text string | source string |
|---|---|
We say that an estimator is efficient if its asymptotic variance aligns with Vcens. 4.2 Semiparametric efficient estimator Based on the efficient influence function, we propose an ATE estimator defined as bτcens-eff n := 1 nPn i=1Scens(Xi, Oi, Yi;bµT,n,i,bνn,i,bπn,i,bgn,i), where bµT,n,i,bνn,i,bπn,iandbgn,iare estimato... | https://arxiv.org/abs/2501.19345v2 |
does not hold in several cases, such as high-dimensional regression settings. In such cases, asymptotic normality can still be established through sample splitting, a technique in this field (Klaassen, 1987), which has been recently refined by Chernozhukov et al. (2018) as cross-fitting. Cross-fitting. Cross-fitting is... | https://arxiv.org/abs/2501.19345v2 |
proposed efficient estimator, this estimator does not use the conditional outcome estimators (Horvitz & Thompson, 1952). If g0andπ0are known, this esti- mator is unbiased. However, it incurs a large asymptotic variance, given as VIPW:= Eh 1−g0(1|X) g0(0|X)2E[Y(1)2|X] π0(1|X)+E[eY2|X] g0(0|X)2π0(1|X)i . Here, it holds... | https://arxiv.org/abs/2501.19345v2 |
setting (Informal)) .Fixα∈(0,1). Forn > 0, consider the case-control setting with sample sizes m, l such that m=αn andl= (1−α)n. If the case-control propensity score e0and the density ratio are known (r0ben,i=e0andbrn,i=r0), and bµT,n,i=bµ(ℓ) T,m,bµU,n,i=bµ(ℓ) U,lare consistent estimators constructed via cross-fitting.... | https://arxiv.org/abs/2501.19345v2 |
Although logistic regression is used, the propensity score model is misspecified, while the expected conditional outcome follows a linear model. Overall, bτcc-eff n demonstrates robust performance in terms of MSE, bias, and coverage ratio. The poor performance of the IPW estimator is attributed to model misspecificatio... | https://arxiv.org/abs/2501.19345v2 |
Data Mining (KDD) , 2008. 2, 5, 6, 8, 12, 21, 27, 28, 30, 47, 49 Jinyong Hahn. On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica , 66(2):315–331, 1998. 19, 30 Negar Hassanpour and Russell Greiner. Learning disentangled representations for coun- terfact... | https://arxiv.org/abs/2501.19345v2 |
theory and empirical processes in causal inference, 2016. arXiv: 1510.04740. 19 Edward H. Kennedy. Efficient nonparametric causal inference with missing exposure informa- tion. The International Journal of Biostatistics , 16(1), 2020. 20, 21, 22 Edward H. Kennedy. Semiparametric doubly robust targeted double machine le... | https://arxiv.org/abs/2501.19345v2 |
and Victor Veitch. Adapting neural networks for the estimation of treatment effects. In International Conference on Neural Information Processing Systems . Curran Associates Inc., 2019. 20 Dan Steinberg and N. Scott Cardell. Estimating logistic regression models when the dependent variable has no variance. Communicatio... | https://arxiv.org/abs/2501.19345v2 |
epidemiology, economics, and machine learning (Imbens & Rubin, 2015). While randomized controlled trials are considered the gold standard, it is extremely important to estimate the ATE in observational studies. In ATE estimation with observational data, one of the basic approaches is to employ the IPW estimator, which ... | https://arxiv.org/abs/2501.19345v2 |
Hirano et al. (2003). This phenomenon is known as the paradox of the propensity score (Henmi & Eguchi, 2004; Kato et al., 2021). 19 CATE estimation is also an important topic related to this study (Heckman et al., 1997). Various methods have been proposed for estimating CATE including methods using neural networks (Joh... | https://arxiv.org/abs/2501.19345v2 |
data. Although the setting is not directly related, integrating insights from both areas could enhance the applicability. 20 A.2 Introduction of PU learning algorithms Another related body of work comes from the literature on PU learning. PU learning is a classification method primarily designed for binary classifiers ... | https://arxiv.org/abs/2501.19345v2 |
contrast, Kennedy (2020) can observe both Di= 1andDi= 0when the label is observed. Our setting is designed to be more suitable for applications in marketing and recommendation systems, where implicit feedback is common4. Note that in our study, we do not explicitly use eOibut instead denote eOiDiby another random varia... | https://arxiv.org/abs/2501.19345v2 |
is complete. C.2 Example ATE estimator in the case-control setting In the case-control setting, PU learning methods have been investigated by Imbens & Lancaster (1996) and du Plessis et al. (2015). In that works, we typically make the following assumption, which corresponds to the SCAR assumption in the censoring setti... | https://arxiv.org/abs/2501.19345v2 |
D.3. Asn→ ∞ , it holds that ben,i−e0 2=op(1)and brn,i−r0 2=op(1). Then, the following consistency result holds. Theorem D.4 (Consistency in the case-control setting) .If Assumption D.3 holds, then bτcc-eff np− →τ0asn→ ∞ . Interestingly, to achieve consistency, it is sufficient to obtain consistent ben,i. Compared to As... | https://arxiv.org/abs/2501.19345v2 |
specific loss functions, including logistic loss, the obtained classifiers can be interpreted as estimators of the probability (Elkan & Noto, 2008; Kato et al., 2019; Kato & Teshima, 2021). E.1 Censoring PU learning In Elkan & Noto (2008), it is assumed that only a fraction of the truly positive instances are labeled a... | https://arxiv.org/abs/2501.19345v2 |
instances. Lethbe a classifier. In conventional supervised learning, the classification risk is defined as: R(h) =e0(1)R+(h,+1) + (1 −e0(1))R−(h,−1), where e0(1) is the prior probability of being positive, and R+(h,+1) and R−(h,−1) denote the risks over the positive and negative distributions, respectively. Specificall... | https://arxiv.org/abs/2501.19345v2 |
on the nuisance parameter estimation in the censoring setting In the censoring setting, by applying the method of Elkan & Noto (2008), we can obtain an estimator of P(D= 1|X) from an estimator of π0(x) =P(O= 1|X). However, our objective is to estimate g0(1|X) =P(D= 1|X, O = 0) rather than P(D= 1|X). Letbκn(1|X) be an e... | https://arxiv.org/abs/2501.19345v2 |
under the parametric submodels as τ(θ):=ZZ y(1)pY(1)(y(1)|x;θ)ζ(x;θ)dy(1)dx−ZZ y(0)pY(0)(y(0)|x;θ)ζ(x;θ)dy(0)dx =ZZ y(1)pY(1)(y(1)|x;θ)ζ(x;θ)dy(1)dx−ZZ y(0)1 g0(0|x)peY(y(0)|x;θ)ζ(x;θ)dy(0)dx +ZZ y(0)g0(1|x) g0(0|x)pY(1)(y(0)|x;θ)ζ(x;θ)dy(0)dx. Them, the derivative is given as ∂τ(θ) ∂θ=Eθh Y(1)SY(1)(Y(1)|X;θ)i −Eθ1 g0... | https://arxiv.org/abs/2501.19345v2 |
i=1Scens(Xi, Oi, Yi;bµT,n,i,bνn,i,bπn,i, g0), 35 where recall that Scens(X, O, Y ;bµT,n,i,bνn,i,bπn,i, g0) =1[O= 1] Y−bµT,n,i(X) bπn,i(1|X)−1[O= 0] Y−bνn,i(X) g0(0|X)bπn,i(0|X)+g0(1|X) 1[O= 1] Y−bµT,n,i(X) g0(0|X)bπn,i(1|X) +bµT,n,i(X)−1 g0(0|X)ν(X) +g0(1|X) g0(0|X)bµT,n,i(X). We have bτcens-eff n =1 nnX i=1Scens... | https://arxiv.org/abs/2501.19345v2 |
g0)| C(ℓ)i −Eh Scens(eX(ℓ) i,eO(ℓ) i,eY(ℓ) i;bµ(ℓ) T,n,bν(ℓ) n,bπ(ℓ) n, g0)| C(ℓ)i! ≥ε| C(ℓ)! ≤n εVar 1 nmX i=1Scens(eX(ℓ) i,eO(ℓ) i,eY(ℓ) i;µT,0, ν0, π0, g0)−1 nmX i=1Scens(eX(ℓ) i,eO(ℓ) i,eY(ℓ) i;bµ(ℓ) T,n,bν(ℓ) n,bπ(ℓ) n, g0) − Eh Scens(eX(ℓ) i,eO(ℓ) i,eY(ℓ) i;µT,0, ν0, π0, g0)| C(ℓ)i −Eh Scens(eX(ℓ) i,eO(ℓ) i,eY(ℓ)... | https://arxiv.org/abs/2501.19345v2 |
(iv) calculate the expected square of the influence function. In the case-control setting, the observations are generated as follows: DT:= XT,j, Yj(1) m j=1, XT,j, Yj(1) ∼pT,0(x, y(1)) = ζT,0(x)pY(1),0(y(1)|x), DU:= Xk, YU,k l k=1, Xk, YU,k ∼pU,0(x, yU) =ζ0(x)pYU,0(yU|x). We derive the efficiency bound by reg... | https://arxiv.org/abs/2501.19345v2 |
e0(0|X)µT,0(X), Then,eΨcc∈ T holds. KProof of Theorem D.7: : Semiparametric efficient ATE estimator under the case-control setting Recall that we have defined the ATE estimators as bτcc-eff n =1 mmX j=1Scc (T)(Xj, Yj;bµ(ℓ) T,n,be(ℓ) n,br(ℓ) n) +1 llX k=1Scc (U)(Xk, Yk;bµ(ℓ) T,n,bµ(ℓ) U,n,be(ℓ) n). We aim to show √n bτ... | https://arxiv.org/abs/2501.19345v2 |
setting, where the covariates Xare drawn from a multivariate normal distribution as X∼ζ0(x), where ζ0(x) is the density of N(0, Ip), pdenotes the number of covariates, and Ipis the ( p×p) identity matrix. We set p= 10. The propensity score is given by g0(1|X) =sigmoid (X⊤β1+X2⊤β2), where X2is the element-wise square of... | https://arxiv.org/abs/2501.19345v2 |
networks are applied. For simplicity, we assume that the class prior e0(1) is known. We consider two cases: ( m, l) = (1000 ,2000) and (2000 ,3000). We conduct 5000 trials 47 and report the empirical mean squared errors (MSEs) and biases for the true ATE, along with the coverage ratio computed from the confidence inter... | https://arxiv.org/abs/2501.19345v2 |
a positive dataset by selecting only the treated units from it. The class prior is set as e0(1) = 0 .1. For each outcome model (response surface A and B), we conduct 1000 trials and report the empirical mean squared errors (MSEs), biases for the true ATE, and the coverage ratio (Cov. ratio) computed from the confidence... | https://arxiv.org/abs/2501.19345v2 |
Carefree multiple testing with e-processes Yury Tavyrikov1, Jelle J. Goeman2, and Rianne de Heide∗3 1Vrije Universiteit Amsterdam 2Leiden University Medical Centre 3University of Twente February 3, 2025 Abstract E-processes enable hypothesis testing with ongoing data collection while maintaining Type I error control. H... | https://arxiv.org/abs/2501.19360v1 |
this with an example. For K= 2e-values, e-BH rejects any hypothesis for which the e-value exceeds 2/α, and both hypotheses if both e-values exceed 1 /α. Now suppose that at time twe have 1/α≤E1 t<2/αandE2 t<1/α. Gathering more data for both e-processes, however, the situation could reverse, and E1 t+1<1/αand 1 /α≤E2 t+... | https://arxiv.org/abs/2501.19360v1 |
e-variables. Thus, this procedure reduces to the standard e-BH procedure on the stopped e-processes {Sk}k∈K, and the FDR guarantee [Wang and Ramdas, 2022] carries over. 2.3 Related work Online multiple testing has been considered both classically with p-values [Fischer et al., 2024] ande-values [Fischer and Ramdas, 202... | https://arxiv.org/abs/2501.19360v1 |
present a counterexample that demonstrates how the e-BH procedure, when applied to the running maxima of e-processes, fails to control the FDR at the nominal level α. The FDR is evaluated numerically. Consider two true null hypotheses with corresponding e-processes E1 tandE2 t, defined as: E1 t=X1 0tY s=1e1 s, E2 t=X2 ... | https://arxiv.org/abs/2501.19360v1 |
FDR-sup at level K0α/K . One way to prove this is to follow the classical proof of Wang and Ramdas [2022], and is detailed in Appendix A. 4.2 Example: e-BH with the adjusted running maximum Two examples of admissible adjusters are A1(E) =E−1−logE log2E, A2(E) =√ E−1. (4) In Figure 1 we see an example of running e-BH on... | https://arxiv.org/abs/2501.19360v1 |
Carefree multiple testing procedures must operate on the running maxima ofe-processes. Such running maxima are not themselves e-processes, but can be turned into e-processes by adjusters. Adjustment is costly, however, in terms of power. Declaration of funding Rianne de Heide’s work was supported by NWO Veni grant numb... | https://arxiv.org/abs/2501.19360v1 |
Using gradient of Lagrangian function to compute efficient channels for the ideal observer Weimin Zhoua, b aWyant College of Optical Sciences, University of Arizona, Tucson, USA bDepartment of Medical Imaging, University of Arizona, Tucson, USA ABSTRACT It is widely accepted that the Bayesian ideal observer (IO) should... | https://arxiv.org/abs/2501.19381v1 |
in this study, I propose a new method for generating efficient channels based on the gradient of the Lagrangian-based loss Further author information: (Send correspondence to Weimin Zhou) Weimin Zhou.: E-mail: weiminzhou@arizona.eduarXiv:2501.19381v1 [eess.SP] 31 Jan 2025 function that was designed to learn the HO. The... | https://arxiv.org/abs/2501.19381v1 |
signal detectability, quantified by the signal-to-noise ratio of test statistics, by solving the following constrained opti- mization problem:5 minimize w1 2 [wTg−wT¯¯g0]2 0+1 2 [wTg−wT¯¯g1]2 1, subject to wT¯¯g1−wT¯¯g0=C,(6) where Cis an arbitrary constant that does not affect the optimal solution. The Lagrangian func... | https://arxiv.org/abs/2501.19381v1 |
sizes, including 1000, 2000, 4000, 6000, 8000, 10000, 20000, 30000, and 40000 total images. Half of the images in each training dataset are signal-absent, and the other half are signal-present. After the channels were produced, the corresponding CHO was subsequently computed on another set of 4000 images that comprised... | https://arxiv.org/abs/2501.19381v1 |
computation time 0.5610 0.8513 1.6240 2.0403 8.0119 11.1468 13.7784 L-grad computation time 0.1722 0.2092 0.3637 0.4900 0.9078 1.3251 1.7663 3.2 SKE detection task with VICTRE Mammography ROIs and spiculated mass A set of signal-absent VICTRE Mammography regions of interest (ROIs) ( https://github.com/DIDSR/ VICTRE_DM_... | https://arxiv.org/abs/2501.19381v1 |
function that was designed to learn the Hotelling observer. Knowledge of the signal and background statistics can readily be incorporated in the proposed L-grad method to produce the efficient channels. The ability of L-grad channels to approximate the ideal linear observer was investigated in two different signal dete... | https://arxiv.org/abs/2501.19381v1 |
Minkowski tensors for voxelized data: robust, asymptotically unbiased estimators D. Hug∗, M.A. Klatt†, and D. Pabst∗ Abstract Minkowski tensors, also known as tensor valuations, provide robust n-point information for a wide range of random spatial structures. Local estimators for voxelized data, however, are unavoidabl... | https://arxiv.org/abs/2502.00092v1 |
5 Isotropic random polytopes 25 5.1 Exact expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Experimental data 28 6.1 Metallic grains . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.00092v1 |
37, 11]. 2 A challenge, however, is the application of Minkowski tensors to voxelized data because the discrete set of available orientations introduces a systematic bias usually even in the limit of infinitely high resolution. For gray-scale data, marching-cube algorithms can turn the voxelized data into triangulated ... | https://arxiv.org/abs/2502.00092v1 |
of nanorough surfaces from [59] (see Section 6.2). 2 Minkowski tensors in integral geometry In this section, we provide a brief introduction to Minkowski tensors (or tensor valuations) and summarize their basic properties. Let Kddenote the set of all compact convex subsets 3 (convex bodies) of Rd. We denote by ⟨·,·⟩and... | https://arxiv.org/abs/2502.00092v1 |
tensor of rank r1+···+rk. Denoting by S(m) the group of bijections of {1, . . . , m }, s0:= 0 and si:=r1+···+ri, fori= 1, . . . , k , it is defined by (a1⊙ ··· ⊙ ak)(x1, . . . , x sk) :=1 sk!X σ∈S(sk)kY i=1ai(xσ(si−1+1), . . . , x σ(si)) forx1, . . . , x sk∈Rd. In this way, the space of symmetric tensors (of arbitrary ... | https://arxiv.org/abs/2502.00092v1 |
Kis a polytope, we refer to [53, Chapter 4.2]. The maps K7→Λj(K,·) are measure-valued, additive (valuations), weakly continuous and for Borel sets α⊆Rdandβ⊆Sd−1they satisfy the covariance condition Λj(gK, gα ×g0β) = Λ j(K, α×β), where g:Rd→Rdis a rigid motion and g0is its rotational part (that is, g(x) =g0(x) +tforx∈Rd... | https://arxiv.org/abs/2502.00092v1 |
In contrast to the real-valued and vector-valued case, these tensor valuations are no longer linearly independent. A study of linear dependencies was initiated by McMullen and completed by Hug, Schneider, Schuster (for precise references and detailed statements, see, e.g., [25, Thms. 2.6 and 2.7]). These general invest... | https://arxiv.org/abs/2502.00092v1 |
Λ d−1(A,·) (3.2) 7 as Borel measures on Rd×Sd−1; see also [48, Corollary 2]. In particular, A7→µd−1(A;·) is additive on Ud. On the other hand, the support measures are in general not defined on the larger domain of finite unions of sets with positive reach. For this reason, we work with the reach measures whenever the ... | https://arxiv.org/abs/2502.00092v1 |
−2}, then ε−1 Zε 0hi(t) dt ≤ωd−i εZε 0td−1−iZ Nor(A)|f(x)|1{rA(x, u)> t}∥µi(A)∥(d(x, u)) dt ≤ωd−i d−i· |f|∂A· ∥µi(A)∥(Nor( A))·εd−i−1→0 as ε→0+, where |f|∂A:= sup {|f(x)|:x∈∂A}<∞and basic properties of the norm | · |on (symmetric) tensors were used, in particular (2.2). Moreover, by the dominated convergence theorem, h... | https://arxiv.org/abs/2502.00092v1 |
rank parameter tuple ( r, s)∈N2 0of a nonempty compact set K⊂Rdis defined by Vr,s R(K) :=Z KRpK(x)r(x−pK(x))sHd(dx), (4.1) where KRis the set of points with distance at most R > 0 from KandpKis the (almost everywhere uniquely defined) metric projection on K. For r=s= 0 the Voronoi tensor with distance parameter Ris the... | https://arxiv.org/abs/2502.00092v1 |
be replaced by dH(K, K 0) ifr=s= 0. Evaluating the integral in (4.3) can be cumbersome and inefficient. Moreover, we do not require a precision beyond the approximation of KviaK0. We, therefore, estimate the Voronoi tensors in (4.3) by considering a random grid η=X z∈a·Zdδz+Ua (4.5) with some scaling parameter a >0 and... | https://arxiv.org/abs/2502.00092v1 |
|bVr,s R(K0, a)− Vr,s R(K)| ≤C′(d, R, ρ, r, s ) dH(K, K 0)1 2+ max( a N(K0),a N(K0)d)! , where C′(d, R, ρ, r, s )is a positive constant, and dH(K, K 0)1 2can be replaced by dH(K, K 0)if r=s= 0. In particular, if (K0(i))i∈Nis a sequence of finite sets in Rdand(ai)i∈Nis a sequence of positive numbers such that K0(i)→Ki... | https://arxiv.org/abs/2502.00092v1 |
implies the required bound, since N(K0)≤2ρ. From the comments after (4.4) and the proof of Theorem 4.3, which yields more explicit information about the constants involved, we get the following consequence. Corollary 4.4. LetK⊂Rdbe a compact set, and let r, s∈N0. Let ρ > 0be such that K⊆B(o, ρ). IfK0⊆B(o, ρ)is a finite... | https://arxiv.org/abs/2502.00092v1 |
objects with a grid. The simulated estimates can be found in Table 2. To achieve good results, however, we needed an athat is significantly smaller than R, which itself must already be small. Since a small value of aconsiderably slows down the computation of the estimator from (4.6), this is a substantial disadvantage ... | https://arxiv.org/abs/2502.00092v1 |
to use more than only d+ 1 equations. However, the basic idea of our algorithm remains to estimate the Minkowski tensors of Kvia the Voronoi tensors of a finite set K0. Here K0can be viewed as a finite approximation of a possibly infinite set K. In practice K0is usually considered to be a finite sample of points in K, ... | https://arxiv.org/abs/2502.00092v1 |
input b. In particular, if Kis a set with positive reach and Rnis less then the reach of K, then the tensors bΦr,s j(K0, a, R ) from (4.15) converge to Φr,s j(K) asbVr,s R(K0, a) converges to Vr,s R(K), for which Theorem 4.3 provides sufficient conditions. Note that the approximating tensors bΦr,s j(K0, a, R ) depend o... | https://arxiv.org/abs/2502.00092v1 |
that the tensors are symmetric and therefore the algorithm does not estimate all the values since some of them are redundant. For a tuple of indices ( i1, . . . , i r+s)∈ {1, . . . , d }r+swith i1≤ ··· ≤ ir+sthe algorithm estimates the value of the tensor corresponding to these indices, but not those corresponding to p... | https://arxiv.org/abs/2502.00092v1 |
the distances Ri+1−Ribetween the values of consecutive radii should not be smaller than the average nearest neighbor distance in the test data. We often used the value n= 50. The choice of the parameter Rnis much more intricate. The theory requires the value of Rnto be smaller than the reach of the original set Kthat i... | https://arxiv.org/abs/2502.00092v1 |
2]×[−a2 2,a2 2]⊂R2, where the values of Φ1,0 k, Φ0,1 kvanish for all k∈ {0,1,2}. Tensor Formula Tensor Formula Tensor Formula Φ0,0 0 1 Φ0,0 1 a1+a2 Φ0,0 2 a1·a2 Φ0,2 01 4π e2 1+e2 2 Φ0,2 11 4π a2e2 1+a1e2 2 Φ0,2 2 0 Φ2,0 01 8 a2 1e2 1+a2 2e2 2 Φ2,0 11 24P2 i=1 a3 i+ 3a2 iaj e2 iΦ2,0 21 24 a3 1a2e2 1+a3 2a1e2 2... | https://arxiv.org/abs/2502.00092v1 |
7] have been used to determine the integral. Ifk=d−1, this specializes to Φr,s d−1(A) =1{r+seven}2ωd+r+s r!s!ωs+1ωr+s+1 ρr+d−1 2 + (−1)sρr+d−1 1 Qr+s 2, and if k=d−1 and r= 0, we obtain Φ0,s d−1(A) =1{seven}2ωd+s s!ω2 s+1 ρd−1 1+ρd−1 2 Qs 2. Table 5 shows simulation results for the 2-dimensional spherical shell wit... | https://arxiv.org/abs/2502.00092v1 |
EVk(Z)<∞. From Z SO(d)(ρv)sν(dρ) =1 ωdZ Sd−1wsHd−1(dw) =1{seven}2ωd+s ωdωs+1Qs 2 (see, e.g., [27, Lemma 7] and the reference given there) we deduce that EΦ0,s k(Z) =1{seven}2ωd+s s!ωdωs+1ωd−k+sQs 2EX F∈Fk(Z)Vk(F)Hd−k−1(N(Z, F)∩Sd−1) =1{seven}2ωd+sωd−k s!ωdωs+1ωd−k+sEVk(Z)·Qs 2, which proves the assertion. We consider t... | https://arxiv.org/abs/2502.00092v1 |
gives the results for the expectation of the tensor Φ0,2 3(Pβ 10,4)i,j. 27 Table 7: Results for 2- and 4-dimensional beta-polytopes Pβ 10,d. We intersected the simulated polytope with a grid of resolution 0.005 (resp. 0.02 for d= 4). The parameter choices were n= 50 and Rn= 1 (resp. Rn= 0.5 for d= 4). We took the avera... | https://arxiv.org/abs/2502.00092v1 |
the expectation Eh Φ0,2 3(Pβ 10,4)i,ji . β i j Exact values Algorithm β i j Exact values Algorithm 1 21 1 0.016. . . 0.010(2) −1 21 1 0.027. . . 0.022(3) 1 22 2 0.016. . . 0.011(1) −1 22 2 0.027. . . 0.020(3) 1 23 3 0.016. . . 0.011(1) −1 23 3 0.027. . . 0.022(4) 1 24 4 0.016. . . 0.014(2) −1 24 4 0.027. . . 0.019(2) T... | https://arxiv.org/abs/2502.00092v1 |
[60]. 6.2 Nanorough surfaces Next, we apply our algorithm to a distinctly different experimental data set, a nanorough surface from [59], measured via atomic force microscopy (AFM). The roughness at the nano scale, i.e., an order of magnitude above atomic resolution, was obtained by a specific etching protocol; see [59... | https://arxiv.org/abs/2502.00092v1 |
was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the SPP 2265, under grant numbers HU 1874/5-1, ME 1361/16-1, KL 3391/2-2, WI 5527/1-1, and LO 418/25-1, as well as by the Helmholtz Association and the DLR via the Helmholtz Young Investigator Group “DataMat”. Referen... | https://arxiv.org/abs/2502.00092v1 |
B¨ ohlke. Characterizing digital microstructures by the Minkowski-based quadratic normal tensor. Mathematical Methods in the Applied Sciences , 46(1):961–985, 2023. 2 [18] J. R. Gott III, C. Park, R. Juszkiewicz, W. E. Bies, D. P. Bennett, F. R. Bouchet, and A. Stebbins. Topology of microwave background fluctuations - ... | https://arxiv.org/abs/2502.00092v1 |
Imaging , volume 2177 of Lecture Notes in Mathematics , pages 385–421. Springer International Publishing, Cham, 2017. 2, 29 [35] M. A. Klatt and K. Mecke. Detecting structured sources in noisy images via Minkowski maps. EPL, 128:60001, 2020. 2 [36] M. A. Klatt, G. E. Schr¨ oder-Turk, and K. Mecke. Anisotropy in finite ... | https://arxiv.org/abs/2502.00092v1 |
Notices of the Royal Astronom- ical Society , 297:355–365, 1998. 2 [53] R. Schneider. Convex Bodies: The Brunn–Minkowski Theory . Cambridge University Press, second expanded edition, 2014. 5 35 [54] R. Schneider. Valuations on convex bodies: the classical basic facts. In Tensor valuations and their applications in stoc... | https://arxiv.org/abs/2502.00092v1 |
for a 2-dimensional rectangle with rounded vertices. Specif- ically, we mean the parallel set of a rectangle. The parallel set of a compact set Kwith parameter r0is the set of all points with a distance less than r0toK. •Table 13: Simulation results for the 3-dimensional rectangle [ −1 2,1 2]×[−1,1]×[−3 2,3 2]. 37 Tabl... | https://arxiv.org/abs/2502.00092v1 |
A Bayesian decision-theoretic approach to sparse estimation By Aihua Li Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, North Carolina 27708, U.S.A. aihua.li@duke.edu Surya T. Tokdar Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, North Carolina 27708, U.S.A... | https://arxiv.org/abs/2502.00126v1 |
shrinkage and selection”. They considered the use of a shrinkage prior in the first stage for computational ease but burdened with the liability that no expression of sparsity is offered by the corresponding posterior. This necessitated a second stage where a loss function is brought into the picture to produce a spars... | https://arxiv.org/abs/2502.00126v1 |
a posterior benchmarking criterion, which has easier interpretation and broader applicability than the criterion used by Hahn & Carvalho (2015). Second, we utilize the two-stage framework to augment prevailing decision-theoretic ap- proaches to Bayesian sparse learning reliant on model selection. A common practice alon... | https://arxiv.org/abs/2502.00126v1 |
methods for both stages are also discussed, along with a new criterion function for choosing the tuning parameter introduced in Section 2.4. Section 3 introduces an adaptive thresholding approach by framing Bayesian model selection by mpm within the two-stage formulation. Section 4 presents results from a numerical stu... | https://arxiv.org/abs/2502.00126v1 |
agnostic to model or prior specification, and whose objectives are articulated through the design of the loss function. Two primary advantages of doing so are (1) the choice of the prior does not need to be constrained by the estimation objective, and (2) multiple estimation tasks can be performed simultaneously under ... | https://arxiv.org/abs/2502.00126v1 |
often forces the lasso estimate to include irrelevant variables with small coefficient estimates and thus an excessive number of false discoveries (e.g., see Song & Liang, 2015 and Fan & Li, 2001). The same issue arises when applying the 𝑙1penalty within the bdframework. One popular solution to the above problem is to... | https://arxiv.org/abs/2502.00126v1 |
mo- tivated by Zou (2006) and Cand `es et al. (2008) who use weights inversely proportional to Sparse estimation by Bayesian decision theory 7 the true signal magnitude. Here 𝜖is a small nonnegative constant introduced to guarantee the existence of the posterior expectation, and to bring in computational stability. No... | https://arxiv.org/abs/2502.00126v1 |
with fixed prior parameters𝜋0=0.5,𝜎2=1. The parameters used for illustration are 𝜆=2 for the𝑙1 penalty,𝜆=6 for the fdpenalty,𝜆=0.01 and𝜖=0.001 for the ispenalty. Each plot shows a 45-degree dotted reference line. 𝜆ˆ𝑤𝑖/2 from the posterior mean ¯𝛽𝑖towards zero. Furthermore, the continuity of the estimate (12... | https://arxiv.org/abs/2502.00126v1 |
posterior expected squared prediction error. We denote this benchmark as E=inf 𝑏∈R𝑝𝐸{∥𝑋𝛽−𝑋𝑏∥2 2|𝑦}=𝐸{∥𝑋𝛽−𝑋¯𝛽∥2 2|𝑦}. Increased sparsity leads to diminished predictive power compared to the dense estimate ¯𝛽, as reflected in the posterior expectation of E𝜆. To determine an appropriate balance, we propose... | https://arxiv.org/abs/2502.00126v1 |
optimal prediction under the orthogonal design (Barbieri & Berger, 2004), making it a popular Bayesian approach to sparse learning. However, it has been observed that the performance of the mpm rule deteriorates in high- dimensional, highly correlated settings. In such cases, the explanatory power is distributed across... | https://arxiv.org/abs/2502.00126v1 |
(𝑗=1,...,𝑝). Thus, the decision rule under the loss (14) boils down to a thresholding approach with the threshold determined by the tuning parameter 𝜆: all and only variables with pips greater than or equal to𝜆are selected. We call ˆ 𝛾apm 𝜆theadaptive probability model ( apm), emphasizing its contrast with mpm wh... | https://arxiv.org/abs/2502.00126v1 |
performs the best. Posterior sampling was performed by using the R package scaleBVS (Zanella & Roberts, 2019) for the spike-and-slab prior (10) and the R package horseshoe (Bhattacharya et al., 2016) for the horseshoe prior (11). Both implementations assume error variance prior 𝑝(𝜎2)∝1/𝜎2. Forbd, the optimization pr... | https://arxiv.org/abs/2502.00126v1 |
FDP TPP MPM 20.00 (0.00) 1.11 (0.12) 0.00 (0.00) 1.00 (0.00) 3.80 (1.46) 19.02 (74.8) 0.01 (0.04) 0.19 (0.07) APM 19.31 (0.46) 2.21 (0.79) 0.00 (0.00) 0.97 (0.02) 7.52 (1.89) 4.87 (1.40) 0.04 (0.07) 0.36 (0.09) BD-𝑙1(SS) 46.34 (7.38) 1.62 (0.56) 0.56 (0.08) 1.00 (0.00) 70.94 (7.20) 4.66 (0.69) 0.82 (0.03) 0.65 (0.10) ... | https://arxiv.org/abs/2502.00126v1 |
for strong predictive performance but high false discovery rate – Sparse estimation by Bayesian decision theory 15 while selecting noticeably fewer but more relevant predictors. On the other hand, if one prioritizes a lower false discovery proportion, then apm would be the most preferable. 5. Double regularization and ... | https://arxiv.org/abs/2502.00126v1 |
Iterative 𝑙1algorithm, despite their similar penalty constructions. While their performance was comparable in low-correlation settings ( 𝜌=0.3),bdwith ispenalty demonstrated a clear advantage in high-correlation settings ( 𝜌=0.95), achieving a significantly better fdp-tpptrade- off. Recall that both methods aim to d... | https://arxiv.org/abs/2502.00126v1 |
𝑘=20. The third row shows the trade-off between fdpandtpp. associated with TRIM32. Thus, this dataset presents a high-dimensional, correlated (as shown in Fig. 8), and sparse estimation problem. In this section, we apply and compare the proposed method as well as existing approaches to this problem. To allow for perfo... | https://arxiv.org/abs/2502.00126v1 |
the potential importance of the following probes: Probes 1389910 at and 1388491 at were selected by all three approaches, suggesting a significant association with TRIM32 expression; probe 1389910 at, in particular, had the largest absolute coefficient estimate. Additionally, probe 1377836 at was picked up by both bdap... | https://arxiv.org/abs/2502.00126v1 |
direction is to examine the convergence properties of our approach relative to the minimax convergence rate established by Donoho et al. (1992), which is a widely recognized standard for assessing sparse estimators in both frequentist (e.g., Abramovich et al., 2006; Su & Cand `es, 2016) and Bayesian (e.g., van der Pas ... | https://arxiv.org/abs/2502.00126v1 |
54, 41–67. Efron, B. (2007). Size, power and false discovery rates. The Annals of Statistics 35. Efron, B. ,Hastie, T. ,Johnstone, I. &Tibshirani, R. (2004). Least angle regression. The Annals of Statistics 32. Efron, B. &Tibshirani, R. (2002). Empirical bayes methods and false discovery rates for microarrays. Genetic ... | https://arxiv.org/abs/2502.00126v1 |
l1-regularization. Journal of the American Statistical Association 110, 1607–1620. Storey, J. D. (2003). The positive false discovery rate: a Bayesian interpretation and the q-value. The Annals of Statistics 31, 2013 – 2035. Storey, J. D. (2011). False discovery rate. International encyclopedia of statistical science 1... | https://arxiv.org/abs/2502.00126v1 |
Supervised Quadratic Feature Analysis: An Information Geometry Approach to Dimensionality Reduction Daniel Herrera-Esposito1Johannes Burge1 Abstract Supervised dimensionality reduction aims to map labeled data to a low-dimensional feature space while maximizing class discriminability. Despite the availability of method... | https://arxiv.org/abs/2502.00168v2 |
we present Supervised Quadratic Feature Analysis (SQFA), a dimensionality reduction method for learning a set of linear features that support quadratic discriminability. To accomplish this aim, SQFA exploits a relation between class discriminability and the information geometry of sym- metric positive definite (SPD) ma... | https://arxiv.org/abs/2502.00168v2 |
reflect the quadratic discriminability of the classes in this space. 2.3.2. dAIIS INDUCED BY THE FISHER INFORMATION METRIC The affine-invariant distance is induced by the Fisher infor- mation metric for zero-mean Gaussian distributions (Atkin- son & Mitchell, 1981), which is a local measure of discrim- inability (see A... | https://arxiv.org/abs/2502.00168v2 |
again provides a proxy for quadratic discriminability (via similar reasoning as with zero-centered Gaussians; see Sec- tion 2.3.2). We refer to this more general method as SQFA. Unfortunately, there is no closed-form expression for the Fisher-Rao distance in this space (Miyamoto et al., 2024). However, a lower bound to... | https://arxiv.org/abs/2502.00168v2 |
prevented by adding a regularization term: Ψi= FTΦiF+Imσ2, where Imis the identity matrix and σ2is a regularization parameter and Φiis the second-moment ma- trix of the raw data for class i. Regularization makes training more stable, but it can mask relevant information in low- variance subspaces. σ2was set manually fo... | https://arxiv.org/abs/2502.00168v2 |
The most popular method of this kind is PCA, which finds the orthonormal linear filters F that maximize the variance of the projected data z=FTx. Although unsupervised methods do not use class labels, some assume that an unobserved low-dimensional latent variable generates the observed raw data. For example, FA assumes... | https://arxiv.org/abs/2502.00168v2 |
hand, learns a single low- dimensional set of features that maximizes second-order differences between all classes simultaneously. Other methods learn features that optimize for more com- plex criteria of class separation, which can be exploited by classifiers like k-Nearest Neighbors (kNN). Wasserstein Dis- criminant ... | https://arxiv.org/abs/2502.00168v2 |
and discriminable class-specific covariances, so these dimensions are favored by SQFA; because these dimensions contain no differences in means and moderate variance, they are non-preferred by PCA and drLDA. Dimensions 3-4 contain differences in the class- specific means, so they are preferred by drLDA, but these dimen... | https://arxiv.org/abs/2502.00168v2 |
SQFA and smSQFA show a more digit-like structure than those learned by other methods (Figure 4), indicating that they can extract meaningful fea- tures from complex data. The similarity between SQFA and smSQFA suggests that, for this dataset, the class means are not very informative. QDA accuracy is higher for the SQFA... | https://arxiv.org/abs/2502.00168v2 |
using the features learned by the different methods. 15 frames. The vertical dimension was averaged out; thus videos can be represented as 2D space-time plots (Figure 6). Each video shows a textured surface moving with one of 41 different speeds (i.e. classes). We learned 8 filters with each method. (See Appendix E for... | https://arxiv.org/abs/2502.00168v2 |
one of the shortcomings of QDA as compared to cLDA, is that it requires estimating a larger number of parameters, making it more prone to overfitting. SQFA has the potential to make QDA more practical by reducing the dimensionality of the data in a way that is useful for quadratic classification. A useful feature of SQ... | https://arxiv.org/abs/2502.00168v2 |
nance in Medicine , 56(2):411–421, 2006. ISSN 1522-2594. doi: 10.1002/mrm.20965. URL https://onlinelibrary.wiley.com/ doi/abs/10.1002/mrm.20965 . eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.20965. Atkinson, C. and Mitchell, A. F. S. Rao’s Distance Mea- sure. Sankhy ¯a: The Indian Journal of Statistics, ... | https://arxiv.org/abs/2502.00168v2 |
https: //www.jneurosci.org/content/early/ 2024/11/25/JNEUROSCI.0490-24.2024 . Pub- lisher: Society for Neuroscience Section: Research Articles. Huang, Z., Wang, R., Shan, S., Li, X., and Chen, X. Log- Euclidean Metric Learning on Symmetric Positive Defi- nite Manifold with Application to Image Set Classifica- tion. In ... | https://arxiv.org/abs/2502.00168v2 |
Machine Intelligence. Moreno-Bote, R., Beck, J., Kanitscheider, I., Pitkow, X., Latham, P., and Pouget, A. Information-limiting 10 SQFA: An Information Geometry Approach to Dimensionality Reduction correlations. Nature Neuroscience , 17(10):1410– 1417, October 2014. ISSN 1546-1726. doi: 10. 1038/nn.3807. URL https://ww... | https://arxiv.org/abs/2502.00168v2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.