Typical error pattern recovery of the Hopfield memory under error-tolerant conditions

We lower-bound the error-correcting capabilities of the Hopfield (1982) network under error-tolerant conditions, given some distorted version of a fundamental memory with a random (typical) error pattern. Our main result is that for any sufficiently small α, and for any ρ<1/2, a Hopfield memory with n neurons can store αn fundamental memories, such that the following is satisfied in probability: a distorted fundamental memory with ρn errors, located at random positions may be corrected up to a residual error rate of exp{-1/2α}