Draft of paper published in:

Context-Dependent Thinning 8

Let us consider some versions of the CDT procedure, their properties and implementations.

4.1. Direct conjunctive thinning of two or more codevectors

Direct conjunctive thinning of binary x and y is implemented as their element-wise conjunction:

z = x ∧ y, (4.1)

where z is thinned and bound result.

The requirement of determinism (section 3.1) holds for the direct conjunctive thinning
procedure. The requirement of variable number of inputs (3.2) is not met, since only two codevectors are
thinned. Overlapping 1s of x and y go to z, therefore the sampling of inputs requirement (3.3) holds.
Since equal number of 1s from x and y enters into z even if x and y are of different density, the
requirement of proportional sampling (3.4) is not fulfilled in general case.

For stochastically independent vectors x and y the density of the resulting vector z is:
p(z) = p(x)p(y) < min(p(x),p(y)) < 1. (4.2)

Here min() selects the smallest of its arguments. Let us note that for correlated x and y the density of 1s
in z depends on the degree of their correlation. Thus p(z) is maintained the same only for independent
codevectors of constant density, and the requirement of uniform low density (3.5) is generally not met.
Since p(z) for sparse vectors is substantially lower than p(x) and p(y), the requirement of density control
(3.6) is not met and recursive construction of bound codevectors is not supported (see also Kanerva,
1998; Sjodin, et. al., 1998). Similarity and binding requirements (3.7-3.10) may be considered as
partially satisfied for two codevectors (see also Table 1).

Table 1. Properties of various versions of thinning procedures. "Yes" means that the property is present,
"No" means that the property is not present, “No-Yes” and "Yes-No" mean that the property is partially
present. See text for details.

Though the operation of direct conjunctive thinning of two codevectors does not meet all
requirements on the CDT procedure, it has been applied by us for encoding of external information, in
particular, for binding of distributed binary codevectors of feature item and its numerical value (Kussul
& Baidyk, 1990; Rachkovskij & Fedoseyeva, 1990; Artykutsa et al., 1991; Kussul, Rachkovskij, &
Baidyk, 1991a, 1991b). The density p of the codevectors of features and numerical values was chosen so
as to provide a specified density p' of the resulting codevector (Table 2, K=2).

Table 2. The density p of K independent codevectors chosen to provide a specified density p' of
codevectors produced by their conjunction.

<<   <   5   6   7   8   9   10   11   >   >>