  We describe a general mathematical framework for $\chi^2$ discriminators in
the context of the compact binary coalescence search. We show that with any
$\chi^2$ is associated a vector bundle over the signal manifold, that is, the
manifold traced out by the signal waveforms in the function space of data
segments. The $\chi^2$ is then defined as the square of the $L_2$ norm of the
data vector projected onto a finite dimensional subspace (the fibre) of the
Hilbert space of data trains and orthogonal to the signal waveform - any such
fibre leads to a $\chi^2$ discriminator and the full vector bundle comprising
the subspaces and the base manifold constitute the $\chi^2$ discriminator. We
show that the $\chi^2$ discriminators used so far in the CBC searches
correspond to different fiber structures constituting different vector bundles
on the same base manifold, namely, the parameter space. The general formulation
indicates procedures to formulate new $\chi^2$s which could be more effective
in discriminating against commonly occurring glitches in the data. It also
shows that no $\chi^2$ with a reasonable degree of freedom is foolproof. It
could also shed light on understanding why the traditional $\chi^2$ works so
well. As an example, we propose a family of ambiguity $\chi^2$ discriminators
that is an alternative to the traditional one. Any such ambiguity $\chi^2$
makes use of the filtered output of the template bank, thus adding negligible
cost to the overall search. We test the performance of ambiguity $\chi^2$ on
simulated data using spinless TaylorF2 waveforms. We show that the ambiguity
$\chi^2$ essentially gives a clean separation between glitches and signals.
Finally, we investigate the effects of mismatch between signal and templates on
the $\chi^2$ and also further indicate how the ambiguity $\chi^2$ can be
generalized to detector networks for coherent observations.
