FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP

51The following lemma describes the asymptotic behavior of P(α,τ), for any α ∈ V(τ). Toclarify, the lemma describes the asymptotic rate of decay of the joint probability of an authenticationgroup with histogram α 0 and an estimation/categorization process corresponding to α 1 ,given that the group and categorization process result in an unpruned set of size|S| = τ n ρ(4.18)for some 0 ≤ τ ≤ ρ. This behavior will be described below as a concise function of the binomialrate-function (see [CT06])The lemma follows.I f (x) ={xlog(xɛ f)+(1−x)log( 1−x1−ɛ f) f ≥ 2xlog( x1−ɛ 1)+(1−x)log( 1−xɛ 1) f = 1.(4.19)Lemma 6wherelog− limN→∞ n/ρ P(α,τ) = ρD(α 0||p)+D(α 0 ||p) = ∑ fρ∑f=1α 0,f log α 0,fp fα 0,f I f(α 1,fα 0,f),is the informational divergence between α 0 and p (see [CT06]).The proof follows soon after. We now proceed with the main result, which averages the outcomein Lemma 6, over all possible authentication groups.Theorem 2 In SBS-based pruning, the size of the remaining set |S|, satisfies the following:logρ∑J(τ) := − limN→∞ n/ρ P(|S| ≈ τn ρ ) = inf ρ α 0,f log α 0,f+α∈V p fFurthermore we have the following.f=1ρ∑f=1α 0,f I f(α 1,fα 0,f). (4.20)Theorem 3 The probability that after pruning, the search space is bigger (resp. smaller) thanτ n ρ , is given for τ ≥ τ 0 byand for τ < τ 0log− limN→∞ n/ρ P(|S| > τn ) = J(τ) (4.21)ρlog− limN→∞ n/ρ P(|S| < τn ) = J(τ). (4.22)ρThe above describe how often we encounter authentication groupsvand feature estimation behaviorthat jointly cause the gain to deviate, by a specific degree, from the common behavior describedin (4.11), i.e., how often the pruning is atypically ineffective or atypically effective. We offer theintuition that the atypical behavior of the pruning gain is dominated by a small set of authenticationgroups, that minimize the expression in Theorem 2. Such minimization was presented inFig. 4.3, and in examples that will follow after the proofs.Please see the Annex B for the proofs.The following examples are meant to provide insight on the statistical behavior of pruning.