6270234
Spectral Relaxation for K-means Clustering
The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.
46294089
1992 Gordon Bell Prize Winners
36271721
Industrial Application Areas of High-Performance Computing
In 1993, a list of the top 500 supercomputer sites worldwide was made available for the first time. Since then, the Top500 list has been published twice a year. The list allows a detailed and well-founded analysis of the state of high-performance computing (HPC). This article summarizes the recent trends in application areas of HPC systems, focusing on the increase in industrial installations and applications.
965932
Editors' Introduction
Not since the publication of Rachel Carson's Silent Spring (1962) has a book about the environment provoked such controversy as Bjorn Lomborg's The Skeptical Environmen-talist (2001). In the four decades since Silent Spring a popular environmental awareness and suspicion of ''chemicals'' and other aspects of modern industrial life has increasingly challenged and, in some cases, displaced scientific and technological optimism as an orthodox view throughout most of the industrialized world. A proliferation of nongovern-mental organizations and scientific bodies repeatedly invoke science to warn us of the potential for irreversible and, possibly, catastrophic environmental decline, unless people on the planet mend their ways. Powerful interests resist this message, increasingly by attacking the science that informs it. In this context, it may seem unsurprising that Lomborg's challenge to contemporary catastrophic views of inexorable global environmental decline has itself attracted criticism from environmentalists. Perhaps what is surprising are the amount of attention, the degree of opprobrium, and the ad hominem nature of the criticism that has been launched at Lomborg for a book that builds upon a tradition of similar arguments. Indeed the central theme of TSE, that some combination of business-as-usual and incremental change will be sufficient for children born today to ''get more food, a better education, a higher standard of living, more leisure time and far more possibi-lities—without the global environment being destroyed'', seems a relatively modest assertion of the cornucopian world view by comparison with some of Lomborg's predecessors, such as deconstructions of environmental concerns over the preceding decade including Ron Bailey's Ecoscam (1993) and Aaron Wildavsky's But is it True? (1995) provoked few serious rebuttals from scientists or environmental activists. The controversy over TSE serves to remind us of the anthropological observation that science is the trump card that we play in disputes about values. In our contemporary society, science has become the medium through which we understand nature. However, science has also become the medium through which we wage political battles not just on the environment, but also over many other contested issues. Hence, the debate over TSE provides a rich case study through which to examine issues associated with the relationships among science, policy, and politics, which observers on the right (e.g., Gough, 2003) and left (e.g., UCS, 2004) of the US political spectrum agree is a critical area for governance. The five papers that comprise this special issue were originally solicited for a symposium titled …
6041481
Parallel preconditioning and approximation inverses on the Connection Machine
The authors present a new approach to preconditioning for very large, sparse, non-symmetric, linear systems. It explicitly computes an approximate inverse to the original matrix that can be applied most efficiently for iterative methods on massively parallel machines. The algorithm and its implementation on the Connection Machine CM-2 are discussed in detail and supported by timings obtained from real problem data.<<ETX>>
26191709
The Implementation of the U.S. High Performance Computing and Communications Program
This paper gives an overview of the federal High Performance Computing and Communication (HPCC) Program. This initiative involves the Defense Advanced Projects Agency (DARPA), the Department of Energy (DOE), the National Aeronautics and Space Administration (NASA), and the National Science Foundation (NSF). Its goal is to advance U.S. science and industrial competitiveness by supporting the development of teraops high performance systems, the software and algorithms required to render those massively parallel systems efficiently usable, and the Gigabit networks to make those machines accessible, as well as to broaden the base of computationally literate scientists. We outline the HPCC effort and the rationale underlying it, and give an overview of some of the “Grand Challenge” work under way at Argonne’s Mathematics and Computer Science Division, as well as the Grand Challenges addressed by NASA’s Computational Aerosciences program.
10043574
Scientific Model-Building as Search in Matrix Spaces
Many reported discovery systems build discrete models of hidden structure, properties, or processes in the diverse fields of biology, chemistry, and physics. We show that the search spaces underlying many well-known systems are remarkably similar when re-interpreted as search in matrix spaces. A small number of matrix types are used to represent the input data and output models. Most of the constraints can be represented as matrix constraints; most notably, conservation laws and their analogues can be represented as matrix equations. Typically, one or more matrix dimensions grow as these systems consider more complex models after simpler models fail, and we introduce a notation to express this. The novel framework of matrix-space search serves to unify previous systems and suggests how at least two of them can be integrated. Our analysis constitutes an advance toward a generalized account of model-building in science.
17442679
Inner Product Spaces for Bayesian Networks
Bayesian networks have become one of the major models used for statistical inference. We study the question whether the decisions computed by a Bayesian network can be represented within a low-dimensional inner product space. We focus on two-label classification tasks over the Boolean domain. As main results we establish upper and lower bounds on the dimension of the inner product space for Bayesian networks with an explicitly given (full or reduced) parameter collection. In particular, these bounds are tight up to a factor of 2. For some nontrivial cases of Bayesian networks we even determine the exact values of this dimension. We further consider logistic autoregressive Bayesian networks and show that every sufficiently expressive inner product space must have dimension at least Ω(n2), where n is the number of network nodes. We also derive the bound 2Ω(n) for an artificial variant of this network, thereby demonstrating the limits of our approach and raising an interesting open question. As a major technical contribution, this work reveals combinatorial and algebraic structures within Bayesian networks such that known methods for the derivation of lower bounds on the dimension of inner product spaces can be brought into play.
31930258
How Many Missing Answers Can Be Tolerated by Query Learners?
We consider the model of exact learning using an equivalence oracle and an incomplete membership oracle. In this model, a random subset of the learners membership queries is left unanswered. Our results are as follows. First, we analyze the obvious method for coping with missing answers: search exhaustively through all possible "answer patterns" associated with the unanswered queries. Thereafter, we present two specific concept classes that are efficiently learnable using an equivalence oracle and a (completely reliable) membership oracle, but are provably not polynomially learnable if the membership oracle becomes slightly incomplete. The first class will demonstrate that the aforementioned method of exhaustively searching through all possible answer patterns cannot be substantially improved in general (despite its apparent simplicity). The second class will demonstrate that the incomplete membership oracle can be rendered useless even if it leaves only a fraction 1/poly(n) of all queries unanswered. Finally, we present a learning algorithm for monotone DNF formulas that can cope with a relatively large fraction of missing answers (more than sixty percent), but is as efficient (in terms of run-time and number of queries) as the classical algorithm whose questions are always answered reliably.
37804897
Directions for Qualitative Reasoning
Sacks & Doyle provide an excellent overview of the fundamental limitations of the SPQR representations for reasoning about the qualitative properties of dynamic systems. We take this opportunity to outline some new directions for qualitative reasoning. In this paper, we provide a rigorous mathematical characterization for the term “qualitative property” in the context of static and dynamic systems. Based on these characterizations, we show that interval representations are well suited for reasoning about the qualitative properties of static systems such as qualitative comparative statics and qualitative stability. Moreover, we also show that symbolic computations help in the derivation of useful global properties of dynamic systems which can be used to guide numerical sampling of differential equations. The integration of symbolic and numeric methods provides a powerful approach for automating the qualitative analysis of differential equations.
6846387
Causality and Model Abstraction
Abstract Much of science and engineering is concerned with characterizing processes by equations describing the relations that hold among parameters of objects and govern their behavior over time. In formal descriptions of processes in terms of parameters and equations, the notion of causality is rarely made explicit. Formal treatments of the foundations of sciences have avoided discussions of causation and spoken only of functional relations among variables. Nevertheless, the notion of causality plays an important role in our understanding of phenomena. Even when we describe the behavior of a system formally in terms of acausal, mathematical relations, we often give an informal, intuitive explanation of why the system behaves the way it does in terms of cause-effect relations. In this paper, we will present an operational definition of causal ordering. The definition allows us to extract causal dependency relations among variables implicit in a model of a system, when a model is represented as a set of acausal, mathematical relations. Our approach is based on the theory of causal ordering first presented by Simon [22]. The paper shows how to use the theory and its extension in reasoning about physical systems. Further, the paper studies the relation of the theory to the problems of model aggregation.