Approximation Theorem Research Articles

UNIPoint: Universally Approximating Point Processes Intensities

Point processes are a useful mathematical tool for describing events over time, and so there are many recent approaches for representing and learning them. One notable open question is how to precisely describe the flexibility of point process models and whether there exists a general model that can represent all point processes. Our work bridges this gap. Focusing on the widely used event intensity function representation of point processes, we provide a proof that a class of learnable functions can universally approximate any valid intensity function. The proof connects the well known Stone-Weierstrass Theorem for function approximation, the uniform density of non-negative continuous functions using a transfer functions, the formulation of the parameters of a piece-wise continuous functions as a dynamic system, and a recurrent neural network implementation for capturing the dynamics. Using these insights, we design and implement UNIPoint, a novel neural point process model, using recurrent neural networks to parameterise sums of basis function upon each event. Evaluations on synthetic and real world datasets show that this simpler representation performs better than Hawkes process variants and more complex neural network-based approaches. We expect this result will provide a practical basis for selecting and tuning models, as well as furthering theoretical work on representational complexity and learnability.

Normal approximation and fourth moment theorems for monochromatic triangles

AbstractGiven a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well‐known fourth‐moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth‐moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

AbstractLet$H^\infty ({\mathbb {D}}\times {\mathbb {N}})$be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk${\mathbb {D}}\subset {{\mathbb C}}$. We show that the dense stable rank of$H^\infty ({\mathbb {D}}\times {\mathbb {N}})$is$1$and, using this fact, prove some nonlinear Runge-type approximation theorems for$H^\infty ({\mathbb {D}}\times {\mathbb {N}})$maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra$H^\infty ({\mathbb {D}})$.

How Deep Is Deep Enough for Deep Belief Network for Approximating Model Predictive Control Law

Deep belief network (DBN) is an effective learning model based on deep learning. It can hierarchically transform the input data via stacked feature detectors. As a predictive model, DBN has shown a promising prospect in model predictive control (MPC). However, its successful application relies seriously on the suitable structure size (the numbers of hidden layers and neurons), which is challenging to determine. In this work, we present a theoretical bound on its minimum structure size in order to accurately approximate a desired control law of MPC, called DBN-MPC. First, according to the Markov assumption, a controlled system is equivalent to a quadratic program, which only depends on the current system state. Second, a universal theorem is proposed to give a bound on the minimum structure size of DBN from the perspective of piecewise affine function analysis. Third, a partial least square regression is used to fine-tune DBN to overcome the problems of local-minimum and time-consuming training process. Finally, we demonstrate the effectiveness of the proposed method through two classical experiments: 1) tracking control of a benchmark dynamical system and 2) temperature control of a practical second-order continuous stirred tank reactor (CSTR) system. The experimental results generally give an answer to the question that how deep is deep enough for DBN to approximate an MPC law. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The proposed deep belief network (DBN)-model predictive control (MPC) scheme presents an answer to the question that how deep is deep enough for DBN, from the perspective of approximating an MPC law. Given an industrial system, the proposed DBN-MPC scheme can give the best tracking control performance with minimal computational complexity through an effective self-growing DBN and optimal approximation theorem of MPC law. In practical applications, it is better for practitioners to obtain accurate historical input and output data. During the operation process, DBN-MPC will be more practical if more accurate initial structure size of DBN is provided in advance because the initial structure size of DBN is the starting point of the dynamic optimization.

Approximation Properties of Generalized λ -Bernstein–Stancu-Type Operators

The present study introduces generalized λ -Bernstein–Stancu-type operators with shifted knots. A Korovkin-type approximation theorem is given, and the rate of convergence of these types of operators is obtained for Lipschitz-type functions. Then, a Voronovskaja-type theorem was given for the asymptotic behavior for these operators. Finally, numerical examples and their graphs were given to demonstrate the convergence of G m , λ α , β f , x to f x with respect to m values.

A new kind of variant of the Kantorovich type modification operators introduced by D. D. Stancu

In the present article we investigate a variant of the Kantorovich type modification defined by Kajla (2018) i.e. we introduce a function ζ(ϰ) in the operators defined by Kajla (2018) s.t. ζ(ϰ) is infinitely differentiable function on [0,1],ζ(0)=0,ζ(1)=1 and ζ′(ϰ)>0,∀ϰ∈[0,1]. We prove an approximation theorem with the help of Bohman–Korovkin’s theorem, and investigate the estimate of the rate of convergence by means of modulus of smoothness and Lipschitz type function for these operators and the approximation of functions with derivatives of bounded variation are also studied. We study an approximation theorem with the help of Bohman–Korovkin’s principle in A−statistical convergence. Finally, by means of a numerical example we illustrate the convergence of these operators to certain functions through graphs with the help of MATHEMATICA and show that a careful choice of the function ζ(ϰ) leads to a better approximation than the Kantorovich type modification defined by Kajla (2018).

Reduction of multisymplectic manifolds

We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.

Approximation by One and Two Variables of the Bernstein-Schurer-Type Operators and Associated GBS Operators on Symmetrical Mobile Interval

In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’sK-functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.

The Direct and Converse Theorems for Best Approximation of Algebraic Polynomial in Lp, α (X)

The direct and inverse algebraic polynomials approximation theorems in weighted spaces of unbounded functions are proved by using modulus of smoothness. Also, we obtain sharp Jackson ( direct ) inequality of algebraic approximation of unbounded functions in terms modulus of smoothness. In addition, constructive characterization of modulus of smoothness are considered.

Universal Approximations of Invariant Maps by Neural Networks

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are both invariant/equivariant and provably complete in the sense of their ability to approximate any continuous invariant/equivariant map. Our contribution is three-fold. First, in the general case of compact groups we propose a construction of a complete invariant/equivariant network using an intermediate polynomial layer. We invoke classical theorems of Hilbert and Weyl to justify and simplify this construction; in particular, we describe an explicit complete ansatz for approximation of permutation-invariant maps. Second, we consider groups of translations and prove several versions of the universal approximation theorem for convolutional networks in the limit of continuous signals on euclidean spaces. Finally, we consider 2D signal transformations equivariant with respect to the group SE(2) of rigid euclidean motions. In this case we introduce the “charge–conserving convnet”—a convnet-like computational model based on the decomposition of the feature space into isotypic representations of SO(2). We prove this model to be a universal approximator for continuous SE(2)—equivariant signal transformations.

Best approximation theorems for almost cyclic contractions

A new class of cyclic mappings, known as almost cyclic contractions, is introduced. Indeed, it is interesting to see that almost cyclic contractions are weaker than cyclic contractions. The main purpose of this article is to explore the existence of a best approximation but not a best proximity point for almost cyclic contractions. Further, it is interesting to observe that a best proximity point theorem for cyclic contractions is deduced from a best approximation theorem for almost cyclic contractions.

Straightforward path to Zernike polynomials

Starting from Weierstrass’ approximation theorem, Zernike polynomials are obtained by a few straightforward steps involving only the recast of the aberration function as a double sum in the polar coordinates followed by the weighted orthogonalization of a power series. The origin of the name Fringe Zernike polynomials is also explained.

Classical and Approximate Theorems of Weighted Space Sampling

We show the same findings in this article, for the authors introduced in 2005 and 2004 that proved the classical sampling theorems, in the space ,and Bulzer show that three variants of the indicative analysis sampling theorem are similar in the significance that one can be shown as a corollary of one of the others. in 2014 [1] and [2], but in this work for any function in the space the space of all functions which are integral by weight function. Consequently, the two sampling theorems in the universal standard are fully identical. The outcome seems to be that our work is successful

Learnability and robustness of shallow neural networks learned by a performance-driven BP and a variant of PSO for edge decision-making

It may not be easy to implement complex AI models in edge devices without strong computing capacity (e.g. GPU). The Universal Approximation Theorem states that a shallow neural network (SNN) can represent any nonlinear function. In this paper, we focus on the learnability and robustness of SNNs, obtained by a greedy tight force heuristic algorithm [a Performance Driven Back-Propagation (PDBP)] and a loose force meta-heuristic algorithm [a variant of particle swarm optimization (VPSO)]. From the engineering prospective, all sensors are well justified for a specific task. Hence, all sensor readings should be strongly correlated to the target, and the structure of an SNN should depend on the dimensions of a problem space. The key findings of the research are summarized as follows: (1) The number of hidden neurons of an SNN depends on the nonlinearity of the training data, and the number of hidden neurons up to the dimension number of a problem space could be enough; (2) The learnability of SNNs, produced by error-driven PDBP, is always better than that of SNNs, optimized by error-driven VPSOs; (3) The performances of SNNs, obtained by PDBPs and VPSOs, do not change much for different training rates; and (4) Comparing with other classic machine learning algorithms, such as C4.5, NB and NN in literature, the SNNs, obtained by accuracy-driven PDBPs, win for all tested data sets, and the improvement percentage is up to 32.86%. Hence, the research could provide valued guidance for the implementation of edge intelligence.

De Branges Type Lemma and Approximation in Weighted Spaces

The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

Finiteness Theorems for Kac–Moody Groups over Non-archimedean Local Fields

Abstract We prove the finiteness of formal analogs of the spherical function (Spherical Finiteness), the ${\textbf c}$-function (Gindikin–Karpelevich Finiteness), and obtain a formal analog of Harish-Chandra’s limit (Approximation Theorem) relating spherical and ${\textbf c}$-function in the setting of $p$-adic Kac–Moody groups. The finiteness theorems imply that the formal analog of the Gindikin–Karpelevich integral is well defined in local Kac–Moody settings. These results extend Braverman–Garland–Kazhdan–Patnaik’s affine Gindikin–Karpelevich finiteness theorems from [ 4] and provide an algebraic analog of the combinatorial results of Gaussent–Rousseau [10] and Hébert [14].

On Best Approximations in Hyperconvex Spaces

In this manuscript, we present further extensions of the best approximation theorem in hyperconvex spaces obtained by Khamsi.

Approximations in globally subanalytic and Denjoy-Carleman classes

Efroymson's Approximation Theorem asserts that if f is a C0 semialgebraic mapping on a C∞ semialgebraic submanifold M of Rn and if ε:M→R is a positive continuous semialgebraic function then there is a C∞ semialgebraic function g:M→R such that |f−g|<ε. We prove a generalization of this result to the globally subanalytic category. Our theorem actually holds in a larger framework since it applies to every function which is definable in a polynomially bounded o-minimal structure (expanding the real field) that admits C∞ cell decomposition. We also establish approximation theorems for Lipschitz and C1 definable functions.

Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

In this paper, we investigate and study the concepts of statistical product convergence and statistical product summability via deferred Cesàro and deferred weighted product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two nice and potentially useful concepts. Also, based upon our proposed concepts, we state and prove a set of new Korovkin‐type approximation theorems for a martingale sequence over a Banach space. Moreover, we demonstrate that our approximation theorems effectively extend and improve most (if not all) of the previously existing results (both in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are quite stronger than the traditional and statistical versions of different theorems existing in the literature. We also suggest a direction for future researches on this subject, which are based upon the basic (or q‐) calculus, but not upon the trivial and inconsequential variations involving the so‐called (p, q)‐calculus.

Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem.

We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols to characterize the QECC performance measured in terms of the worst-case entanglement fidelity. The theorem is applicable to a large class of decoherence models, including erasure and depolarizing noise. Our approach is unorthodox, as instead of following the established path of utilizing QECCs to mitigate noise in quantum metrological protocols, we apply methods of quantum metrology to explore the limitations of QECCs.