Clustering alternatives in preference-approvals via novel pseudometrics

Preference-approval structures combine preference rankings and approval voting for declaring opinions over a set of alternatives. In this paper, we propose a new procedure for clustering alternatives in order to reduce the complexity of the preferenceapproval space and provide a more accessible interpretation of data. To that end, we present a new family of pseudometrics on the set of alternatives that take into account voters’ preferences via preference-approvals. To obtain clusters, we use the Ranked k-medoids (RKM) partitioning algorithm, which takes as input the similarities between pairs of alternatives based on the proposed pseudometrics. Finally, using non-metric multidimensional scaling, clusters are represented in 2-dimensional space

Formal Results Regarding Metric-Space Techniques for the Study of Astrophysical Maps

We extend a newly developed formal system for the description of astrophysical maps. In this formalism, we consider the difference between maps to be the distance between elements of a pseudometric space (the space of all such maps). This ansatz allows us to measure quantitatively the difference between any two maps and to order the space of all maps. For each physical characteristic of interest, this technique assigns an ``output'' function to each map; the difference between the maps is then determined from the difference between their corresponding output functions. In this present study, we show that the results of this procedure are invariant under a class of transformations of the maps and the domains of the maps. In addition, we study the propagation of errors (observational uncertainties) through this formalism. We show that the uncertainties in the output functions can be controlled provided that the signal to noise ratios in the original astrophysical maps are sufficiently high. The results of this paper thus increase the effectiveness of this formal system for the description, classification, and analysis of astrophysical maps.Comment: 30 pages Plain Tex, submitted to ApJ, UM-AC 93-1

Metric Stabilization of Invariants for Topological Persistence

Rank or the minimal number of generators is a natural invariant attached to any n-dimensional persistent vector space. However, rank is highly unstable. Building an algorithmic framework for stabilizing the rank in one-dimensional persistence and proving its usefulness in concrete data analysis are the main objectives of this thesis. Studied stabilization process relies on choosing a pseudometric between tame persistent vector spaces. This allows to minimize the rank of a persistent vector space in larger and larger neighbourhoods around it with respect to the chosen pseudometric. The result is the stable rank invariant, a simple non-increasing function from non-negative reals to non-negative reals. We show how the needed pseudometrics arise from so called persistence contours. Contour is a certain function system which can be generated very efficiently and in implementable way by integrating a so called density function from non-negative reals to strictly positive reals. We prove an algorithmic way of computing the stable rank invariant with respect to a chosen contour. The result of the theoretical development is an embedding theorem showing that persistent vector spaces embed into Lebesgue measurable functions through stable rank. The success of persistent homology in data analysis has been largely due to the barcode decomposition and its efficient computation. One result of this thesis is that the barcode decomposition can be proved using the monotonicity of the rank with respect to taking a subspace of persistent vector space. This property of the rank only holds in one-dimensional case. We claim that rank is more fundamental for persistence and barcode is but a technical artifact of its properties. Even though barcode is a powerful tool, progress in persistence theory requires invariants generalizing to multi-dimensional persistence and not relying on decomposition theorems. Recent years have seen active research around mapping barcodes to some representation that enables statistics of results from persistent homology analysis and connects naturally to machine learning algorithms. Our embedding theorem shows that the stable rank provides a connection to machine learning. One of our main results is the full applicability of our pipeline in practical data analysis. We demonstrate how choosing an appropriate contour can enhance results of supervised learning. Contour can also be seen to act as a form of feature selection on the bar decomposition

Combining Sources of Description for Approximating Music Similarity Ratings

In this paper, we compare the effectiveness of basic acoustic features and genre annotations when adapting a music similarity model to user ratings. We use the Metric Learning to Rank algorithm to learn a Mahalanobis metric from comparative similarity ratings in in the MagnaTagATune database. Using common formats for feature data, our approach can easily be transferred to other existing databases. Our results show that genre data allow more effective learning of a metric than simple audio features, but a combination of both feature sets clearly outperforms either individual set

Topological structures of complex belief systems

The concepts of substantive beliefs and derived beliefs are defined, a set of substantive beliefs S like open set and the neighborhood of an element substantive belief. A semantic operation of conjunction is defined with a structure of an Abelian group. Mathematical structures exist such as poset beliefs and join-semilattttice beliefs. A metric space of beliefs and the distance of belief depending on the believer are defined. The concepts of closed and opened ball are defined. S′ is defined as subgroup of the metric space of beliefs Σ and S′ is a totally limited set. The term s is defined (substantive belief) in terms of closing of S′. It is deduced that Σ is paracompact due to Stone's Theorem. The pseudometric space of beliefs is defined to show how the metric of the nonbelieving subject has a topological space like a nonmaterial abstract ideal space formed in the mind of the believing subject, fulfilling the conditions of Kuratowski axioms of closure. To establish patterns of materialization of beliefs we are going to consider that these have defined mathematical structures. This will allow us to understand better cultural processes of text, architecture, norms, and education that are forms or the materialization of an ideology. This materialization is the conversion by means of certain mathematical correspondences, of an abstract set whose elements are beliefs or ideas, in an impure set whose elements are material or energetic. Text is a materialization of ideology