Graphs for Pattern Recognition

5Inequality systems, committees, (hyper)graphs, and alternative covers

A committee of a rank n infeasible system (2.26),

S := {⟨ai , x⟩ > 0: ai , x ∈ R n; ‖ai‖ = 1, i ∈ [m]; i1 ≠ i2 ⇒ ai1 ≠ −ai2} ,

of homogeneous strict linear inequalities over the real Euclidean space R n, introduced on page 33, is defined as a finite set of vectors K ⊂ R n − {0} satisfying the relation

|{x ∈ K: ⟨ai , x⟩ > 0}| > |K| , for every vector ai, i ∈ [m].

As earlier, we will denote by J := {Js ⊂ [m] : s ∈ [q]} the family of the multi-indices of all MFSs of the system .

Among the techniques that are used in the committee method is the extracting minimal infeasible and ...