ranP= span/parenleftBig C1/2 prwi, i≤r/parenrightBig , it holds that P2=P, so thatPis indeed a projectorofrankat most r. Let (/tildewideAry,/tildewideCr) denote the posteriormeanand covariancefor the model (29) with the g... | https://arxiv.org/abs/2503.24209v1 |
likelihood -informed subspace methods. Bernoulli , 28(4), 2022. [11] T. Cui, X. T. Tong, and O. Zahm. Prior normalization for certified likelihood-informed subspace detection of Bayesian inverse problems. Inverse Problems , 38(12):124002, 2022. [12] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions ... | https://arxiv.org/abs/2503.24209v1 |
Comput. , 39(5):S167–S196, 2017. [33] A. Spantini, A. Solonen, T. Cui, J. Martin, L. Tenorio, and Y. Mar zouk. Optimal low-rank approx- imations of Bayesian linear inverse problems. SIAM J. Sci. Comput. , 37(6):A2451–A2487, 2015. [34] A. M. Stuart. Inverse problems: A Bayesian perspective. Acta Numer. , 19:451–559, 201... | https://arxiv.org/abs/2503.24209v1 |
Estimating a graph’s spectrum via random Kirchhoff forests Simon B ARTHELM ´E1Fabienne C ASTELL2Alexandre G AUDILLI `ERE3Clothilde M ELOT2Matteo Q UATTROPANI4 Nicolas T REMBLAY1,5 1CNRS, Univ Grenoble-Alpes, Grenoble-INP, GIPSA-lab, Grenoble, France 2Aix-Marseille Univ, I2M, Marseille, France 3CNRS & Aix-Marseille Univ... | https://arxiv.org/abs/2503.24236v1 |
the discrete measure µ=1 nX iδλi. (2) To estimate the c.d.f. c(τ), the standard approach is to first estimate moments of the empirical distribution (the estimation step) then to reconstruct the cumulative density function from the estimated mo- ments (the reconstruction step). In the existing literature, the estimation... | https://arxiv.org/abs/2503.24236v1 |
2 Forests for moment estimation In what follows we assume that Lis the Laplacian matrix of a weigh- ted, undirected graph G= (V,E)withnnodes and |E|edges : L=D−A (5) where Ais the adjacency matrix and Dthe degree matrix. See [ 3] for how to extend this to diagonally-dominant L. We denote by µ(f) the expectation of func... | https://arxiv.org/abs/2503.24236v1 |
small (typically l= 3in our experiments). All these expectations yield information about µ, and can be used for reconstruction. To estimate these quantities, we use the following fact : define Kq=q(qI+L)−1;h(q, k) =1 nTr(Kk q) (10) and for a given KF ϕq, the following matrix : M(ϕq) = [ I(rϕq(i) =j)]i∈V,j∈V (11) where ... | https://arxiv.org/abs/2503.24236v1 |
q+λi(15) then : h(q, k) =1 nXq q+λik =1 nX xk i (16) which are the classical moments of the empirical distribution of the xi’s. We note this measure νq=1 nX iδxi (17) Since we have access to the moments of νq, we can apply the standard toolset of classical moment theory to form an approximation. In our numerical simu... | https://arxiv.org/abs/2503.24236v1 |
i−mi2≤1) where viis an upper-bound on the variance of the estimate mi. When solving eq. 20, we interrupt the optimisation once the moments corres- ponding to the iterate βtare in B. In addition, a vector of estimated moments mcan fall outside the set of valid moments Ml([a, b]) [18]. For instance, for all probability ... | https://arxiv.org/abs/2503.24236v1 |
step, but close enough to keep this term 3 n= 1000 n= 5000 n= 100002d Grid 1101001000 time102 100error 1101001000 time102 100error 1101001000 time102 100error Sparse ER 1 10 100 1000 time102 100error 1 10 100 1000 time102 100error 1 10 100 1000 time102 100error Sparse BA 1 10 100 1000 time102 100error 1 10 100 1000 tim... | https://arxiv.org/abs/2503.24236v1 |
of each method, the time to reach a given error is linear in the number of edges4|E|forpoly andslq andsublinear in|E|(in fact, linear inn) for our method – which is, we believe, remarkable. 5 Concluding remarks This is a first communication on a new and promising class of methods to estimate a graph’s spectrum. Whereas... | https://arxiv.org/abs/2503.24236v1 |
Non-Asymptotic Analysis of Classical Spectrum Estimators for L-mixing Time-series Data with Unknown Means Yuping Zheng and Andrew Lamperski Abstract — Spectral estimation is an important tool in time se- ries analysis, with applications including economics, astronomy, and climatology. The asymptotic theory for non-para... | https://arxiv.org/abs/2504.00217v1 |
class of L-mixing processes quantifies the decay of dependencies of stochastic processes over time and was first introduced in [18]. Other related work uses the theory of L-mixing processes to study stochastic optimization algorithms with time-correlated data streams [19], [20]. Time dependencies have also been describ... | https://arxiv.org/abs/2504.00217v1 |
estimators. Both methods rely on decomposing the data into segments of Mdata points and working with Fourier transform estimates. For streamlined notation, the sample mean and Fourier transform estimates for the ith data segment are denoted by: ¯yi=1 MM−1X k=0y[iK+k] ˆyi(s) =M−1X k=0wk(s)y[iK+k]. The Bartlett method ha... | https://arxiv.org/abs/2504.00217v1 |
describes how L-mixing properties of the original data sequence induce L-mixing properties on the vectors and matrices used in the spectral estimation algorithms. Lemma 1: If y[k]is L-mixing with respect to (F,F+), then for all s∈R,˜y(s)and˜y(s)˜y⋆(s)are L-mixing with respect to (G,G+)where Gi=FiK+M−1for all i∈N. Furth... | https://arxiv.org/abs/2504.00217v1 |
by ∥Φ(s)−¯Φ∥2≤2Mq(y)X |k|≥Mγ2(|k|,y)+ 2Mq(y)X |k|<Mγ2(|k|,y)M−1X i=|k|vi−|k|vi ∥v∥2 2. IV. P ROOF OF SUPPORTING LEMMAS A. Proof of Lemma 1 Knowing that y[k]−µis zero-mean and then applying Theorem 1 gives ∥ˆyi−hµ∥L2q ≤ M−1X k=0wk(s)(y[i]−µ) L2q ≤2q 2(2q−1)M2q(y−µ)Γd,2q(y−µ)vuutM−1X k=0wk(s)2. We can show that for all q... | https://arxiv.org/abs/2504.00217v1 |
ˆyi−hˆµkand then expanding the terms. Now, we bound the L2q-norm of the four terms in (6) separately. For the first term, Theorem 1 gives 1 kk−1X i=0 (ˆyi−hµ)(ˆyi−hµ)⋆−¯Φ L2q ≤2q 2(2q−1)M2q(˜y˜y∗−¯Φ)Γd,2q(˜y˜y⋆−¯Φ)vuutk−1X i=01 k2 ≤4q (2q−1)M2q(˜y˜y⋆)Γd,2q(˜y˜y⋆)1√ k. (7) Note that similar to (3), applying the triang... | https://arxiv.org/abs/2504.00217v1 |
is δ= 0.72. There- fore, we have the following L-mixing statistics: Γd,4q(y)≤ 4Gmax1 1−(1−δ)1 4q≤4Gmax δ4qandM4q(y)≤Gmax. 100101102103104105106107 Iteration10−1010−810−610−410−2100102104106 Error Theoretical Bound(a)Bartlett Estimator .M= 5, L= 107 100101102103104105106107 Iteration10−1010−810−610−410−2100102104106Erro... | https://arxiv.org/abs/2504.00217v1 |
, IEEE, 2019, pp. 5655–5661. [10] A. Tsiamis and G. J. Pappas, “Finite sample analysis of stochastic system identification,” in 2019 IEEE 58th Conference on Decision and Control (CDC) , IEEE, 2019, pp. 3648–3654. [11] B. Lee and A. Lamperski, “Non-asymptotic closed-loop system identification using autoregressive proces... | https://arxiv.org/abs/2504.00217v1 |
arXiv:2504.00243v1 [math.ST] 31 Mar 2025Non-parametric cure models through extreme-value tail estimation∗ Jan Beirlant Department of Mathematics, KU Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium e-mail:jan.beirlant@kuleuven.be Martin Bladt Department of Mathematical Sciences, University of Copenh agen, Denmark e-... | https://arxiv.org/abs/2504.00243v1 |
distributions . . . . . . . 8 3 Asymptotic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Finite-sample behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6 Conclusion . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2504.00243v1 |
proportion of right-censored individuals can be c lassified as susceptible, coming one step closer to the true cure rate. /Non-parametric cure models through extreme-value tail es timation 3 We adopt the general idea of using EVT to extrapolate beyond the c ensoring support. General reviews on EVT can be found in [ 6], ... | https://arxiv.org/abs/2504.00243v1 |
mechanism, respectively. In the case of insufficient follow-up , i.e.τ0> τc (and hence τc<∞), [9] and [11] proposed to do a simple improvement of pn considering ˆFatZn,nand at two additional points below, which results in a correction of pn, though it is of course not possible to obtain consistency. The methods we propos... | https://arxiv.org/abs/2504.00243v1 |
can use the substitute ˆFn p(Zn−j+1,n), j= 1,...,k, and we perform the least-squares regression optimization to estim ate the pa- rameters βandp: SSp(β,p) =k/summationdisplay j=1 −log1−ˆFn(Zn−j+1,n) p 1−ˆFn(Zn−k,n) p−βlogZn−j+1,n Zn−k,n 2 .(2.5) The resulting estimator of pis denoted by ˆ pP k. The slope parameter ... | https://arxiv.org/abs/2504.00243v1 |
for when there is sufficient follow-up, while under insufficient follow-up they are conve rging to F(τc) =pF0(τc), unless assuming τc→ ∞. 2.2. Cure rate estimation using the peaks-over-threshold ( POT) method An alternative approach to targeting specific distributions as abov e, is to target specific max-domains of attractio... | https://arxiv.org/abs/2504.00243v1 |
cure models through extreme-value tail es timation 10 The last term p−p0(τc) =p¯F0,∗(τc) in (3.3) leads to bias terms which are smaller for lighter tailed distributions ¯F0,∗such as for Weibull distributed data compared to Pareto data. Concerning the term in ˆFn(y)−F(y), Theorem 3.14 in [ 16] states that the empirical ... | https://arxiv.org/abs/2504.00243v1 |
taken as k=⌊n/5⌋. 5. The Weibull distribution, with df F(x) = 1−exp(−xa),x≥0,a= 0.5, and insufficient follow-up with Gthe df of a uniform random variable on (0,6). The sample fraction is taken as k=⌊n/5⌋. 6. The Weibull distribution with df Fas in the previous case and more insufficient follow-up with Gthe df of a uniform ... | https://arxiv.org/abs/2504.00243v1 |
the Kaplan-Meier survival function, the cure rate estimates ˆ pG kand ˆpF kjointly with the corresponding goodness-of-fit plots, each atk/n= 0.5,0.1 and 0.025. The Gumbel plot shows the better fit at the large kwhile for smaller kno clear favorable model appears. In Figure 4in a similar way we present the ˆ pP k, ˆpW kan... | https://arxiv.org/abs/2504.00243v1 |
nF EstimatorLog squared errorp = 0.95 −0.050.000.050.10 pnpyp^ kPp^ nF EstimatorBiasp = 0.9 −0.06−0.030.000.03 pnpyp^ kPp^ nF EstimatorBiasp = 0.95 −16−12−8−4 pnpyp^ kPp^ nF EstimatorLog squared errorp = 0.9 −15−10−5 pnpyp^ kPp^ nF EstimatorLog squared errorp = 0.95 −0.10.00.1 pnpyp^ kPp^ nF EstimatorBiasp = 0.9 −0.15−... | https://arxiv.org/abs/2504.00243v1 |
0.000.250.500.751.00 1 2 3 4 TimeSurvivalPareto dataCure rate estimates 0.000.250.500.751.00 0 1 2 3 4 TimeSurvivalWeibull dataCure rate estimates 0.000.250.500.751.00 0 1 2 3 4 TimeSurvivalLognormal dataCure rate estimates 0.00.51.0 0 1 2 xyPareto dataPareto G−o−F diagnostic 024 0.0 0.3 0.6 0.9 xyWeibull dataPareto G−... | https://arxiv.org/abs/2504.00243v1 |
pn(thick solid). The pG(n,ε)for this specific data is very close to the latter, and not distinguishable in the plot. The horiz ontal line shows the value of ˆpG n for the original (unmodified) data. 6. Conclusion In this paper, we have introduced a non-parametric cure model th at integrates extreme value tail estimation ... | https://arxiv.org/abs/2504.00243v1 |
use the above approximation of log( Zn−j+1,n/Zn−k,n) and that 1 kk/summationdisplay j=1/parenleftig 1−V−γc j,k/parenrightig2 =2γ2 c (1−γc)(1−2γc)(1+op(1)) ask→ ∞andkUk+1,n/n= 1+op(1)) ask,n→ ∞. Moreover we approximate Yn(Zn−j+1,n) by (1−F)(Zn−j+1,n)Z(Zn−j+1,n) (j= 1,...,k) based on the almost sure convergence of Ynto... | https://arxiv.org/abs/2504.00243v1 |
n−1/parenleftign k/parenrightig1+γc1 kk/summationdisplay j=1/parenleftigg 1−/parenleftbiggj k+1/parenrightbigg−γc/parenrightigg h1+γc/parenleftbiggk+1 j/parenrightbigg =O/parenleftbigg1 k/parenrightbigg =o/parenleftbigg k−1/parenleftign k/parenrightig−γc/parenrightbigg . This finishes the proof. /Non-parametric cu... | https://arxiv.org/abs/2504.00243v1 |
GRAPHICAL MODELS AND EFFICIENT INFERENCE METHODS FOR MULTIVARIATE PHASE PROBABILITY DISTRIBUTIONS A P REPRINT Andrew S. Perley∗ Department of Bioengineering Stanford University Stanford, CA 94305 aperley@stanford.edu Todd P. Coleman Department of Bioengineering Stanford University Stanford, CA 94305 toddcol@stanford.ed... | https://arxiv.org/abs/2504.00459v1 |
and uncover interactions among oscillating elements has profound implications. For example, identifying spatiotemporal patterns in neuronal oscillations can shed light on how spontaneous neural processes inter-relate in the resting state as well as how traveling waves mediate information transfer during motor tasks and... | https://arxiv.org/abs/2504.00459v1 |
by Shah et al, has also extended this estimator to bounded continuous real random variables whose pairwise interaction terms can be described by a Kronecker product [ 23,24]. Within the specific context of multivariate phase interactions, Cadieu and Koepsell proposed a distribution and method to solve this problem usin... | https://arxiv.org/abs/2504.00459v1 |
according toPXandPYrespectively. Definition 2.2 (Mutual Information) .Consider random variables X, Y with joint density fX,Y(x, y). The mutual information between XandYis defined as follows: I(X;Y) =Z xD(PY|X=x∥PY)fX(x)dx (3a) =Z x,yfX,Y(x, y) logfX,Y(x, y) fX(x)fY(y)dxdy. (3b) It is often interpreted as the amount of ... | https://arxiv.org/abs/2504.00459v1 |
time. Now suppose we are recording a wave at discretely sampled locations, {(xi, yi)}p i=1. Given a wave described by Eq. (6), it now suffices to understand the discrete analog to the derivatives - the phase differences. Thus, we must have an explicit representation of the phase differences in our model given that our ... | https://arxiv.org/abs/2504.00459v1 |
distribution as desired in Section 2.3. Lemma 2.1. For any random vector distributed according to (8),κij= 0is equivalent to conditional independence between YiandYjgiven all nodes except for iandj Proof. For any integer u∈V, define y−uto be the vector of observations on nodes excluding node u:y−u= yj:j∈V\ {u}. By defi... | https://arxiv.org/abs/2504.00459v1 |
where for (i, j)∈E ϕij,c(y) = cos( yj−yi) (12) ϕij,s(y) = sin( yj−yi). (13) Thus, in accordance with Definition 2.5, we desire to show that the following quantity X (i,j)∈Eaij,ccos(yj−yi) +aij,ssin(yj−yi), (14) cannot equal a constant for all y∈ Ypfor any possible avector of coefficients for this model to be of a minim... | https://arxiv.org/abs/2504.00459v1 |
Relative Entropy and Mutual Information of a Pair of Oscillators Consider the wave model in the case of two oscillators, i.e., p= 2. We compute the pairwise mutual information under this model and show that under this formulation, we have a closed-form solution. We first show the following results on the two-dimensiona... | https://arxiv.org/abs/2504.00459v1 |
property. The middle graph violates the acyclic property of trees and the right graph violates the connected property of trees. we must also encode information about the probability distribution. We do this using the following factorization of the joint distribution fT Y(y) =dY i=1fYi|Yj(i)(yi|yj(i)), (24) where for tr... | https://arxiv.org/abs/2504.00459v1 |
and then use a maximum spanning tree algorithm to obtain the dependence tree. The approximated joint density can then be computed as ˆfY(y) =fˆT Y(y)nY (i,j)∈ˆTfYj|Yi(yj|yi). (28) 4 Interaction Screening for Inference of the Full Graphical Model In the previous section, we described a general procedure to efficiently i... | https://arxiv.org/abs/2504.00459v1 |
screening approach [ 23], which leverages general conditions for convergence of high dimensional M-estimators [ 33]. We first state the two conditions and then the proposed theorem that quantitatively shows the utility of this method. Condition 4.1. Letλbe the penalty parameter for an ℓ1regularized M-estimator describe... | https://arxiv.org/abs/2504.00459v1 |
First we provide some notation that will aid in making the proofs succinct. Definition 4.1. For any (l, m)∈E, define θ∗ l,m≜ θ∗ lm,c, θ∗ lm,sT(38) tl,m≜[cos(ym−yl),sin(ym−yl)]T(39) and note as a consequence that ⟨θ∗ l,m, tl,m⟩=θ∗ lm,ccos(ym−yl) +θ∗ lm,ssin(ym−yl) Moreover, from (35) and (36) it follows that Xij,c(θ∗ ... | https://arxiv.org/abs/2504.00459v1 |
by analogous argument. We again note similarities to [ 23, Lemma 2], where they take advantage of the symmetry of the sum in the exponent with respect a negation of the random variable (e.g., σij− → −σij). Here, we take advantage of the symmetry of periodic functions in that an integral over any period is equivalent. L... | https://arxiv.org/abs/2504.00459v1 |
we would like to think about the remainder of the Taylor expansion as some quadratic term and use properties of its Hessian to bound it. In the following lemma, we show that it is indeed bounded by a quadratic. 15 Graphical Models for Multivariate Phase Relationships A P REPRINT Lemma 4.5. Define Hn u, the empirical co... | https://arxiv.org/abs/2504.00459v1 |
ϕ E ϕ⊤≻0. (67) From (67) we note that Eh ϕϕ⊤i is positive definite since E ϕ E ϕ⊤is an outer product and thus positive semi- definite. Now, we are in a position to reason about Hu. LetHfull=Eh ϕϕ⊤i . Since tuis the subset of the entries of ϕthat only pertain to entries that involve the node u,Hu is a principal m... | https://arxiv.org/abs/2504.00459v1 |
data are generated according to the model. Then, investigate examples of the algorithm’s performance in settings with wave-like data. 18 Graphical Models for Multivariate Phase Relationships A P REPRINT 5.1 Model Performance and Computation Speed Figure 2: (a) Model Performance Comparison between the Chow-Liu Algorithm... | https://arxiv.org/abs/2504.00459v1 |
node in the grid is connected to its neighbors above, below, to the left, and to the right of itself. If there are no immediate neighbors in any of the directions, then the edge wraps around to the other side. This implies that each node has degree 4. We set κij= 1, as in the previous experiment, and generate samples v... | https://arxiv.org/abs/2504.00459v1 |
To evaluate the model, we use the classical M-ary hypothesis testing framework, where the goal is to classify M= 16 equally spaced wave propagation directions based on noisy plane wave data. We generate the data by first setting up an 8×8grid of (x, y)locations for measurements. The phase data is defined as: ϕ(x, y, t ... | https://arxiv.org/abs/2504.00459v1 |
1×1unit grid. We let the radial wave have a large semi-major axis and small semi-minor axis compared to the grid size. This is to allow for a diverging wave if the center is on the grid and a unidirectional wave if the center is off-grid. More concretely, we generate the phase data according to ϕ(x, y, t ) =Krq 0.09 [(... | https://arxiv.org/abs/2504.00459v1 |
log-likelihood, and τis the decision threshold. Since the partition function is only a function of the parameters, and not the data, we can rewrite this as 1 nnX k=1 X (i,j)∈Eκ1,ijcos(y(k) j−y(k) i−µ1,ij)−X (i,j)∈Eκ0,ijcos(y(k) j−y(k) i−µ0,ij) ˆH=1 ≷ ˆH=0τ−Z(κ1, µ1) +Z(κ0, µ0) ˆH=1 ≷ ˆH=0α, which is simply a shifte... | https://arxiv.org/abs/2504.00459v1 |
hypothesis testing problem, we demonstrate further the tradeoffs between Chow-Liu and the ISO by noting that the ROC performance of the ISO estimator outclassed that of the Chow-Liu, albeit being slower (as shown in the previous experiment). This may be because using the full wave model allows for more flexibility in t... | https://arxiv.org/abs/2504.00459v1 |
estimation approach include exploring the connection of our ISO problem formulation with distributionally robust estimation, as has been done with covariance estimation and graphical lasso [43, 44, 45]. References [1]J. Polo Jr. and A. Lakhtakia, “Surface electromagnetic waves: A review,” Laser & Photonics Reviews , vo... | https://arxiv.org/abs/2504.00459v1 |
Berestycki and F. Hamel, “Generalized travelling waves for reaction-diusion equations,” Contemp. Math. , vol. 446, 01 2007. [15] S. Kim, A. Sengupta, and B. Arnold, “A multivariate circular distribution with applications to the protein structure prediction problem,” Journal of Multivariate Analysis , vol. 143, 10 2015.... | https://arxiv.org/abs/2504.00459v1 |
Nguyen, G. B. Benigno, J. Ðoàn, J. Miná ˇc, T. J. Sejnowski, and L. E. Muller, “Analytical prediction of specific spatiotemporal patterns in nonlinear oscillator networks with distance-dependent time delays,” Physical Review Research , vol. 5, no. 1, p. 013159, 2023. [32] S. Romano and V . Zagrebnov, “On the XY model a... | https://arxiv.org/abs/2504.00459v1 |
Power comparison of sequential testing by betting procedures. Amaury Durand1,2and Olivier Wintenberger2 1´Electricit´ e de France R&D, Bd Gaspard Monge, 91120 Palaiseau, France. 2Sorbonne Universit´ e, 4 Place Jussieu, 75005 Paris, France. April 2, 2025 Abstract In this paper, we derive power guarantees of some sequent... | https://arxiv.org/abs/2504.00593v1 |
inference (SAVI), which includes tests and confidence sequences, has been rapidly growing in recent years and we refer the reader to [Ramdas et al., 2022a] and [Ramdas and Wang, 2024] for recent surveys. One of the key tools to derive safe anytime valid confi- dence sequences is time-uniform concentration bounds which ... | https://arxiv.org/abs/2504.00593v1 |
of e-power introduced in [Vovk and Wang, 2024]. Intuitively, maximizing the the growth rate should provide optimal power guarantees under the alternative Qand thus it is an appropriate notion of power (see also [Ramdas and Wang, 2024, Section 2.7] for more formal justification). The problem lies in the choice of Qwhich... | https://arxiv.org/abs/2504.00593v1 |
inf {n≥1 : inf k≥nuk≥x}. Proof. The first inequality is a consequence of the Borell-Cantelli theorem and the second comes from the relation E[τα] =P n≥1P(τα> n) =P n≥1P(logWn≤log (1 /α)). We focus on showing that (6) holds under some alternatives in a time-varying setting where the distribution of the observation chang... | https://arxiv.org/abs/2504.00593v1 |
any z >0. Finally, we let ( mn)n≥1and ( vn)n≥1be two nonnegative sequences and consider the alternatives hypotheses H1:ϱ1:=X n≥1P(∥µn∥2< m n)<+∞, (9) and for p∈ {1,···,+∞}, H2,p:ϱ2,p:=X n≥1P ∥µn∥p< m norνn,p> vn <+∞. (10) In the next sections, we define the test supermartingales and provide power guarantees under H1o... | https://arxiv.org/abs/2504.00593v1 |
with the Hoeffding test supermartingale of Section 2.1.1 and define, for all α∈(0,1), the rejection time at level αby τH α:= inf n∈N:WH n≥1/α . We rely on the following non-restrictive assumption on the betting strategy. Assumption 2.2. For any process (Xt)t∈N, the betting strategy (λt)t≥1constructed using (Xt)t∈N sat... | https://arxiv.org/abs/2504.00593v1 |
the Capital test supermartingale of Section 2.1.2. We let (e 1,···,e2d) be such that (e 1,···,ed) is the canonical basis of Rdand e d+i=−eifor all i= 1,···, d. Theorem 2.8. Assume that the regret Rn:= max γ∈ΓlogLC n(γ)−logWC nof the betting strategy (γn)n≥1satisfies ρ:=P n≥1P(Rn> rn)<+∞for some nonnegative sequence (rn... | https://arxiv.org/abs/2504.00593v1 |
moment properties under additionnal variance contraints in the alternative. While Theorems 2.7 and 2.9 are the more restrictive for mn, they have the advantage of providing dimension free bounds and the ability to consider an alternative on the euclidean norm. On the other side, Theorem 2.8 considers an alternative on ... | https://arxiv.org/abs/2504.00593v1 |
If vn=vn−1withv >0, then lim inf n→+∞logWC,EWA n n1−a≥ϵmP-a.s., and E τC,EWA α ≤ O linlog1 ϵm(1−a) +log(d/α) +ϵ2v ϵm1 1−a! . 4. If vn=vlog(n)/nwithv >0, then lim inf n→+∞logWC,EWA n n1−a≥ϵmP-a.s., and E τC,EWA α ≤ O linlog1 +ϵ2v ϵm(1−a) +log(d/α) ϵm1 1−a! . 11 It is remarkable to consider rates n−awith a≥... | https://arxiv.org/abs/2504.00593v1 |
∞i ≤B2. In this case, Corollary 2.13 recovers similar power guarantees as Corol- lary 2.10 and so does Assertion 1 of Corollary 2.12 with an additional O(dlog(d)) dependence on the dimension. Corollary 2.11, on the other hand, only applies when the second moment decreases at least as fast as the mean. It seems that the... | https://arxiv.org/abs/2504.00593v1 |
and lower bounds obtained in Propositions 2.3 to 2.6 remain valid and the following composite counterparts of Theorems 2.7 and 2.8 hold. Theorem 3.2. Assume that the regret Rn:= max λ≥0logLH n(λ)−logWH nof the betting strategy (λn)n≥1satisfies ρ:=X n≥1P(Rn> rn)<+∞, for some nonnegative sequence (rn)n≥1. Let (mn)n≥1. be... | https://arxiv.org/abs/2504.00593v1 |
γn)n≥1, we define the Capital test supermartingales as LC n(γ) =nY t=1(1 +γgt(Xt)) and WC n=nY t=1(1 +γtgt(Xt)), n∈N. (24) The following proposition holds. Proposition 3.6. Relation (24) defines two test supermartingales for H0of(21) if we take Γ = [−1/2,1/2]and for H− 0of(22) if we take Γ = [0 ,1/2]. Proof. The proof ... | https://arxiv.org/abs/2504.00593v1 |
case of bounded functionals, the test supermartingales proposed in their Lemmas 3.1 and 3.2 reduce to the Capital test supermartingale of Section 3.2 up to some rescaling of the functions mandℓ. Transposing their results to the setting of Section 3.2, Theorem 4.2 and Proposition 4.3 of [Casgrain et al., 2024] guarantee... | https://arxiv.org/abs/2504.00593v1 |
0.9 and we have ∥µn∥∞=∥µn∥2=mn:=m nnX t=1t−aand νn,∞≤νn,2≤vn:=1 nnX t=1 m2t−2a+t−2b 25 . 19 In the stationary case where a=b= 0, we have mn=mandvn=v:=m2+ 1/25 and our theoretical bounds give E τH,FTL α ≤ O linlog1 m2 +log(1/α) m2 E τC,EWA α ≤ O linlog1 (ϵm−4ϵ2v)+ +log(d/α) (ϵm−4ϵ2v)+ E τC,ONS α ≤ O lin... | https://arxiv.org/abs/2504.00593v1 |
andm= 0.4,d= 5. 21 Figure 4: Truncated Rejection time for Experiment 2. 5.2 Comparison of binary forecasters In this section, we reproduce the experiment of Section 4.2 in [Henzi and Ziegel, 2021] to compare their testing procedure with ours. We generate Zt=ϵt+θP4 j=1ϵt−jand take Yt=1{Zt>0}. The two forecasters in comp... | https://arxiv.org/abs/2504.00593v1 |
properties of the alter- native, particularly for betting strategies with low regret. Upper bounds on averaged stopping times and extensive numerical experiments do not yield conclusive comparisons between the EWA, ONS, and 2-steps betting strategies. In summary, ONS demonstrates the highest robustness to alternatives ... | https://arxiv.org/abs/2504.00593v1 |
Statistics , pages 1027–1047. [Manole and Ramdas, 2023] Manole, T. and Ramdas, A. (2023). Martingale methods for sequential estimation of convex functionals and divergences. IEEE Transactions on Information Theory , pages 1–1. 24 [Noether, 1955] Noether, G. E. (1955). On a Theorem of Pitman. The Annals of Mathematical ... | https://arxiv.org/abs/2504.00593v1 |
Statistical Hypotheses. The Annals of Mathematical Statistics , 16(2):117 – 186. [Wald and Wolfowitz, 1948] Wald, A. and Wolfowitz, J. (1948). Optimum Character of the Sequential Probability Ratio Test. The Annals of Mathematical Statistics , 19(3):326 – 339. [Wang and Ramdas, 2022] Wang, H. and Ramdas, A. (2022). Cato... | https://arxiv.org/abs/2504.00593v1 |
translated. Lemma A.4. Let(ηn)n≥1be constructed with the online gradient ascent (OGA) algorithm with gra- dient steps1 B2√ t. That is ηt+1= ΠBd 1/2B ηt+Xt B2√ t . Then for all n≥1, with probability at least 1−1/n2, Sn:= sup η∈Bd 1/2BnX t=1Et−1 η⊤Xt −nX t=1Et−1 η⊤ tXt ≤√n(1 + 2p log(n)) 27 Algorithm 1 Online Newto... | https://arxiv.org/abs/2504.00593v1 |
Et−1 γ⊤ kXt −γ⊤ kXt−(γ⊤ kXt)2−Et−1 (γ⊤ kXt)2 ≤1. On the other hand, Applying Lemma A.3 of [Cesa-Bianchi and Lugosi, 2006] with s= 1/2 and X= 4(γ⊤ kXt)2∈[0,1] yields Et−1 exp 2(γ⊤ kXt)2−3Et−1 (γ⊤ kXt)2 ≤1, where we have used that 4( e1/2−1)≤3. Hence, the Cauchy-Schwarz inequality and the inequalities of the ... | https://arxiv.org/abs/2504.00593v1 |
Hence letting En:={logWn≥un}, we have shown that P(Ec n∩Gn∩Bn,k∩Cn∩Dn,k) = 0, for all n≥1 and k∈ {1,···,2d}. Finally, we get X n≥1P(Ec n)≤ρ+ϱ+X n≥1P(Ec n∩Gn∩Cn∩Dn)≤ρ+ϱ+X n≥1P 2d[ k=1Ec n∩Gn∩Cn∩Dn,k! =ρ+ϱ+X n≥1P 2d[ k=1Ec n∩Bc n,k∩Gn∩Cn∩Dn,k! ≤ρ+ϱ+X n≥1P 2d[ k=1Bc n,k! ≤ρ+ϱ+π2 6. 31 This concludes the proof of the first... | https://arxiv.org/abs/2504.00593v1 |
linlogr+ 4 ϵm(1−a) +2 x+ 2 log(2 d) + 4ϵ2v ϵm!1/(1−a) . •Ifvn=vlog(n)/n, then un≥ϵmn1−a− 4ϵ2v+r+ 4 log(n)−2 log(2 d) and Lemma B.3 gives ℵ ≤& 2 linlog(r+ 4 + 4 ϵ2v) ϵm(1−a) +2(x+ 2 log(2 d)) ϵm1/(1−a)' . Proof of Corollary 2.12. We have un≥u(1) n∧u(2) nwith u(1) n=mn1−a 4B−(r+ 4) log( n)−2 log(2 d) andu(2... | https://arxiv.org/abs/2504.00593v1 |
6. This concludes the proof. Proof of Theorem 3.7. Let us denote Zt=gt(Xt). Using the fact that log(1 + x)≥x−x2for any x≥ −1/2, we have, for all n≥1, log(Wn)≥max γ∈Γ nX t=1γZt−nX t=1(γZt)2! − R n. Define for all n≥1 and γ∈Γ, Bn(γ) :=(nX t=1(γZt−(γZt)2)≥nX t=1Et−1[γZt]−4nX t=1Et−1 (γZt)2 −4 log( n)) , Cn:=( sup g∈G1 n... | https://arxiv.org/abs/2504.00593v1 |
arXiv:2504.00919v1 [math.ST] 1 Apr 2025Nonparametric spectral density estimation using interact ive mechanisms under local differential privacy Cristina Butucea1Karolina Klockmann2 2,3Tatyana Krivobokova3 Abstract We address the problem of nonparametric estimation of the sp ectral density for a centered stationary Gauss... | https://arxiv.org/abs/2504.00919v1 |
ensuring that the original data remains private. While individuals maintain control ov er their original data, some level of interaction among the nindividuals may be permitted. LDP offers stronger privacy guarantees than DP at the expense of a lower utility of the data for analysis. In this paper, we focus on local diff... | https://arxiv.org/abs/2504.00919v1 |
is symmetric at zero. Since the absolute summability of the sequence o f covariances ( σj)j∈Zimplies its square summability, the covariance function can be recoveredf rom the spectral density function by the inverse Fourier transform. We observe the sequence ( X1,...,X n) drawn from the time series ( Xt)t∈Z. In this pa... | https://arxiv.org/abs/2504.00919v1 |
of sequen- tial data, see the survey by Zhang et al. (2022a). Privatization of correlated, but non-sequential data, with multiple observations has been extensively studied in nume rous works; see for example (Zhu et al., 2014) and the references therein. In the context of covariance estimation, Amin et al. (2019) studi... | https://arxiv.org/abs/2504.00919v1 |
2s+1,iff∈Ws,∞(L0,L), E|ˆf(ω)−f(ω)|2≤c2(nα2)−2s−1 2s,iff∈Ws,2(L), wherec,c1,c2>0 are constants depending only on M1>0 ands, L, L 0, respectively. It is important to note that these privacy mechanisms depend on th e indexjand, respectively, the frequency ω, such that they cannot be directly generalized for estimating fgl... | https://arxiv.org/abs/2504.00919v1 |
Z1,...,Znare successively obtained by using a conditional distribution Qi(· |Xi,Z1,...,Z i−1), i= 1,...,n. Letα∈(0,∞). We say that a sequence of conditional distributions ( Qi)i=1,...,nprovides α−local differential privacy ( α−LDP) or that Z1,...,Znareα−local differentially private views of X1,...,Xnif sup Z1,...,Zi−1,Xi... | https://arxiv.org/abs/2504.00919v1 |
c3>0depends on s, LandM, as defined in (2). In particular, to estimate σj, only square-summability of the Fourier coefficients, but no furthe r smoothness assumption on fis required. The result for ˆ σNI jholds for j∈ {−(n−1),...,(n−1)} fixed or slowly increasing with nas long as j=O(n1/2).If a different normalization facto... | https://arxiv.org/abs/2504.00919v1 |
˜ τn>0, and we build /tildewideZiby adding a Laplace distributed random variable ˜ξiwith˜ξK+1,...,˜ξni.i.d.∼Lap(4˜τn/α), i.e., /tildewideZi=/tildewideVi+˜ξi,where/tildewideVi= (Vi∧˜τn)∨(−˜τn). (8) 11 The proposed mechanism is sequentially interactive and satisfies the definition of α−local differ- ential privacy as shown ... | https://arxiv.org/abs/2504.00919v1 |
{−B,B}K+1| /an}⌊ra⌋ketle{tz,/tildewideYi/an}⌊ra⌋ketri}ht>0 or/parenleftBig /an}⌊ra⌋ketle{tz,/tildewideYi/an}⌊ra⌋ketri}ht= 0 and ˜ z1=B ˜τn/tildewideYi,0/parenrightBig/bracerightBig ifTi= 1, Uniform/braceleftBig z∈ {−B,B}K+1| /an}⌊ra⌋ketle{tz,/tildewideYi/an}⌊ra⌋ketri}ht<0 or/parenleftBig /an}⌊ra⌋ketle{tz,/tildewideYi/a... | https://arxiv.org/abs/2504.00919v1 |
There fore, we develop a new information-theoretic inequality to hold for arbitra ry multivariate likelihoods of the joint vector X1:n= (X1,...,Xn). The Fisher information of the parameter θcontained in a sample Z1:n= (Z1,...,Zn) also satisfies a chain rule: I(Z1:n) =I(Z1)+I(Z2|Z1)+...I(Zn|Z1:(n−1)), with, for all ifrom... | https://arxiv.org/abs/2504.00919v1 |
to see that E[˜σNI j] =1 nn−j/summationdisplay i=1E[XiXi+j] =1 nn−j/summationdisplay i=1σj=n−j nσj. As 0≤j≤R < nandR=O(n1/2), it follows ( E[˜σNI j]−σj)2=j2 n2σ2 j=O(n−1). The hidden constant does not depend on jsinceσ2 j≤M1by assumption. Let ¯ σj=1 n/summationtextn−j i=1XiXi+jbe the biased estimator built from the ori... | https://arxiv.org/abs/2504.00919v1 |
>0 depends on s, L, iff∈Ws,2(L), and additionally on L0, iff∈ Ws,∞(L0,L). To continue with the variance, define the estimator without trunca tion and without added Laplace noise, i.e., ¯fm(ω) =1 2πnm/summationdisplay j=−mn−|j|/summationdisplay i=1XiXi+|j|exp(−ijω). Then, ˜fNI m(ω) =1 2πnm/summationdisplay j=−mn−|j|/summ... | https://arxiv.org/abs/2504.00919v1 |
> j)}, Ac=n/uniondisplay i=1{/tildewideXi/ne}ationslash=Xior (/tildewiderWi,j/ne}ationslash=Wi,j)I(i > j)}. 24 L2risk on the event A If the event Aoccurs, then ˆ σjcoincides with the untruncated estimator ˜ σj, i.e., ˜σj=1 n−jn/summationdisplay i=j+1Xi·(Xi−j+ξi−j)+˜ξi. In particular, E[(ˆσj−σj)2] =E[IA(ˆσj−σj)2]+E[IAc(... | https://arxiv.org/abs/2504.00919v1 |
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