Sample-Optimal Density Estimation in Nearly-Linear Time

Deceptive path planning (DPP) aims to find a path that minimizes the probability of the observer identifying the real goal of the observed before it reaches. It is important for addressing issues such as public safety, strategic path planning, and logistics route privacy protection. Existing traditional methods often rely on “dissimulation”—hiding the truth—to obscure paths while ignoring the time constraints. Building upon the theory of probabilistic goal recognition based on cost difference, we proposed a DPP method, DPP_Q, based on count-based Q-learning for solving the DPP problems in discrete path-planning domains under specific time constraints. Furthermore, to extend this method to continuous domains, we proposed a new model of probabilistic goal recognition called the Approximate Goal Recognition Model (AGRM) and verified its feasibility in discrete path-planning domains. Finally, we also proposed a DPP method based on proximal policy optimization for continuous path-planning domains under specific time constraints called DPP_PPO. DPP methods like DPP_Q and DPP_PPO are types of research that have not yet been explored in the field of path planning. Experimental results show that, in discrete domains, compared to traditional methods, DPP_Q exhibits better effectiveness in enhancing the average deceptiveness of paths. (Improved on average by 12.53% compared to traditional methods). In continuous domains, DPP_PPO shows significant advantages over random walk methods. Both DPP_Q and DPP_PPO demonstrate good applicability in path-planning domains with uncomplicated obstacles.

Solving parametric piecewise polynomial systems

In smooth support vector machine (SSVM), the plus function must be approximated by some smooth function, and the approximate error will affect the classification ability. This paper studies the smooth approximation to the plus function by piecewise polynomials. First, some standard piecewise polynomial smooth approximation problems are formulated. Then, the existence and uniqueness of solution for these problems are proved and the analytic solutions are achieved. The comparison between the results in this paper and the previous ones shows that the piecewise polynomial functions in this paper achieve better approximation to the plus function

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In practice, we are interested in approximating a function f(x) near x = a by a polynomial of degree M. Other issues involved in approximation are how good the approximation is and over what interval can we expect the approximation to be good. In this paper we present approximation that is realized as piecewise polynomial function where in each interval ƒ(x) is approximated by a Taylor polynomial of low order. The piecewise polynomial approximation can be effectively implemented using Farrow structure.

B-splines caught interest of many engineering applications due to their merits of being flexible and provide a large degree of differentiability and cost/quality trade off relationship. However they have less impact with continuous time applications as they are constructed from piecewise polynomials. On the other hand, Exponential spline polynomials (E-splines) represent the best smooth transition between continuous and discrete domains as they are made of exponential segments. In this paper we present a technique for utilizing E-splines in image compression and de-noising applications. This technique is based upon sub-band decomposition of the image through an E-spline based perfect reconstruction (PR) system. Different thresholdings are applied on the decomposition layers for de-noising purposes. Due to the selective nature of E-spline based decomposition, the performance of our E-spline based de-noising technique outperforms all other literature techniques.

<abstract><p>The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially relevant procedure. GD type optimization schemes can be regarded as temporal discretization methods for the gradient flow (GF) differential equations associated to the considered optimization problem and, in view of this, it seems to be a natural direction of research to <italic>first aim to develop a mathematical convergence theory for time-continuous GF differential equations</italic> and, thereafter, to aim to extend such a time-continuous convergence theory to implementable time-discrete GD type optimization methods. In this article we establish two basic results for GF differential equations in the training of fully-connected feedforward ANNs with one hidden layer and ReLU activation. In the first main result of this article we establish in the training of such ANNs under the assumption that the probability distribution of the input data of the considered supervised learning problem is absolutely continuous with a bounded density function that every GF differential equation admits for every initial value a solution which is also unique among a suitable class of solutions. In the second main result of this article we prove in the training of such ANNs under the assumption that the target function and the density function of the probability distribution of the input data are piecewise polynomial that every non-divergent GF trajectory converges with an appropriate rate of convergence to a critical point and that the risk of the non-divergent GF trajectory converges with rate 1 to the risk of the critical point. We establish this result by proving that the considered risk function is <italic>semialgebraic</italic> and, consequently, satisfies the <italic>Kurdyka-Łojasiewicz inequality</italic>, which allows us to show convergence of every non-divergent GF trajectory.</p></abstract>

Discrete domain swarm robotics is an emerging and challenging field of research. Unlike continuous domain where working place of the robots is a two-dimensional plane, in the discrete domain their working place is modelled by a graph. Robots are deployed on the nodes of a given graph, and they are allowed to move only along the edges of that graph. Consequently, the models used in the continuous domain are not always applicable in the discrete domain. Exemplified by the gathering problem, this article critically reviews models, assumptions, and approaches that have been proposed in solving the problems in the discrete domain.

The paper is concerned with some characteristics of approximation by piecewise polynomial functions with nonfixed knots in various integral metrics Lp(a,b), 0 <p ≤∞ .

A multi-level discontinuous Galerkin method for solving the stationary Navier–Stokes equations

Stochastic Satisfiability Modulo Theories (SSMT) [1] is a quantitative extension of classical Satisfiability Modulo Theories (SMT) inspired by stochastic logics. It extends SMT by the usual as well as randomized quantifiers, facilitating capture of stochastic game properties in the logic, like reachability analysis of hybrid-state Markov decision processes. Solving for SSMT formulae with quantification over finite and thus discrete domain has been addressed by Tino Teige et al. [2]. In this paper, we extend their work to SSMT over continuous quantifier domains (CSSMT) in order to enable capture of continuous disturbances and uncertainty in hybrid systems. We extend the semantics of SSMT and introduce a corresponding solving procedure. A simple case study is pursued to demonstrate applicability of our framework to reachability problems in hybrid systems.

The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F -spaces N 0 p , n {\mathbf {N}}_0^{p,n} (introduced for this purpose in [2]). We obtain two o -results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in N 0 p , n {\mathbf {N}}_0^{p,n} we define a derivative D n , σ f {D^{n,\sigma }}f in L σ , σ = ( n + p − 1 ) − 1 {L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}} , which agrees with D n f {D^n}f for smooth f , and prove several properties of D n , σ {D^{n,\sigma }} .

In previous works, we derived stationary density distribution functions (DDF) where the local equilibrium is determined by imposing the maximum entropy (MaxEnt) principle, under the scale invariance restrictions, and the minimum entropy production theorem. In this paper we demonstrate that it is possible to reobtain these DDF solely from the MaxEnt principle subject to suitable scale invariant restrictions in all the variables. For the sake of concreteness, we analyse the example of ohmic, fully ionized, tokamak-plasmas, in the weak-collisional transport regime. In this case we show that it is possible to reinterpret the stationary distribution function in terms of the Prigogine distribution function where the logarithm of the DDF is directly linked to the entropy production of the plasma. This leads to the suggestive idea that also the stationary neoclassical distribution functions, for magnetically confined plasmas in the collisional transport regimes, may be derived solely by the MaxEnt principle.

This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.