The method of Structural Risk Minimization refers to tuning the capacity of the classifier to the available amount of training data. This capacity is influenced by several factors, including: (1) properties of the input space, (2) nature and structure of the classifier, and (3) learning algorithm. Actions based on these three factors are combined here to control the capacity of linear classifie...

Learning is posed as a problem of function estimation, for which two principles of solution are considered: empirical risk minimization and structural risk minimization. These two principles are applied to two different statements of the function estimation problem: global and local. Systematic improvements in prediction power are illustrated in application to zip-code recognition.

We extend the VC theory of statistical learning to data dependent spaces of classifiers. This theory can be viewed as a decomposition of classifier design into two components; the first component is a restriction to a data dependent hypothesis class and the second is empirical risk minimization within that class. We define a measure of complexity for data dependent hypothesis classes and provid...

Given a probability measure P and a reference measure μ, one is often interested in the minimum μ-measure set with P -measure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P , and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samp...

We examine the NFA minimization problem in terms of atomic NFA’s, that is, NFA’s in which the right language of every state is a union of atoms, where the atoms of a regular language are non-empty intersections of complemented and uncomplemented left quotients of the language. We characterize all reduced atomic NFA’s of a given language, that is, those NFA’s that have no equivalent states. Usin...

Data warehouses (DWs) can become inconsistent when some dimensional constraints are not satisfied by the dimension instances. In this paper, we present preliminary results about the effects of the violation of partitioning constraints in homogeneous dimension instances over aggregation queries, and in particular over the summarizability property (SUMM) of the DWs. We are interested in finding w...

We consider the classical statistical learning/regression problem, when the value of a real random variable Y is to be predicted based on the observation of another random variableX . Given a class of functions F and a sample of independent copies of (X,Y ), one needs to choose a function f̂ from F such that f̂(X) approximates Y as well as possible, in the mean-squared sense. We introduce a new p...

Let F be a class of measurable functions f :S 7→ [0,1] defined on a probability space (S,A, P ). Given a sample (X1, . . . ,Xn) of i.i.d. random variables taking values in S with common distribution P , let Pn denote the empirical measure based on (X1, . . . ,Xn). We study an empirical risk minimization problem Pnf →min, f ∈ F . Given a solution f̂n of this problem, the goal is to obtain very ge...

Data warehouses (DWs) can become inconsistent when some dimensional constraints are not satisfied by the dimension instances. In this paper, we present preliminary results about the effects of the violation of partitioning constraints in homogeneous dimension instances over aggregation queries, and in particular over the summarizability property (SUMM) of the DWs. We are interested in finding w...