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When a plain Monte Carlo estimate on n samples has variance σ 2 / n , then scrambled digital nets attain a variance that is o (1 / n) as n → ∞. For finite n and an adversarially selected integrand, the variance of a scrambled (t , m , s) -net can be at most Γ σ 2 / n for a maximal gain coefficient Γ < ∞. The most widely used digital nets and sequences are those of Sobol'. It was previously known that Γ ⩽ 2 t 3 s for any nets in base 2. For digital nets, Dick and Pillichshammer (2010) obtained the bound 2 t + s. In this paper we study digital nets in base 2 and show that Γ ⩽ 2 t + s − 1 for such nets. This bound is a simple, but apparently unnoticed, consequence of a microstructure analysis by Niederreiter and Pirsic in 2001. We obtain a sharper bound that is smaller than this for some digital nets. Our main finding is that all nonzero gain coefficients must be powers of two. A consequence of this latter fact is a simplified algorithm for computing gain coefficients of digital nets in base 2. [ABSTRACT FROM AUTHOR]

Computing large Riemann–Roch spaces for plane projective curves still constitutes a major algorithmic and practical challenge. Seminal applications concern the construction of arbitrarily large algebraic geometry error correcting codes over alphabets with bounded cardinality. Nowadays such codes are increasingly involved in new areas of computer science such as cryptographic protocols and "interactive oracle proofs". In this paper, we design a new probabilistic algorithm of Las Vegas type for computing Riemann–Roch spaces of smooth divisors, in characteristic zero, and with expected complexity exponent 2.373 (a feasible exponent for linear algebra) in terms of the input size. [ABSTRACT FROM AUTHOR]

Functional linear model is a fruitfully applied general framework for regression problems, including those with intrinsically infinite-dimensional data. Online gradient descent methods, despite their evidenced power of processing online or large-sized data, are not well studied for learning with functional data. In this paper, we study reproducing kernel-based online learning algorithms for functional data, and derive convergence rates for the expected excess prediction risk under both online and finite-horizon settings of step-sizes respectively. It is well understood that nontrivial uniform convergence rates for the estimation task depend on the regularity of the slope function. Surprisingly, the convergence rates we derive for the prediction task can assume no regularity from slope. Our analysis reveals the intrinsic difference between the estimation task and the prediction task in functional data learning. [ABSTRACT FROM AUTHOR]

We study approximation of multivariate periodic functions from Besov and Triebel–Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the L q -norm for functions from the Besov spaces B p , θ s (T d) and the Triebel–Lizorkin spaces F p , θ s (T d) for all s > 0 and admissible 1 ≤ p , θ ≤ ∞ as well as provide analogues of the Littlewood–Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators. [ABSTRACT FROM AUTHOR]

The efficient evaluation of multivariate polynomials at many points is an important operation for polynomial system solving. Kedlaya and Umans have recently devised a theoretically efficient algorithm for this task when the coefficients are integers or when they lie in a finite field. In this paper, we assume that the set of points where we need to evaluate is fixed and "sufficiently generic". Under these restrictions, we present a quasi-optimal algorithm for multi-point evaluation over general fields. We also present a quasi-optimal algorithm for the opposite interpolation task. [ABSTRACT FROM AUTHOR]

In this paper we study exponential tractability of multivariate approximation problem for weighted Korobov spaces in the worst case setting. The considered algorithms are constructive and use the class Λ std of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic algorithm was obtained by Keller-Gehrig in 1985 and involves logarithmic factors. Our algorithm computes more generally the determinant of a univariate polynomial matrix in reduced form, and relies on new subroutines for transforming shifted reduced matrices into shifted weak Popov matrices, and shifted weak Popov matrices into shifted Popov matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study tractability of L 2 -approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class Λ all consisting of all continuous linear functionals and from the class Λ std , which only consists of function evaluations. We provide necessary and sufficient conditions on the weights of the function space for quasi-polynomial tractability, uniform weak tractability, weak tractability and (σ , τ) -weak tractability. Together with the already known results for strong polynomial and polynomial tractability, our findings provide a complete picture of the weight conditions for all current standard notions of tractability. [ABSTRACT FROM AUTHOR]