Decision bireducts and decision reducts – a comparison

The Problem of Finding the Simplest Classifier Ensemble is NP-Hard – A Rough-Set-Inspired Formulation Based on Decision Bireducts

“…It requires encoding of each input graph G to its corresponding decision table A G . This encoding is analogous to those that were utilized for other (bi)redect-related optimization problems [2,8]. In particular, as it could be already noticed earlier in Fig.…”

“…Every decision bireduct (X, B) may be understood as a pair consisting of an irreducible subset of attributes that can be evaluated by means of an non-extendable subset of objects for which it provides good classification. It was shown in [8] that X is actually the set-theoretic sum of objects supporting deterministic rules using the values of attributes in B to describe the values of d. Figure 1 displays some bireducts derived for decision table A with objects U = {u 1 , ..., u 6 } and conditional attributes A = {a 1 , a 2 , a 3 }. For instance, consider the pair B = {a 1 , a 2 } and X = {u 2 , ..., u 6 }.…”

“…The rough literature provides a great number of theoretical works on computational complexity of optimization problems focused on deriving the simplest possible decision models from the data [2]. Let us refer to a recent comparative study reflecting both decision bireducts and so-called approximate reducts with this respect [8]. By "the simplest" one can mean (bi)reducts involving the minimal amounts of attributes, generating minimal amounts of decision rules, having the minimal information entropy, etc.…”

“…For example, in case of rough-set-inspired approaches to knowledge discovery, it refers to computation of diverse decision reducts, i.e., irreducible (the smaller -the better) subsets of attributes that are sufficient to determine decision labels. If we want to put this idea together with diversification of cases which are classified correctly/wrongly by particular models, we can rely on ensembles of so-called decision bireducts [7,8], whereby irreducibility of a subset of attributes is combined with non-extendability of a subset of objects (cases), for which those attributes let us form valid classification rules.…”

Representative Set of Objects in Rough Sets Based on Galois Connections

“…This kind of reduction has been seldom considered by the research community, which has mainly focused on attribute reduction [2,5,12,18]. Some examples of the study of object reductions are [15], which analyses the reduction of objects oriented to keep the original attribute reducts and [1,13,16,22,23] that reducts objects and attributes in parallel. The present paper is oriented in a different way than the existing approaches dealing with object reduction.…”