In cases where the Cesaro limit does not exist this function can actually be defined as the Banach limit of the indicator functions, which is an extension of this limit.
42.
Now generalize the concept of set indicator function by releasing the constraint that the values are 0 and 1 only and allow the values 2, 3, 4 and so on.
43.
Let " x i " be the set membership indicator function for feature " f i "; then the above can be rewritten as an optimization problem:
44.
Where I ( . ) is the indicator function, H _ m is the homography transformation from I _ 0 to I _ m, and \ epsilon is the threshold.
45.
In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domain " D ".
46.
Where \ Pr ( A \ mid \ Sigma ) denotes the conditional expectation of the indicator function of the event A, \ chi _ A, given the sigma algebra \ Sigma.
47.
This means 2 ?! 1 ( since indicator functions are Riemann-integrable ), and 1 ?! 2 for " f " being an indicator function of an interval.
48.
This means 2 ?! 1 ( since indicator functions are Riemann-integrable ), and 1 ?! 2 for " f " being an indicator function of an interval.
49.
Thus, the softmax function can be used to construct a weighted average that behaves as a smooth function ( which can be conveniently differentiated, etc . ) and which approximates the indicator function
50.
This may be achieved by choosing " S " randomly, or by choosing the indicator function of " S " to be the concatenation of all finite binary sequences.
