The following problem relates to the phenomenom of spontaneous recovery of
memory in linear associative neural networks (although this may not be
obvious from the form of the problem below - interested readers can obtain
details of this interpretation from me).
Consider an nD space containing two linear manifolds L1 and L2 of dimension
naD and nbD, where n=na+nb, (typically na=nb=n/2). The intersection of L1
and L2 defines a point, O, which we define to be the origin. The angle
between L1 and L2 is alpha. Now choose a unit vector P at random (i.e.
with uniform pdf) from the surface of the unit hypershere. We define the
distance of P from L1 as p=|P-L1|. Now define P' as the orthogonal
projection of P onto L2, and define the distance of P' from L1 as
q=|P'-L1|. Define d=p-q.
The key questions are:
1) What is the expected probability <rho> that p>q as a function of na and
nb, where this expectation is taken over alpha? (As stated above,
typically na=nb=n/2).
2) What is the expected value <d> of d=p-q?
Subsidiary questions that contribute to the above question are
A What is the probability rho that p>q as a function of na, nb and alpha?
B What is the expected value of p-q as a function of na, nb and alpha?
Notes:
In 2D, the probability rho that p>q is 1-[atan(2*tan alpha)/pi],
and the expected value is <rho>=0.68.
Numerical simulations suggest that <rho> increases with n, as shown in the
table below.
n na=nb <rho> <p-q>
2 1 0.68 0.176
4 2 0.75 0.176
6 3 0.79 0.186
8 4 0.83 0.200
10 5 0.85 0.190
12 6 0.87 0.192
14 7 0.89 0.196
16 8 0.91 0.192
18 9 0.92 0.199
20 10 0.93 0.200
comments and suggestions welcome,
thanks in advance,
Jim Stone.
Dr Jim Stone,
Psychology Department,
Sheffield University, UK.
http://www.shef.ac.uk/~pc1jvs/
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