Up
Previous
Next
Optimal mutual information quantization is NP-complete
Brendan Mumey and Tomas Gedeon
Montana State University
We consider the computational complexity of quantizing a joint (X,Y)
probability distribution in order to maximize the mutual information
of the quantization. We show that, in general, this problem is
NP-complete via reductions from various forms of the PARTITION
problem. In recent years, the problem of optimal quantization of the
mutual information between two random variables has received increased
attention. We note the application to discovery of neural coding
scheme in early sensory system of the cricket as well as the central
role of quantization of mutual information in the closely related
"Information bottleneck" optimization scheme. The latter method has
been applied to a variety of coding applications. In the application
to neural coding, one models the input to the system as a random
variable X and the output as a random variable Y. To simplify the
situation, as well as to acknowledge inherent discreteness of
collected data, X and Y are assumed to be discrete random variables.
Mutual information is always nonnegative and tells us, roughly, how
well we can predict X by observing Y, and vice versa. Since noise is
omnipresent in neural processing, sensory systems must be robust. As
such, they must represent similar stimuli in a similar way and then
react to similar internal representations of the outside stimuli in a
consistent way. This leads to a conclusion that individual input and
output patterns are not important for understanding neural function,
but rather classes of input stimuli and classes of output patterns and
their correspondence is the key. Therefore we are led to a problem of
optimal assignment of input and output patterns to classes, where the
optimality is judged on how much original mutual information is still
present between the collections of classes. More specifically, we will
consider two different quantization methods one-sided quantization and
two-sided (joint) quantization. In a one-sided quantization, we
quantize (cluster) only one variable, let us say Y, to a space of
finitely many abstract classes, YN, where N represents the number of
classes. In two-side quantization, both variables are simultaneously
quantized. In both cases, we have shown that finding the quantization
that maximizes mutual information is NP-complete. While it has been
speculated that theses two problems (and some associated quantization
problems) were NP-complete, to the authors' knowledge our work is the
first formal demonstration that this is so.