Non-uniform Probability Research Articles

A New Sampling Scheme for an Improved Monitoring of the Process Mean

ABSTRACT This article introduces an innovative sampling scheme, the median sampling (MS), utilizing individual observations over time to efficiently estimate the mean of a process characterized by a symmetric (non-uniform) probability distribution. The mean estimator based on MS is not only unbiased but also boasts enhanced precision compared to its simple random sampling-based counterpart. Moreover, a new EWMA chart based on the mean estimator within the MS scheme is proposed. The performance of the EWMA charts, derived from both simple random sampling (SRS) and MS schemes, is evaluated using the metrics of steady-state average run-length and average number of items-to-signal. The findings underscore the superiority of the EWMA-MS chart over the EWMA-SRS chart. Additionally, as the magnitude of ranking errors escalates, the behavior of the EWMA-MS chart converges toward that of the EWMA-SRS chart. The practical implementation of the newly introduced EWMA chart is demonstrated through an illustrative example.

FAST: Filter-Adapted Spatio-Temporal Sampling for Real-Time Rendering

Stochastic sampling techniques are ubiquitous in real-time rendering, where performance constraints force the use of low sample counts, leading to noisy intermediate results. To remove this noise, the post-processing step of temporal and spatial denoising is an integral part of the real-time graphics pipeline. The main insight presented in this paper is that we can optimize the samples used in stochastic sampling such that the post-processing error is minimized. The core of our method is an analytical loss function which measures post-filtering error for a class of integrands --- multidimensional Heaviside functions. These integrands are an approximation of the discontinuous functions commonly found in rendering. Our analysis applies to arbitrary spatial and spatiotemporal filters, scalar and vector sample values, and uniform and non-uniform probability distributions. We show that the spectrum of Monte Carlo noise resulting from our sampling method is adapted to the shape of the filter, resulting in less noisy final images. We demonstrate improvements over state-of-the-art sampling methods in three representative rendering tasks: ambient occlusion, volumetric ray-marching, and color image dithering. Common use noise textures, and noise generation code is available at https://github.com/electronicarts/fastnoise.

Actor Prioritized Experience Replay (Abstract Reprint)

A widely-studied deep reinforcement learning (RL) technique known as Prioritized Experience Replay (PER) allows agents to learn from transitions sampled with non-uniform probability proportional to their temporal-difference (TD) error. Although it has been shown that PER is one of the most crucial components for the overall performance of deep RL methods in discrete action domains, many empirical studies indicate that it considerably underperforms off-policy actor-critic algorithms. We theoretically show that actor networks cannot be effectively trained with transitions that have large TD errors. As a result, the approximate policy gradient computed under the Q-network diverges from the actual gradient computed under the optimal Q-function. Motivated by this, we introduce a novel experience replay sampling framework for actor-critic methods, which also regards issues with stability and recent findings behind the poor empirical performance of PER. The introduced algorithm suggests a new branch of improvements to PER and schedules effective and efficient training for both actor and critic networks. An extensive set of experiments verifies our theoretical findings, showing that our method outperforms competing approaches and achieves state-of-the-art results over the standard off-policy actor-critic algorithms.

Analytical modeling of the electrical conductivity of CNT-filled polymer nanocomposites

Electrical conductivity of most polymeric insulators can be drastically enhanced by introducing a small volume fraction [Formula: see text] of conductive nanofillers. These nanocomposites find wide-ranging engineering applications from cellular metamaterials to strain sensors. In this work, we present a mathematical model to predict the effective electrical conductivity of carbon nanotubes (CNTs)/polymer nanocomposites accounting for the conductivity, dimensions, volume fraction, and alignment of the CNTs. Eshelby’s classical equivalent inclusion method (EIM) is generalized to account for electron-hopping—a key mechanism of electron transport across CNTs, and is validated with experimental data. Two measurements, namely, the limit angle of filler orientation and the probability distribution function, are used to control the alignment of CNTs within the composites. Our simulations show that decreasing the angle from a uniformly random distribution to a fully aligned state significantly reduces the transverse electrical conductivity, while the longitudinal conductivity shows less sensitivity to angle variation. Moreover, it is observed that distributing CNTs with non-uniform probability distribution functions results in an increase in longitudinal conductivity and a decrease in transverse conductivity, with these differences becoming more pronounced as the volume fraction of CNTs is increased. A reduction in CNT length decreases the effective electrical conductivity due to the reduced number of available conductive pathways while reducing CNT diameter increases the conductivity.

Grid-Based Non-Uniform Probabilistic Roadmap-Based AGV Path Planning in Narrow Passages and Complex Environments

In this paper, we propose a Grid-based Non-uniform Probability Density Sampling Probabilistic Roadmap algorithm (GN-PRM) in response to the challenges of difficult sampling in narrow passages and low-probability map generation in traditional Probabilistic Roadmap algorithms (PRM). The improved algorithm incorporates grid-based processing for map segmentation, employing non-uniform probability density sampling based on the different attributes of each block to enhance sampling probability in narrow passages. Additionally, considering the computational cost and frequent ineffective searches in traditional PRM algorithms during pathfinding, this paper optimizes the time required for query route planning by altering connection strategies to improve the algorithm’s runtime. Finally, the simulation results indicate that, with a reduction of over 50% in undirected line segments and a reduction of over 85% in runtime, the GN-PRM algorithm achieves a 100% success rate in complex planning scenarios with a sampling point value of K = 500. In comparison, the traditional PRM algorithm has a success rate of no more than 10%, with a sampling point value of K = 500.

Optimizing Sequential Decisions: Enhancements to the Brickman Principle with Cumulative Punishment and Probability Adjustments

Introduction Determining the optimal stopping point in sequential decision-making scenarios is crucial for maximizing rewards and minimizing costs. Traditional models like the original Brickman Principle often simplify this process by assuming fixed critical values and equal probabilities at each decision stage. These assumptions may not accurately reflect real-world complexities, where costs can be cumulative and probabilities variable. Objective This work seeks to enhance the Brickman Principle by including cumulative punishment elements and non-uniform probability distributions, therefore improving its capacity to accurately represent the intricacies of real-world decision-making. Methods Through a rigorous experimental study, we evaluate the impact of these modifications on optimal stopping rules and expected profits. Results In line with Prospect Theory's emphasis on loss aversion, the results reveal a distinct pattern of risk-averse behavior, with most participants choosing to stop sooner in the sequence to avoid growing fines. Furthermore, we saw substantial variation in both the termination points and anticipated earnings across participants, suggesting that individual disparities in risk tolerance and decision-making approaches are crucial in influencing results

Deep Adaptive Quadruplet Hashing With Probability Sampling for Large-Scale Image Retrieval

With the preferable efficiency in storage and computation, hashing has shown potential application in large-scale multimedia retrieval. Compared with traditional hashing algorithms using hand-crafted characteristics, deep hashing inherits the representational capacity of deep neural networks to jointly learn semantic features and hash functions, encoding raw data into compact binary codes with significant discrimination. Generally, most of the current multi-wise hashing methods view the similarity margins between image pairs as constant values in training process. When the distance between sample pairs exceeds the fixed margin, the hashing network would not learn anything. Besides, available hashing methods commonly introduce the random sampling strategy to build training batches and ignore the sample distribution, which is harmful to parameter optimization. In this paper, we propose a novel Deep Adaptive Quadruplet Hashing with probability sampling (DAQH) for discriminative binary code learning. Specifically, with exploring the distribution relationship of raw samples, a non-uniform probability sampling strategy is proposed to build more informative and representative training batches, while maintaining the diversity of training samples. By introducing the prior similarity of sample pairs to calculate corresponding margins, an adaptive margin quadruplet loss is designed to dynamically preserve the underlying semantic relationships with its neighbors. To tune the attributes of binary codes, by combining quadruple regularization and orthogonality optimization, binary code constraint is developed to make the learned embedding with significant discrimination. Extensive experimental results on various benchmark datasets demonstrate our proposed DAQH framework achieves state-of-the-art visual similarity search performance.

Actor Prioritized Experience Replay

A widely-studied deep reinforcement learning (RL) technique known as Prioritized Experience Replay (PER) allows agents to learn from transitions sampled with non-uniform probability proportional to their temporal-difference (TD) error. Although it has been shown that PER is one of the most crucial components for the overall performance of deep RL methods in discrete action domains, many empirical studies indicate that it considerably underperforms off-policy actor-critic algorithms. We theoretically show that actor networks cannot be effectively trained with transitions that have large TD errors. As a result, the approximate policy gradient computed under the Q-network diverges from the actual gradient computed under the optimal Q-function. Motivated by this, we introduce a novel experience replay sampling framework for actor-critic methods, which also regards issues with stability and recent findings behind the poor empirical performance of PER. The introduced algorithm suggests a new branch of improvements to PER and schedules effective and efficient training for both actor and critic networks. An extensive set of experiments verifies our theoretical findings, showing that our method outperforms competing approaches and achieves state-of-the-art results over the standard off-policy actor-critic algorithms.

Asymptotic Probability Expansions for Random Elements in a Hilbert space

In this article, we approach a class of problems in probability theory, namely, the asymptotic expansion of probability. We consider an independent, identically distributed, and normalized stochastic process in a separable Hilbert space H, and associate it with the normalized partial sum.As a result, we built on the ball with a fixed center asymptotic expansion of non-uniform probabilities; our conditions on the moments are minimal, and the dependency of estimates on the covariance operator is expressed with the terms of the eigenvalue series. Likewise, the covariance operators of the random elements do not coincide. In the open ball set with fixed center a and radius , we estimate the optimal result of the Berry-Esseen type of the remainder, and the terms of the probability by the Fourier method.

Effect of porosity and pore size distribution on elastic modulus of foams

In this study, we investigate the impact of pore distribution and size on the mechanical response of elastic porous materials. Two-dimensional porous media with convex porosity are considered (e.g., foams and sponges). These structures are numerically generated by subtracting randomly dispersed circular holes from a solid square domain. The in-plane position vector of pore centers is given by a uniform probability density function (PDF). The pore diameters are, instead, drawn from a non-uniform two-parameter continuous probability distribution. The influence of the void ratio, i.e., the volume fraction of voids, and the parameters of the diameter PDF are studied over a wide range of values, including the percolation threshold, where the effective elastic modulus of the material abruptly drops to zero. This transition takes place at a critical value of the void ratio, which turns out to be unaffected by the parameters of the pore size distribution. Our results demonstrate that the elastic response of these structures is primarily governed by the void ratio, with the correlation length playing a secondary role. While the pore size distribution may affect the correlation length, it has a minimal impact on the foam’s overall behavior.

Non-Abelian sandpile automata with height restrictions.

We have studied the properties of a sandpile automata under the constraint of height restriction of sand columns. In this sandpile, an active site transfers a grain to a neighboring site if and only if the height of the sand column at the destination site is less than a preassigned value n_{c}. This sandpile was studied by Dickman etal. [Phys. Rev. E 66, 016111 (2002)1063-651X10.1103/PhysRevE.66.016111] in a conserved system with a fixed number of sand grains. In contrast, we have studied the avalanche dynamics of the driven sandpile under the open boundary conditions. The deterministic dynamics of the Bak, Tang, and Wiesenfeld (BTW) sandpile under the height restriction is found to be non-Abelian. Using numerical results, we argue that the steady states of the sandpile are exactly the recurrent states of the BTW sandpile, but occur with nonuniform probabilities. A detailed analysis of the cluster size distributions indicates that the associated exponent values are likely to be different from those of the BTW sandpile. The other differences include that the drop number distribution decays as a power law, and the largest avalanche size grows as the fourth power of the system size.

PopArt: Ranked Testing Efficiency

Too often, programmers are under pressure to maximize their confidence in the correctness of their code with a tight testing budget. Should they spend some of that budget on finding “interesting” inputs or spend their entire testing budget on test executions? Work on testing efficiency has explored two competing approaches to answer this question: systematic partition testing ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ST</small> ), which defines a testing partition and tests its parts, and random testing ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RT</small> ), which directly samples inputs with replacement. A consensus as to which is better when has yet to emerge. We present Probability Ordered Partition Testing ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PopArt</small> ), a new systematic partition-based testing strategy that visits the parts of a testing partition in decreasing probability order and in doing so leverages any non-uniformity over that partition. We show how to construct a homogeneous testing partition, a requirement for systematic testing, by using an executable oracle and the path partition. A program's path partition is a naturally occurring testing partition that is usually skewed for the simple reason that some paths execute more frequently than others. To confirm this conventional wisdom, we instrument programs from the Codeflaws repository and find that 80% of them have a skewed path probability distribution. <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PopArt</small> visits the parts of a testing partition in decreasing probability order. We then compare <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PopArt</small> with <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RT</small> to characterise the configuration space in which each is more efficient. We show that, when simulating Codeflaws, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PopArt</small> outperforms <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RT</small> after <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$100{,}000$</tex-math></inline-formula> executions. Our results reaffirm <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RT</small> 's power for very small testing budgets but also show that for any application requiring high (above 90%) probability-weighted coverage <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PopArt</small> should be preferred. In such cases, despite paying more for each test execution, we prove that <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PopArt</small> outperforms <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RT</small> : it traverses parts whose cumulative probability bounds that of random testing, showing that sampling without replacement pays for itself, given a nonuniform probability over a testing partition.

Regularization oversampling for classification tasks: To exploit what you do not know

In numerous binary classification tasks, the two groups of instances are not equally represented, which often implies that the training data lack sufficient information to model the minority class correctly. Furthermore, many traditional classification models make arbitrarily overconfident predictions outside the range of the training data. These issues severely impact the deployment and usefulness of these models in real life. In this paper, we propose the boundary regularizing out-of-distribution (BROOD) sampler, which adds artificial data points on the edge of the training data. By exploiting these artificial samples, we are able to regularize the decision surface of discriminative machine learning models and make more prudent predictions. Next, it is crucial to correctly classify many positive instances in a limited pool of instances that can be investigated with the available resources. By smartly assigning predetermined nonuniform class probabilities outside the training data, we can emphasize certain data regions and improve classifier performance on various material classification metrics. The good performance of the proposed methodology is illustrated in a case study that consists of both benchmark balanced and imbalanced classification data sets.

Model constraints independent optimal subsampling probabilities for softmax regression

A prevailing method to alleviate the computational cost is to perform analysis on a subsample of the full data. Optimal subsampling algorithm utilizes non-uniform subsampling probabilities, derived through minimizing the asymptotic mean squared error of the subsample estimator, to acquire a higher estimation efficiency for a given subsample size. The optimal subsampling probabilities for softmax regression have been studied under the baseline constraint which treats one dimension of the multivariate response differently from other dimensions. In this paper, we show that different model constraints lead to different optimal subsampling probabilities, and the summation constraint corresponds to a better subsampling strategy than the baseline constraint in terms of balancing the responses among all categories. Furthermore, we derive the asymptotic distribution of the mean squared prediction error, and minimize its asymptotic expectation to define the optimal subsampling probabilities that are invariant to model constraints. Simulations and a real data example are provided to show the effectiveness of the proposed optimal subsampling probabilities.

Sampling With Replacement vs Poisson Sampling: A Comparative Study in Optimal Subsampling

Faced with massive data, subsampling is a commonly used technique to improve computational efficiency, and using nonuniform subsampling probabilities is an effective approach to improve estimation efficiency. For computational efficiency, subsampling is often implemented with replacement or through Poisson subsampling. However, no rigorous investigation has been performed to study the difference between the two subsampling procedures such as their estimation efficiency and computational convenience. This paper performs a comparative study on these two different sampling procedures. In the context of maximizing a general target function, we first derive asymptotic distributions for estimators obtained from the two sampling procedures. The results show that the Poisson subsampling may have a higher estimation efficiency. Based on the asymptotic distributions for both subsampling with replacement and Poisson subsampling, we derive optimal subsampling probabilities that minimize the variance functions of the subsampling estimators. These subsampling probabilities further reveal the similarities and differences between subsampling with replacement and Poisson subsampling. The theoretical characterizations and comparisons on the two subsampling procedures provide guidance to select a more appropriate subsampling approach in practice. Furthermore, practically implementable algorithms are proposed based on the optimal structural results, which are evaluated through both theoretical and empirical analyses.

GRASP algorithms for the unrelated parallel machines scheduling problem with additional resources during processing and setups

ABSTRACT This paper addresses an unrelated parallel machines scheduling problem with the need of additional resources during the processing of the jobs, as well as during the setups that machines need between the processing of any two jobs. This problem is highly complex, and therefore in this paper we propose several constructive heuristics to solve it. To improve the performance of these heuristics, we propose several variations, including randomisation with different probability distributions and a local search phase, having this way GRASP algorithms. The results of extensive experiments over randomly generated instances show several findings on the different parameters that characterise our constructive algorithms. In particular, we highlight the fact that non-uniform probability distributions might be advisable for choosing elements of a restricted candidate list in GRASP algorithms.

Improving accuracy of expected frequency of uncertain roles based on efficient ensembling

This study tackles the problem of extracting the node roles in uncertain graphs based on network motifs. Uncertain graphs are useful for modeling information diffusion phenomena because the presence or absence of edges is stochastically determined. In such an uncertain graph, the node role also changes stochastically according to the presence or absence of edges, so approximate calculation using a huge number of samplings is common. However, the calculation load is very large, even for a small graph. We propose a method to extract uncertain node roles with high accuracy and high speed by ensembling a large number of sampled graphs and efficiently searching for all other transitionable roles. This method provides highly accurate results compared to simple sampling and ensembling methods that do not consider the transition to other roles. In our evaluation experiment, we use real-world graphs artificially assigned uniform and non-uniform edge existence probabilities. The results show that the proposed method outperforms an existing method previously reported by the authors, which is the basis of the proposed method, as well as another current method based on the state-of-the-art algorithm, in terms of efficiency and accuracy.

Time-Division Multiplexing Ising Computer Using Single Stochastic Magnetic Tunneling Junction

A magnetic tunneling junction (MTJ) with very low energy barrier, which shows controllable stochastic property, is recently proposed to constitute the probability bit (P-Bit). With the P-Bit, Ising computation, which is a general and powerful computing platform for both conventional and non-conventional problems, is implemented. However, such hardware emulation suffers from the severe intrinsic variations of a stochastic MTJ device. The nonuniform probability switching curves of different stochastic MTJs hinder straightforward expansion of Ising computer to a large P-Bits array for solving more complicated problems. In this work, we propose a novel Ising computer using a single stochastic MTJ device. With the utilization of the time-division multiplexing (TDM) technology, a “Compute–Read–Switch” scheme is proposed, so that the single stochastic MTJ can act as multiple P-Bits during Ising computation. A design guide for the current magnitude and time duration of the “Compute–Read–Switch” scheme is investigated and provided. Furthermore, NOT and XOR logic gates are implemented with our proposal. An accuracy rate as high as 69% for the integer factorization is also achieved, which is comparable with conventional Ising computer. Meanwhile, the TDM Ising computer avoids the calibration process, which is mandatory in the conventional Ising computer.

Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P(i)∼i−β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P(k)∼k−δ, where δ=1+1/β. This property has been observed in most real-world SAT instances. For β=0, our model extends classical random SAT instances. We prove the existence of a SAT–UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β&lt;1/2 when the clause/variable ratio is m/n=1−2β(1−β)2. We also prove that scale-free random k-SAT instances are unsatisfiable with a high probability when the number of clauses exceeds ω(n(1−β)k). The proof of this result suggests that, when β&gt;1−1/k, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes.

EMF-Aware Probabilistic Shaping Design for Hardware-Distorted Communication Systems

The fifth-generation cellular network requires dense installation of radio base stations (BS) to support the ever-increasing demands of high throughput and coverage. The ongoing deployment has triggered some health concerns among the community. To address this uncertainty, we propose an EMF-aware probabilistic shaping design for hardware-distorted communication systems. The proposed scheme aims to minimize human exposure to radio frequency (RF) radiations while achieving the target throughput using probabilistic shaping. The joint optimization of the transmit power and nonuniform symbol probabilities is a non–convex optimization problem. Therefore, we employ alternate optimization and successive convex approximation to solve the subsequent problems. Our findings reveal a significant reduction in the users’ exposure to EMF while achieving the requisite quality of service with the help of probabilistic shaping in a hardware-distorted communication system.