Semiring induced valuation algebras: Exact and approximate local computation algorithms

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra.There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.

New insights on postglacial colonization in western Europe: the phylogeography of the Leisler’s bat (Nyctalus leisleri)

Despite recent advances in the understanding of the interplay between a dynamic physical environment and phylogeography in Europe, the origins of contemporary Irish biota remain uncertain. Current thinking is that Ireland was colonized post-glacially from southern European refugia, following the end of the last glacial maximum(LGM), some 20 000 years BP. The Leisler’s bat (Nyctalus leisleri), one of the few native Irish mammal species, is widely distributed throughout Europe but, with the exception of Ireland, is generally rare and considered vulnerable. We investigate the origins and phylogeographic relationships of Irish populations in relation to those across Europe, including the closely related species N. azoreum. We use a combination of approaches, including mitochondrial and nuclear DNA markers, in addition to approximate Bayesian computation and palaeo-climatic species distribution modelling. Molecular analyses revealed two distinct and diverse European mitochondrialDNAlineages,which probably diverged in separate glacial refugia. Awestern lineage, restricted to Ireland, Britain and the Azores, comprises Irish and British N. leisleri and N. azoreum specimens; an eastern lineage is distributed throughout mainland Europe. Palaeo-climatic projections indicate suitable habitats during the LGM, including known glacial refugia, in addition to potential novel cryptic refugia along the western fringe of Europe. These results may be applicable to populations of many species.

Unique mitochondrial DNA lineages in Irish stickleback populations: cryptic refugium or rapid recolonization?

Repeated recolonization of freshwater environments following Pleistocene glaciations has played a major role in the evolution and adaptation of anadromous taxa. Located at the western fringe of Europe, Ireland and Britain were likely recolonized rapidly by anadromous fishes from the North Atlantic following the last glacial maximum (LGM). While the presence of unique mitochondrial haplotypes in Ireland suggests that a cryptic northern refugium may have played a role in recolonization, no explicit test of this hypothesis has been conducted. The three-spined stickleback is native and ubiquitous to aquatic ecosystems throughout Ireland, making it an excellent model species with which to examine the biogeographical history of anadromous fishes in the region. We used mitochondrial and microsatellite markers to examine the presence of divergent evolutionary lineages and to assess broad-scale patterns of geographical clustering among postglacially isolated populations. Our results confirm that Ireland is a region of secondary contact for divergent mitochondrial lineages and that endemic haplotypes occur in populations in Central and Southern Ireland. To test whether a putative Irish lineage arose from a cryptic Irish refugium, we used approximate Bayesian computation (ABC). However, we found no support for this hypothesis. Instead, the Irish lineage likely diverged from the European lineage as a result of postglacial isolation of freshwater populations by rising sea levels. These findings emphasize the need to rigorously test biogeographical hypothesis and contribute further evidence that postglacial processes may have shaped genetic diversity in temperate fauna.

Advances in Learning Bayesian Networks of Bounded Treewidth

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

QRD and SVD processor design based on an approximate rotations algorithm

A silicon implementation of the Approximate Rotations algorithm capable of carrying the computational load of algorithms such as QRD and SVD, within the real-time realisation of applications such as Adaptive Beamforming, is described. A modification to the original Approximate Rotations algorithm to simplify the method of optimal angle selection is proposed. Analysis shows that fewer iterations of the Approximate Rotations algorithm are required compared with the conventional CORDIC algorithm to achieve similar degrees of accuracy. The silicon design studies undertaken provide direct practical evidence of superior performance with the Approximate Rotations algorithm, requiring approximately 40% of the total computation time of the conventional CORDIC algorithm, for a similar silicon area cost. © 2004 IEEE.

A Wavelet Approach to the Selective Computation of Eigenvalues for Small Signal Stability Analysis

This paper proposes a method to assess the small signal stability of a power system network by selective determination of the modal eigenvalues. This uses an accelerating polynomial transform, designed using approximate eigenvalues
obtained from a wavelet approximation. Application to the IEEE 14 bus network model produced computational savings of 20%,over the QR algorithm.

Significance Driven Computation on Next-Generation Unreliable Platforms

In this paper, we propose a design paradigm for energy efficient and variation-aware operation of next-generation multicore heterogeneous platforms. The main idea behind the proposed approach lies on the observation that not all operations are equally important in shaping the output quality of various applications and of the overall system. Based on such an observation, we suggest that all levels of the software design stack, including the programming model, compiler, operating system (OS) and run-time system should identify the critical tasks and ensure correct operation of such tasks by assigning them to dynamically adjusted reliable cores/units. Specifically, based on error rates and operating conditions identified by a sense-and-adapt (SeA) unit, the OS selects and sets the right mode of operation of the overall system. The run-time system identifies the critical/less-critical tasks based on special directives and schedules them to the appropriate units that are dynamically adjusted for highly-accurate/approximate operation by tuning their voltage/frequency. Units that execute less significant operations can operate at voltages less than what is required for correct operation and consume less power, if required, since such tasks do not need to be always exact as opposed to the critical ones. Such scheme can lead to energy efficient and reliable operation, while reducing the design cost and overheads of conventional circuit/micro-architecture level techniques.

Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

Approximate credal network updating by linear programming with applications to decision making

Credal nets are probabilistic graphical models which extend Bayesian nets to cope with sets of distributions. An algorithm for approximate credal network updating is presented. The problem in its general formulation is a multilinear optimization task, which can be linearized by an appropriate rule for fixing all the local models apart from those of a single variable. This simple idea can be iterated and quickly leads to accurate inferences. A transformation is also derived to reduce decision making in credal networks based on the maximality criterion to updating. The decision task is proved to have the same complexity of standard inference, being NPPP-complete for general credal nets and NP-complete for polytrees. Similar results are derived for the E-admissibility criterion. Numerical experiments confirm a good performance of the method.

Generalized loopy 2U: A new algorithm for approximate inference in credal networks

Credal networks generalize Bayesian networks by relaxing the requirement of precision of probabilities. Credal networks are considerably more expressive than Bayesian networks, but this makes belief updating NP-hard even on polytrees. We develop a new efficient algorithm for approximate belief updating in credal networks. The algorithm is based on an important representation result we prove for general credal networks: that any credal network can be equivalently reformulated as a credal network with binary variables; moreover, the transformation, which is considerably more complex than in the Bayesian case, can be implemented in polynomial time. The equivalent binary credal network is then updated by L2U, a loopy approximate algorithm for binary credal networks. Overall, we generalize L2U to non-binary credal networks, obtaining a scalable algorithm for the general case, which is approximate only because of its loopy nature. The accuracy of the inferences with respect to other state-of-the-art algorithms is evaluated by extensive numerical tests.

Generalized Loopy 2U: A New Algorithm for Approximate Inference in Credal Networks

Credal nets generalize Bayesian nets by relaxing the requirement of precision of probabilities. Credal nets are considerably more expressive than Bayesian nets, but this makes belief updating NP-hard even on polytrees. We develop a new efficient algorithm for approximate belief updating in credal nets. The algorithm is based on an important representation result we prove for general credal nets: that any credal net can be equivalently reformulated as a credal net with binary variables; moreover, the transformation, which is considerably more complex than in the Bayesian case, can be implemented in polynomial time. The equivalent binary credal net is updated by L2U, a loopy approximate algorithm for binary credal nets. Thus, we generalize L2U to non-binary credal nets, obtaining an accurate and scalable algorithm for the general case, which is approximate only because of its loopy nature. The accuracy of the inferences is evaluated by empirical tests.

Propositional and Relational Bayesian Networks Associated with Imprecise and Qualitative Probabilistic Assessments

This paper investigates a representation language with flexibility inspired by probabilistic logic and compactness inspired by relational Bayesian networks. The goal is to handle propositional and first-order constructs together with precise, imprecise, indeterminate and qualitative probabilistic assessments. The paper shows how this can be achieved through the theory of credal networks. New exact and approximate inference algorithms based on multilinear programming and iterated/loopy propagation of interval probabilities are presented; their superior performance, compared to existing ones, is shown empirically.

Exploiting Significance of Computations for Energy-Constrained Approximate Computing

Approximate execution is a viable technique for environments with energy constraints, provided that applications are given the mechanisms to produce outputs of the highest possible quality within the available energy budget. This paper introduces a framework for energy-constrained execution with controlled and graceful quality loss. A simple programming model allows developers to structure the computation in different tasks, and to express the relative importance of these tasks for the quality of the end result. For non-significant tasks, the developer can also supply less costly, approximate versions. The target energy consumption for a given execution is specified when the application is launched. A significance-aware runtime system employs an application-specific analytical energy model to decide how many cores to use for the execution, the operating frequency for these cores, as well as the degree of task approximation, so as to maximize the quality of the output while meeting the user-specified energy constraints. Evaluation on a dual-socket 16-core Intel platform using 9 benchmark kernels shows that the proposed framework picks the optimal configuration with high accuracy. Also, a comparison with loop perforation (a well-known compile-time approximation technique), shows that the proposed framework results in significantly higher quality for the same energy budget.

Learning Bayesian Networks with Bounded Tree-Width via Guided Search

Bounding the tree-width of a Bayesian network can reduce the chance of overfitting, and allows exact inference to be performed efficiently. Several existing algorithms tackle the problem of learning bounded tree-width Bayesian networks by learning from k-trees as super-structures, but they do not scale to large domains and/or large tree-width. We propose a guided search algorithm to find k-trees with maximum Informative scores, which is a measure of quality for the k-tree in yielding good Bayesian networks. The algorithm achieves close to optimal performance compared to exact solutions in small domains, and can discover better networks than existing approximate methods can in large domains. It also provides an optimal elimination order of variables that guarantees small complexity for later runs of exact inference. Comparisons with well-known approaches in terms of learning and inference accuracy illustrate its capabilities.

Local Computation in Credal Networks

The goal of this contribution is to discuss local computation in credal networks — graphical models that can represent imprecise and indeterminate probability values. We analyze the inference problem in credal networks, discuss how inference algorithms can benefit from local computation, and suggest that local computation can be particularly important in approximate inference algorithms.