Low-dimensional Grid Research Articles

High-Dimensional Dynamic Stochastic Model Representation

We propose a generic and scalable method for computing global solutions of nonlinear, high-dimensional dynamic stochastic economic models. First, within an MPI–TBB parallel time-iteration framework, we approximate economic policy functions using an adaptive, high-dimensional model representation scheme, combined with adaptive sparse grids. With increasing dimensions, the number of points in this efficiently-chosen combination of low-dimensional grids grows much more slowly than standard tensor product grids, sparse grids, or even adaptive sparse grids. Moreover, the adaptivity within the individual component functions adds an additional layer of sparsity, since grid points are added only where they are most needed — that is to say, in regions of the computational domain with steep gradients or at non-differentiabilities. Second, we introduce a performant vectorization scheme of the interpolation compute kernel. Third, we validate our claims with numerical experiments conducted on “Piz Daint (Cray XC50) at the Swiss National Super-computing Center. We observe significant speedups over the state-of-the-art techniques, and almost ideal strong scaling up to at least 1, 000 compute nodes. Fourth, to demonstrate the broad applicability of our method, we compute global solutions to two different versions of a dynamic stochastic economic model: a high-dimensional international real business cycle model with capital adjustment costs, and with or without irreversible investment. We solve these models up to 300 continuous state variables globally.

The Need for Structure in Quantum LDPC Codes

The existence of quantum LDPC codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information. Despite years of research good quantum LDPC codes are not known to exist, but at the very least it is known they cannot be defined on very regular topologies, like low-dimensional grids. In this work we establish a complementary result, showing that good quantum CSS codes which are sparsely generated require “structure” in the local terms that constrain the code space so as not to be “too-random” in a well-defined sense. To show this, we prove a weak converse to a theorem of Krasikov and Litsyn on weight distributions of classical codes due to which may be of independent interest: subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight, tantamount to having a non-negligible “approximate” minimal distance. While they may not have a large minimal non-zero weight, they still have very few words of low Hamming weight.

An approximate nearest neighbors search algorithm for low-dimensional grid locations

We propose a new algorithm for the problem of approximate nearest neighbors (ANN) search in a regularly spaced low-dimensional grid for interpolation applications. It associates every sampled point to its nearest interpolation location, and then expands its influence to neighborhood locations in the grid, until the desired number of sampled points is achieved on every grid location. Our approach makes use of knowledge on the regular grid spacing to avoid measuring the distance between sampled points and grid locations. We compared our approach with four different state-of-the-art ANN algorithms in a large set of computational experiments. In general, our approach requires low computational effort, especially for cases with high density of sampled points, while the observed error is not significantly different. At the end, a case study is shown, where the ionosphere dynamics is predicted daily using samples from a mathematical model, which runs in parallel at 56 different longitude coordinates, providing sampled points not well distributed that follow Earth’s magnetic field-lines. Our approach overcomes the comparative algorithms when the ratio between the number of sampled points and grid locations is over 2849:1.

On Approximation of Reserve Factors Dependency on Loads for Composite Stiffened Panels

We present two level approach to build accurate approximations for Reserve Factors dependency on loads for composite stiffened panels. Such dependency is continuous non-smooth function with complex form plateaux regions (i.e. regions where function has zero gradient), defined on low dimensional grids. The main problem that arises if one tries to construct global approximation in such case is the occurrence of Gibbs effect (i.e. harmonic oscillations of prediction) near the borders of plateaux that may significantly deteriorate approximation quality. Viable existing solution: approximation based on linear triangular interpolation avoids oscillations, but unlike the proposed approach it provides model that is not smooth outside plateaux regions and generally requires larger sample size to achieve same accuracy of approximation.

Visual query processing for efficient image retrieval using a SOM-based filter-refinement scheme

Visual query processing is one of the new issues for content-based image retrieval. In this paper, we propose (i) a filter-refinement scheme based on a modified form of the self-organizing map and (ii) a new interactive approach for similarity matching in image retrieval based on visual query processing. Specifically, we first propose a new local membership function, which preserves the relationships between the input feature vectors of the images and their neighboring weight vectors, to project the high dimensional input feature vectors to a low dimensional grid. Then, all the input feature vectors are mapped and visualized in the 2D grid. The feature vector of the query image is mapped and visualized in the 2D grid as well. The users not only can visualize the locations of the query image and the image data in the database, but also visualize the locations of the relevant and irrelevant images. Next, the users retrieve the candidates from the 2D grid interactively through visual query processing in the filter phase. Finally, the query results are obtained from the candidates by performing similarity ranking in the original feature space during the refinement phase. In order to accelerate the query process, we use a hierarchical tree to index the weight vectors of the self-organizing map (SOM) units to reduce the computation cost for finding the best matching unit. Our experiments show that (i) the proposed approach works well on both synthetic datasets and image data, (ii) the proposed visual query processing approach is more efficient than conventional approaches and can enhance the overall interactive experience through fast feedback, and (iii) the filter-refinement scheme makes our proposed approach more robust than conventional approaches.

Water quality comprehensive evaluation method for large water distribution network based on clustering analysis

In order to evaluate water quality for a large water distribution network comprehensively, a two-stage classification method was used and the clustering methods, self-organizing map (SOM), K-means method and fuzzy c-mean (FCM), were represented. With these clustering methods, the pipes of a large real water distribution network were divided into some groups considering one or more water quality indicators synchronously. The water quality indicators of residual chlorine, water age, THMs, TAAs, TOC and BDOC are used in this paper. Residual chlorine and water age are two main water quality indicators. THMs and TAAs can represents the disinfection byproducts information. And TOC and BDOC are used to represents biological stability. According to the clustering results, the status of water quality of the water network was analysed. The results showed that the classification of SOM could express the comprehensive water quality in a water distribution network (WDN) directly and vividly by high-dimension water quality indicator projection to a low dimensional topology grid and that two-stage classification method has higher efficiency in comparison to the traditional clustering method. Water quality comprehensive evaluation was of significance for locating water quality monitoring, water network rehabilitation and expansion.

The use of artificial neural networks to study fatty acids in neuropsychiatric disorders

BackgroundThe range of the fatty acids has been largely investigated in the plasma and erythrocytes of patients suffering from neuropsychiatric disorders. In this paper we investigate, for the first time, whether the study of the platelet fatty acids from such patients may be facilitated by means of artificial neural networks.MethodsVenous blood samples were taken from 84 patients with a DSM-IV-TR diagnosis of major depressive disorder and from 60 normal control subjects without a history of clinical depression. Platelet levels of the following 11 fatty acids were analyzed using one-way analysis of variance: C14:0, C16:0, C16:1, C18:0, C18:1 n-9, C18:1 n-7, C18:2 n-6, C18:3 n-3, C20:3 n-3, C20:4 n-6 and C22:6 n-3. The results were then entered into a wide variety of different artificial neural networks.ResultsAll the artificial neural networks tested gave essentially the same result. However, one type of artificial neural network, the self-organizing map, gave superior information by allowing the results to be described in a two-dimensional plane with potentially informative border areas. A series of repeated and independent self-organizing map simulations, with the input parameters being changed each time, led to the finding that the best discriminant map was that obtained by inclusion of just three fatty acids.ConclusionOur results confirm that artificial neural networks may be used to analyze platelet fatty acids in neuropsychiatric disorder. Furthermore, they show that the self-organizing map, an unsupervised competitive-learning network algorithm which forms a nonlinear projection of a high-dimensional data manifold on a regular, low-dimensional grid, is an optimal type of artificial neural network to use for this task.

Clustering of volcanic ash arising from different fragmentation mechanisms using Kohonen self-organizing maps

In this study, we present the visualization and clustering capabilities of self-organizing maps (SOM) for analyzing high-dimensional data. We used SOM because they implement an orderly mapping of a high-dimensional distribution onto a regular low-dimensional grid. We used surface texture parameters of volcanic ash that arose from different fragmentation mechanisms as input data. We found that SOM cluster 13-dimensional data more accurately than conventional statistical classifiers. The component planes constructed by SOM are more successful than statistical tests in determining the distinctive parameters.

Classification of above burden profile using SOM and k-means

Blast furnace operators mostly depend on the measurements taken from the inputs and outputs in the control of the process to interpret the information available for vertical and radial distribution of variables, such as, heat load on the boundary and the distribution of the gas across the section of the furnace. Gas distribution inside the furnace is largely dependent on the distribution of the burden inside the furnace and its estimation is a challenge to the blast furnace operators. However, the measurement from the above burden probes gives some insight to the radial gas distribution in the shaft. Design of methods for automatic classification of above burden probe (ABP) profile is thus warranted. The self-organising map (SOM) is an excellent tool in the exploratory phase of data mining. It projects input space on prototypes of a low dimensional regular grid that can be effectively utilised to visualise and explore properties of the data. When the number of SOM units is large, to facilitate quantitative analysis of the map and the data, similar units need to be grouped, i.e. clustered. In the present investigation, a model for classification, visualisation and interpretation of ABP profiles has been developed by two stage procedure, i.e. SOM followed by k-means clustering. This classifier has the potential to be a useful tool for operator guidance in daily practice.

Generalizing Self-Organizing Map for Categorical Data

The self-organizing map (SOM) is an unsupervised neural network which projects high-dimensional data onto a low-dimensional grid and visually reveals the topological order of the original data. Self-organizing maps have been successfully applied to many fields, including engineering and business domains. However, the conventional SOM training algorithm handles only numeric data. Categorical data are usually converted to a set of binary data before training of an SOM takes place. If a simple transformation scheme is adopted, the similarity information embedded between categorical values may be lost. Consequently, the trained SOM is unable to reflect the correct topological order. This paper proposes a generalized self-organizing map model that offers an intuitive method of specifying the similarity between categorical values via distance hierarchies and, hence, enables the direct process of categorical values during training. In fact, distance hierarchy unifies the distance computation of both numeric and categorical values. The unification is done by mapping the values to distance hierarchies and then measuring the distance in the hierarchies. Experiments on synthetic and real datasets were conducted, and the results demonstrated the effectiveness of the generalized SOM model.

SOM에서 개체의 시각화

다변량 자료를 분석하는 데 있어서 관측 개체들의 분포적 양태를 파악하는 것은 자료 특성의 이해에 도움이 될 뿐만 아니라 이후 모형화 과정에도 큰 도움을 준다. 이를 위하여 다변량자료의 저차원 시각화에 대한 많은 연구가 진행되어 왔다. 그 중 하나가 코호넨(T. Kohonen)의 자기조직화지도(Self-Organizing Map; SOM)이다. SOM은 저차원 그리드 공간에 고차원 다변량 자료를 축약하여 시각적으로 나타내는 비지도 학습법의 일종으로 최근 들어 통계 분석자들이 많은 관심을 가지고 있는 분야이다. 그러나 SOM은 개체공간의 연속형으로 표현되는 개체를 저차원 그리드 공간에 승자노드에 의해 비연속적으로 표현한다는 단점을 지니고 있다. 본 논문에서는 SOM을 통계적 목적으로 사용하기 위해 요구되는 그리드 공간에 개체를 연속적으로 표현하는 방법들을 제안하고 환용 예를 제시 하고자 한다. Exploring distributional patterns of multivariate data is very essential in understanding the characteristics of given data set, as well as in building plausible models for the data. For that purpose, low-dimensional visualization methods have been developed by many researchers along various directions. As one of methods, Kohonen's SOM (Self-Organizing Map) is prominent. SOM compresses the volume of the data, yields abstraction from the data and offers visual display on low-dimensional grids. Although it is proven quite effective, it has one undesirable property: SOM's display is discrete. In this study, we propose two techniques for enhancing quality of SOM's display, so that SOM's display becomes continuous. The proposed methods are demonstrated in two numerical examples.

Smoothly distributed fuzzy c-means: a new self-organizing map

This paper presents a new self-organizing map algorithm. Unlike the well-known method of Kohonen, the new algorithm corresponds to the optimization of an unambiguously defined cost function. It consists of a modified version of the widely used fuzzy c-means functional, where the code vectors are distributed on a regular low-dimensional grid, and a penalization term is added in order to guarantee a smooth distribution for the values of the code vectors on the grid. The mapping properties of the new method, similar to those of Kohonen's algorithm, are illustrated with several data sets. Computer programs (source code and executables) and data are available upon request to the authors.