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Rdefined by ˜R(u) = sup v∈(0,u)R(v)for each u∈(0,1). For each n∈N, let ˜Yn=F(Yn). Then (˜Yn)is iid with quantile function ˜R. Proof. Plainly ( ˜Yn) is iid, so it suffices to show that ˜Ynhas quantile function ˜R. Let Ube a random variable uniformly distributed on (0 ,1). To show that ˜Ynhas quantile function ˜R, it suf... | https://arxiv.org/abs/2502.01254v2 |
converges in outer probability to zero because ( ζ∗ n) is a bootstrap version in outer probability of ( ζn) w.r.t the convergence in (5.6). Therefore, ( ζ∗ n) is also a bootstrap version in outer probability of ( ζn) w.r.t the convergence in (5.4). This proves (i). Next we prove (ii). Define the sequence of random vari... | https://arxiv.org/abs/2502.01254v2 |
differentiable at θ:= (F, θ2) tangentially to (SF⊗L1(0,1))⟨D[−∞,∞]⊗L1(0,1)⟩. Its quasi-Hadamard derivative ˙ψθ: SF⊗L1(0,1)→L1(0,1)is given by ˙ψθ(g, h) =g(F−1)◦θ2+h. The last equality defines a continuous, linear, and ball measurable extension of ˙ψθto all of D[−∞,∞]⊗L1(0,1). Proof. The map ( g, h)7→g(F−1)◦θ2+hfrom D[−... | https://arxiv.org/abs/2502.01254v2 |
equality F−1(F(xi)) =xi, we obtain 1 nnX i=11 xi≤F−1(F(z)) =1 nnX i=11(xi≤z) for all z∈[−∞,∞]. If each xiis replaced with Xithen the last equality becomes Fn(F−1(F(z))) = Fn(z). Thus claim (i) follows from the fact that F−1(F(Xi)) = Xifor all i∈Na.s., a conse- quence of Lemma B.1(iv). An obvious modification to the p... | https://arxiv.org/abs/2502.01254v2 |
we now have an∥Rn−R∥1≤√nx∥Fnx−F∥∞+√ny∥Hny−H∥∞. Thus we are done if we can show that the sequences of random variables√n∥Fn−F∥∞ and√n∥Hn−H∥∞,n∈N, are uniformly integrable. But this is well-known: by applying the Dvoretzky-Kiefer-Wolfowitz inequality (see e.g. van der Vaart, 1998, p. 268) we obtain E √n∥Fn−F∥∞2=Z∞ 02tP ... | https://arxiv.org/abs/2502.01254v2 |
This justifies using the dominated convergence theorem to write EZb a|Z(u)|du= lim k→∞EZbk ak|Z(u)|du. (5.23) We have ER1 0|B(u)|dR(u)<∞by Lemma 4.3(ii) because Ris uniformly bounded. This justifies using the dominated convergence theorem to write EZ [a,b]|B(u)|dR(u) = lim k→∞EZ [ak,bk)|B(u)|dR(u). (5.24) Plainly, we a... | https://arxiv.org/abs/2502.01254v2 |
curve Rndefined in Section 5. Let Y1,ny, . . . , Y ny,ny be the ascending order statistics for Y1, . . . , Y ny. The map Rn: Ω→L1(0,1) sends each ω∈Ω to the function in L1(0,1) that sends each u∈(0,1) to 1 nxnxX i=1nyX j=11(Xi(ω)≤Yj,ny(ω)) 1((j−1)/ny< u≤j/ny). CONVERGENCE IN DISTRIBUTION OF QUANTILE AND P-P PROCESSES 3... | https://arxiv.org/abs/2502.01254v2 |
space. Equip L1 loc(−∞,∞) with the metric d1 loc(g, h) =∞X n=11 2nRn −n|g(x)−h(x)|dx 1 +Rn −n|g(x)−h(x)|dx. Lemma 5.17 in Meise and Vogt (1997) establishes that the last equality defines a valid metric, and that under this metric L1 loc(−∞,∞) is complete and separable. Proof of Lemma A.1. We will apply Theorem 15.1 in ... | https://arxiv.org/abs/2502.01254v2 |
θ: (0,1)→Rtakes values only in [0 ,1]. This assumption is made without loss of generality because θ(u)∈ran(F) for a.e. u∈(0,1) by the definition of LF, and because the vector g(F−1)◦θis unaffected by modifications to θon a null set. We write g(F−1(θ)) rather than g(F−1)◦θin what follows. Fix a function h∈L1(0,1). Since... | https://arxiv.org/abs/2502.01254v2 |
satisfiesR1 0|h(u)|dQ(u)<∞or is nonnegative. Then Z [a,b)h(u)dQ(u) =ZQ(b) Q(a)h(F(x))dxfor all a, b∈(0,1)satisfying a < b, andZ1 0h(u)dQ(u) =ZβF αFh(F(x))dx. It is straightforward to prove Lemma B.2 using Theorem 3.6.1 in Bogachev (2007) and Lemma B.1(i). We omit the details. See also Theorem 6.14 in Leoni (2017). CONV... | https://arxiv.org/abs/2502.01254v2 |
Society. [DOI] Mason, D.M. (1984). Weak convergence of the weighted empirical quantile process in L2(0,1).Annals of Probability , 12(1):243–55. [DOI] Meise, R., Vogt, D. (1997). Introduction to Functional Analysis . Oxford University Press. [DOI] Pollard, D. (1984). Convergence of Stochastic Processes . Springer. [DOI]... | https://arxiv.org/abs/2502.01254v2 |
Inference of Half Logistic Geometric Distribution Based on Generalized Order Statistics Neetu Guptaa, S. K. Neogya, Qazi J. Azhadb1, Bhagwati Devic aIndian Statistical Institute, Sanswal Marg, New Delhi. bDepartment of Mathematics, Shiv Nadar Institution of Eminence, Dadri, India cDepartment of Statistics, Central Univ... | https://arxiv.org/abs/2502.01255v1 |
̸=−1 −ln (1−x), m =−1 and gm(x) =hm(x)−hm(0), x∈[0,1). Since its emergence, numerous researchers have included the concept of gosin their work such as recurrence relation of moments, estimation of parameters, distributional properties for various distributions etc. (see Aboeleneen (2010), Burkschat (2009), Gupta and Ja... | https://arxiv.org/abs/2502.01255v1 |
steam flow and precipitation (in hydrology), and the distribution of wealth or income (in economics). Having remarkable properties and applications in various areas, some of the researchers argued of the left-hand limit of this distribution approach to negative infinity. They found it inappropriate in many situations o... | https://arxiv.org/abs/2502.01255v1 |
model of gos. In Section [4], approximate Bayes estimators are obtained with the aid of Lindley approximation and Markov chain Monte Carlo method. In Section [5], a simulation study is reported for the sub model of gosi.e., order statistics with detailed analysis. In section [6], two real data applications are given fr... | https://arxiv.org/abs/2502.01255v1 |
0etx¯F(x)γr+1gr−1 mF(x)dx After integrating Z(x) by parts, we obtain Z(x) =γr t[Mr,n,m,k (t)−Mr−1,n,m,k (t)] In similar way, integrating Z∗(x), we obtain Z∗(x) =γn+1,k−m r C∗ t[Mr,n+1,m,k−m(t)−Mr−1,n+1,m,k−m(t)] where C∗=cr−1 cn+1,k−m r−1. Substituting Z(x) and Z∗(x) in (2.8), we obtain tMr,n,m,k (t) =γr[Mr,n,m,k (t)... | https://arxiv.org/abs/2502.01255v1 |
(s−r−1) !Z∞ 0et1x¯F(x)mf(x) ×gr−1 m(F(x))I(x)dx. (3.7) I(x) =Z∞ xet2y[hmF(y)−hmF(x)]s−r−1¯F(y)γs−1f(y)dy or, making use of (1.9) in I(x), I(x) =W(x)−2−θ 2 W∗(x) (3.8) where, W(x) =Z∞ xet2y[hmF(y)−hmF(x)]s−r−1¯F(y)γsdy Integrating by parts , we have s≥r+ 1 W(x) =γs t2Z∞ xet2y[hmF(y)−hmF(x)]s−r−1¯F(y)γs−1f(y)dy... | https://arxiv.org/abs/2502.01255v1 |
two parameter gamma distribution and is defined as π(θ) =ba Γaθa−1e−bθ, a, b > 0. (4.5) 9 Now, the joint posterior density of θis obtained by using (4.1) and (4.5), and is given as π(θ|x)∝kθa2(m+1)(n−1)+k n−1Y j=1γj nY i=1 θ+ (2−θ)e−xi−(m+2)(k+1)! ×exp" −n−1X i=1xi(m+ 1)−kxn−bθ# ,0< θ < 1. (4.6) It is observed t... | https://arxiv.org/abs/2502.01255v1 |
, θT). Now, after employing algorithm and obtaining samples of θ,we calculate the Bayes estimators ofθas,δθ SEL=1 TPT i=1θi,δθ LINEX =−1 cln1 TPT i=1e−cθi andδθ GE=1 TPT i=1(θi)−c−1/c under SELF, LINEX and GE loss functions, respectively. 5 Simulation Study In this section, a simulation study is conducted to monito... | https://arxiv.org/abs/2502.01255v1 |
0.003981 0.005776 0.665214 0.065214 0.018427 Prior II θ nLindley MCMC AE AB MSE AE AB MSE 0.3 10 0.368686 0.068686 0.011495 0.425618 0.125618 0.030661 0.3 20 0.326848 0.026848 0.004064 0.391349 0.091349 0.021955 0.3 30 0.312646 0.012646 0.002435 0.370522 0.070522 0.016353 0.3 40 0.305899 0.005899 0.001785 0.356750 0.05... | https://arxiv.org/abs/2502.01255v1 |
0.000007 0.590195 -0.009805 0.010757 c=1 0.3 10 0.323031 0.023031 0.000531 0.430095 0.130095 0.040656 0.312836 0.012836 0.000165 0.361615 0.061615 0.018591 0.3 30 0.308239 0.008239 0.000068 0.365679 0.065679 0.018885 0.304887 0.004887 0.000024 0.340646 0.040645 0.012062 0.3 50 0.306126 0.006126 0.000038 0.341443 0.0414... | https://arxiv.org/abs/2502.01255v1 |
in the field of reliability, we have considered a lifetime data of traction motors. Jung and Bai (2007) investigated the eligibility of warranty claim of the same dataset by assuming that a two dimensional warranty has been provided by the manufacture. The data set is taken from the maintenance records of a type of loc... | https://arxiv.org/abs/2502.01255v1 |
Bayes estimator of unknown quantity of HLG distribution. We have considered the case of order statistics to discuss the behaviour of derived estimators. It has been observed that the Bayes estimators based on Lindley approximation are perform- ing better than MCMC estimators. Inference can be made that as we increase t... | https://arxiv.org/abs/2502.01255v1 |
Simulation and Computation , pages 1–19. Arshad, M., Khandelwal, N., and Azhad, Q. J. (2022). Stress–strength reliability estimation in a system with p-type non-identical multicomponents from prhr family based on records. Journal of Statistical Computation and Simulation , pages 1–32. Azhad, Q. J., Arshad, M., and Khan... | https://arxiv.org/abs/2502.01255v1 |
of Applied Statistical Analysis , 6(2). Singh, B., Khan, R., and Khan, M. A. (2021). Exact moments of generalized pareto distribution based on generalized order statistics and characterizations. Pakistan Journal of Statistics and Operation Research , pages 213–226. Verhulst, P. (1845). La loi d’accroissement de la popu... | https://arxiv.org/abs/2502.01255v1 |
arXiv:2502.01271v1 [math.ST] 3 Feb 2025On tail dependence parameters for non-continuous and autocorrelated margins Victory Idowu January 2025 Abstract Tail dependence plays an essential role in the characteriza tion of joint extreme events in multivariate data. However, most standar d tail depen- dence parameters assum... | https://arxiv.org/abs/2502.01271v1 |
shrinking tail regions. In particular, our approach has the inter- pretation of quantifying tail dependence as the limit over the contin uous ranges of the copula’s domain. This paper proceeds as follows. In Section 2, we provide a brief over view of key concepts from copula theory and tail dependence. In Section 3 we ... | https://arxiv.org/abs/2502.01271v1 |
can define the simplified standard Bivariate Regular Variation [2, 3]. Continuing from Definition (4), let X,Ybe a non-negative pair of random variables with right-continuous cd fF andU=UX=UY. We say there is standard bivariate variation if the following limit exists, lim x→∞xP/parenleftbigg(X,Y) U(x)∈B/parenrightbigg =ν(... | https://arxiv.org/abs/2502.01271v1 |
of Fon itself and V˜Cbe its corresponding volume. ˜λXi+h|Xi= lim t→1−V˜C([t,1]×[t,1]) 1−t 5 Lemma 3.2. ˜λXi+h|Xiis well defined. Proof.Proof is immediate from Theorem 3.1 by setting ˜C=C. We can show the equivalence of ˜λUin terms of regular variation on VC. Theorem 3.2. For arbitrary w,z∈R+, letB= [w,∞)×[z,∞). LetCbe a... | https://arxiv.org/abs/2502.01271v1 |
WRAPPED GAUSSIAN ON THE MANIFOLD OF SYMMETRIC POSITIVE DEFINITE MATRICES THIBAULT de SURREL LAMSADE, CNRS, PSL University Paris-Dauphine, France FABIEN LOTTE Inria center at the University of Bordeaux / LaBRI, France SYLVAIN CHEVALLIER TAU, LISN, University Paris-Saclay, France FLORIAN YGER LITIS, INSA de Rouen-Normand... | https://arxiv.org/abs/2502.01512v3 |
their theoretical properties in Section 4. In Section 5, we develop a Maximum Likelihood Estimator for parameter estimation and validate it with synthetic experiments. Building on this foundation, Section 6 revisits existing classifiers onPdthrough a probabilistic lens and introduces novel classifiers based on wrapped ... | https://arxiv.org/abs/2502.01512v3 |
parameters from a finite number of samples. Finally, we use our wrapped Gaussian to build a framework that unifies and generalizes classification on SPD matrices, and propose new classifiers. This application shows the potential of our wrapped Gaussian to become a generic, flexible and powerful tool for manifold-based ... | https://arxiv.org/abs/2502.01512v3 |
Pd, we define the vectorization at p: Vect p:u∈TpPd7→Vect Id(p−1/2up−1/2). One of the important property of Vect pis that it is an isometry between (TpPd,⟨·,·⟩p)and (Rd(d+1)/2,⟨·,·⟩2). More information on this vectorization can be found in Section 3.3.3.3. of Pennec (2020) or in Appendix A. 4.Wrapped Gaussian on the ma... | https://arxiv.org/abs/2502.01512v3 |
(2020) or Appendix C). Proposition 4.3. The Jacobian determinant of the exponential map at the identity ExpIdis: ∀u∈TIdPd, JId(u) = 2d(d−1)/2Y i<jsinhλi(u)−λj(u) 2 λi(u)−λj(u) where the λi(u)are the eigenvalues of u. Then, one can use the previous formula to compute the Jacobian determinant of the exponential map at ... | https://arxiv.org/abs/2502.01512v3 |
a mean, or exponential barycenter, of a probability distribution αonPdis defined as a point ¯p∈ PdsatisfyingR PdLog¯p(x)dα(x) = 0 .For wrapped Gaussians, one has the following result: Proposition 4.8. A mean of WG(p; 0,Σ)is p. The proof is straightforward from the definition of the wrapped Gaussian WG(p; 0,Σ). 1More in... | https://arxiv.org/abs/2502.01512v3 |
µminis minimal in the sens of∥ · ∥ 2. We call it the minimal representative . One is able, given a tuple of parameters θ= (p, µ,Σ), to compute the minimal representative of the class [θ]of equivalent tuples of parameters using the following proposition: Proposition 4.14. Letθ= (p, µ,Σ)∈Θbe parameters. Then, the minimal... | https://arxiv.org/abs/2502.01512v3 |
that the covariance matrix Σis diagonal, which reduces the number of coefficients to O(d2). This assumption implies independent entries of the SPD matrices and will be used in Section 6.4. In other works, such as in Chevallier et al. (2022) or in Chevallier and Guigui (2020), the authors use the method of moments (see ... | https://arxiv.org/abs/2502.01512v3 |
on Pdcan be integrated into a probabilistic framework. We also introduce new classifiers based on the wrapped Gaussian and conduct experiments on real-world data from various applications. 6.1.Classifiers used for SPD matrices. MDM.TheMinimum Distance to Mean (MDM) algorithm described in Barachant et al. (2010) is a po... | https://arxiv.org/abs/2502.01512v3 |
back from the Euclidean space which is not the case of the AIRM metric we use in our work. Several deep learning approaches have been proposed to classify SPD matrices Huang and Van Gool (2017); Brooks et al. (2019); Nguyen (2021). However, most of these approaches distort the geometry of the manifold and are out of th... | https://arxiv.org/abs/2502.01512v3 |
of this classifier where each class has its own covariance matrix Σk:αk=WG(pk;µk,Σk).We call this classifier the Heterogeneous Wrapped Discriminant Analysis (He-WDA). In that case, an MLE is optimized on each class individually, as in Section 5: ∀k∈ {1, ..., K},(pk, µk,Σk)∈argmax p,µ,ΣNkY i=1fp;µ,Σ(xk i). 6.4.Experimen... | https://arxiv.org/abs/2502.01512v3 |
other hand, estimates a single covariance matrix for all the classes and is less sensitive to the number of samples per class. 7.Conclusion In this work, we present a generalization of non-isotropic multivariate Gaussians on the manifold of SPD matrices: Wrapped Gaussians , and we give some theoretical properties. We s... | https://arxiv.org/abs/2502.01512v3 |
clearly unethical use, e.g., for manipulation. References Aristimunha, B., Carrara, I., Guetschel, P., Sedlar, S., Rodrigues, P., Sosulski, J., Narayanan, D., Bjareholt, E., Quentin, B., Schirrmeister, R. T., Kalunga, E., Darmet, L., Gregoire, C., Abdul Hussain, A., Gatti, R., Goncharenko, V., Thielen, J., Moreau, T., ... | https://arxiv.org/abs/2502.01512v3 |
Guigui, N. (2020). Wrapped statistical models on manifolds: Motivations, the case SE(n), and generalization to symmetric spaces. In Joint Structures and Common Foundations of Statistical Physics, Information Geometry and Inference for Learning , Les Houches, France. Chevallier, E., Li, D., Lu, Y., and Dunson, D. (2022)... | https://arxiv.org/abs/2502.01512v3 |
A., Linnik, I., and Rao, C. (1973). Characterization Problems in Mathematical Statistics . Probability and Statistics Series. Wiley. Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis , 88(2):365–411. Lee, J. M. (2018). Introduction t... | https://arxiv.org/abs/2502.01512v3 |
brain-computer communication. Proceedings of the IEEE , 89(7):1123–1134. Rußwurm, M., Pelletier, C., Zollner, M., Lefèvre, S., and Körner, M. (2020). Breizhcrops: A time series dataset for crop type mapping. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences ISPRS (2020) . Sai... | https://arxiv.org/abs/2502.01512v3 |
one can unveil an orthonormal basis ofTpPd. One can see that the tangent space at the identity TIdPdis the classical Euclidean space Sdequipped with the Frobenius inner product. Therefore, one can easily build an orthonormal basis of TIdPdand then, transport it to the other tangent spaces. Proposition A.1 (Orthonormal ... | https://arxiv.org/abs/2502.01512v3 |
at Equation (3) and the expression of the AIRM distance given at Equation (2): ∥Vect p(Logpu)∥2 2=∥Logpu∥2 p=∥p−1/2Logpu p−1/2∥F=∥log(p−1/2up−1/2)∥F=δ(p, u)2. □ Finally, let us give a consequence of the previous property on the Jacobian of the vectorization: asVect pis an isometry, there is no volume change viathe vect... | https://arxiv.org/abs/2502.01512v3 |
formula, one can compute Ψ(EId,ij)fori≤j. Now, one needs to compute the coefficients of Ψ(EId,ij)in the basis (Eexp(u),kl)k≤l. As the basis (Eexp(u),kl)k≤lis orthonormal, one simply needs to compute the dot product between Ψ(EId,ij)and one element of the basis to 20 get the corresponding coefficient. For k≤l, one has: ... | https://arxiv.org/abs/2502.01512v3 |
p(p1/2log(y)p1/2) =Vect Id(p−1/2p1/2log(y)p1/2p−1/2) =Vect Id(log(y)) andLogId= log, we have, E[φ(Y)] =Z Pdφ(y)1p (2π)ddet Σexp −1 2(Vect Id(LogId(y))−µ)⊤Σ−1(Vect Id(LogId(y))−µ) |JId(log(y))|dvol(y). This shows us that Y∼WG(Id;µ,Σ). (2)Let now Y=Expp(Logp−Vect−1 p(µ)). We want to show that Y∼WG(p; 0d(d+1)/2,Σ). Letφ... | https://arxiv.org/abs/2502.01512v3 |
CLT. The previous version of the wrapped Central Limit Theorem is centered around the identity matrix Id. However, one can generalize this theorem to any point p∈ Pd. For this, we need to introduce a generalized logarithmic product ⊙pbetween two points q1, q2∈ Pd: q1⊙pq2= Expp(Logpq1+ Logpq2). In the same way as before... | https://arxiv.org/abs/2502.01512v3 |
=λi p−1/2Logp(x)p−1/2 −λj p−1/2Logp(x)p−1/2 and therefore, this leads to: JId(u) =JId p−1/2Logp(x)p−1/2 =Jp Logp(x) . So the denominator of the density ˜fis the same as the denominator of the density fand therefore, the two densities are equal. □ Remark G.2.The function γ:t7→etpis actually the geodesic with ini... | https://arxiv.org/abs/2502.01512v3 |
function generates a random SPD matrix by generating a random matrix Aand then computing exp((¯X+s∗A)⊤(¯X+s∗A))where ¯Xandsare parameters chosen by the user. We set ¯X= 0.1Idands= 1. •Forµ⋆, we generate a random vector of size d(d+1)/2with values in [0,0.1]. •ForΣ⋆, we generate a random SPD matrix using the same functi... | https://arxiv.org/abs/2502.01512v3 |
all subject except one and tested on this last subject. •AirQuality : This dataset is from the Beijing Municipal Monitoring Center. It is a dataset of air quality monitored from 34 different sites in Beijing, China Hua et al. (2021). For each site, six atmospheric pollutants where recorded every hour: CO, NO 2, O3, PM ... | https://arxiv.org/abs/2502.01512v3 |
Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery Filip Kovačević∗Yihan Zhang†Marco Mondelli‡ February 4, 2025 Abstract Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections wi... | https://arxiv.org/abs/2502.01583v1 |
to design, efficient to compute, and effective in practice [ CCFM21 ]. However, such class of methods remains understudied for multi-index models, with existing results falling short of producing exact asymptotics [CM20] and being restricted to special cases, such as single-index (for which p= 1, corresponding to gener... | https://arxiv.org/abs/2502.01583v1 |
[ APVZ14 ,CM20], with the latter work proposing a spectral warm start that requires a sample size n≳d(log(d))deg(q), where deg(q) denotes the degree of the link. The area has witnessed a renewed interest in recent years, due to the connection of multi-index models with two-layer neural networks, and a quickly growing l... | https://arxiv.org/abs/2502.01583v1 |
a mix of tools from random matrix theory and the theory of AMP. In contrast, our approach is purely random matrix theoretic, and it allows us to handle a general class of multi-index models with arbitrary correlation among the signals. 3 Preliminaries Notation. Given a positive integer n, we use the shorthand [n] :={1,... | https://arxiv.org/abs/2502.01583v1 |
outputs the eigenvectors corresponding to the plargest eigenvalues of D, i.e., vD 1,···, vD p . We now make a simplifying assumption on W∗without loss of generality. Note that, for any W∗subject to Assumption (A2), by orthogonal invariance of the random design matrix A, the law of |⟨vD i,W∗v⟩| ∥W∗v∥2:i∈[p], v∈Sd−1 ... | https://arxiv.org/abs/2502.01583v1 |
[MM19]. Our second result characterizes the asymptotic performance, in terms of overlaps, of the eigen- vectors corresponding to the spectral outliers of D. 6 Theorem 4.2. In the setting of Theorem 4.1, let αk=αk+1=···=αk+m−1be solutions to (4.2) of multiplicity m, i.e., αk−1> α k> α k+m−2whenever k≥2, k+m−2≤j. Let E∞ ... | https://arxiv.org/abs/2502.01583v1 |
weak recovery at a fixed aspect ratio δ. Then, define the optimal weak recovery threshold as δc:= inf{δ∈]0,∞[:Tδ̸=∅}. (4.7) Finally, for random variables (s, y)∈Rp×Rjointly distributed according to (3.4), we use p(y|s)to denote the conditional density of ygiven s. Theorem 4.3. The optimal weak recovery threshold δcequa... | https://arxiv.org/abs/2502.01583v1 |
same for every j∈ {1,2}. As there is a single outlier in the model, we only plot vD 1, w∗ 1 and vD 1, w∗ 2 . In Figure 2, we consider a two-component mixed phase retrieval model: q(ξ1, ξ2, ε) =|ξε|, where the mixing variable εis{1,2}-valued with P(ε= 1) = 1−P(ε= 2) = 0.6. The prior distribution on w∗ 1, w∗ 2is chosen t... | https://arxiv.org/abs/2502.01583v1 |
and, for all eigenvectors va iofa, it holds that qva i̸= 0(this is the case for the matrix Dnin (6.1) as showed in Lemma A.2 in Appendix A). We start with a characterization of the eigenvalues λD 1, . . . , λD p, and note that λD isolves det(D−λIp) = deta−λIp q⊤ q P −λId−p = 0. For1≤i≤d−p, consider a function Li:R\... | https://arxiv.org/abs/2502.01583v1 |
be convenient for the asymptotic analysis of Section 6.2. Lemma 6.3. Let us assume that λ > λP 1. Then, for the diagonal elements of R(λ), it holds that R(λ)i,i=ai,i+1 L−1 i(λ), (6.8) 12 d dλR(λ)i,i=−1 L−1 i(λ)2L′ i(L−1 i(λ)). (6.9) Moreover, for the off-diagonal elements, we have 2R(λ)i,j= 2ai,j+1 L−1 i,j(λ)−1 L−1 ... | https://arxiv.org/abs/2502.01583v1 |
remaining p−jeigenvalues. As (4.2)has only jsolutions by assumption, it follows that ζδ(α)−λ∞ i(α) =˜L∞ i(λ∞ i(α))−λ∞ i(α) = 0 has no solutions for α > τandi > j. Denoting µ=λ∞ i(α), it further holds that ˜L∞ i(µ)−µ= 0 has no solutions for µ∈]λa∞ i, t∞ i[. Since lim µ→λa∞ i˜L∞ i(µ)−µ= +∞, it must be that ˜L∞ i(µ)−µ >0f... | https://arxiv.org/abs/2502.01583v1 |
of Proposition 6.2 are satisfied as in the proof of Proposition 6.4. Thus, it holds that hi=˜hiq 1−˜h⊤ id dλR(λD i)˜hi, (6.20) where ˜hi=hi ∥hi∥2is the unit norm eigenvector of R(λD i). Note that the vectors ˜hiare orthogonal. Furthermore, Proposition 6.4 gives that R(λD k)a.s.− − →R∞(αk),d dλR(λD k)a.s.− − →1 ζ′ δ(αk)... | https://arxiv.org/abs/2502.01583v1 |
This also implies 1 +ε > ΠE∞ k˜hi 2≥1−ε, hence ˜hi−h∞ i 2= ˜hi−ΠE∞ k(˜hi) ΠE∞ k(˜hi) 2 2 = ˜hi−ΠE∞ k(˜hi) ΠE∞ k(˜hi) 2+˜hi1− ΠE∞ k(˜hi) 2 ΠE∞ k(˜hi) 2 2 ≤ε 1−ε+ε 1−ε<4ε. Thus, (6.21) implies that, for large enough d, ˜h⊤ id dλR(λD k)˜hi−1 ζ′ δ(αk)h∞ i⊤d dαR∞(αk)h∞ i 2< c·ε, for some constant cindependent of ε. Plugging... | https://arxiv.org/abs/2502.01583v1 |
to statistical gaps in learning a two-layers neural network. J. Stat. Mech. Theory Exp. , (12):124023, 51, 2019. 2, 19 [APVZ14] Alexandr Andoni, Rina Panigrahy, Gregory Valiant, and Li Zhang. Learning sparse poly- nomial functions. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms , pa... | https://arxiv.org/abs/2502.01583v1 |
Alex Damian, Eshaan Nichani, Rong Ge, and Jason D Lee. Smoothing the landscape boosts the signal for sgd: Optimal sample complexity for learning single index models. InAdvances in Neural Information Processing Systems , volume 36, pages 752–784. Curran Associates, Inc., 2023. 2 [DPVLB24] Alex Damian, Loucas Pillaud-Viv... | https://arxiv.org/abs/2502.01583v1 |
2024. 2 [NJW01] Andrew Ng, Michael Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems , volume 14. MIT Press, 2001. 3 [OSSW24] Kazusato Oko, Yujin Song, Taiji Suzuki, and Denny Wu. Learning sum of diverse features: computational hardness and e... | https://arxiv.org/abs/2502.01583v1 |
large n, there are almost surely at least pelements in the array [z1, z2, . . .]that are non-zero. This follows from Assumption (A5) that P(Z= 0) <1, as done in the proof of [ LL20, Proposition 3.2]. Now, the i-th column of ZSis obtained by scaling a standard pdimensional Gaussian by zi. As the columns of Sare almost s... | https://arxiv.org/abs/2502.01583v1 |
a similar manner as before, we have that λj(Mµ1) =λj((Πr+ Πr⊥)Mµ1(Πr+ Πr⊥)) ≥λj1 2ΠrMµ1Πr+ Πr⊥Mµ1Πr⊥ +λd−p1 2ΠrMµ1Πr+ Πr⊥Mµ1Πr+ Π rMµ1Πr⊥ ≥λj1 2ΠrMµ1Πr+ Πr⊥Mµ1Πr⊥ −εµ1, where as µ1→λa i+it holds that εµ1→0with the same arguments as above. Note that each of the eigenvectors of1 2ΠrMµ1Πrcorresponding to a non-zero ... | https://arxiv.org/abs/2502.01583v1 |
the only if part of the statement. We denote the eigenvector corresponding toλD iasvD i=hi gi , where hi∈Rp,g∈Rd−p. It follows that DvD i=a q⊤ q Phi gi =λa jhi gi . Splitting this equation into pandd−pcoordinates gives ahi+q⊤gi=λa jhi, (A.6) qhi+Pgi=λa jgi. (A.7) Since (a−λa jIp)is singular, its SVD decompositi... | https://arxiv.org/abs/2502.01583v1 |
by checking the conditions Lk(µ) =µorlim µ→λa j+Lk(µ) =λa j, which are all covered by considering ˜Lm(µ)form∈[d−p]. As˜L1(µ)≥˜L2(µ)≥ ··· ≥ ˜Lp(µ)≥ ··· ≥ ˜Ld−p(µ)andλD 1≥λD 2≥. . . λD p, it must be that the solution to (6.4) is exactly the i-th eigenvalue of the matrix D, and the proof is complete. ■ We conclude this ap... | https://arxiv.org/abs/2502.01583v1 |
prove that (B.5)hasjsolutions for α >max{zi}, it equivalent to prove that, for each i, λi(β′ i)> µ > λ i(β′′ i), (B.6) 31 for some β′′ i> β′ i>maxzi. Note that, for any fixed α, it holds that 1 nnX i=1αzisis⊤ i α−zia.s.− − →Eαzss⊤ α−z =R∞(α), (B.7) due to the law of large numbers. Due to the continuity of eigenvalues... | https://arxiv.org/abs/2502.01583v1 |
that, for large enough n, (B.5)has no more than jsolution for α >max{zi}. As stated in (B.10)and(B.11), it holds that limα→∞λ∞ i(α)> µforis.t.1≤i≤kand that limα→τ+λ∞ i(α)< µforis.t.j+1+ k≤i≤p. Thus, using the same argument as before, we also have that, for large enough nand any α > τ,λi(α)> µ foris.t.1≤i≤kandλi(α)< µfo... | https://arxiv.org/abs/2502.01583v1 |
in this interval of µ, the equation λ∞ i(α) =µ 34 has no solutions in α > τ. Let us examine the equation (B.2) in Lemma B.1: det(µIp−R∞(α)) =pY l=1(µ−λ∞ l(α)) = 0 . The previous equation has a solution µ−λ∞ l(α) = 0, as long as λa∞ l< µ < t∞ l, (B.16) due to the monotonicity of each λ∞ l(α). As λa∞ i··· ≤ λa∞ k+1≤µ < λ... | https://arxiv.org/abs/2502.01583v1 |
of the reparametrized signal. Without loss of generality, we can assume that qis permutation invariant in the first mcoordinates, i.e., q(t1, . . . , t m, tm+1. . . , t p, ε) =q(tπ(1), . . . , t π(m), tm+1. . . , t p, ε), for any permutation π: [m]→[m]. Let Ebe the span of {wi−wi+1:i∈[m−1]}. Note that E has dimension m... | https://arxiv.org/abs/2502.01583v1 |
Thus, α1is the largest solution to det(ζδ(α)Ip−R∞(α)) = 0 , or equivalently λ1(R∞(α)) =ζδ(α). This means that, for (4.2) to have solutions larger than ¯λδ, there has to exist α1> τsuch that max ∥u∥2=1u⊤R∞(α1)u=ζδ(α1), 38 ζ′ δ(α1)>0, or equivalently max ∥u∥2=1u⊤R∞(¯λδ)u > ζ δ(¯λδ). (D.1) This follows from the fact that ... | https://arxiv.org/abs/2502.01583v1 |
follows that √ δ−(√ δ−p δc)· T∗(y)≥p δc, and T∗ δ(y) =√δc· T∗(y)√ δ−(√ δ−√δc)· T∗(y)≤√δcT∗(y)√δc≤1. Furthermore, for T∗(y)̸= 0, we have T∗ δ(y) =√δc√ δ T∗(y)−(√ δ−√δc), and it holds that√ δ T∗(y)−(√ δ−√δc)∈]− ∞,−(√ δ−√δc)[S]√δc,+∞[.Thus, for T∗(y)̸= 0, T∗ δ(y)≥ −√δc√ δ−√δc, whereas T∗ δ(y) = 0forT∗(y) = 0. This proves ... | https://arxiv.org/abs/2502.01583v1 |
The Shape of Generalization through the Lens of Norm-based Capacity Control Yichen Wang∗Yudong Chen†Lorenzo Rosasco‡Fanghui Liu§ Abstract Understanding how the test risk scales with model complexity is a central question in machine learning. Classical theory is challenged by the learning curves observed for large over-... | https://arxiv.org/abs/2502.01585v2 |
learned model is peaked around n≈d.Right: The test error against the norm of the learnt model. The color bar indicate the number of parameters and the arrows indicates the direction of increasing model size. Their relationship are closer to the convention wisdom than to a double descent. Setup: We consider a linear reg... | https://arxiv.org/abs/2502.01585v2 |
and Rakhlin (2020); Wang et al. (2022); Belkin et al. (2018); Zhang et al. (2021); Nakkiran et al. (2021). Empirical observations on the learning curve under norm-based capacity have been discussed in the lecture notes (Ng and Ma, 2023, Fig. 8.12), as shown in Fig. 1(a): when changing the model capacity from model size... | https://arxiv.org/abs/2502.01585v2 |
corresponding relationship between RandN, allowing a precise characterization, i.e. Our target N(X, f∗, λ) = (1 + O(n−1/2) +O(p−1/2))·N(Σ, f∗, λ∗) =⇒R=g(N) for some function g. The main results are given by Table 1 for linear regression and Table 2 for RFMs, which covers linear ridge regression and random features ridg... | https://arxiv.org/abs/2502.01585v2 |
the model size. •Controlling norm-based capacity can be achieved by the tuned regularization parameter λ:Norm-based capacity appears less intuitive used in practice when compared to model size. Our results demonstrate that the norm decreases monotonically with increas- ingλ, and in both under- and over-parameterized re... | https://arxiv.org/abs/2502.01585v2 |
provides powerful mathematical tools to precisely characterize the relationship between the test risk Randn, p, d via deterministic equivalence, in an asymptotic regime ( n, p, d → ∞ , Mei and Montanari 2022; Ghorbani et al. 2021; Wu and Xu 2020; Xiao et al. 2022; Bach 2024), or non-asymptotic regime (Hastie et al., 20... | https://arxiv.org/abs/2502.01585v2 |
the feature matrix can be denoted by Z=1√pGFT∈Rn×p. Note that fhas covariance matrix E[ffT] =Λ, and we further introduce bΛF:=Ez[zzT|F] =1 pFFT∈Rp×p. 6 Assuming that f∗∈L2(µx) admits f∗(x) =P k≥1θ∗,kψk(x), we have a bias-variance decomposi- tion of the excess risk RRFM:=Eε θ∗−1√pFTba 2 2= θ∗−1√pFTEε[ba] 2 2+ Tr bΛFCov... | https://arxiv.org/abs/2502.01585v2 |
hold for min- ℓ2-norm interpolator are given by Corollary D.4. We show that the solution λnto the self-consistent equation Tr(Σ(Σ+λnI)−1)∼ncan be obtained from the variance VLS N,0=σ2/λn. We remark that, by checking Eq. (2.1) and Eq. (3.1), norm-based capacity suffices to characterize generalization while effective dim... | https://arxiv.org/abs/2502.01585v2 |
In the under-parameterized regime, the test error RLS 0is a linear function of the norm NLS 0. In the over-parameterized regime, RLS 0andNLS 0formulates a rectangular hyperbola: RLS 0decreases with NLS 0ifNLS 0<∥β∗∥2 2−σ2while RLS 0increases with NLS 0ifNLS 0>∥β∗∥2 2−σ2. 4 Main results on RFMs via deterministic equival... | https://arxiv.org/abs/2502.01585v2 |
risk VRFM Rscaled by the factorn σ2. Therefore, it first increases and then decreases with p, reaching a peak near at the interpolation threshold ( p=n). The above two metrics offer a variance-based measure of model capacity: they capture the variance component of test risk but contain no information about the target f... | https://arxiv.org/abs/2502.01585v2 |
which will provide a more detailed description and understanding on learning curves. Reshape bias-variance trade-offs and double descent: We plot the bias and variance components of the test risk over model size pand norm, see Fig. 3(a) and Fig. 3(b), respectively. Note that, our theory (shown in curve) can precisely p... | https://arxiv.org/abs/2502.01585v2 |
size por fix pand constrain the weight norm, The latter approach is mathe- matically equivalent to tuning the regularization parameter λin random feature ridge regression, as ev- idenced by the equivalence to the constrained optimization problem: minβ∥y−Za∥2s.t.∥a∥2= B. This yields a ridge-type solution: ba= (Z⊤Z+λI)−1... | https://arxiv.org/abs/2502.01585v2 |
Under power-law, we need to handle the self-consistent equations to approximate the infinite summation. We have the following approximation. Corollary 4.6 (Relationship for min- ℓ2norm interpolator under power law) .Under Assumption 4.5, the deterministic equivalents RRFM 0andNRFM 0admit2the following relationship with... | https://arxiv.org/abs/2502.01585v2 |
provide a possible way to study distribution shift and out-of-distribution (OOD) (Patil et al., 2024a) with a precise estimation, which requires the deterministic equivalence of Tr(A(X⊤X+λI)−1B(X⊤X+λI)−1) for two matrices AandB. Acknowledgment We thank for Denny Wu’s discussion on asymptotic deterministic equivalence a... | https://arxiv.org/abs/2502.01585v2 |
International Conference on Learning Representations . Efron, B. (1986). How biased is the apparent error rate of a prediction rule? Journal of the American statistical Association 81461–470. Efron, B. (2004). The estimation of prediction error: covariance penalties and cross-validation. Journal of the American Statist... | https://arxiv.org/abs/2502.01585v2 |
124003. Nakkiran, P. ,Venkat, P. ,Kakade, S. andMa, T. (2020). Optimal regularization can mitigate double descent. arXiv preprint arXiv:2003.01897 . Neyshabur, B. ,Tomioka, R. andSrebro, N. (2014). In search of the real inductive bias: On the role of implicit regularization in deep learning. arXiv preprint arXiv:1412.6... | https://arxiv.org/abs/2502.01585v2 |
15. 19 Contents A Notations 21 B Preliminary and background 22 B.1 Asymptotic deterministic equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 B.2 Deterministic equivalence for ridge regression . . . . . . . . . . . . . . . . . . . . . . 23 B.3 Deterministic equivalence for random feature ridge regres... | https://arxiv.org/abs/2502.01585v2 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 G Discussion 70 G.1 Discussion on the shape of the generalization curve in Fig. 1 . . . . . . . . . . . . . . 70 G.2 Discussion on approaches to modifying the norm . . . . . . . . . . . . . . . . . . . . 71 G.3 Discussion with other mo... | https://arxiv.org/abs/2502.01585v2 |
T - An integral operator defined by ( Tf)(w) :=R Rdφ(x;w)f(x)dµx,∀f∈L2(µx) V - The image of T ξk R Thek-th eigenvalue of T, defined by T=P∞ k=1ξkψkϕ∗ k ψk - Thek-th eigenfunction of Tin the space L2(µx), defined by the decomposition T=P∞ k=1ξkψkϕ∗ k ϕk - Thek-th eigenfunction of Tin the space V, defined by the decompos... | https://arxiv.org/abs/2502.01585v2 |
∗ λ2Tr(A(Σ+λ∗)−1B(Σ+λ∗)−1) +λ2 ∗ λ2Tr(A(Σ+λ∗)−2Σ)·Tr(B(Σ+λ∗)−2Σ)·1 n−df2(λ∗).(B.5) Proposition B.5. (Bach, 2024, Restatement of Proposition 2) Assume (A1) ,(A2) ,(A3) , we consider AandBwith bounded operator norm, admitting the convergence of the empirical measures, i.e.,Pd i=1v⊤ iAvi·δσi→νAandPd i=1v⊤ iBvi·δσi→νBwith ... | https://arxiv.org/abs/2502.01585v2 |
a brief introduction here for self-completeness. More details can be found in Cheng and Montanari (2022); Misiakiewicz and Saeed (2024); Defilippis et al. (2024). Given x∈Rdwith d∈N, the associated covariance matrix is given by Σ=E[xx⊤]. We denote the eigenvalue of Σin non-increasing order as σ1≥σ2≥σ3≥ ··· ≥ σd. We int... | https://arxiv.org/abs/2502.01585v2 |
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