The value of the energy change 
can be
used to measure the (local) stability of 
with
respect to a certain configuration 
. Ideally, we want
to be very low and 
to be very high, such that 
is
very large, for all 
. In such a situation, 
is expected to be a stable minimum where the stability is said
w.r.t. perturbations in the observation and w.r.t. the local minimum
problem with the minimization algorithm.
The smaller is 
, the larger is the chance
with which a perturbation to the observation will cause 
to become negative to violate the correctness. When
, 
no longer corresponds
to the global minimum. Moreover, we assume that configurations f whose
energies are slightly higher than 
are
possibly local energy minima at which an energy minimization
algorithm is most likely to get stuck.
Therefore, the energy difference, i.e.
the local stabilities should be
enlarged. One may define the global stability as the sum of all 
. For reasons to be explained later,
instability, instead of stability, is used for
evaluating 
.
The local instability for a correct estimate 
is
defined as
where 
. It is ``local'' because it considers only one
It is desirable to choose 
such that the value of
is small for all 
. Therefore, we
defined the following global p-instability of 
where 
. The total global p-instability of 
is
In the limit as 
, we have (This is because for the p-norm defined by 
, we have 
.)
is due solely to f having the smallest 
or largest
value.
Unlike the global stability definition, the global instability treats
each item in the following manner: those f having smaller 
(larger 
) values affect
in a more significant way. For p=2, for example,
the partial derivative is
where 
takes the linear form
(7.2). The smaller the 
is, the more it affects 
. This is desirable because
such f are more likely than the others to violate the correctness,
because their 
values are small, and
should be more influential in determining 
.
