Estimating Maximally Probable Constrained Relations by Mathematical Programming

Estimating a constrained relation is a fundamental problem in machine learning. Special cases are classification (the problem of estimating a map from a set of to-be-classified elements to a set of labels), clustering (the problem of estimating an equivalence relation on a set) and ranking (the problem of estimating a linear order on a set). We contribute a family of probability measures on the set of all relations between two finite, non-empty sets, which offers a joint abstraction of multi-label classification, correlation clustering and ranking by linear ordering. Estimating (learning) a maximally probable measure, given (a training set of) related and unrelated pairs, is a convex optimization problem. Estimating (inferring) a maximally probable relation, given a measure, is a 01-linear program. It is solved in linear time for maps. It is NP-hard for equivalence relations and linear orders. Practical solutions for all three cases are shown in experiments with real data. Finally, estimating a maximally probable measure and relation jointly is posed as a mixed-integer nonlinear program. This formulation suggests a mathematical programming approach to semi-supervised learning.Comment: 16 page

Justifying Information-Geometric Causal Inference

Information Geometric Causal Inference (IGCI) is a new approach to distinguish between cause and effect for two variables. It is based on an independence assumption between input distribution and causal mechanism that can be phrased in terms of orthogonality in information space. We describe two intuitive reinterpretations of this approach that makes IGCI more accessible to a broader audience. Moreover, we show that the described independence is related to the hypothesis that unsupervised learning and semi-supervised learning only works for predicting the cause from the effect and not vice versa.Comment: 3 Figure

Modal Similarity

Just as Boolean rules define Boolean categories, the Boolean operators define higher-order Boolean categories referred to as modal categories. We examine the similarity order between these categories and the standard category of logical identity (i.e. the modal category defined by the biconditional or equivalence operator). Our goal is 4-fold: first, to introduce a similarity measure for determining this similarity order; second, to show that such a measure is a good predictor of the similarity assessment behaviour observed in our experiment involving key modal categories; third, to argue that as far as the modal categories are concerned, configural similarity assessment may be componential or analytical in nature; and lastly, to draw attention to the intimate interplay that may exist between deductive judgments, similarity assessment and categorisation

An analysis of word embedding spaces and regularities

Word embeddings are widely use in several applications due to their ability to capture semantic relationships between words as relations between vectors in high dimensional spaces. One of the main problems to obtain the information is to deal with the phenomena known as the Curse of Dimensionality, the fact that some intuitive results for well known distances are not valid in high dimensional contexts. In this thesis we explore the problem to distinguish between synonyms or antonyms pairs of words and non-related pairs of words attending just to the distance between the words of the pair. We considerer several norms and explore the problem in the two principal kinds of embeddings, GloVe and Word2Vec

Graph ambiguity

In this paper, we propose a rigorous way to define the concept of ambiguity in the domain of graphs. In past studies, the classical definition of ambiguity has been derived starting from fuzzy set and fuzzy information theories. Our aim is to show that also in the domain of the graphs it is possible to derive a formulation able to capture the same semantic and mathematical concept. To strengthen the theoretical results, we discuss the application of the graph ambiguity concept to the graph classification setting, conceiving a new kind of inexact graph matching procedure. The results prove that the graph ambiguity concept is a characterizing and discriminative property of graphs. (C) 2013 Elsevier B.V. All rights reserved