Comments on "On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension"

Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in $\mathbb{Z}^n$. In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in $\mathbb{R}^n$. We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors.Comment: 7 pages, 1 figure, 3 tables. arXiv admin note: substantial text overlap with arXiv:1008.487

The Physics of Communicability in Complex Networks

A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and applied to a wide variety of real-world networks in recent years. Several such communicability functions are reviewed in this paper. It is emphasized that communication and correlation in networks can take place through many more routes than the shortest paths, a fact that may not have been sufficiently appreciated in previously proposed correlation measures. In contrast to these, the communicability measures reviewed in this paper are defined by taking into account all possible routes between two nodes, assigning smaller weights to longer ones. This point of view naturally leads to the definition of communicability in terms of matrix functions, such as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either the adjacency matrix or the graph Laplacian associated with the network. Considerable insight on communicability can be gained by modeling a network as a system of oscillators and deriving physical interpretations, both classical and quantum-mechanical, of various communicability functions. Applications of communicability measures to the analysis of complex systems are illustrated on a variety of biological, physical and social networks. The last part of the paper is devoted to a review of the notion of locality in complex networks and to computational aspects that by exploiting sparsity can greatly reduce the computational efforts for the calculation of communicability functions for large networks.Comment: Review Article. 90 pages, 14 figures. Contents: Introduction; Communicability in Networks; Physical Analogies; Comparing Communicability Functions; Communicability and the Analysis of Networks; Communicability and Localization in Complex Networks; Computability of Communicability Functions; Conclusions and Prespective

Distance Transform Computation for Digital Distance Functions

International audienceIn image processing, the distancetransform (DT), in which each object grid point is assigned the distance to the closest background grid point, is a powerful and often used tool. In this paper, distancefunctions defined as minimal cost-paths are used and a number of algorithms that can be used to compute the DT are presented. We give proofs of the correctness of the algorithms

A walk in the noncommutative garden

This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the ``Tehran program'') of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.Comment: 106 pages, LaTeX, 23 figure

A Formal Model of Ambiguity and its Applications in Machine Translation

Systems that process natural language must cope with and resolve ambiguity. In this dissertation, a model of language processing is advocated in which multiple inputs and multiple analyses of inputs are considered concurrently and a single analysis is only a last resort. Compared to conventional models, this approach can be understood as replacing single-element inputs and outputs with weighted sets of inputs and outputs. Although processing components must deal with sets (rather than individual elements), constraints are imposed on the elements of these sets, and the representations from existing models may be reused. However, to deal efficiently with large (or infinite) sets, compact representations of sets that share structure between elements, such as weighted finite-state transducers and synchronous context-free grammars, are necessary. These representations and algorithms for manipulating them are discussed in depth in depth. To establish the effectiveness and tractability of the proposed processing model, it is applied to several problems in machine translation. Starting with spoken language translation, it is shown that translating a set of transcription hypotheses yields better translations compared to a baseline in which a single (1-best) transcription hypothesis is selected and then translated, independent of the translation model formalism used. More subtle forms of ambiguity that arise even in text-only translation (such as decisions conventionally made during system development about how to preprocess text) are then discussed, and it is shown that the ambiguity-preserving paradigm can be employed in these cases as well, again leading to improved translation quality. A model for supervised learning that learns from training data where sets (rather than single elements) of correct labels are provided for each training instance and use it to learn a model of compound word segmentation is also introduced, which is used as a preprocessing step in machine translation

Off-critical local height probabilities on a plane and critical partition functions on a cylinder

We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a $4 N$-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of $N$, the winding number of the spiral, and $\tau$, the departure from criticality of the model, and observe that the result depends only on the product $N \, \tau$. In the limit $N \rightarrow 1$, $\tau \rightarrow \tau_0$, such that $\tau_0$ is finite, we recover the off-critical local height probability on a plane, $\tau_0$-away from criticality. In the limit $N \rightarrow \infty$, $\tau \rightarrow 0$, such that $N \, \tau = \tau_0$ is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio $\tau_0$. We conclude that the off-critical local height probability on a plane, $\tau_0$-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio $\tau_0$, in agreement with a result of Saleur and Bauer.Comment: 28 page