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´λhpγqTrpAn,Mpωqq n`λλ1λn λ1`λnďnlogp5{p4γ2qq `t M. Then, using the note notation cλ“1{p1´hpγqq, we have xAn,Mpωqv,vy ě1 1´hpγq„ ´hpγqTrpAn,Mpωqq n`λ1λn λ1`λn´nlogp5{p4γ2qq `t λM . 30 Whenvruns overSn´1, the left-hand side becomes λminpAn,Mpωqq “infvPΘxAn,Mpωqv,vy, while the right-hand side is independent of v.Therefo... | https://arxiv.org/abs/2502.20368v1 |
bound the total variation distance, we resort to Pinsker’ s inequality (see, e.g., [ Tsy08 , Lemma 2.5]). It applies to probabilities on general measura ble spaces, including finite and infinite-dimensional spaces. Lemma C.2 (Pinsker’s inequality) LetP0andP1be two probability measures defined on the same measurable space ... | https://arxiv.org/abs/2502.20368v1 |
explicit description ofPφis as follows. 34 • When Yis finite-dimensional (i.e., Y“span ty1,...,yNu),Pφis a measure on FN: PφpANq:“PpAφ Nq,@ANPFN. • When Yis infinite-dimensional, Pφis a measure on F8, determined by PφpANq:“PpAφ Nq,@ANPFN,@Ně1. In particular, we define the restricted measures of PφonFNas Pφ,N:“Pφˇˇ FN,i.e.... | https://arxiv.org/abs/2502.20368v1 |
section shows that the logistic distribution εonRN, which is non-Gaussian, satisfies As- sumption 2.2. Example D.1 Letεbe anRN-valued random variable with i.i.d. logistic-distributed entries, each has a probability density function ppxq “e´x p1`e´xq2. Then,εsatisfies Assumption 2.2. Proof of Example D.1.We start with the... | https://arxiv.org/abs/2502.20368v1 |
biography of Billingsley by Steve Kop pes. [BM18] Gilles Blanchard and Nicole Mucke. Optimal rates for re gularization of statistical inverse learning problems. Foundations of Computational Mathematics , 18(4):971–1013, 2018. [BMS22] Krishnakumar Balasubramanian, Hans-Georg Müller, a nd Bharath K Sriperumbudur. Uni- fie... | https://arxiv.org/abs/2502.20368v1 |
Statistics . Dover books on intermediate and advanced mathematics. Peter Smith, 1978. [LAY23] Fei Lu, Qingci An, and Yue Yu. Nonparametric learnin g of kernels in nonlocal operators. Journal of Peridynamics and Nonlocal Modeling , pages 1–24, 2023. [LC73] Lucien Le Cam. Convergence of estimates under dimens ionality re... | https://arxiv.org/abs/2502.20368v1 |
pages 3394–3402. PMLR, 2021. [Tsy08] Alexandre B. Tsybakov. Introduction to Nonparametric Estimation . Springer New York, NY, 1st edition, 2008. [VdV00] Aad W Van der Vaart. Asymptotic statistics , volume 3. Cambridge university press, 2000. [Ver18] Roman Vershynin. High-dimensional probability , volume 47 of Cambridge... | https://arxiv.org/abs/2502.20368v1 |
Robust statistical inference for accelerated life-tests with one-shot devices under log-logistic distributions Mar´ ıa Gonz´ alez1, Mar´ ıa Jaenada1and Leandro Pardo1 1Department of Statistics and O.R., Complutense University of Madrid, Madrid, Spain. Abstract A one-shot device is a unit that operates only once, after ... | https://arxiv.org/abs/2502.20467v1 |
ALTs shortens the life of products by increasing the levels of certain stressors influencing the degradation of the product. Some example of stress factors are air pressure, temperature, humidity and voltage, that can artificially controlled in an laboratory. However, the main purpose of reliability analysis is underst... | https://arxiv.org/abs/2502.20467v1 |
Section 6 provides a numerical analysis, including Monte Carlo simulations and estimations based on a real dataset. 2 Model description and maximum likelihood estimator Let us assume that the one-shot device data from a reliability testing experiment are stratified into Itesting conditions with Kidevices placed under t... | https://arxiv.org/abs/2502.20467v1 |
for the hazard and survival functions, making it more straightforward to use in practical applications. Because the primary goal of ALT is to extrapolate results under normal operating conditions, for ALT tests it is necessary to relate the lifetime distribution of the devices the stress levels at which units 3 Figure ... | https://arxiv.org/abs/2502.20467v1 |
observed in the previous section between the MLE and the Kullback-Leibler diver- gence suggests defining a class of estimators using a divergence measure distinct from the Kullback- Leibler divergence, and ideally, these estimators would address the lack of robustness associated with the MLE. Given the good performance... | https://arxiv.org/abs/2502.20467v1 |
where Mi= β2 ixixiT−logτi αiβ2 ixixiT −logτi αiβ2 ixixiT logτi αiβi2 xixiT! . Proof. (See Appendix: Section A.3) Corollary 4 The asymptotic distribution of the MLE of θ,ˆθγ=0, is given by: √ K(ˆθγ=0−θ0)L→ K→∞N 02(J+1),IF(θ0)−1 , where IF(θ) =IX i=1KiMi(Rαi,βi(τi,xi)Fαi,βi(τi,xi))2 Fαi,βi(τi,xi)−1+Rαi,βi(τi,xi)−1 ... | https://arxiv.org/abs/2502.20467v1 |
. . , G i0−1⊗1T Ki0−1, Gi0,ϵ⊗1T Ki0, Gi0+1⊗1T Ki0+1, . . . , G I⊗1T KI), where Gi0,ϵis the distribution function associated with the probability mass function pi0,ϵ,k(y) = (1 −ϵ)πi0(y,θ0) +ϵδti0,k(y), 8 andδti0,k(y) =yδ(1) ti0,k+ (1−y)δ(2) ti0,k,with δ(1) ti0,kbeing the degenerate function at point ( i0, k),δ(2) ti0,k=... | https://arxiv.org/abs/2502.20467v1 |
The main advantage of these robust versions is their resilience to outliers, ensuring that decisions are not strongly influenced by atypical observations. 5.1 Wald-Type Test Statistics Definition 8 The Wald-type test statistics for testing (16) based on the WMDPDE is defined as WK(ˆθγ) =Km(ˆθγ)T M(ˆθγ)TΣ(ˆθγ)M(ˆθγ)−1... | https://arxiv.org/abs/2502.20467v1 |
γ(θ0)−Qγ(θ0)M(θ0)TJ−1 γ(θ0) and Qγ(θ0) =J−1 γ(θ0)M(θ0) M(θ0)TJ−1 γ(θ0)M(θ0)−1 . Corollary 12 Forγ= 0, we have the MLE, and in this case, the asymptotic distribution of ˜θγ=0is obtained considering the results of Corollary 4. It follows that √ K(˜θγ=0−θ0)L→ K→∞N 02(J+1),S(θ0) , where S(θ0) =IF(θ0)−1 I−M(θ0) M(θ0)T... | https://arxiv.org/abs/2502.20467v1 |
b0, b1) = (1 ,−0.5,0.8,0.4), meaning that failures in each testing condition are generated 13 Table 3: Values of αiandβifor each Testing Condition. Condition ( i) αi βi 1 2.718282 2.225541 2 2.117000 2.718282 3 1.648721 3.320117 4 2.718282 2.225541 5 2.117000 2.718282 6 1.648721 3.320117 7 2.718282 2.225541 8 2.117000 ... | https://arxiv.org/abs/2502.20467v1 |
However, beyond 20% contamination, the MLE (γ= 0) begins to diverge to higher levels. In contrast, the level remains more stable for higher values ofγ. This reveals that WMDPDEs with higher γvalues are less sensitive to sample contamination, 15 Figure 3: RMSE (bottom left panel), Empirical Level (top left) and Empirica... | https://arxiv.org/abs/2502.20467v1 |
18.3781 0.1 -11.0716 4421.303 5.0544 -1440.1036 68.5913 32.8659 17.9939 0.2 -11.1037 4430.991 5.0508 -1439.4333 68.7738 32.8853 17.9770 0.3 -10.4249 4213.497 3.7787 -1026.4958 57.6527 31.4461 18.5268 0.4 -10.0718 4101.951 3.5032 -935.2010 54.9180 31.0031 18.7098 0.5 -11.0531 4415.347 5.0611 -1439.0802 66.9270 32.3657 1... | https://arxiv.org/abs/2502.20467v1 |
Engineering Research Council of Canada (of the first author) through an Individual Discovery Grant (No. 20013416) and a Departmental Collaboration Scholarship of M. Gonz´ alez. . M. Jaenada and L. Pardo are members of the Interdisciplinary Mathematics Institute (IMI). References [1] Baghel, S. and Mondal, S. (2023) Rob... | https://arxiv.org/abs/2502.20467v1 |
. [17] Balakrishnan, N., & Ling, M. H. (2023). Accelerated life testing data analyses for one-shot devices. In H. Pham (Ed.), Springer handbook of engineering statistics (2nd ed., pp. 1039-1057). London: Springer. [18] Balakrishnan, N., Castilla, E., Jaenada, M., and Pardo, L. (2023a). Robust inference for nonde- struc... | https://arxiv.org/abs/2502.20467v1 |
we arrive at IX i=1Ki Fθ(τi,xi)γ−1 Fθ(τi,xi)−ni Ki + (1−Fθ(τi,xi))γ−1 −1 +Fθ(τi,xi) +Ki−ni Ki∂Fθ(τi,xi) ∂θ =02(J+1) or what is equivalent, IX i=1Ki Fθ(τi,xi)−ni Ki Fθ(τi,xi)γ−1+ (1−Fθ(τi,xi))γ−1∂Fθ(τi,xi) ∂θ =02(J+1). Based on this result, and by replacing∂Fαi,βi(τi,xi) ∂ajfrom equation (29), we obtain:... | https://arxiv.org/abs/2502.20467v1 |
αi)2 (τβi i+αβi i)4·β2 ix2 ij =α2βi i (τβi i+αβi i)2·τ2βi i (τβi i+αβi i)2·β2 ix2 ij −logτi αi·β2 ix2 ij −logτi αi·β2 ix2 ij(logτi αi·βi)2x2 ij =Fαi,βi(τi,xi)2·Rαi,βi(τi,xi)2·β2 ix2 ij −logτi αi·β2 ix2 ij −logτi αi·β2 ix2 ij(logτi αi·βi)2x2 ij =Fαi,βi(τi,xi)2·Rαi,βi(τi,xi)2·Mi where Mi= β2 ixixiT−logτi αiβ2 i... | https://arxiv.org/abs/2502.20467v1 |
Characterizing the Training-Conditional Coverage of Full Conformal Inference in High Dimensions Isaac Gibbs∗†Emmanuel J. Cand` es‡ Abstract We study the coverage properties of full conformal regression in the proportional asymptotic regime where the ratio of the dimension and the sample size converges to a constant. In... | https://arxiv.org/abs/2502.20579v1 |
for even moderately complex models it is well-known that overfitting yields incomparable training and test errors. ∗Code for reproducing the experiments in this article is available at https://github.com/isgibbs/high-dim-fc . †Department of Statistics, University of California, Berkeley. Email: igibbs@berkeley.edu ‡Dep... | https://arxiv.org/abs/2502.20579v1 |
For example, a diagnostic model fit on historical patient records is typically deployed to predict the outcomes of many new individuals. Similarly, an insurance company may set the rates of many customers based on a single model of claim rates and risk factors built on outcomes from previous years. In these scenarios, ... | https://arxiv.org/abs/2502.20579v1 |
and empirical results, we demonstrate that our results can be extended to other popular methods, such as the full conformal LASSO and full conformal quantile regression, that do not directly meet our assumptions. Interestingly, we find that while the full conformal correction is critical in fitting the regression, it i... | https://arxiv.org/abs/2502.20579v1 |
achieves exact training-conditional coverage. Moreover, by leveraging additional information about the model fitting procedure, we will derive a much more precise characterization of the behaviour of the full conformal residuals, estimated quantile function, and coverage properties of alternative methods. 3 Setting and... | https://arxiv.org/abs/2502.20579v1 |
First, note that to determine if Yn+1∈ˆCfullwe just need to consider the case y=Yn+1. Thus, to ease notation we will let ˆβ:=ˆβYn+1denote the regression coefficients obtained when the model is fit using the full dataset, {(Xi, Yi)}n+1 i=1. We let ˆβ(j)denote the same quantity when the j-th sample is left out of the fit... | https://arxiv.org/abs/2502.20579v1 |
much more than stability. As a starting point, let us consider the behaviour of the ( n+1)-st residual, Yn+1−X⊤ n+1ˆβ. Ignoring asymptotically negligible terms, the calculations above give us the representation, Yn+1−X⊤ n+1ˆβ≈ 1−1 n+ 1X⊤ n+11 n+ 1X⊤X+τId−1 Xn+1! (Yn+1−X⊤ n+1ˆβ(n+1)). The first term in this product is... | https://arxiv.org/abs/2502.20579v1 |
1 n+ 1n+1X i=1ψ(Yi−X⊤ iˆβ)P→E ψ1 1 + 2 λ2c∞(ϵ+λN∞Z) . Moreover, Quantile 1−α,1 n+ 1n+1X i=1δ|Yi−X⊤ iˆβ|! P→Quantile 1−α, 1 1 + 2 λ2c∞(ϵ+λN∞Z) . With these preliminary results in hand, we are now ready to state the main result of this section. In particular, we show that the training-conditional coverage of full ... | https://arxiv.org/abs/2502.20579v1 |
More precisely, we will study a class of convex losses with three bounded derivatives. Interest in these functions comes from their application in high-dimensional robust regression and this class includes, for example, smooth approximations of the Huber function (see Section 2.3.5 of El Karoui [2018] for details). Our... | https://arxiv.org/abs/2502.20579v1 |
the function prox( f) :R→R defined by prox( f)(x) := argminvf(v) +1 2(x−v)2. Then, leveraging the results of El Karoui [2013], we obtain the following asymptotic characterization of the residuals. Lemma 6. Under Assumptions 1-3 above, there exists a constant c∞>0such that (Yn+1−X⊤ n+1ˆβ)−prox( λ2 n+1c∞ℓ) Yn+1−X⊤ n+1ˆβ... | https://arxiv.org/abs/2502.20579v1 |
property that there exists τ0>0such that P(τrand.≥ τ0)→1. Then, P Yn+1∈ˆCfull| {(Xi, Yi)}n i=1P→1−α. The proof of Theorem 4 is conceptually quite similar to that of Theorems 2 and 3 above. In particular, we show that the asymptotic approximations of the residuals and quantile function that we obtained in the previous... | https://arxiv.org/abs/2502.20579v1 |
accurate at even moderate sample sizes and dimensions. Overall, these results demonstrate the generality of our approach and its potential applicability to a variety of other regression methods. 5.1 The full conformal LASSO We begin by characterizing the behaviour of the full conformal LASSO. Our results will rely heav... | https://arxiv.org/abs/2502.20579v1 |
+1 n+ 1ℓα(y−β0−X⊤ n+1β), (5.2) †A detailed description of this system can be found in Section 3.2.3 of Bean et al. [2012]. 12 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 d/n0.120.140.160.180.200.220.240.26Miscoverage τ 0.5 1.0 2.0 100 200 300 400 500 600 700 800 Number of Training Points0.0500.0750.1000.1250.1500.1750... | https://arxiv.org/abs/2502.20579v1 |
1≤i≤nandvn+1=y−β0−X⊤ n+1β, the quantile regression (5.2) can be rewritten as minimize (β0,β)∈Rd+1,v∈Rn+11 n+ 1n+1X i=1ℓα(vi) subject to vi=Yi−β0−X⊤ iβ,∀1≤i≤n, vn+1=y−β0−X⊤ n+1β. Then, letting η∈Rn+1denote the dual variables for these equality constraints, standard results in convex ‡Note that since ˆCQRis one-sided we ... | https://arxiv.org/abs/2502.20579v1 |
justifying a system of three equations for the values of ( c∞, N∞, β0,∞). Most critically, these equations imply that Pϵ−β0,∞+λN∞Z−prox( λ2 n+1c∞ℓα)(ϵ−β0,∞+λN∞Z) λ2 n+1c∞< U = 1−α, and thus informally suggest that P(Yn+1∈ˆCQR dual| {(Xi, Yi)}n i=1)P→1−α. Finally, combining these equations with the previous approximat... | https://arxiv.org/abs/2502.20579v1 |
Yu Bai, Song Mei, Huan Wang, and Caiming Xiong. Understanding the under-coverage bias in uncertainty estimation. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems , 2021. URL https://openreview.net/forum?id=te8iyHjbPQd . Derek Bean, Peter Bickel,... | https://arxiv.org/abs/2502.20579v1 |
, 110(36):14557–14562, 2013. doi: 10.1073/pnas.1307842110. URL https://www.pnas.org/doi/abs/10.1073/pnas.1307842110 . M. Ledoux. The Concentration of Measure Phenomenon . Mathematical surveys and monographs. American Mathematical Society, 2001. ISBN 9780821837924. URL https://books.google.com/books?id=mCX_ cWL6rqwC . J... | https://arxiv.org/abs/2502.20579v1 |
URL https://eprints.soton.ac.uk/258960/ . Vladimir Vovk, Alex Gammerman, and Glenn Shafer. Algorithmic Learning in a Random World . Springer- Verlag, Berlin, Heidelberg, 2005. ISBN 0387001522. Martin J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint . Cambridge Series in Statistical and Probabilist... | https://arxiv.org/abs/2502.20579v1 |
. Combining these results proves claim 1. To prove claim 2 note that, by Lemma 8 E" τ n+ 1X⊤ 11 n+ 1X⊤X+τId−1 ˆβ(n+1) # ≤E1 n+ 1∥X1∥2∥ˆβ(n+1)∥2 ≤1 n+ 1E[∥X1∥2 2]1/2E" 1 τnnX i=1Y2 i#1/2 =O1√n . B.2 Proof of Lemma 3 Proof of Lemma 3. LetY1:n= (Y1, . . . , Y n)Tandϵ1:n= (ϵ1, . . . , ϵ n)T. By standard formula for r... | https://arxiv.org/abs/2502.20579v1 |
case ( I+A)−1=I−(I+A)−1A), tr 1 nX⊤ 1:nX1:n1 nX⊤ 1:nX1:n+τId−21 nX⊤ 1:nX1:n! = tr Id−τ1 nX⊤ 1:nX1:n+τId−1!2 = tr ( Id)−2τtr 1 nX⊤ 1:nX1:n+τId−1! +τ2tr 1 nX⊤ 1:nX1:n+τId−2! , while by similar calculations, tr X1:n1 nX⊤ 1:nX1:n+τId−21 nX⊤ 1:n! = tr 1 nX⊤ 1:nX1:n+τId−21 nX⊤ 1:nX1:n! = tr 1 nX⊤ 1:nX1:n+τ... | https://arxiv.org/abs/2502.20579v1 |
show that f∞has a unique root on {a+bi∈C:a >0, b < 0}. To prove that f∞has a unique root, we repeat the arguments of Rubio and Mestre [2011] for demon- strating the uniqueness of the root of fn. Namely, suppose that ˜ e1,˜e2∈ {a+bi∈C:a >0, b < 0}are both roots of f∞. For ease of notation, let x∞(˜ei) := 1 /˜ei−z=Eh λ2 ... | https://arxiv.org/abs/2502.20579v1 |
for e′ ∞(z), one simply notes that Eλ2e∞(z) 1 +γλ2e∞(z) −1 +ze∞(z) = 0 = ⇒Eλ2 (1 +γλ2e∞(z))2 e′ ∞(z) +e∞(z) +ze′ ∞(z) = 0 , where here we have used the fact thatλ2 (1+γλ2e∞(z))2is bounded (recall that Re( e∞(z))>0) to swap the derivative in zwith the expectation. With the previous two lemmas in hand, we are now rea... | https://arxiv.org/abs/2502.20579v1 |
direct computation shows that E 1 n+ 1W⊤ n+11 n+ 1X⊤ 1:nX1:n+τId−1 Wn+1−1 n+ 1tr 1 n+ 1X⊤ 1:nX1:n+τId−1!!2 =E[(W2 1−1)2] (n+ 1)2dX i=1E 1 n+ 1X⊤ 1:nX1:n+τId−1 ii!2 +E[W2 1]2 (n+ 1)2X i̸=jE 1 n+ 1X⊤ 1:nX1:n+τId−1 ij!2 ≤E[W4 11] (n+ 1)2dX i=1dX j=1E 1 n+ 1X⊤ 1:nX1:n+τId−1 ij!2 =E[W4 11] ... | https://arxiv.org/abs/2502.20579v1 |
denominator and letting X·d:= (X1d, . . . , X (n+1)d) be the dthcolumn vector of the covariates, we have that 1 n+ 1n+1X i=1X2 id−1 n+ 1n+1X i=1XidX⊤ i,−d1 n+ 1X⊤ (d)X(d)+τId−1−11 n+ 1n+1X i=1XidXi,−d =1 n+ 1X⊤ ·d In−1 n+ 1X(d)1 n+ 1X⊤ (d)X(d)+τId−1−1 X⊤ (d)! X·d ≥0, where here we have used the fact that Id−1 n+1X(... | https://arxiv.org/abs/2502.20579v1 |
in hand we are ready to prove Lemma 12 Proof of Lemma 12. We will first show that the second part of the lemma follows from the first. Let ˆβ(i,j) denote the fitted coefficients when both the data points iandjare removed from the dataset. Then, by Lemma 1, E" ψ1 1 + 2 λ2 ic∞ Yi−X⊤ iˆβ(i) ψ 1 1 + 2 λ2 jc∞ Yj−X⊤ jˆβ... | https://arxiv.org/abs/2502.20579v1 |
denote a sample from N(0,1)⊗Pλ⊗Pϵ. C.2 Proof of Theorem 3 The proof of Theorem 3 will make extensive use of the following results of El Karoui [2018]. Theorem 6. Under Assumptions 1-3 from Section 4.2, we have that for any fixed k∈Nand1≤i≤n+ 1, 1. sup 1≤i≤n+1sup j̸=i|R(i) j−Rj| ≤OLkpolylog( n) n1/2 . 2. sup 1≤i≤n+1 R... | https://arxiv.org/abs/2502.20579v1 |
theorem we have that E" 1 n+ 1n+1X i=1ψ(Ri)# =E[ψ(Ri)] =E[ψ(prox( λ2 iciℓ)(R(i) i))] + o(1) =E[ψ(prox( λ2 ic∞ℓ))(R(i) i)] +o(1) =E[ψ(prox( λ2 ic∞ℓ)(ϵ+λiN∞Z))] + o(1), 34 as desired. To evaluate the variance note that by the exchangeability of the residuals Var 1 n+ 1n+1X i=1ψ(Ri)! =E[ψ(Ri)]2 n+ 1+(n+ 1)n (n+ 1)2(E[ψ(R1... | https://arxiv.org/abs/2502.20579v1 |
increasing inverse function, prox( cℓ)−1:R→R. Using this fact, we compute that P(|prox( λ2˜cℓ)(ϵ+λ˜NZ)| ≤x+δ) =P(−x−δ≤prox( λ2˜cℓ)(ϵ+λ˜NZ)≤x+δ) =E[P(−x−δ≤prox( λ2˜cℓ)(ϵ+λ˜NZ)≤x+δ|λ, ϵ)] =E Pprox( λ2˜c)−1(−x−δ)−ϵ ˜N≤λZ≤prox( λ2˜c)−1(x+δ)−ϵ ˜N|λ, ϵ >E Pprox( λ2˜c)−1(−x)−ϵ ˜N≤λZ≤prox( λ2˜c)−1(x)−ϵ N∞|λ, ϵ =P(|prox... | https://arxiv.org/abs/2502.20579v1 |
by Lemma 31 we have that for any fixed c≥0, d dxprox( cℓ)(x) =1 1 +cℓ′′(prox( cℓ)(x)). SinceP(λ >0) = 1, P(ϵ+λN∞Z= 0) = 0, and prox( cℓ)(0) = 0, this immediately implies that P(|prox ( λ2c∞ℓ)(ϵ+λN∞Z)|<|ϵ+λN∞Z)|) = 1 . Moreover, since the cumulative distribution functions of |ϵ+λN∞Z|and|prox( λ2c∞ℓ)(ϵ+λN∞Z)|are strictly... | https://arxiv.org/abs/2502.20579v1 |
used the fact that ˆβ(τ1) minimizes β7→1 n+1Pn+1 i=1ℓ(Yi− X⊤ iβ) +τ1∥β∥2. Our next lemma shows that the regression coefficients decrease in norm as the penalty, τ, increases. Lemma 21. The function τ7→ ∥ˆβ(τ)∥2is non-increasing. Proof. For ease of notation let L(β) :=1 n+ 1n+1X i=1ℓ(Yi−X⊤ iβ). Fix any τ2, τ1≥0. By defi... | https://arxiv.org/abs/2502.20579v1 |
Lemmas 3.5 and 3.6 of El Karoui [2018] we have the deterministic bound, Rn+1(τ)−prox( cn+1(τ)ℓ)(R(n+1) n+1(τ)) ≤∥Xn+1∥2 2 τ2(n+ 1) 1 nnX i=1XiX⊤ i 2 opsup i<n+1|X⊤ i(Sn+1(τ) + 2τId)−1Xn+1| n+ 1. Now, by well-known results on the operator norm of the empirical covariance, ∥1 n+1Pn+1 i=1XiX⊤ i∥op≤ ∥Pλ∥2 ∞∥1 n+1Pn+1 i=1Wi... | https://arxiv.org/abs/2502.20579v1 |
the pointwise convergence of cn+1(τ) toλ2 n+1c∞(τ) also guarantees uniform convergence. Unfortunately, cn+1(τ) is somewhat delicate and thus it will be useful to go through an intermediate quantity that is easier to control. In particular, we will proceed in two steps, first bounding sup τ≥τ0 cn+1(τ)−λ2 n+1 n+ 1tr 1... | https://arxiv.org/abs/2502.20579v1 |
≤|τ1−τ2| 2τ2 0 ∥ℓ′′′∥∞ ∥Pλ∥2 ∞ n+ 1nX i=1WiW⊤ i op4(∥ˆβ(τ0)−β∗∥2+∥β∗∥2 2) τ0vuut1 n+ 1nX i=1∥Xi∥4 2 (n+ 1)2+ 1 , where the second last inequality above uses Lemmas 20 and 21 to control ∥ˆβ(τ1)−ˆβ(τ2)∥2. Now, all the stochastic elements above converge in probability to constants. Thus, we find that there exists a ... | https://arxiv.org/abs/2502.20579v1 |
i=1XiX⊤ i op≤ ∥Pλ∥2 ∞ 1 n+1Pn+1 i=1WiW⊤ i opP→ ∥Pλ∥2 ∞(1 +√γ)2(see Theorem 3.1 of Yin et al. [1988]), while part 4 of Theorem 6 and our assumptions on β∗imply that ∥ˆβ(τ0)∥2 2≤ 2(∥ˆβ(τ0)−β∗∥2 2+∥β∗∥2 2)P→2N∞(τ0)2+ 2E[(√ dβ∗ i)2]). So, in particular, we find that there exists L > 0 such that with probability converging ... | https://arxiv.org/abs/2502.20579v1 |
2 or Theorem 3 hold. Then, for all τ0>0 there exists a constant C >0such that for all τ∈[τ0,∞]anda < b , P(prox( λ2c∞(τ)ℓ)(ϵ+λN∞(τ)Z)∈[a, b])≤C|a−b|. Proof. Recall that by assumption ϵhas a bounded density, fϵ. Using standard formula for convolutions of random variables, we find that for any fixed values of λandτ, the ... | https://arxiv.org/abs/2502.20579v1 |
require an additional bound on R(n+1) n+1(τrand.) (see the arguments at the beginning of Lemma 24). 48 Now, since g(τrand.) takes on only finitely many values, we may apply the results of Lemma 12 and part 5 of Theorem 6 regarding the convergence in distribution of ( R(n+1) n+1(g(τ)), λ) for τfixed along with standard ... | https://arxiv.org/abs/2502.20579v1 |
Z, λ, ϵ )∼N(0,1)⊗Pλ⊗Pϵ. It remains to derive an equation for the intercept. To do this, recall that in the main text we conjectured the approximation ˆηi=R(i) i−prox( λ2 n+1c∞ℓα)(R(i) i) λ2 n+1c∞. Now, the first-order condition for the intercept in the min-max formulation of quantile regression (see (5.3)) necessitates... | https://arxiv.org/abs/2502.20579v1 |
the pointwise limit of fn we clearly have that fisL-Lipschitz. Thus, we find that on the event E, lim sup n→∞∥fn(x)−f(x)∥B≤lim sup n→∞∥fn(xk)−fn(x)∥B+∥f(xk)−f(x)∥B+∥fn(xk)−f(xk)∥B ≤2L∥xk−x∥A+ lim sup n→∞∥fn(xk)−f(xk)∥B≤ϵ, and sending ϵ→0 we find that on the event E, lim n→∞fn(x) =f(x). Since the choice of xwas arbitrar... | https://arxiv.org/abs/2502.20579v1 |
Minimax Optimal Kernel Two-Sample Tests with Random Features Soumya Mukherjee and Bharath K. Sriperumbudur Department of Statistics Pennsylvania State University, University Park, PA 16802, USA. {szm6510,bks18}@psu.edu March 3, 2025 Abstract Reproducing Kernel Hilbert Space (RKHS) embedding of probability distributions... | https://arxiv.org/abs/2502.20755v1 |
with respect to an appropriately defined class of alternatives. The spectral-regularized approach represents a significant advancement in two-sample testing. Instead of relying solely on the difference between the mean embeddings of the two distributions, this method incorporates the regularized covariance operator-wei... | https://arxiv.org/abs/2502.20755v1 |
(RFF) to enhance the computational efficiency of kernel two-sample testing. More recently, Choi and Kim (2024) investigate their RFF-MMD test, examining the trade-offs between computational efficiency and the statistical power of the test. While Choi and Kim (2024) focus on accelerating the classical MMD test, our appr... | https://arxiv.org/abs/2502.20755v1 |
Hilbert-Schmidt and operator norms of S, respectively. For x, y∈H, x⊗Hyis an element of the tensor product space of H⊗Hwhich can also be seen as an operator from H→Has(x⊗Hy)z=x⟨y, z⟩Hfor any z∈H. To enhance the readability of the paper, all relevant proofs of results presented in the main paper are relegated to Section... | https://arxiv.org/abs/2502.20755v1 |
sis chosen to regularize/modify the spectrum of Min a certain way. For any λ >0, choosing s(x) =gλ(x) = ( x+λ)−1and with Irepresenting the identity operator, we define the regularized covariance operator ΣPQ,λasΣPQ,λ :=gλ(ΣPQ) = (Σ PQ+λI)−1. The operators TPQ,λ,ΣPQ,λ,landTPQ,λ,lare defined analogously, with Ilplaying t... | https://arxiv.org/abs/2502.20755v1 |
of the classical MMD test statistic, a method later analyzed from a theoretical perspective by Choi and Kim (2024). In the specific case when the kernel Kis translation invariant on Rdi.e. K(x, y) =υ(x−y), x, y∈Rd for some continuous positive definite function υ, Bochner’s theorem (Wendland (2004), Theorem 6.6) provide... | https://arxiv.org/abs/2502.20755v1 |
the Sobolev ball of order sand fixed radius in Rd. To attain minimax optimality, the tests require at least l≥min{N, M}4d 4s+drandom Fourier features, resulting in a computational complexity of O((N+M) min{N, M}4d 4s+dd). 3.2 Spectral Regularized MMD Test Despite the widespread popularity and elegant theoretical proper... | https://arxiv.org/abs/2502.20755v1 |
the covariance operator ΣPQ in addition to the discrepancy between the mean embeddings µPandµQ. An alternative expression forηλ(P, Q), which will be useful for constructing a statistical estimator for the same, is given by ηλ(P, Q) =Z X4D g1/2 λ(ΣPQ)(K(·, x)−K(·, y)), g1/2 λ ΣPQ)(K(·, x′)−K(·, y′)E HdP(x)dP(x′)dQ(y)d... | https://arxiv.org/abs/2502.20755v1 |
respect to the kernel K, one can define the approximate spectral regularized discrepancy ηλ,lwith respect to the approximate kernel Klas ηλ,l= g1/2 λ(ΣPQ,l) (µQ,l−µP,l) 2 Hl 10 and our primary goal is to construct a test of equality of PandQbased on a statistical estimator ofηλ,l, which is ˆηλ,l. Thus, ˆηλ,lcan be view... | https://arxiv.org/abs/2502.20755v1 |
with the eigenvalues and eigenfunctions being indexed in decreasing order of magnitude of the eigenvalues. We assume in this paper that the index set I is countable, which implies that limi→∞λi= 0. The second assumption regarding the form of the reproducing kernel Kis as follows: (A1)The reproducing kernel Kcorrespondi... | https://arxiv.org/abs/2502.20755v1 |
features land the regularization parameter λ >0that ensures that the test achieves a given Type-II error bound 4δ >0. Theorem 2. (Type-II error analysis/Separation boundary of RFF-based Oracle Test) Suppose that Assumptions (A0),(A1),(A2),(A3),(A5)and(B)hold true. Let the number of samples ssplit from X1:NandY1:Mfor es... | https://arxiv.org/abs/2502.20755v1 |
β )∈Nsuch that, for any choice of N+M≥k(α, δ, θ, β ), the power of the level- αtest of H0:P=Qagainst H1:P̸=Qproposed in Theorem 1 with ˆηλ,las the test statistic and γ=4√ 3(C1+C2)√α 1 n+1 m" 4q 2κlog8 α√ λl+16κlog8 α λl+ 2√ 2N2(λ)# as the critical threshold is at least 14 1−4δover the class of ∆NM-separated alternativ... | https://arxiv.org/abs/2502.20755v1 |
now fully data-driven. We begin by describing the concept underlying the permutation test. The RFF-based test statistic defined in (19), just like its exact kernel-based counterpart in Hagrass et al. (2024b), involves sample splitting resulting in three sets of independent samples, (Xi)n i=1i.i.d.∼P,(Yj)m j=1i.i.d.∼Q, ... | https://arxiv.org/abs/2502.20755v1 |
define ˜α= (w−˜w)αand let C∗be an absolute positive constant as defined in Lemma 16. Then, for any 0< δ≤1, provided (N+M)≥maxn 32κd2 δ,2C∗log(2 ˜α) (1−d2)√ δo ,N2(dθ∆1 2θ N,M)≥1, the number of randomly selected permutations B≥log(2 min{δ,α(1−w−˜w)}) 2 ˜w2α2and∆N,Mand number of random features lsatisfy the following con... | https://arxiv.org/abs/2502.20755v1 |
andµQ,lrespectively satisfy n, m≥2. For any α,wand ˜wsuch that 0≤α≤1,0<˜w < w <1 2 and0≤(w−˜w)α < e−1, consider the level- αtest of H0:P=Qagainst H1:P̸=Qproposed in Theorem 3 with ˆηλ,las the test statistic and ˆqB,λ,l 1−wαas the critical threshold. Further, assume that sup θ>0sup (P,Q)∈P T−θ PQu L2(R)<∞. Finally, defi... | https://arxiv.org/abs/2502.20755v1 |
the optimal choice of the regularization parameter that leads to a minimal optimal RFF-based oracle test or permutation test (as defined in Theorem 1 and Theorem 3, respectively) asλ∗. Assume that there exists a positive constant λLandb∈Nsuch that λL≤λ∗≤λU, where λU= 2bλL. Let us define Λ:= λ∈R:λ= 2iλL, i= 0,1, . . . ... | https://arxiv.org/abs/2502.20755v1 |
i.e. λi≍e−τiforτ >0,λL=r3log(N+M) N+M,λU=minn r4, e−1,1 2∥ΣPQ∥L∞(H))o for some constants r3, r4>0, the separation boundary achieves the following rate of decay ∆N,M=c(˜α, δ, θ ) max(log(N+M) N+M2θ ,p log(N+M) log log( N+M) N+M) and the number of random features satisfies l≳maxn (N+M)1 2θ∗[log(N+M)], N+Mo withc(˜α, δ,... | https://arxiv.org/abs/2502.20755v1 |
|Λ|as in Section 4.5. Similarly, define the potential choices of the kernel Kand its cardinality |K|. Provided the number of permutations B≥|Λ|2|K|2 2 ˜w2α2log(2|Λ||K| α(1−w−˜w)) and|K|<∞, the level- αcritical region for testing H0:P=QvsH1:P̸=Qis given by S (λ,K)∈Λ×K ˆη(K) λ,l≥qB,λ,l 1−wα |Λ||K|,K i.e, PH0 [ (λ,K)∈... | https://arxiv.org/abs/2502.20755v1 |
5 Computational complexity of test statistics The primary focus of the current paper is to show that the use of Random Fourier Feature sampling reduces the computational complexity of the spectral regularized MMD test without compromising the statistical efficiency of the test, provided the number of random features li... | https://arxiv.org/abs/2502.20755v1 |
θT l . Define MX=XTΘT, MY=YTΘTandMZ=ZTΘT. Next, set Φ(X) =K(0,0)√ lPT l cos(MX) sin(MX)T,Φ(Y) = K(0,0)√ lPT l cos(MY) sin(MY)TandΦ(Z) =K(0,0)√ lPT l cos(MZ) sin(MZ)Twhere Plis the 2l×2lcolumn-interleaving permutation matrix Pl= e1,2lel+1,2le2,2lel+2,2l···el,2le2l,2l , andei,2lis the standard basis vector in ... | https://arxiv.org/abs/2502.20755v1 |
. .ˆα2l]. 10:Construct the matrix G=V L1/2VT, where L1/2= q gλ(ˆλ1) ...q gλ(ˆλ2l) . 11:Compute the matrices Ψ(X) =GΦ(X)andΨ(Y) =GΦ(Y). 12:Compute the vectors vX,i= Ψ( X)ei,nfori= 1, . . . , nandvY,j= Ψ( Y)ej.mforj= 1, . . . , m, where {ei,n}n i=1and{ej,m}m j=1are standard basis vectors RnandRm, respectively. ... | https://arxiv.org/abs/2502.20755v1 |
large enough for both the “exact” Adaptive Test and the RFF-based Kernel Adaptive Test. However, it is empirically observed that the number of permutation Brequired for achieving the specified Type-I error control is a bit higher for the RFF-based Kernel Adaptive Test compared to the “exact” Adaptive Test. Despite this... | https://arxiv.org/abs/2502.20755v1 |
of random Fourier features (around 7 or 9) is sufficient to ensure that the power of the RFF-based Kernel Adaptive Test is nearly as high as the “exact” Adaptive test. Most importantly, based on Figure 2 and Table 1, the RFF-based Kernel Adaptive Test compensates more than adequately for the slight loss in power by tak... | https://arxiv.org/abs/2502.20755v1 |
Cauchy distribution with median µand identity scale. Here, we consider the class of median-shifted alternatives and we use the choices µ= 0,0.05,0.1,0.3,0.5,0.7,1as the value of the median shift for our experiments. We consider the sample size to be N=M= 500and data dimensions to be d= 1,10,20,50,100. We choose s= 50. ... | https://arxiv.org/abs/2502.20755v1 |
beB= 550, while the no. of permutations for “exact” Adaptive Test is B′= 350. We consider two sets of experiments: one using the Gaussian kernel and the other using the Laplace kernel. 33 Figure 7: Empirical power for MNIST experiments using Gaussian kernel 6.4.1 Results using Gaussian kernel From Figure 7, we can obse... | https://arxiv.org/abs/2502.20755v1 |
methods and their computational versus statistical tradeoffs remains an intriguing avenue for future research. References R. A. Adams and J. J. F. Fournier. Sobolev Spaces . Academic Press, 2003. M. Albert, B. Laurent, A. Marrel, and A. Meynaoui. Adaptive test of independence based on HSIC measures. The Annals of Stati... | https://arxiv.org/abs/2502.20755v1 |
for homogeneity with kernel fisher discriminant analysis. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems , volume 20. Curran Associates, Inc., 2007. W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American statisti... | https://arxiv.org/abs/2502.20755v1 |
strong topologies. Bernoulli , 22(3):1839 – 1893, 2016. B. K. Sriperumbudur, K. Fukumizu, A. Gretton, G. R. G. Lanckriet, and B. Schölkopf. Kernel choice and classifiability for RKHS embeddings of probability distributions. In Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, editors, Advances i... | https://arxiv.org/abs/2502.20755v1 |
this section. In addition, we also provide the statements and proofs of the main propositions in this section. A.2.1 Proof of Theorem 1 Let us define γ1,l:=2√ 6(C1+C2)N2,l(λ)√ δ 1 n+1 m and let us set δ=α 2.Then, we have that PH0{ˆηλ,l≤γ1} ≥PH0{{ˆηλ,l≤γ1,l} ∩ {γ1,l≤γ1}} ≥1−PH0{ˆηλ,l≥γ1,l} −PH0{γ1,l≥γ1} (a) ≥1−2δ = 1−α... | https://arxiv.org/abs/2502.20755v1 |
s=d1N=d2Nfor estimating the covariance operator ΣPQ,las stated in Theorem 2, we have that m≤n≤D′mwhere D′=D−d2 1−d2≥1is a constant. Therefore, using Lemma (11)under Assumptions (A2)and(A3)and provided the events E1andE2occur simultaneously, we observe that T2≤T1and consequently, the occurrence of the event (E′)c={γ < T... | https://arxiv.org/abs/2502.20755v1 |
∆1+2θ 2θ N,M N2(dθ∆1 2θ N,M)≳d−1 θ×1 δ(N+M)2. Therefore, the conditions specified in the statement of Theorem 2 are sufficient for (31),(32), (33), (38) and (39) and the proof is complete. A.2.3 Proof of Corollary 1 We assume the setting of Theorem 2 is valid. Under polynomial decay of the eigenvalues of ΣPQ i.e.λi≍i−β... | https://arxiv.org/abs/2502.20755v1 |
δ, θ, β )×1≳maxn 7,10o . Further, c′(α, δ, θ, β )×1≳3if l≳h N+M log(N+M)iβ+1 β. Note that, under the condition θ≤1 2−1 4β, we have that max (N+M)2β−1 1+4βθ,(N+M)2(2β−1) 1+2β+4βθ,h N+M log(N+M)iβ+1 β =h N+M log(N+M)iβ+1 β. Therefore, provided l≳h N+M log(N+M)iβ+1 β, (46) reduces to ∆N,M=c(α, δ, θ, β )log(N+M) N+M2θ ... | https://arxiv.org/abs/2502.20755v1 |
h log(N+M) N+Mi2θ ≳h log(l) li2θ ifl≳N+M. Therefore, provided l≳N+M, (56) reduces to ∆N,M=c(α, δ, θ )log(N+M) N+M2θ (59) , where c(α, δ, θ )is a positive constant that depends on α,δandθas defined in (58). Further, Condition 11 in Theorem 2 is also satisfied. A.2.5 Proof of Theorem 3 Conditional on θ1:l, the kernel K... | https://arxiv.org/abs/2502.20755v1 |
2)≤δ. (63) Following the proof of Proposition 2, specifically the proof of (125), we have that, if n, m≥2, 140κ slog32κs 1−√ 1−δ≤λ≤1 2∥ΣPQ∥L∞(H)andl≥max( 2 log2 1−√ 1−δ,128κ2log2 1−√1−δ ∥ΣPQ∥2 L∞(H)) =L(2δ,1 2), then, PH1(Ec 3) =P(Ec 3)≤δ. (64) Now, under Assumption (B)and the choice of the sample splitting size s=d1N=... | https://arxiv.org/abs/2502.20755v1 |
(62), (63), (64), (65), (70) and (72). Let us define c1=sup θ>0sup (P,Q)∈P T−θ PQu L2(R)which is assumed to be finite and dθ= 1 16c2 11 2θ. Since (P, Q)∈ Punder H1, we have that ∥u∥2 L2(R)≥∆N,M. Consequently, the choice λ=dθ∆1 2θ N,M(74) implies that ∥u∥2 L2(R)≥16λ2θ T−θ PQu 2 L2(R)holds. 50 Usingthechoiceof λasgiven... | https://arxiv.org/abs/2502.20755v1 |
d4θ 1+4θ θ d4βθ 1+4βθ θ,d2θ 1+2θ θ d4βθ 1+4βθ θ,d4θ 3+4θ θ d2+4βθ 1+2β+4βθ θ,dθ 1+θ θ d2+4βθ 1+2β+4βθ θ,1) × max [log(1 ˜α)]4×log(4 δ) δ22 1+4θ ([log(1 ˜α)]4 δ2)2β 1+4βθ, (log(1 ˜α))2×log(4 δ) δ2 3+4θ δ−2β 1+2β+4βθ,1 is a constant that depends only on θandβ. Now, we consider two scenarios based... | https://arxiv.org/abs/2502.20755v1 |
λ=dθ∆1 2θ N,M≳maxn 1,2,4o (87) where 1=log(N+M) (N+M),2=log(2 δ) land 4=log(l δ) l. Now, 4≳2andlog(l)≳log(l δ)ifl≥2. Further, if ∆N,M≳maxn 1√ 2θ,1o ×[log(1 ˜α)]4 δ2×√ log(N+M) N+M, then condition 7 is satisfied, while condition 10 is satisfied if ∆N,M≳max( 1 (1 2+θ)θ (1+2θ),1) ×d−2θ 1+2θ θ×[log(1 ˜α)]4θ 1+2θ δ2θ 1+2θ×[... | https://arxiv.org/abs/2502.20755v1 |
˜w2α2maxn log(2 log( N+M) α(1−w−˜w)),log(2 δ)o , the conditions 1 to 10 as specified in Theorem 4 reduce to dθ∆1 2θ N,M≳maxn 1,2,3,4,5,6,7,8,9,10o (94) where 1=log(N+M) (N+M),2=log(2 δ) l,3=hlog(2 δ) liβ β+1,4=log(l δ) l, 5=d4θ 1+4θ θ"h log(log(N+M) ˜α)i4 ×q log(4 δ) δ2×1√ l(N+M)#2 1+4θ , 6=d2θ 1+2θ θh log(log(N+M) ˜α... | https://arxiv.org/abs/2502.20755v1 |
1+2β+4βθ,1 2θ(97) is a positive constant that depends on ˜α,δ,θandβ. Finally, Condition 11 in Theorem 4 is also satisfied. Case II: Suppose θ∗≤θ <1 2−1 4β. Then, we have that c∗(˜α, δ, θ, β )×1≳maxn 7,10o . Further, c∗(˜α, δ, θ, β )×1≳3if l≳h N+M log(N+M)iβ+1 β. Note that, under the condition θ≤1 2−1 4βandN... | https://arxiv.org/abs/2502.20755v1 |
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