Data-Efficient Generative Adversarial Nets (DE-GANs) have become more and more popular in recent years. Existing methods apply data augmentation, noise injection and pre-trained models to maximumly increase the number of training samples thus improving the training of DE-GANs. However, none of these methods considers the sample quality during training, which can also significantly influence the training of DE-GANs. Focusing on sample quality during training, in this paper, we are the first to incorporate discriminator rejection sampling (DRS) into the training process and introduce a novel method, called quality aware dynamic discriminator rejection sampling (QADDRS). Specifically, QADDRS consists of two steps: (1) the sample quality aware step, which aims to obtain the sorted critic scores, i.e., the ordered discriminator outputs, on real/fake samples in the current training stage; (2) the dynamic rejection step that obtains dynamic rejection number N, where N is controlled by the overfitting degree of discriminator (D) during training. When updating the parameters of D, the N high critic score real samples and the N low critic score fake samples in the minibatch are rejected dynamically based on the overfitting degree of D. As a result, QAD-DRS can avoid D becoming overly confident in distinguishing both real and fake samples, thereby alleviating the over-fitting of D issue during training. Extensive experiments on several datasets demonstrate that integrating QADDRS into different DE-GANs can achieve better performance and deliver state-of-the-art results. Codes are available at https://github.com/zzhang05/QADDRS.

Diffusion-based generative models (diffusion models) often require a large amount of data to train a score-based model that learns the score function of the data distribution through denoising score matching. However, collecting and cleaning such data can be expensive, time-consuming, and even infeasible. In this paper, we present a novel theoretical insight for diffusion models that two factors, i.e., the denoiser function hypothesis space and the number of training samples, can affect the denoising score matching error of all training samples. Based on this theoretical insight, it is evident that minimizing the total denoising score matching error is challenging within the denoiser function hypothesis space in existing methods, when training diffusion models with limited data. To address this, we propose a new diffusion model called Limited Data Diffusion (LD-Diffusion), which consists of two main components: a compressing model and a novel mixed augmentation with fixed probability (MAFP) strategy. Specifically, the compressing model can constrain the complexity of the denoiser function hypothesis space and MAFP can effectively increase the training samples by providing more informative guidance than existing data augmentation methods in the compressed hypothesis space. Extensive experiments on several datasets demonstrate that LD-Diffusion can achieve better performance compared to other diffusion models. Codes are available at https://github.com/zzhang05/LD-Diffusion.

Abstract To monitor clients potentially bypassing position limits, business units employ the disjoint sets principle to identify potential client correlations based on account profiles. The key challenge lies in computing disjoint sets for large-scale topological graphs (with millions of nodes and billions of edges) in a short response time. In this article, we propose a multi-DAG indexing algorithm, namely the Hierarchical Isomerism Distributed Equivalent (HIDE) union find. First, in large-scale topological graphs, we utilize two new topological structures: the equivalent sub-Directed Acyclic Graph (sub-DAG) and the hierarchical isomerism topological graph, to reduce the number of edges and nodes in the multi-DAG indexing merging process. Then, our HIDE union find is proposed to achieve computable splitting across temporal and spatial spans. HIDE union find is theoretically proven to ensure correctness and universality. Experimental validation demonstrates that HIDE union find outperforms previous methods in large-scale topological graphs. The results indicate that HIDE union find achieves response times 100 to 200 times faster than those of the current leading methods.

In this research, we present self-learn your key latent (SLYKLatent), a novel approach for enhancing gaze estimation by addressing appearance instability challenges in datasets due to aleatoric uncertainties, covariant shifts, and test domain generalization. SLYKLatent utilizes self-supervised learning for initial training with facial expression datasets, followed by refinement with a patch-based tribranch network and an inverse explained variance weighted training loss function. Our evaluation on benchmark datasets achieves a 10.98% improvement on Gaze360, supersedes the top result with 3.83% improvement on MPIIFaceGaze, and leads on a subset of ETH-XGaze by 11.59%, surpassing existing methods by significant margins. In addition, adaptability tests on RAF-DB and Affectnet show 86.4% and 60.9% accuracies, respectively. Ablation studies confirm the effectiveness of SLYKLatent's novel components. This approach has strong potential in human–robot interaction.

Temporal link prediction is one of the most important tasks for predicting time-varying links by capturing dynamics within complex networks. However, it suffers from difficulties such as vulnerability to adversarial attacks and inadaptation to distinct evolutionary patterns. In this article, we propose a robust temporal link prediction architecture via stable gated models with reinforcement learning (SAGE-RL) consisting of a state encoding network (SEN) and a self-adaptive policy network (SPN). The former is utilized to capture network dynamics, while the latter helps the former adapt to distinct evolutionary patterns across various time periods. Within the SEN, a novel stable gate is introduced to ensure multiple spatiotemporal dependency paths and defend against adversarial attacks. An SPN is proposed to select different SEN instances by approximating the optimal action function, thereby adapting to various evolutionary patterns to learn the robust temporal and structural features from dynamic complex networks. It is proven that SAGE-LR with integral Lipschitz graph convolution is stable to relative perturbations in dynamic complex networks. With the aid of extensive experiments on five real-world graph benchmarks, SAGE-LR is shown to substantially outperform current state-of-the-art approaches in terms of precision and stability of temporal link prediction and ability to successfully defend against various attacks. We also implement the temporal link prediction in shipping transaction networks, which forecast effectively its potential transaction risks.

Recently, many studies have highlighted that training Generative Adversarial Networks (GANs) with limited data suffers from the overfitting of the discriminator (D). Existing studies mitigate the overfitting of D by employing data augmentation, model regularization, or pre-trained models. Despite the success of existing methods in training GANs with limited data, noise injection is another plausible, complementary, yet not well-explored approach to alleviate the overfitting of D issue. In this paper, we propose a simple yet effective method called Dual Adaptive Noise Injection (DANI), to further improve the training of GANs with limited data. Specifically, DANI consists of two adaptive strategies: adaptive injection probability and adaptive noise strength. For the adaptive injection probability, Gaussian noise is injected into both real and fake images for generator (G) and D with a probability p, respectively, where the probability p is controlled by the overfitting degree of D. For the adaptive noise strength, the Gaussian noise is produced by applying the adaptive forward diffusion process to both real and fake images, respectively. As a result, DANI can effectively increase the overlap between the distributions of real and fake data during training, thus alleviating the overfitting of D issue. Extensive experiments on several commonly-used datasets with both StyleGAN2 and FastGAN backbones demonstrate that DANI can further improve the training of GANs with limited data and achieve state-of-the-art results compared with other methods. Codes are available at https://github.com/zzhang05/DANI.

Data augmentation (DA) has shown its effectiveness in training Data-Efficient GANs (DE-GANs). However, applying DA in DE-GANs results in transforming the distributions of generated data and real data to augmented distributions of generated data and real data. This augmentation process could produce some out-of-distribution samples, known as the leaking of augmentations problem, which is highly undesirable in DE-GANs training. Although some methods propose "leaking-free" DAs for DE-GANs, we theoretically and practically argue that the leaking of augmentations problem still exists in these methods. To alleviate the leaking of augmentations in DE-GANs, in this paper, we propose a simple yet effective method called adaptive negative data augmentation (ANDA) for DE-GANs, with a negligible computational cost increase. Specifically, ANDA adaptively augments the augmented distribution of generated data using the augmented distribution of negative real data, where the negative real data is produced by applying negative data augmentation (NDA) on the real data. In this case, potential leaking samples can be presented as "fake" instances to the discriminator adaptively, which avoids the generator (G) learning such samples, thus resulting in better performance. Extensive experiments on several datasets with different DE-GANs demonstrate that ANDA can effectively alleviate the leaking of augmentations problem during training and achieve better performance. Codes are available at https://github.com/zzhang05/ANDA

Generative adversarial networks (GANs) have achieved great success and become more and more popular in recent years. However, understanding of the min-max game in GANs training is still limited. In this paper, we first utilize information game theory to analyze the min-max game in GANs and introduce a new viewpoint on the GANs training that the min-max game in existing GANs is unfair during training, leading to sub-optimal convergence. To tackle this, we propose a novel GAN called Information Gap GAN (IGGAN), which consists of one generator (G) and two discriminators (D1and D2). Specifically, we apply different data augmentation methods to D1and D2, respectively. The information gap between different data augmentation methods can change the information received by each player in the min-max game and lead to all three players G, D1and D2in IGGAN obtaining incomplete information, which improves the fairness of the min-max game, yielding better convergence. We conduct extensive experiments for large-scale and limited data settings on several common datasets with two backbones, i.e., BigGAN and StyleGAN2. The results demonstrate that IGGAN can achieve a higher Inception Score (IS) and a lower Fréchet Inception Distance (FID) compared with other GANs. Codes are available at https://github.com/zzhang05/IGGAN

In advanced manufacturing, strict safety guarantees are required to allow humans and robots to work together in a shared workspace. One of the challenges in this application field is the variety and unpredictability of human behavior, leading to potential dangers for human coworkers. This paper presents a novel control framework by adopting safety-critical control and uncertainty estimation for human-robot collaboration. Additionally, to select the shortest path during collaboration, a novel quadratic penalty method is presented. The innovation of the proposed approach is that the proposed controller will prevent the robot from violating any safety constraints even in cases where humans move accidentally in a collaboration task. This is implemented by the combination of a time-varying integral barrier Lyapunov function (TVIBLF) and an adaptive exponential control barrier function (AECBF) to achieve a flexible mode switch between path tracking and collision avoidance with guaranteed closed-loop system stability. The performance of our approach is demonstrated in simulation studies on a 7-DOF robot manipulator. Additionally, a comparison between the tasks involving static and dynamic targets is provided.Note to Practitioners—This research addresses the need to improve the safety of robots interacting with humans when performing collaborative tasks. Existing safety-critical control (SCC) approaches do not adequately monitor and continuously limit the state of the robot in Cartesian space, which results in a risk of injury to human operators if there is unexpected behavior during collaboration. Additionally, existing SCC approaches only consider system uncertainty for a single task (i.e. path tracking only or collision avoidance only). These problems limit the applicability of SCC techniques to manufacturing cobots. We address these problems by developing a controller that accounts for uncertainty in robot dynamics, guarantees that the robot end-effector remains within a constrained task space, and continuously modifies its motion in real-time to avoid dynamic obstacles that violate this space. We employ a machine learning approach to estimate the unknown uncertainties in real-time, allowing them to be incorporated within the controller design. The designed controller selects the shortest path for collision avoidance at each sample instant in order to minimize the total motion of the robot.

We develop a new framework for trajectory planning on predefined paths, for general N-link manipulators. Different from previous approaches generating open-loop minimum time controllers or pre-tuned motion profiles by time-scaling, we establish analytic algorithms that recover all initial conditions that can be driven to the desirable target set while adhering to environment constraints. More technologically relevant, we characterise families of corresponding safe state-feedback controllers with several desirable properties. A key enabler in our framework is the introduction of a state feedback template, that induces ordering properties between trajectories of the resulting closed-loop system. The proposed structure allows working on the nonlinear system directly in both the analysis and synthesis problems. Both offline computations and online implementation are scalable with respect to the number of links of the manipulator. The results can potentially be used in a series of challenging problems: Numerical experiments on a commercial robotic manipulator demonstrate that efficient online implementation is possible.

Various unsupervised greedy selection methods have been proposed as computationally tractable approximations to the NP-hard subset selection problem. These methods rely on sequentially selecting the variables that best improve performance with respect to a selection criterion. Theoretical results exist that provide performance bounds and enable ‘lazy greedy’ efficient implementations for selection criteria that satisfy a diminishing returns property known as submodularity. Recently, the authors introduced Forward Selection Component Analysis (FSCA) which uses variance explained as its selection criterion. While variance explained is not a submodular criterion, FSCA has been shown to be highly effective for applications such as measurement plan optimization. Motivated by the desire to achieve a more computationally efficient and scalable algorithm implementation, in this paper a ‘lazy’ implementation of FSCA (L-FSCA) is proposed, which, although not equivalent to FSCA due to the absence of submodularity, has the potential to yield comparable performance while being up to an order of magnitude faster to compute. The efficacy of L-FSCA is demonstrated by performing a systematic comparison with FSCA and five other unsupervised variable selection methods from the literature using simulated and real-world case studies. Experimental results confirm that L-FSCA yields almost identical performance to FSCA while reducing computation time by between 22% and 94% for the case studies considered.

Predicting the future trajectories of pedestrians is of increasing importance for many applications such as autonomous driving and social robots. Nevertheless, current trajectory prediction models suffer from limitations such as lack of diversity in candidate trajectories, poor accuracy, and instability. In this paper, we propose a novel Sequence Entropy Energy-based Model named SEEM, which consists of a generator network and an energy network. Within SEEM we optimize the sequence entropy by taking advantage of the local variational inference of f-divergence estimation to maximize the mutual information across the generator in order to cover all modes of the trajectory distribution, thereby ensuring SEEM achieves full diversity in candidate trajectory generation. Then, we introduce a probability distribution clipping mechanism to draw samples towards regions of high probability in the trajectory latent space, while our energy network determines which trajectory is most representative of the ground truth. This dual approach is our so-called all-then-one strategy. Finally, a zero-centered potential energy regularization is proposed to ensure stability and convergence of the training process. Through experiments on both synthetic and public benchmark datasets, SEEM is shown to substantially outperform the current state-of-the-art approaches in terms of diversity, accuracy and stability of pedestrian trajectory prediction.

Reducing dimensionality is a key preprocessing step in many data analysis applications to address the negative effects of the curse of dimensionality and collinearity on model performance and computational complexity, to denoise the data or to reduce storage requirements. Moreover, in many applications it is desirable to reduce the input dimensions by choosing a subset of variables that best represents the entire set without any a priori information available. Unsupervised variable selection techniques provide a solution to this second problem. An autoencoder, if properly regularized, can solve both unsupervised dimensionality reduction and variable selection, but the training of large neural networks can be prohibitive in time sensitive applications. We present an approach called Recovery of Linear Components (RLC), which serves as a middle ground between linear and non-linear dimensionality reduction techniques, reducing autoencoder training times while enhancing performance over purely linear techniques. With the aid of synthetic and real world case studies, we show that the RLC, when compared with an autoencoder of similar complexity, shows higher accuracy, similar robustness to overfitting, and faster training times. Additionally, at the cost of a relatively small increase in computational complexity, RLC is shown to outperform the current state-of-the-art for a semiconductor manufacturing wafer measurement site optimization application.

3D skeleton-based motion prediction and activity recognition are two interwoven tasks in human behaviour analysis. In this work, we propose a motion context modeling methodology that provides a new way to combine the advantages of both graph convolutional neural networks and recurrent neural networks for joint human motion prediction and activity recognition. Our approach is based on using an LSTM encoder-decoder and a non-local feature extraction attention mechanism to model the spatial correlation of human skeleton data and temporal correlation among motion frames. The proposed network can easily include two output branches, one for Activity Recognition and one for Future Motion Prediction, which can be jointly trained for enhanced performance. Experimental results on Human 3.6M, CMU Mocap and NTU RGB-D datasets show that our proposed approach provides the best prediction capability among baseline LSTM-based methods, while achieving comparable performance to other state-of-the-art methods.

Metrology, which plays an important role in ensuring production quality in modern manufacturing industries, incurs substantial costs both in terms of the infrastructure required and the time needed to perform measurements. In particular, in the semiconductor manufacturing industry, measuring fundamental quantities on different sites of a wafer surface is associated with increased production time. To increase metrology efficiency, a typical strategy is to limit the number of sites measured and to exploit statistical models (soft sensing) to reconstruct the wafer profile. Moreover, for quality reasons, spatial dynamic sampling strategies may be employed to ensure that all regions of a wafer surface are checked periodically during production. In this paper, we propose a new sampling strategy, called induced start dynamic sampling (ISDS), which adapts greedy feature selection algorithms to the spatial dynamic sampling problem, such that the number of measured sites at each process run is minimized while achieving good wafer profile reconstruction accuracy and process visibility. The superiority of the proposed strategy with respect to the state of the art is demonstrated using both simulated data and an industrial chemical vapor deposition case study.

In semiconductor manufacturing, metrology is generally a high cost nonvalue-added operation that significantly impacts on cycle time. As such, reducing wafer metrology continues to be a major target in semiconductor manufacturing efficiency initiatives. A novel data-driven spatial dynamic sampling methodology is presented that minimizes the number of sites that need to be measured across a wafer surface while maintaining an acceptable level of wafer profile reconstruction accuracy. The methodology is based on analyzing historical metrology data using forward selection component analysis (FSCA) to determine, from a set of candidate wafer sites, the minimum set of sites that need to be monitored in order to reconstruct the full wafer profile using statistical regression techniques. Dynamic sampling is then implemented by clustering unmeasured sites in accordance with their similarity to the FSCA selected sites and temporally selecting a different sample from each cluster. In this way, the risk of not detecting previously unseen process behavior is mitigated. We demonstrate the efficacy of the proposed methodology using both simulation studies and metrology data from a semiconductor manufacturing process.

A comprehensive knowledge system reveals the intangible insights hidden in an information system by integrating information from multiple data sources in a synthetical manner. In this paper, we present a variable precision reduction theory, underpinned by two new concepts: 1) distribution tables and 2) genealogical binary trees. Sufficient and necessary conditions to extract comprehensive knowledge from a given information system are also presented and proven. A complete variable precision reduction algorithm is proposed, in which we introduce four important strategies, namely, distribution table abstracting, attribute rank dynamic updating, hierarchical binary classifying, and genealogical tree pruning. The completeness of our algorithm is proven theoretically and its superiority to existing methods for obtaining complete reducts is demonstrated experimentally. Finally, having obtaining the complete reduct set, we demonstrate how the relationships between the complete reduct set and the comprehensive knowledge system can be visualized in a double-layer lattice structure using Hasse diagrams.

Principal Component Analysis (PCA) is a powerful and widely used tool for dimensionality reduction. However, the principal components generated are linear combinations of all the original variables and this often makes interpreting results and root-cause analysis difficult. Forward Selection Component Analysis (FSCA) is a recent technique that overcomes this difficulty by performing variable selection and dimensionality reduction at the same time. This paper provides, for the first time, a detailed presentation of the FSCA algorithm, and introduces a number of new variants of FSCA that incorporate a refinement step to improve performance. We then show different applications of FSCA and compare the performance of the different variants with PCA and Sparse PCA. The results demonstrate the efficacy of FSCA as a low information loss dimensionality reduction and variable selection technique and the improved performance achievable through the inclusion of a refinement step.

In many applications, and especially those where batch processes are involved, a target scalar output of interest is often dependent on one or more time series of data. With the exponential growth in data logging in modern industries, such time series are increasingly available for statistical modeling in soft sensing applications. In order to exploit time-series data for predictive modeling, it is necessary to summarize the information they contain as a set of features to use as model regressors. Typically this is done in an unsupervised fashion using simple techniques such as computing statistical moments, principal components or wavelet decompositions, often leading to significant information loss, and hence suboptimal predictive models. In this paper, a functional learning paradigm is exploited in a supervised fashion to derive continuous smooth estimates of time-series data (yielding aggregated local information), while simultaneously estimating a continuous shape function yielding optimal predictions. The proposed supervised aggregative feature extraction (SAFE) methodology can be extended to support nonlinear predictive models by embedding the functional learning framework in a reproducing kernel Hilbert spaces (RKHSs) setting. SAFE has a number of attractive features including closed-form solution and the ability to explicitly incorporate first- and second-order derivative information. Using simulation studies and a practical semiconductor manufacturing case study, we highlight the strengths of the new methodology with respect to standard unsupervised feature extraction approaches.

In this paper, a multiple classifier machine learning (ML) methodology for predictive maintenance (PdM) is presented. PdM is a prominent strategy for dealing with maintenance issues given the increasing need to minimize downtime and associated costs. One of the challenges with PdM is generating the so-called “health factors,” or quantitative indicators, of the status of a system associated with a given maintenance issue, and determining their relationship to operating costs and failure risk. The proposed PdM methodology allows dynamical decision rules to be adopted for maintenance management, and can be used with high-dimensional and censored data problems. This is achieved by training multiple classification modules with different prediction horizons to provide different performance tradeoffs in terms of frequency of unexpected breaks and unexploited lifetime, and then employing this information in an operating cost-based maintenance decision system to minimize expected costs. The effectiveness of the methodology is demonstrated using a simulated example and a benchmark semiconductor manufacturing maintenance problem.

Smart management of maintenances has become fundamental in manufacturing environments in order to decrease downtime and costs associated with failures. Predictive Maintenance (PdM) systems based on Machine Learning (ML) techniques have the possibility with low added costs of drastically decrease failures-related expenses; given the increase of availability of data and capabilities of ML tools, PdM systems are becoming really popular, especially in semiconductor manufacturing. A PdM module based on Classification methods is presented here for the prediction of integral type faults that are related to machine usage and stress of equipment parts. The module has been applied to an important class of semiconductor processes, ion-implantation, for the prediction of ion-source tungsten filament breaks. The PdM has been tested on a real production dataset.

Artifact removal from physiological signals is an essential component of the biosignal processing pipeline. The need for powerful and robust methods for this process has become particularly acute as healthcare technology deployment undergoes transition from the current hospital-centric setting toward a wearable and ubiquitous monitoring environment. Currently, determining the relative efficacy and performance of the multiple artifact removal techniques available on real world data can be problematic, due to incomplete information on the uncorrupted desired signal. The majority of techniques are presently evaluated using simulated data, and therefore, the quality of the conclusions is contingent on the fidelity of the model used. Consequently, in the biomedical signal processing community, there is considerable focus on the generation and validation of appropriate signal models for use in artifact suppression. Most approaches rely on mathematical models which capture suitable approximations to the signal dynamics or underlying physiology and, therefore, introduce some uncertainty to subsequent predictions of algorithm performance. This paper describes a more empirical approach to the modeling of the desired signal that we demonstrate for functional brain monitoring tasks which allows for the procurement of a "ground truth" signal which is highly correlated to a true desired signal that has been contaminated with artifacts. The availability of this "ground truth," together with the corrupted signal, can then aid in determining the efficacy of selected artifact removal techniques. A number of commonly implemented artifact removal techniques were evaluated using the described methodology to validate the proposed novel test platform.

The combination of reducing birth rate and increasing life expectancy continues to drive the demographic shift toward an aging population. This, in turn, places an ever-increasing burden on healthcare due to the increasing prevalence of patients with chronic illnesses and the reducing income-generating population base needed to sustain them. The need to urgently address this healthcare "time bomb" has accelerated the growth in ubiquitous, pervasive, distributed healthcare technologies. The current move from hospital-centric healthcare toward in-home health assessment is aimed at alleviating the burden on healthcare professionals, the health care system and caregivers. This shift will also further increase the comfort for the patient. Advances in signal acquisition, data storage and communication provide for the collection of reliable and useful in-home physiological data. Artifacts, arising from environmental, experimental and physiological factors, degrade signal quality and render the affected part of the signal useless. The magnitude and frequency of these artifacts significantly increases when data collection is moved from the clinic into the home. Signal processing advances have brought about significant improvement in artifact removal over the past few years. This paper reviews the physiological signals most likely to be recorded in the home, documenting the artifacts which occur most frequently and which have the largest degrading effect. A detailed analysis of current artifact removal techniques will then be presented. An evaluation of the advantages and disadvantages of each of the proposed artifact detection and removal techniques, with particular application to the personal healthcare domain, is provided.

This paper deals with reinforcement learning for process modeling and control using a model-free, action- dependent adaptive critic (ADAC). A new modified recursive Levenberg Marquardt (RLM) training algorithm, called temporal difference RLM, is developed to improve the ADAC performance. Novel application results for a simulated continuously-stirred-tank-reactor process are included to show the superiority of the new algorithm to conventional temporal-difference stochastic backpropagation.

This paper presents a new hybrid optimization strategy for training feedforward neural networks. The algorithm combines gradient-based optimization of nonlinear weights with singular value decomposition (SVD) computation of linear weights in one integrated routine. It is described for the multilayer perceptron (MLP) and radial basis function (RBF) networks and then extended to the local model network (LMN), a new feedforward structure in which a global nonlinear model is constructed from a set of locally valid submodels. Simulation results are presented demonstrating the superiority of the new hybrid training scheme compared to second-order gradient methods. It is particularly effective for the LMN architecture where the linear to nonlinear parameter ratio is large.

Various approaches to the parallel implementation of second-order gradient-based multilayer perceptron training algorithms are described. Two main classes of algorithm are defined involving Hessian and conjugate gradient-based methods. The limited- and full-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithms are selected as representative examples and used to show that the step size and gradient calculations are critical components. For larger problems the matrix calculations in the full-memory algorithm are also significant. Various strategies are considered for parallelization, the best of which is implemented on parallel virtual machine (PVM) and transputer-based architectures. Results from a range of problems are used to demonstrate the performance achievable with each architecture. The transputer implementation is found to give excellent speed-ups but the problem size is limited by memory constraints. The speed-ups achievable with the PVM implementation are much poorer because of inefficient communication, but memory is not a difficulty.