Circumcentering approximate reflections for solving the convex feasibility problem

The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP), and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.

Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction

In this paper, we suggested and analyzed a new higher-order iterative algorithm for solving nonlinear equation $$g(x)=0$$ g ( x ) = 0 , $$g:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ g : R ⟶ R , which is free from derivative by using the approximate version of the first derivative, and we studied the basins of attraction for the proposed iterative algorithm to find complex roots of complex functions $$g:{\mathbb {C}}\longrightarrow {\mathbb {C}}$$ g : C ⟶ C . To show the effectiveness of the proposed algorithm for the real and the complex domains, the numerical results for the considered examples are given and graphically clarified. The basins of attraction of the existing methods and our algorithm are offered and compared to clarify their performance. The proposed algorithm satisfied the condition such that $$|x_{m}-\alpha |<1.0 \times 10^{-15}$$ | x m - α | < 1.0 × 10 - 15 , as well as the maximum number of iterations is less than or equal to 3, so the proposed algorithm can be applied to efficiently solve numerous type non-linear equations.

DIAGNOSIS OF CREATIVE THINKING IN MIDDLE SCHOOL STUDENTS AND ITS ASSISTANCE IN THE PROCESS OF PERFORMING CREATIVE ACTIVITIES

It is often claimed that it is impossible to teach a person to think creatively, but this is not at all the case. Both children and adults can become true “creatives”; the main thing is to achieve a certain level of skill in the ability to act and think creatively. Today, there are many tests, games, exercises that allow you to develop creativity, and this is easier for children than for adults. Is it possible to identify a «gifted child»? This article presents an approximate version of testing and analyzing ready-made data on diagnosing the level of creativity of a child.

Generalization of Vaidya’s Metric for A Radiating Star to Rotating Star

Schwarzschild's external solution of Einstein’s gravitational field equations in the general theory of relativity for a static star has been generalized by Vaidya [1], taking into account the radiation of the star. Here, we generalize Vaidya’s metric to a star that is rotating and radiating. Although, there is a famous Kerr solution [2] for a rotating star, but here is a simple solution for a rotating star which may be termed as a zero approximate version of the Kerr solution. Results are discussed.

Statistically Efficient, Polynomial-Time Algorithms for Combinatorial Semi-Bandits

We consider combinatorial semi-bandits over a set of arms X \subset \0,1\ ^d where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound R(T) = O( d (łn m)^2 (łn T) / Δ_\min ) after T rounds, where m = \max_x \in X 1^\top x. However, ESCB it has computational complexity O(|X|), which is typically exponential in d, and cannot be used in large dimensions. We propose the first algorithm that is both computationally and statistically efficient for this problem with regret R(T) = O( d (łn m)^2 (łn T) / Δ_\min ) and computational asymptotic complexity O(δ_T^-1 poly(d)), where δ_T is a function which vanishes arbitrarily slowly. Our approach involves carefully designing AESCB, an approximate version of ESCB with the same regret guarantees. We show that, whenever budgeted linear maximization over X can be solved up to a given approximation ratio, AESCB is implementable in polynomial time O(δ_T^-1 poly(d)) by repeatedly maximizing a linear function over X subject to a linear budget constraint, and showing how to solve these maximization problems efficiently.

An approximate version of Jackson’s conjecture

A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

Faster DBSCAN and HDBSCAN in Low-Dimensional Euclidean Spaces

We present a new algorithm for the widely used density-based clustering method dbscan. For a set of [Formula: see text] points in [Formula: see text] our algorithm computes the dbscan-clustering in [Formula: see text] time, irrespective of the scale parameter [Formula: see text] (and assuming the second parameter MinPts is set to a fixed constant, as is the case in practice). Experiments show that the new algorithm is not only fast in theory, but that a slightly simplified version is competitive in practice and much less sensitive to the choice of [Formula: see text] than the original dbscan algorithm. We also present an [Formula: see text] randomized algorithm for hdbscan in the plane — hdbscan is a hierarchical version of dbscan introduced recently — and we show how to compute an approximate version of hdbscan in near-linear time in any fixed dimension.

Relaxation of monotone coupling conditions: Poisson approximation and beyond

It is well known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also state explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.