Measuring the similarity or distance between two sets of points in a metric space is an important problem in machine learning and has also applications in other disciplines e.g. in computational geometry, philosophy of science, methods for updating or changing theories, . . . . Recently Eiter and Mannila have proposed a new measure which is computable in polynomial time. However, it is not a distance function in the mathematical sense because it does not satisfy the triangle inequality.

Many learning systems, e.g. systems based on clustering and instance based learning systems, need a measure for the distance between objects. Adequate measures are available for attribute value learners. In recent years there is a growing interest in first order learners, however existing proposals for distances between non-ground atoms have some drawbacks. In this paper we develop a new measure for the distance between nonground atoms. 1 Introduction In learning systems based on clustering (e.g. C0.5 [3], KBG [1]) and in instance based learning (e.g. [9, ch.4], RIBL [6]), a measure of the distance between objects is an essential component. Good measures exist for distances between objects in an attribute value representation (see e.g. [9, ch. 4]). Recently there is a growing interest in using more expressive first order representations of objects and in upgrading propositional learning systems into first order learning systems (e.g. TILDE [2], ICL [5] and CLAUDIEN [4]). Some ad-hoc s...

Abstract. It is well known that many hard tasks considered in machine learning and data mining can be solved in an rather simple and robust way with an instance- and distance-based approach. In this paper we present another difficult task: learning, from large numbers of performances by concert pianists, to play music expressively. We model the problem as a multi-level decomposition and prediction task. Motivated by structural characteristics of such a task, we propose a new relational distance measure that is a rather straightforward combination of two existing measures. Empirical evaluation shows that our approach is in general viable and our algorithm, named DISTALL, is indeed able to produce musically interesting results. The experiments also provide evidence of the success of ILP in a complex domain such as music performance: it is shown that our instance-based learner operating on structured, relational data outperforms a propositional k-NN algorithm.

. The principles of instance based function learning are presented. In IBFL one is given a set of positive examples of a functional predicate. These examples are true ground facts that illustrate the input output behaviour of the predicate. The purpose is then to predict the output of the predicate given a new input. Further assumptions are that there is no background theory and that the inputs and outputs of the predicate consist of structured terms. IBFL is a novel technique that addresses this problem and that combines ideas from instance based learning, first order distances and analogical or case based reasoning. We also argue that IBFL is especially useful when there is a need for handling complex and deeply nested terms. Though we present the technique in isolation, it might be more useful as a component of a larger system to deal e.g. with the logic, language and learning challenge. 1

In the present paper we propose a consistent way to integrate syntactical least general generalizations (lgg's) with semantic evaluation of the hypotheses. For this purpose we use two different relations on the hypothesis space -- a constructive one, used to generate lgg's and a semantic one giving the coverage-based evaluation of the lgg. These two relations jointly implement a semantic distance measure. The formal background for this is a height-based definition of a semi-distance in a join semi-lattice. We use some basic results from lattice theory and introduce a family of language independent coverage-based height functions.

The paper presents an approach to inductive machine learning based on a consistent integration of the generalization-based (such as inductive learning from examples) and metric-based (such as agglomerative clustering) approaches. The approach stems from the natural idea (formally studied within lattice theory) to estimate the similarity between two objects in a hierarchical structure by the distances to their closest common parent. The hierarchies used are subsumption lattices induced by generalization operatiors (e.g. lgg) commonly used in inductive learning. Using some results from the theory the paper defines a unified framework for solving basic inductive learning tasks. An algorithm for this purpose is proposed and its performance is illustrated by examples. Introduction Inductive learning addresses mainly classification tasks where a series of training examples (instances) are supplied to the learning system and the latter builds an intensional or extensional rep...

Abstract. Several activities related to semantically annotated resources can be enabled by a notion of similarity, spanning from clustering to retrieval, matchmaking and other forms of inductive reasoning. We propose the definition of a family of semi-distances over the set of objects in a knowledge base which can be used in these activities. In the line of works on distance-induction on clausal spaces, the family is parameterized on a committee of concepts expressed with clauses. Hence, we also present a method based on the idea of simulated annealing to be used to optimize the choice of the best concept committee. 1

Abstract. Distance-based methods have been a successful family of machine learning techniques since the inception of the discipline. Basically, the classification or clustering of a new individual is determined by the distance to one or more prototypes. From a comprehensibility point of view, this is not especially problematic in propositional learning where prototypes can be regarded as a good generalisation (pattern) of a group of elements. However, for scenarios with structured data, this is no longer the case. In recent work, we developed a framework to determine whether a pattern computed by a generalisation operator is consistent w.r.t. a distance. In this way, we can determine which patterns can provide a good representation of a group of individuals belonging to a metric space. In this work, we apply this framework to analyse and define minimal distance-based generalisation operators (mg operators) for first-order data. We show that Plotkin’s lgg is a mg operator for atoms under the distance introduced by J. Ramon, M. Bruynooghe and W. Van Laer. We also show that this is not the case for clauses with the distance introduced by J. Ramon and M. Bruynooghe. Consequently, we introduce a new mg operator for clauses, which could be used as a base to adapt existing bottom-up methods in ILP. 1

. In the present paper we use the approach of height functions to defining a semi-distance measure between Horn clauses. This appraoch is already discussed elsewhere in the framework of propositional and simple first order languages (atoms). Hereafter we prove its applicability for Horn clauses. We use some basic results from lattice theory and introduce a family of language independent coverage-based height functions. Then we show how these results apply to Horn clauses. We also show an example of conceptual clustering of first order atoms, where the hypotheses are Horn clauses. 1 Introduction Almost all approaches to inductive learning are based on generalization and/or specialization hierarchies. These hierarchies represent the hypothesis space which in most cases is a partially ordered set under some generality ordering. The properties of partially ordered sets are well studied in lattice theory. One concept from this theory is mostly used in inductive learning -- this is...

Abstract. In this chapter, we describe a view of statistical learning in the inductive logic programming setting based on kernel methods. The relational representation of data and background knowledge are used to form a kernel function, enabling us to subsequently apply a number of kernel-based statistical learning algorithms. Different representational frameworks and associated algorithms are explored in this chapter. In kernels on Prolog proof trees, the representation of an example is obtained by recording the execution trace of a program expressing background knowledge. In declarative kernels, features are directly associated with mereotopological relations. Finally, in kFOIL, features correspond to the truth values of clauses dynamically generated by a greedy search algorithm guided by the empirical risk. 1