Average Case Setting Research Articles

Tractability of Multivariate Approximation Problem on Euler and Wiener Integrated Processes

This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters {rd}d∈N. We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT.

Polynomial-time trace reconstruction in the smoothed complexity model

In the trace reconstruction problem , an unknown source string x ∈ {0, 1} n is sent through a probabilistic deletion channel which independently deletes each bit with probability δ and concatenates the surviving bits, yielding a trace of x . The problem is to reconstruct x given independent traces. This problem has received much attention in recent years both in the worst-case setting where x may be an arbitrary string in {0, 1} n [DOS19, NP17, HHP18, HL20, Cha21a, Cha21b] and in the average-case setting where x is drawn uniformly at random from {0, 1} n [PZ17, HPP18, HL20, Cha21a, Cha21b]. This paper studies trace reconstruction in the smoothed analysis setting, in which a “worst-case” string x worst is chosen arbitrarily from {0, 1} n , and then a perturbed version x of x worst is formed by independently replacing each coordinate by a uniform random bit with probability σ . The problem is to reconstruct x given independent traces from it. Our main result is an algorithm which, for any constant perturbation rate 0 < σ < 1 and any constant deletion rate 0 < δ < 1, uses poly( n ) running time and traces and succeeds with high probability in reconstructing the string x. This stands in contrast with the worst-case version of the problem, for which \(\text{exp}(\tilde{O}(n^{1/5})) \) is the best known time and sample complexity [Cha21b]. Our approach is based on reconstructing x from the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new poly( n )-time procedure for reconstructing the multiset of all O (log n )-length subwords of any source string x ∈ {0, 1} n given access to traces of x .

Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.

On the power of standard information for tractability for L2-approximation in the average case setting

We study multivariate approximation in the average case setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Λstd and Λall for the absolute error criterion, and show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by Novak and Woźniakowski (2012) [25].

Worst-case versus average-case design for estimation from partial pairwise comparisons

Pairwise comparison data arises in many domains, including tournament rankings, web search and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutation-based models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two computationally efficient estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. We also provide explicit classes of graphs for which the rates achieved by these estimators are optimal. Our results are corroborated by simulations on multiple comparison topologies.

On the power of standard information for [formula omitted]-approximation in the average case setting

This paper studies the equivalence of tractability for classes of Λstd of function evaluations and Λall of all continuous linear functionals for L2-approximation in the average-case setting. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability, in the sense that there exist algorithms using function values at some deterministic points which enjoy the same tractability properties as those using optimally chosen linear functionals. Moreover, for the absolute error criterion, we also show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability under certain assumptions on the trace of the covariance operators. In particular, we give a partial answer to three problems raised by E. Novak and H. Woźniakowski in the book “Tractability of multivariate problems” (Open problems 116-118, EMS Tracts in Mathematics, vol.18, Zürich, 2012), proving that the same results hold under weaker conditions. Finally, we apply our results to the tensor product L2-approximation.

Decomposition of Additive Random Fields

We consider in this work an additive random field on [0, 1]d, which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they arise in the theory of intersections and self-intersections of Brownian processes, in the problems concerning small ball probabilities, and in the finite-rank approximation problems with arbitrarily large parametric dimension d. In problems of the last kind, the spectral characteristics of the covariance operator play a key role. For a given additive random field, the dependence of eigenvalues of its covariance operator on eigenvalues of the covariance operator of the marginal processes is quite simple, provided that the identical 1 is an eigenvector of the latter operator. In the opposite case, the dependence is complex, and, therefore, it is hard to study these random fields. Here, summands of the decomposition of the random field into the sum of its integral and its centered version are orthogonal in L2([0, 1]d), but, in general, they are correlated. In the present paper, we propose another interesting decomposition for random fields (it was discovered by the authors while resolving finite-rank approximation problems in the average-case setting). In the obtained decomposition, the summands are orthogonal in L2([0, 1]d) and are uncorrelated. Moreover, for large d, they are respectively close to the integral and to the centered version of the random field with small relative mean square errors.

EC-[formula omitted]-weak tractability of multivariate linear problems in the average case setting

We study EC-(s,t)-weak tractability of multivariate linear problems in the average case setting. This paper extends earlier work in the worst case setting. The parameters s≥0 and t≥0 allow us to study the information complexity n(ε,d) of a d-variate problem with respect to different powers of lnε−1, corresponding to the bits of accuracy, and d. We consider the absolute and normalized error criteria. In particular, a multivariate problem is EC-(s,t)-weakly tractable iff limd+ε−1→∞lnn(ε,d)∕[dt+lnsε−1]=0. We deal with general linear problems and linear tensor product problems. We show necessary and sufficient conditions for EC-(s,t)-weak tractability. In the case of general linear problem these conditions are matching. For linear tensor product problems, we also show matching conditions with the exception of some cases where s>1, in general.

A note about EC-[formula omitted]-weak tractability of multivariate approximation with analytic Korobov kernels

This paper is devoted to discussing multivariate approximation problems with analytic Korobov kernels in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential convergence-(s,t)-weak tractability (EC-(s,t)-WT) under the absolute or normalized error criterion. We completely solve the problem by filling the remaining gaps left open on EC-(s,t)-WT. That is, we obtain necessary and sufficient conditions for EC-(s,t)-WT for 0<min(s,t)<1 and max(s,t)≤1 in the worst case setting and for s,t>0 except s=t=1 in the average case setting.

(S, T)-Weak Tractability of Multivariate Linear Problems in the Average Case Setting

The purpose of this article is to investigate (s, t)-weak tractability of multivariate linear problems in the average case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. Generally, we obtained matching necessary and sufficient conditions for (s, t)-weak tractability in terms of the corresponding non-increasing sequence of eigenvalues. Specifically, we discussed (s, t)-weak tractability of linear tensor product problems and obtained necessary and sufficient conditions in terms of the corresponding one-dimensional problem. As an example of applications, we discussed also (s, t)-weak tractability of a multivariate approximation problem.

Average case tractability of multivariate approximation with Gaussian kernels

We consider d-variate approximation problems in the average case setting in the weighted L2 space with the standard Gaussian weight. This space is equipped with a zero-mean Gaussian measure whose covariance kernel is a product of univariate Gaussian kernels with non-increasing positive shape parameters γj2 for j=1,2,…,d. We study the average case error of algorithms that use finitely many arbitrary continuous linear functionals. We define the information complexity n(ε,d) to be the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most ε. We investigate different notions of tractability or exponential convergence-tractability (EC-tractability), and get necessary and sufficient conditions in terms of the shape parameters. In particular, for fixed s>0 and t∈(0,1) we obtain that the sufficient and necessary condition on γj2 for which limd+ε−1→∞n(ε,d)ε−s+dt=0holds is limj→∞j1−tγj2ln+γj−2=0,where ln+x=max(1,lnx).

Average case tractability of a multivariate approximation problem

Many authors have studied exponentially-convergent tractability (EC-tractability) in the worst case setting. Here, we study EC-tractability in the average case setting. Our problem is multivariate approximation over the space of continuous real functions equipped with a zero-mean Gaussian measure whose covariance kernel is given as a Korobov kernel. We obtain necessary and sufficient conditions for certain kinds of tractability, includingEC-tractability.

Average sampling numbers of multivariate periodic function spaces with a Gaussian measure

In this paper, we study average sampling numbers of the multivariate periodic function space L̊2 with a Gaussian measure μ in the Lq metric for 1≤q≤∞, and obtain their asymptotical orders, where the Cameron–Martin space of the measure μ is an anisotropic periodic Sobolev space. Moreover, we show that in the average case setting, the Lagrange interpolating operators are asymptotically optimal linear algorithms in the Lq metric for all 1≤q≤∞. This is different from the situation in the worst case setting, where the Lagrange interpolating operators are not asymptotically optimal linear algorithms in the Lq metric for q=1 or ∞.

Approximation of additive random fields based on standard information: Average case and probabilistic settings

We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main question we are concerned with in this paper is “How much do we loose by considering standard information algorithms against those using general linear information?” For both types of the fields, the error of linear algorithms has been studied in great detail; however, the power of standard information was not addressed so far, which we do here. Our main result is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one. The results are obtained by randomization techniques.

Exponential convergence-tractability of general linear problems in the average case setting

We study d-variate general linear problems defined over Hilbert spaces in the average case setting. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. We obtain the necessary and sufficient conditions for Exponential Convergence-(Strong) Polynomial Tractability, Quasi-Polynomial Tractability and (Uniform) Weak Tractability. All results are in terms of the eigenvalues of the corresponding covariance operators.

Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting

Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting

Average errors for Kantorovitch operators on r-fold integrated Wiener space

For weighted approximation in Lp-norm, we determine strongly asymptotic orders for simultaneous approximation average errors of sequence of Kantorovitch operators on r-fold integrated Wiener space. These results show the saturation properties of the Kantorovitch operators in the average case setting.

Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting

In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the Lq space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the Lq space for 2 &lt; q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the Lq space for 1 ≤ q &lt; ∞. Also, in the Lq space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

Uniform weak tractability of multivariate problems with increasing smoothness

We study d-variate approximation problems with varying regularity with respect to successive variables. The varying regularity is described by a sequence of real numbers {rk}k∈N satisfying 0≤r1≤r2≤r3≤⋯. We mainly consider algorithms that use finitely many continuous linear functionals. In the worst case setting we study approximation problems defined over suitable Korobov and Sobolev spaces. In the average case setting we study approximation problems defined over the space of continuous functions C([0,1]d) equipped with a zero-mean Gaussian measure whose covariance operator is given by an Euler or Wiener integrated process. We establish necessary and sufficient conditions on uniform weak tractability of those problems in terms of their regularity parameters {rk}k∈N.

Tractability of linear problems defined over Hilbert spaces

We study d-variate approximation problems in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. In the worst case setting, we obtain necessary and sufficient conditions for quasi-polynomial tractability and uniform weak tractability. Furthermore, we give an estimate of the exponent of quasi-polynomial tractability which cannot be improved in general. In the average case setting, we obtain necessary and sufficient conditions for uniform weak tractability. As applications we discuss some examples.