text string | source string |
|---|---|
−Eν[g]≥ −ϵ.Then for all x∈Rd, we have: logρ(x) ν(x)=−g(x)−logEν[e−g]≤0 +ϵ=ϵ. Therefore, R∞(ρ∥ν)≤ϵ, as claimed. Next, since g(x)≥0, we have Eν[e−g]≤1. Since |g′(x)|= 1/ηalmost everywhere for |x| ≤a (except for xwhich are integer multiples of η) and g′(x) = 0 for|x|> a, and since g(x)≤1, we have Eν[(g′)2e−g] =1 η2Za −ae−... | https://arxiv.org/abs/2502.05623v1 |
(33) Using relation (32a) and integration by parts, we can write the relative Fisher information as: FI(ρt∥νt) =Eρt ∥∇ft− ∇gt∥2 =Eρt ∥∇ft∥2+∥∇gt∥2−2⟨∇ft,∇gt⟩ =Eρt ∥∇ft∥2 +Eρt ∥∇gt∥2 −2Eρt[∆gt]. (34) We compute the time derivative of each term above, spelling out the computations below explicitly, and color codi... | https://arxiv.org/abs/2502.05623v1 |
the third term in (34), using the Fokker- Planck equation (29a) and the formula (31), and integration by parts, to get: d dt(−2Eρt[∆gt]) =−2Z Rd(∂tρt) ∆gtdx−2Z Rdρt∆(∂tgt)dx = 2Z Rd∇ ·(ρtbt) (∆gt)dx−cZ Rd(∆ρt) (∆gt)dx (40a) + 2Z Rdρt∆(⟨∇gt, bt⟩ − ∇ · bt)dx (40b) −cZ Rdρt∆∆gtdx+cZ Rdρt∆∥∇gt∥2dx. (40c) We calculate the t... | https://arxiv.org/abs/2502.05623v1 |
formula from Lemma 2 simplifies to: d dtFI(ρt∥νt) =−Eρt" ∇2logρt νt 2 HS# −2Eρt" ∇logρt νt 2 (−∇2logνt)# . (43) The first term in the right-hand side above is the second-order Fisher information, which is non- negative: Eρt ∇2logρt νt 2 HS ≥0. In parts (i), (ii), and (iv) below we drop this first term, and only consi... | https://arxiv.org/abs/2502.05623v1 |
=FI(ρ0∥ν0) (1 +αt)2exp2tL2 αt+ 1+8L√ t√αt+ 1 . 32 MIXING TIME OF THE PROXIMAL SAMPLER IN RELATIVE FISHER INFORMATION VIA SDPI F.3. Detail for Counterexample of DPI along the Gaussian Channel The following is a detailed computation for Example 3. See Figure 1 for an illustration. Proposition 10 Letρ0=N(0,1)onR. LetM≥2... | https://arxiv.org/abs/2502.05623v1 |
yields the desired bound: FI(ρt∥νt)≤exp −2Zt 0λsds FI(ρ0∥ν0) =γ2e−2γt (α+e−2γt(γ−α))2FI(ρ0∥ν0). 34 MIXING TIME OF THE PROXIMAL SAMPLER IN RELATIVE FISHER INFORMATION VIA SDPI Part (ii): Since ρ0andν0are symmetric, along the OU channel (9) to N(0, γ−1I), the solutions ρtandνtare also symmetric, as can be seen from the... | https://arxiv.org/abs/2502.05623v1 |
Theorem 6 (Convergence of Proximal Sampler in Relative Fisher Information) Proof [Proof of Theorem 6] As we reviewed in Section 4.2, we use stochastic process interpretations of the forward and backward steps of the Proximal Sampler. (1) Forward step: The first step of the Proximal sampling algorithm is a convolution w... | https://arxiv.org/abs/2502.05623v1 |
(1 +αη)2k≤dL (1 +α dL)2k≤dLexp −αk dL where in the last inequality we use the bound 1 +c≥ec/2which holds for 0≤c=α dL≤1. Then to reach FI(ρX k∥ν)≤ε, it suffices to set dLexp −αk dL ≤ε, or equivalently, we run the Proximal Sampler for k≥dL αlogdL εiterations, as claimed. 37 WIBISONO Acknowledgments. The author thank... | https://arxiv.org/abs/2502.05623v1 |
distributions in one dimension. In Conference on Learning Theory , pages 2041–2059. PMLR, 2022. Sinho Chewi, Patrik Gerber, Holden Lee, and Chen Lu. Fisher information lower bounds for sam- pling. In International Conference on Algorithmic Learning Theory , pages 375–410. PMLR, 2023. Sinho Chewi, Murat A Erdogdu, Mufan... | https://arxiv.org/abs/2502.05623v1 |
Mitra, and Andre Wibisono. Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Φ-Mutual Information. arXiv preprint arXiv:2402.17067v2 , 2024. Yi-An Ma, Niladri S Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L Bartlett, and Michael I Jordan. Is there an analog of Nest... | https://arxiv.org/abs/2502.05623v1 |
Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation Spencer Compton and Gregory Valiant Stanford University {comptons, valiant}@stanford.edu Abstract LeCam’stwo-pointtestingmethodyieldsperhapsthesimplestlowerboundforestimatingthe mean of a distribution: roughly, if it is impossible to well-di... | https://arxiv.org/abs/2502.05730v2 |
have mostly so far been treated on a more case-by-case basis. Examples. Let us showcase some instances that illustrate interesting behaviors for adaptive location estimation. •Fornsamples from a Gaussian N(µ, σ2), the sample mean/median both incur optimal error of|µ−ˆµ|= Θ(σ√n). •For the uniform distribution Unif( µ−1µ... | https://arxiv.org/abs/2502.05730v2 |
not leveraged by the other estimators, and for Neven larger than our simulation then we expect sample median/mean to improve beyond sample midrange and close the gap with our estimator (recall our earlier discussion of Fig. 1e). Finally, in Fig. 2f, we observe a sharp improvement in performance when Nis large enough th... | https://arxiv.org/abs/2502.05730v2 |
combinations of translations of symmetric distributions need not be symmetric. We will study the shape-constraints on Dunder which it is possible to attain error |µ−ˆµ| ≤ polylog( n)·ωD(polylog( n) n)(our formal statement of results will add dependence on a failure prob- ability δ). This roughly corresponds to error th... | https://arxiv.org/abs/2502.05730v2 |
of p(x)/q(x)≥τ for a well-chosen threshold parameter τ≥0. Roughly, if PandQare easy to distinguish from n samples, then T(P)andT(Q)are easy to distinguish from ˜O(n)samples. From their results, it becomes clear that our key question is essentially resolved if the appropriate likelihood threshold channel can be simulate... | https://arxiv.org/abs/2502.05730v2 |
unequal to 1in regions that are spaced apart. We carefully study a family of step distributions with different step widths, and show this mixture family is indistinguishable from a triangle distribution (which has a worse two-point testing rate). Attainability for location estimation. On the other hand, we show the two... | https://arxiv.org/abs/2502.05730v2 |
From the details of our construction, these modified distributions should not actually be much easier to estimate than by using the sample midrange for error Θ(1 n), but the two-point testing lower bound will deceptively look much more favorable. 1.3 Related Work Asymptotic setting. Location estimation and adaptive loc... | https://arxiv.org/abs/2502.05730v2 |
µ−1, µ+ 1)and ˜O(n−2/3)for the semicircle distribution. Their results also extend to the regression setting. In their discussion, they remark how this approach is unable to leverage discontinuities in the interior of the support, such as in Fig. 1d, which our results will encompass. Additional related work. For example... | https://arxiv.org/abs/2502.05730v2 |
all values of ˆµ, a, b, but we will delay this concern. Notably, it is not clear how good of an estimate ˆµmust be if it passes these interval tests. For arbitrary symmetric distributions, a ˆµpassing interval tests can indeed be a poor estimate. Surprisingly, we will show that for mixtures of log-concave distributions... | https://arxiv.org/abs/2502.05730v2 |
[PJL23]; preservation of Hellinger distance) .For any p, q∈∆k, there exists a T*∈ Tthreshsuch that the following holds: 1≤d2 h(p, q) d2 h(T*p,T*q)≤1800 min {k, k′}, (2) where k′= log(4 /d2 h(p, q)). We remark on some properties of this result. Note that properties 2-4 simultaneously hold for p, qor after exchanging p, ... | https://arxiv.org/abs/2502.05730v2 |
compute: q Prx∼p[T’(x) = 1] −q Prx∼q[T’(x) = 1]2 ≥ q Prx∼p[T’(x) = 1] −r 1 1 +α·δPrx∼p[T’(x) = 1]!2 Recall for this case, δ= 1: = 1−r 1 1 +α!2 ·Prx∼p[T’(x) = 1] ≥ 1−r 1 1 +α!2 ·β·Prx∼p[T*(x) = 1] Observe that Prx∼p[T*(x) = 1] = p′and use p′≥d2 h(p, q)/4: ≥ 1−r 1 1 +α!2 ·β·d2 h(p, q) 4 ≥ 1−r 1−α 22 ·β·d2 h(p, q) 4 ≥... | https://arxiv.org/abs/2502.05730v2 |
(Piecewise-constant decomposition of log-concave densities; implicit in Lemma 27 of [CDSS14]) .Letqbe a log-concave distribution over R. For any 0< δ≤1 2, there exists a function ˜qwhich is a piecewise-constant function over Rconsisting of O(log(1 δ))pieces. The function ˜q approximates qin the sense that ˜q(x)≤q(x)for... | https://arxiv.org/abs/2502.05730v2 |
2 ≤p(x) 2+k·κ·˜p(x) 2 Using κ≤1 k: ≤p(x) 2+˜p(x) 2 =⇒˜p(x)≥p(x) 2·(1 +1 2)≥p(x) 4 16 We will show that ˜p(x)is valid for most of the mass of p, and that these valid regions correspond to a small number of disjoint intervals: Claim 2.9. Ifκ≤1 k, then PrX∼p[˜p(x)is invalid ]≤O(δ κ) Proof.LetS⊂Rbe the values of xwhere ˜p(... | https://arxiv.org/abs/2502.05730v2 |
domain increases by at most 1. In total, the region where ˜pis valid is the union of ≤ |D ˜p|+k≤O(klog(1 δ))disjoint intervals. Our last component will introduce our approximation for t, defined with respect to an approxi- mation of each ti: Lemma 2.11 (˜tidecomposition) .For any log-concave distribution q, there exist... | https://arxiv.org/abs/2502.05730v2 |
sufficiently well-approximates t. Case 1: p(x+ ∆)≥1 16p(x). We will drop from the summation t(x)the indices corresponding to unsupported components of the mixture, and components for which tiis small; we claim that this does not affect the value of t(x)significantly: Remark 2.15.P i∈Kunsupp (x)wipi(x)−wipi(x+∆) p(x+∆)≤... | https://arxiv.org/abs/2502.05730v2 |
of intervals. Finally, our proof of Lemma 2.5 concludes by considering all intervals satisfying the conditions of Claim 2.14: x≥0, all˜pi(x)are constant, ˜p(x) 21 isκ-valid, and all ˜ti(x)are constant. Recall that we seek to find a collection of rdisjoint intervals I=I1∪···∪ Irwhere: (i) r=O(klog(n)), (ii) PrX∼p[x∈I]≥Ω... | https://arxiv.org/abs/2502.05730v2 |
≥β·Prx∼p[T*(x) = 1]for0< β≤1. If we invoke Lemma 2.5 with τmin≤τ∗and use τ=τ∗, then all intervals will satisfy t(x)≥ Ω(1)·τ∗. Recall by Remark 2.3 (3) that τ∗≥r d2 h(pµ,pµ−∆) 104 log(4 /d2 h(pµ,pµ−∆)). So, we may set τmin= minr d2 h(pµ,pµ−∆) 104 log(4 /d2 h(pµ,pµ−∆)),1 k , and thus we approximate with α= Ω(1). Also, ... | https://arxiv.org/abs/2502.05730v2 |
is to show is that if we choose γcorrectly, then with high probability, µwill pass all tests with the empirical samples, and all bad ˆµwill fail some test with the empirical samples: Theorem 2.20. Suppose pis a distribution that is a centered/symmetric mixture of klog-concave distributions. There exists some universal ... | https://arxiv.org/abs/2502.05730v2 |
at µ, our claim follows by: p ρ(µ−b, µ−a)−p ρ(µ+a, µ+b) ≤ p ρ(µ−b, µ−a)−p E[ρ(µ−b, µ−a)] + p ρ(µ+a, µ+b)−p E[ρ(µ+a, µ+b)] +|p E[ρ(µ−b, µ−a)]−p E[ρ(µ+a, µ+b)]| = p ρ(µ−b, µ−a)−p E[ρ(µ−b, µ−a)] + p ρ(µ+a, µ+b)−p E[ρ(µ+a, µ+b)] Using Claim 2.22: ≤O(1)·p log(2n/δ) 26 Let us set γ=Cγ·p log(2n/δ)for the value of Cγyielded by... | https://arxiv.org/abs/2502.05730v2 |
by Theorem 2.20 if γ=Cγlog(2/δ)then at least one ˆµwill pass all tests, and all ˆµ that pass the test satisfy the desired condition on |µ−ˆµ|. Since at least one ˆµwill pass all tests, then the modified algorithm will choose a value of γwhere γ≤Cγlog(2/δ). Moreover, the set of ˆµ that pass the tests with this γwill be ... | https://arxiv.org/abs/2502.05730v2 |
expec- tations: p E[ρ(ˆµ−b,ˆµ−a)]−p E[ρ(ˆµ+a,ˆµ+b)] −p ρ(ˆµ−b,ˆµ−a)−p ρ(ˆµ+a,ˆµ+b) = p E[ρ(ˆµ−b,ˆµ−a)]−p ρ(ˆµ−b,ˆµ−a) +p ρ(ˆµ+a,ˆµ+b)−p E[ρ(ˆµ+a,ˆµ+b)] ≤ p E[ρ(ˆµ−b,ˆµ−a)]−p ρ(ˆµ−b,ˆµ−a) + p ρ(ˆµ+a,ˆµ+b)−p E[ρ(ˆµ+a,ˆµ+b)] By Claim 2.22: ≤O(1)·p log(2n/δ) Hence, for sufficiently large Cγ, ifp ρ(ˆµ−b,ˆµ−a)−p ρ(ˆ... | https://arxiv.org/abs/2502.05730v2 |
and ending at r. 30 We use the decomposition of Claim 2.30 to consider two tests,7[a, m 1]and[m0, b], where both contain 2isamples and we are hoping one test will nearly be a good 2i-heavy test. Moving forward, we will show that the decomposition does yield a good test: Claim 2.31. Consider a subset Sof the domain, and... | https://arxiv.org/abs/2502.05730v2 |
left interval has much more samples. Without loss of generality, we focus on the former: Lemma 2.32. BiggestLowerBound([ X1, . . . , X n], γ, ℓ)computes the rightmost ˆµwhere an ℓ-heavy test (with the heavier side being on the right) centered at ˆµfails with parameter γ. Proof.Recall that such an ℓ-heavy test will have... | https://arxiv.org/abs/2502.05730v2 |
to be matched with any remaining non-dominated right intervals, so we may remove it from LeftStack . Finally, in Line 26, we consider matching the Xi-right interval with the top left interval in LeftStack (if there is one). This left interval from the top of the stack is long enough to match with the Xi-right interval,... | https://arxiv.org/abs/2502.05730v2 |
ℓsamples we have yet seen. 8:fori∈ {n−ℓ+ 1, . . . , 1}do 9: RightLength[ i]←Xi+ℓ−1−Xi 10: ifRightLength[ i]<ShortestConsidered then 11: ShortestConsidered ←RightLength[ i] 12: NonDominatedRightOption[ i]←True 13: LeftLength ←[] ▷LeftLength[ i]will be the length of the longest interval ending at Xi (non-inclusive) that ... | https://arxiv.org/abs/2502.05730v2 |
account for non-deterministic estimators. Proof.Let us define some relevant distributions in terms of a sample size n≥1, and parameter 0< ε < 1where1 2εis an integer. Definition 3.1 (Triangle Distribution) . Tri(x) =( 1− |x| |x| ≤1 0 |x|>1 Before we define step distributions, let us define a helper function sw(x)which ... | https://arxiv.org/abs/2502.05730v2 |
= x |x|<1or|x| ≥3 2 x−1 1≤x <3 2 x+ 1−3 2< x≤ −1 Claim 3.8. dTV(Tri⊗n,Rand-Step⊗n)≤dTV(Mod-Tri⊗n,Rand-Mod-Step⊗n) Proof.Note how Tri = h(Mod-Tri) andStepv=h(Mod-Stepv). Thus, dTV(Tri⊗n,Rand-Step⊗n) = dTV(h(Mod-Tri)⊗n, h(Rand-Mod-Step)⊗n)≤dTV(Mod-Tri⊗n,Rand-Mod-Step⊗n)bydata-processing inequality. Now,webound dTV(... | https://arxiv.org/abs/2502.05730v2 |
3 andln(1−x)≤ −x, then this is valid for 0< ε≤1 2: ≤ −ε−ε2+ε 2·2bi−ε 1 +ε−2bi+ε 2+ε2 2 ·2ai−ε 1 +ε−2ai+ε 2+ε2 2 ·−2ai−ε 1 +ε+ 2ai +ε 2+ε2 ·−ε−2bi 1 +ε+ 2bi+ 21 2+ε2 · 2ε−2ε2+8ε3 3 Note that all terms other than the last are non-positive, as 0< a i, bi≤ε 2. Now, we use1 1+z= 1−z 1+z≥(1−z)forz≥0. ≤ −ε−ε2+ε 2·(2... | https://arxiv.org/abs/2502.05730v2 |
2·s 1− 1−2·Z1 2·∆tri(√x−p x−∆tri)2dx+Z2∆tri 0xdxn ≤√ 2·s 1− 1−2·Z1 2·∆tri(∆tri/p x−∆tri)2dx+Z2∆tri 0xdxn ≤√ 2·q 1− 1−2· ∆2 tri·ln(1/∆tri) + 2·∆2 trin We will choose a sufficiently small Cwhere ln(1/∆tri)≥1, as it enforced by n≥2andC≤√ 2 e: ≤√ 2·s 1− 1−6·C2 log(n)·n·ln(p log(n)·n/C)n 41 Using log(n)≤n: ≤√ ... | https://arxiv.org/abs/2502.05730v2 |
the constraint that1 2εis an integer. For sufficiently large n, there will be a satisfying value of εwhere1 ε∈[ n C2/5,2· n C2/5]. Given Corollary 3.11, then it is sufficient to show: Cp nlog(n)≥ν·ωStepv1 n0 It is our goal to see how large1 n0can be while satisfying this inequality. If we later set parameters such ... | https://arxiv.org/abs/2502.05730v2 |
that we choose to use a different subroutine for pairwise comparisons. The pairwise comparison of Birgé [B+13] (see Theorem 32.8 and Remark 32.2 in [PW25] for discussion) would suffice for our theorem statement, although our different pairwise test enables a running time of ˜O(n3/2)instead of ˜O(n2)(see remarks at end ... | https://arxiv.org/abs/2502.05730v2 |
over ntest samples, and then show how this implies likelihood tests with PXi∗will perform well. Lemma 4.3. There exists a constant 0< C test<1such that if ntest≜⌊Ctest·n log(n/δ)⌋then dTV(P⊗ntestµ , P⊗ntest Xi∗)≤0.01. Proof. dTV(P⊗ntestµ , P⊗ntest Xi∗) ≤√ 2·q d2 h(P⊗ntestµ , P⊗ntest Xi∗) =√ 2·q 1−(1−d2 h(Pµ, PXi∗))ntes... | https://arxiv.org/abs/2502.05730v2 |
follows. Otherwise, if no sample is undefeated, let a sample’s “radius” be the distance from its farthest loss. By Claim 4.6, Xi∗will have radius at most ∆1+ 2∆∗≤3∆∗. For sake of contradiction, suppose Xj′/∈[µ−4∆∗, µ+ 4∆∗]. Then, Xi∗must beat it, yet their distance is >3∆∗, so this is impossible. Thus, our algorithm in... | https://arxiv.org/abs/2502.05730v2 |
any number of samples n, there is a symmetric distribution where any estimator ˆθ(X)will incur error arbitrarily larger than the two-point testing rate (even incurring error worse than ωD(C)for a constant C > 0, which is a much weaker goal than the typical ωD(˜Θ(1 n))): Theorem 1.7. For any positive integer nand positi... | https://arxiv.org/abs/2502.05730v2 |
each batch contains constant probability mass, and that the distribution is symmetric. We formally define the distribution: Definition 5.1 (Modified symmetric uniform distribution) .Letvbe a vector in {0,1}T, then Dv is the distribution: Dv(x) = vi |x| ∈[i·1 T, i·1 T+1 2T)fori∈ {0, . . . , T −1} 1−vi|x| ∈[i·1 T+1... | https://arxiv.org/abs/2502.05730v2 |
probability. We may now conveniently bound the probability: Eq.(17)≤ 2·T 5+ 1 · max |θ|∈{1 T,2 T,...,⌈T 5⌉·1 T+1 T}·PrDv∼FD ⌈T/2⌉X k=1Bern1 2 ·1 2T ≤1 10 ≤2T 5+ 4 ·2·exp −2·(1/80)2 ⌈T 2⌉(1/2T)2! ≤2T 5+ 4 ·exp−T 800 For sufficiently large T, this quantity is upper bounded by1 4(or any chosen consta... | https://arxiv.org/abs/2502.05730v2 |
Dv,1, . . . , D v,m, where Dv,i(x)≜Dv(x−θi). We prove the crucial property required to use the probabilistic method to invoke Lemma 5.4: Lemma 5.5. PrDv∼FDh miniPrX∼D⊗n v,ih (max j̸=iD⊗n v,j(x))≥D⊗n v,i(x)i ≥1 2i ≥9 10 54 Proof.Note that the constant1 2in the lemma statement could be an arbitrary constant in (0,1). Let... | https://arxiv.org/abs/2502.05730v2 |
v,i(x)|E −1 Observe how the event Ewas not actually affected by the realization of Dv, it was only affected by which segments of length1 Thad samples realized within them, and these have the same joint probabilities for all Dv∈ F D. Moreover, by definition of E, each sample is within the potential support of each Dv,j... | https://arxiv.org/abs/2502.05730v2 |
of continuity. We would like to thank Tselil Schramm for helpful technical discussions and feedback. This work was supported by the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program, Tselil Schramm’s NSF CAREER Grant no. 2143246, and Gregory Valiant’s Simons Foundation Investigator Award and NS... | https://arxiv.org/abs/2502.05730v2 |
Luc Devroye, Silvio Lattanzi, Gábor Lugosi, and Nikita Zhivotovskiy. On mean estima- tion for heteroscedastic random variables. In Annales del’Institut HenriPoincare (B) Probabilites etstatistiques, volume 59, pages 1–20. Institut Henri Poincaré, 2023. [DR24] John C Duchi and Feng Ruan. The right complexity measure in ... | https://arxiv.org/abs/2502.05730v2 |
location parameter. The annalsofStatistics, pages 267–284, 1975. [VC15] Vladimir N Vapnik and A Ya Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. In Measures ofcomplexity: festschrift for alexeychervonenkis, pages 11–30. Springer, 2015. [VdV00] Aad W Van der Vaart. As... | https://arxiv.org/abs/2502.05730v2 |
arXiv:2502.05969v1 [stat.ML] 9 Feb 2025Asymptotic FDR Control with Model-X Knockoffs: Is Moments Matching Sufficient?∗ Yingying Fan1Lan Gao2Jinchi Lv1Xiaocong Xu1 University of Southern California1and University of Tennessee2 February 9, 2025 Abstract We propose a unified theoretical framework for studying the r obustness ... | https://arxiv.org/abs/2502.05969v1 |
variables in implementing the model-X knockoffs. For differen tiation, we name the resulting model-X knockoff variables /hatwideXas theapproximate knockoff variables since they correspond to the working distribution /hatwideFrather than the true distribution F, and the cor- 2 responding knockoffs inference method as the app... | https://arxiv.org/abs/2502.05969v1 |
(2006) andBarber and Cand` es (2015). Here,Xis categorical data, which is ev- idently non-Gaussian. Despite the distributional misspecification, t he results indicate that, in almost all cases, the FDR is controlled near the prespecified level q= 0.2, as summarized in Table 1. Additional details are provided in Section 4... | https://arxiv.org/abs/2502.05969v1 |
generator. We delve into specific constructions of knoc koff statistics based on the first two moments matching and verify their asymptotic FDR contro l property in Section 3. Section4presents several simulation and real data examples. We discuss so me implications and extensions of our work in Section 5. All technical pr... | https://arxiv.org/abs/2502.05969v1 |
FDR control is given by Tq≡min/braceleftigg t∈/braceleftbig |/hatwiderW1|,···,|/hatwiderWp|/bracerightbig :#/braceleftbig j:/hatwiderWj≤ −t/bracerightbig #/braceleftbig j:/hatwiderWj≥t/bracerightbig ∨1≤q/bracerightigg . It is seen that the only distinction between the approximate knocko ffs procedure and the model-X k... | https://arxiv.org/abs/2502.05969v1 |
may differ, these conditions share essen tial conceptual similarities with the earlier work. In the subsequent sec tion, we will demonstrate how these conditions can be verified for specific examples of /hatwiderWj’s. Condition (1), on the other hand, is novel. It requires that the kno ckoff statistics /hatwiderWj exhibit ... | https://arxiv.org/abs/2502.05969v1 |
12 We are now ready to discuss the major distinctions of our study with the work by Fan et al. (2025). Our Theorem 1provides a unified framework for verifying the robust- ness of the knockoffs procedure. At a high level, Fan et al. (2025) proved the robustness of the knockoffs procedure through verifying conditions (1)–(3... | https://arxiv.org/abs/2502.05969v1 |
knockoffs generation process. 3 Two-moment-based knockoff statistics We consider three concrete constructions of the knockoff statis tics in this section and verify that they can satisfy conditions (1)–(3) in Theorem 1and thus achieves the asymptotic FDR control. It is important to note that in all examples in this sectio... | https://arxiv.org/abs/2502.05969v1 |
Clearly, /hatwidewj= 0 for each j∈ H0. For knockoff statistics, one expects large and positive signals for 16 j∈ H1, cf.Barber and Cand` es (2015);Cand` es et al. (2018). Specifically, we impose the following conditions on the class of approximate knockoff statistics: (C3) For a large enough constant C >0,an≡#{j∈ H1:/hatw... | https://arxiv.org/abs/2502.05969v1 |
small enough constant c>0. 2.(C1’)and(C3’)hold for some q≥3,(C2)and(C4)hold, andplogn=o(√n). Then for any q∈(0,1), conditions (1)–(3) in Theorem 1are satisfied for αn=P−1/parenleftbigqan 2p/parenrightbig . Consequently, we have limsup n→∞FDR≤q. Remark 1. In the heavy-tailed setting, additional requirements are i mposed ... | https://arxiv.org/abs/2502.05969v1 |
that ξ= (ξ1,···,ξn)⊤∈Rndenotes the noise vector. Given the original data matrixXand the knockoffs data matrix /hatwiderXin (2), let/hatwideZ= [X,/hatwiderX]∈Rn×2p. The knockoff statistic based on the OLS estimator is defined as /hatwiderWj≡ |/hatwideβLS j|−|/hatwideβLS j+p|, j∈[p] with/hatwideβLS≡1√n/parenleftbigg/hatwide... | https://arxiv.org/abs/2502.05969v1 |
statistics. Proposition 2. Assume that conditions (O1)–(O3)are satisfied. Then there exists some constantC=C(M,τ,cL)>0such that for any j∈[p]andp≥C,P(|/hatwiderWj−√n|β∗,j|| ≥ C√logp)≤Cp−10. The signal strength |β∗,j|significantly influences the quality of the knockoff statistics. To ensure robust performance, we impose the... | https://arxiv.org/abs/2502.05969v1 |
sub-Gaussian. Th e rows of/hatwideQ are i.i.d., with mean-zero, unit-variance, and uniformly sub-Gaussian entries. More- over, the restricted eigenvalue property holds for /hatwideZ/hatwideZ⊤/nwith high probability: there exist some large enough constants C1,C2,C3>0 such that P/parenleftbigg min /bardblδ/bardbl0≤C1sδ⊤/... | https://arxiv.org/abs/2502.05969v1 |
and to focus on the main contributions of this paper, we omit furthe r technical details. From a broader perspective, the above procedure indicates that establishing asymptotic FDR control does notnecessarily require coupling the approximate knockoffs matrix with the ideal knockoffs matrix; in some cases, coupling the ap... | https://arxiv.org/abs/2502.05969v1 |
0.180 (0.289) 0.197 (0.721) DL0.153 (0.338) 0.194 (0.998) 0.186 (0.929) 0.185 (1) 0.7MC 0.143 (0.068) 0.184 (0.264) 0.164 (0.155) 0.200 (0.542) DL0.156 (0.326) 0.194 (0.996) 0.192 (0.906) 0.186 (1) 4.2 Real data application In this section, we provide details of the real data results present ed in Table 1in the In- tro... | https://arxiv.org/abs/2502.05969v1 |
effective FDR control in practice . Finally, we emphasize that the relatively low power is also because of the small subsampling s ize of 200. 5 Discussions We have developed a unified theoretical framework for studying th e asymptotic FDR con- trol of the approximate knockoffs procedure that generates kn ockoff variables ... | https://arxiv.org/abs/2502.05969v1 |
with knockoffs inference. J. Amer. Statist. Assoc. , 115(532):1822–1834. Fang, X. and Koike, Y. (2023). From p-Wasserstein bounds to mod erate deviations. Elec- tronic Journal of Probability , 28:1–52. 33 Fukumizu, K.,Gretton, A., Sun, X., andSch¨ olkopf, B.(2007). Ker nelmeasuresofconditional dependence. Advances in Ne... | https://arxiv.org/abs/2502.05969v1 |
For vectors x= (xi) andy= (yi)∈Rn, let/a\}bracketle{tx,y/a\}bracketri}ht ≡x⊤y=/summationtext i∈[n]xiyi. We use{ek}to denote the canonical basis of the Euclidean space, whose dimens ionality should be self-clear from the context. For a random variable X, we use PX,EX(respec- tively,PX,EX) to indicate that the probabilit... | https://arxiv.org/abs/2502.05969v1 |
this range. As will be clear later, under additional conditions ( C3) and (C4), it can be shown that Tqis localized within this domain for some αn≍an/pwith asymptotic probability 1. Consequently, this range is sufficient for achieving the goal of asymptotic FDR control. Approximation for indicator functions As outlined i... | https://arxiv.org/abs/2502.05969v1 |
localization ofTq. To maintain this localization, we need the signal strength to increas e as the tails of the entries in the knockoffs data matrix become heavier, as formaliz ed in condition ( C3’). Lemma 3. Assume that conditions (C1’)and(C3’)hold for some q≥3, and conditions (C2)and(C4)hold. Then for any q∈(0,1), ifp... | https://arxiv.org/abs/2502.05969v1 |
7. Assume that conditions (O1)–(O3)hold. For any j∈ H0, it holds that for W∈ {/hatwiderW,−/hatwiderW}, there exists some constants C=C(M,τ,cL)>0such that for n≥Cand t∈(0,n1/7/C), /vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbig Wj≥t/parenrightbig Pj(t)−1/vextendsingle/vextendsingle/vextendsingle/v... | https://arxiv.org/abs/2502.05969v1 |
the proof of Lemma 8estab- lishes the following localization of Tq. Lemma 11. Assume that conditions (O3)–(O5)and(L1)hold. Then for any q∈(0,1), if s/radicalbig bnlogp/n=o(1), we have P−1/parenleftbigqan 2p/parenrightbig =O(log1/2p). Furthermore, under the additional assumption that bnslog7(np)/√n=o(1), we have P/paren... | https://arxiv.org/abs/2502.05969v1 |
(D1)and(D2)hold. Then if logp=o(√n), there exists some constant C=C(M)>0such that for any j∈[p], P/parenleftig/vextendsingle/vextendsingle/hatwiderWj−/hatwidewj/vextendsingle/vextendsingle≥Cδn/parenrightig ≤p−10+exp(−nE2Y2/C), where/hatwidewj≡√n(|EXjY|−|E/hatwideXjY|)/E1/2Y2andδn≡logp E1/2Y2+log2p√ nEY2. With slight ... | https://arxiv.org/abs/2502.05969v1 |
−x⊤Λx 2/parenrightbigg dx2dx1 /integraltext∞ 0/integraltext∞ x1+texp/parenleftbigg −x⊤Λx 2/parenrightbigg dx2dx1. Obviously, f(0) = 0. Taking the derivative yields that f′(ε) =/integraltext∞ 0exp/parenleftig −/parenleftbig (x1+t−ε)2Λ11+2(x1+t−ε)x1Λ12+x2 1Λ22/parenrightbig /2/parenrightig dx1 /integraltext∞ 0/integral... | https://arxiv.org/abs/2502.05969v1 |
Var/parenleftig/summationdisplay j∈H01{Wj≥t}/parenrightig ≤C/braceleftbigg(1+t3)log3(np)√n/parenleftbigg/summationdisplay j∈H0P/parenleftig Wj≥t/parenrightig/parenrightbigg2 +p−10/bracerightbigg . Proof of Lemma 12. We show only the bound for W=/hatwiderW, while the other cases can be dealt with similarly. Observe ... | https://arxiv.org/abs/2502.05969v1 |
H0/summationdisplay j∈H0Pj(2Cδn) =1+o(1) |H0|/summationdisplay j∈H0P/parenleftbig/hatwiderWj≤ −2Cδn/parenrightbig ≤qan 2p, which entails that P−1/parenleftbigqan 2p/parenrightbig ≤2Cδn=O(logp). Thus, on event E1, it holds that /summationdisplay j∈[p]1/braceleftbigg /hatwiderWj>P−1/parenleftbiggqan 2p/parenrightbigg/bra... | https://arxiv.org/abs/2502.05969v1 |
The conclusion follows from an application of the triangle inequality, wh ich completes the proof of Proposition 4. E.2.3 Proof of Lemma 3 The proof follows similar arguments as those for Lemmas 1and2, so we only sketch the differences here. We will first prove the following variance bound: fo rW∈ {/hatwiderW,−/hatwiderW... | https://arxiv.org/abs/2502.05969v1 |
/vextendsingle/vextendsingleE/a\}bracketle{tu,/parenleftbig Z⊤Z+ηI2p/parenrightbig−1v/a\}bracketri}ht−E/a\}bracketle{tu,/parenleftbig/hatwideZ⊤/hatwideZ+ηI2p/parenrightbig−1v/a\}bracketri}ht/vextendsingle/vextendsingle≤C/parenleftbigglog3/2n√n+η−1n−10/parenrightbigg . Here,Z≡(Z1,·,···,Zn,·)⊤∈Rn×2pwith{√nZi,·}i∈[n]i.i.d... | https://arxiv.org/abs/2502.05969v1 |
of Theorem 7 We need the following aprior estimates to prove Theorem 7. Recall that for j∈[2p], /hatwideβLS j=√nβau ∗,j+e⊤ j(/hatwideZ⊤/hatwideZ)−1/hatwideZ⊤ξ. 38 Lemma 16. Assume that conditions (O1)–(O3)hold. Then there exists some constant C=C(M,τ,cL)>0such that for n≥C, P/parenleftig max i∈[n],j∈[2p]|e⊤ j(/hatwide... | https://arxiv.org/abs/2502.05969v1 |
=/parenleftigg 1 2π/radicalig det(/hatwideΣ(j) n)−1 2π/radicalig det(Σ(j) n)/parenrightigg ·/integraldisplay∞ −∞/integraldisplay|y|+t −(|y|+t)ρ((x,y)⊤;Σ(j);−1 n)dxdy ≤Clog2n√n·H(Σ(j) n,Σ(j) n). Combining the estimates for |H(/hatwideΣ(j) n,/hatwideΣ(j) n)−H(/hatwideΣ(j) n,Σ(j) n)|and|H(/hatwideΣ(j) n,Σ(j) n)−H(Σ(j)... | https://arxiv.org/abs/2502.05969v1 |
first rewrite that /vextendsingle/vextendsingle/vextendsingle/vextendsingle√n/hatwidezj,i /a\}bracketle{t/hatwidezj,/hatwideZ·,j/a\}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|/hatwidezj,i| /bardbl/hatwidezj/bardbl·√n/bardbl/hatwidezj/bardbl |/a\}bracketle{t/hatwidezj,/hatwideZ·,j/a\}bracketri}... | https://arxiv.org/abs/2502.05969v1 |
a care- ful bookkeeping of errors arising from the distributional approxim ation and the indicator function approximation. For brevity, the details are omitted. E.5 Proofs for Section D E.5.1 Proof of Proposition 5 The proof follows the same structure as that of Proposition 1, with minor modifications; we omit repetitiv... | https://arxiv.org/abs/2502.05969v1 |
(∪j∈[n]Bj)c, and/tildewidef(X)≡ E(f(X)|1{X∈Bj}) ifX∈Bj. Then we can deduce that E/tildewidef(X) =E/tildewidef(X)1/braceleftig X∈/parenleftig/uniondisplay j∈[n]Bj/parenrightigc/bracerightig +E/tildewidef(X)1/braceleftig X∈/uniondisplay j∈[n]Bj/bracerightig 52 =Ef(X)1/braceleftig X∈/parenleftig/uniondisplay j∈[n]... | https://arxiv.org/abs/2502.05969v1 |
arXiv:2502.06002v1 [math.ST] 9 Feb 2025Fixed-strength spherical designs Travis Dillon Abstract Aspherical t-design is a finite subset Xof the unit sphere such that every polynomial of degree at mos t thas the same average over Xas it does over the entire sphere. Determining the minimum po ssible size of spherical design... | https://arxiv.org/abs/2502.06002v1 |
designs. The earliest bounds on the sizes of spher ical designs were established by Delsarte, Goethals, and Seidel [ 17], by extending Delsarte’s linear programming method for as sociation schemes and coding theory [ 16]. Theorem 1.3 (Delsarte, Goethals, Seidel [ 17]).The number of points in a spherical t-design in Rdi... | https://arxiv.org/abs/2502.06002v1 |
well, then (X,w)is called a signed design. 2 Fixed-strength spherical designs Travis Dillon While unweighted designs are delicate, weighted designs ar e easier to construct and are often amenable to linear-algebraic techniques. In fact, it takes no more than a page and a half to prove a version of Theorem 1.4 for weigh... | https://arxiv.org/abs/2502.06002v1 |
functions on Sd−1given by polynomial formulas. (Since the poly- nomialsx2 1+···+x2 dand1have the same values on Sd−1, they represent the same element of Pt, not two separate elements.) The beginning of Section A.2 outlines a proof that dim(Pt) =/parenleftbigd+t−1 d−1/parenrightbig +/parenleftbigd+t−2 d−1/parenrightbig ... | https://arxiv.org/abs/2502.06002v1 |
there is a finite set Yof real numbers so that the first tmoments of ψare equal to the first tmoments of the uniform distribution on Y. 2.3. optimal constructions for small t Proposition 2.7.The vertices of a cross-polytope form a spherical 2-design in Rdwith2dpoints. Proof . We first evaluate the spherical integrals of al... | https://arxiv.org/abs/2502.06002v1 |
of Yand♦dto get the correct averages. We claim that the multiset X:=Y∪4·♦d is a spherical 4-design with |X|= 4d(d+2). To show this, we just check the averages of monomials over th e vectors inX. First, 1 |X|/summationdisplay x∈Xx4 i=1 4d(d+2)/parenleftbigg/summationdisplay x∈Yx4 i+4·/summationdisplay x∈♦dx4 i/parenrigh... | https://arxiv.org/abs/2502.06002v1 |
set Xso that the polynomials |x|2javerage correctly. To do this, we find real numbers r1,...,r t+1such that ˆX=/uniontextt+1 i=1riXis a Gaussian design. (Here, riis a scaling factor, so riX={rix:x∈X}.) This approach has the advantage that/summationtext x∈ˆXw(x)f(x) = 0 for any functionfsuch that/summationtext x∈Xw(x)f(x... | https://arxiv.org/abs/2502.06002v1 |
t-design in Rk. Corollary 3.6.If there is a weighted spherical t-design in RdwithNpoints, then there is a weighted sphericalt-design in Rk, for eachk≤d, with at most 2(t+1)Npoints. Proof . Use Proposition 3.2 to convert to a Gaussian design with (t+1)Npoints; project the design to Rk using Lemma 3.5; then convert back ... | https://arxiv.org/abs/2502.06002v1 |
this lower bound. To improve the upper bound, we will instead construct small G aussian designs directly using k-wise independent sets, then transfer them to spherical designs u sing Proposition 3.4. (Here and later in the paper, we use {{·}}to denote multisets.) Our first goal is to construct a small no nbinaryt-wise i... | https://arxiv.org/abs/2502.06002v1 |
Dillon 5. optimal signed designs The starting point for this section is in attempting to overc ome the failed opening strategy of the previous section, where we claimed that transforming a partial desig nXby judicious coordinate negations cannot produce a smaller design. That’s true if we try to use coordin ate negatio... | https://arxiv.org/abs/2502.06002v1 |
0 for any weight function. To finish the proof, we therefore need to show that m∈span/parenleftbig b(a) :a∈Rt/parenrightbig . Then, by choosing a set A⊂Rtof at most |P|points such that m∈span/parenleftbig b(a) :a∈A/parenrightbig , we obtain a 2t-design with at most pt2tdt points, where ptis the number of partitions of i... | https://arxiv.org/abs/2502.06002v1 |
spherical and Gaussian Po 2t-designs in RdwithO(dt)points. This is a strong statement, since Po 2tis an infinite-dimensional vector space, and an indication t hat the monomials with all even degrees are the driving force behind the lower bound of Ωt(dt)for the size of a spherical 2t-design. In the proof of Theorem 5.1, ... | https://arxiv.org/abs/2502.06002v1 |
polynomials is that they are positive definite kernels : For any finite point set X⊂Sd−1andk≥0, we have /summationdisplay x,y∈XQd k/parenleftbig /an}⌊ra⌋ke⊔le{⊔x,y/an}⌊ra⌋ke⊔ri}h⊔/parenrightbig ≥0. Lemma 6.3.A setX⊆Sd−1is anε-approximate spherical t-design if and only if 1 |X|2/summationdisplay x,y∈XQd ≤t/parenleftbig /a... | https://arxiv.org/abs/2502.06002v1 |
perspective will provide a different definition of approximate designs. Given a vector x∈Rd, the entries ofx⊗tcorrespond to evaluations of monomials: (x⊗t)i1,i2,...,it=/producttextt i=1xαi. A weighted set (X,w)is therefore a t-design if and only if Ex∼w[x⊗k] =Ev∼µ[v⊗k], whereµis the uniform distribution on the sphere, fo... | https://arxiv.org/abs/2502.06002v1 |
a polynomial R=/summationtext y∈Yw(y)evy∈conv(Y)with/⌊ard⌊lR−Q/⌊ard⌊l2≤ε, and /vextendsingle/vextendsingle/vextendsingle/summationdisplay y∈Yw(y)f(x)−/integraldisplay Sd−1f(x)dµ/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/an}⌊ra⌋ke⊔le{⊔R,f/an}⌊ra⌋ke⊔ri}h⊔−/an}⌊ra⌋ke⊔le{⊔Q,f/an}⌊ra⌋ke⊔ri}h⊔/ve... | https://arxiv.org/abs/2502.06002v1 |
Fixed-strength spherical designs Travis Dillon The upper bounds in Section 6 produce weighted approximate d esigns, so the question of optimal un- weighted approximate designs also remains open. Question 4.What is the size of the smallest unweighted L2- or tensor-approximate t-designs in Rd? In another direction, all t... | https://arxiv.org/abs/2502.06002v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.