one examines the aforementioned applications, one invariably finds that the most general settings for each of them would fall under this situation. The restriction to equidimensional subspaces thus somewhat limits the utility of these applications. For example, the principal subspaces of two matrices $A$ and $B$ for a given noise level would typically be of different dimensions, since there is no reason to expect the number of singular values of $A$ above a given threshold to be the same as that of $B$.

As such one may also find many applications that involve distances between subspaces of different dimensions: numerical linear algebra [@BES; @SS], information retrieval [@ref2a; @text], facial recognition [@ref2b; @face], image classification [@ref2a; @ref2b], motion segmentation [@motion; @ref2c; @ref2d], <span style="font-variant:small-caps;">eeg</span> signal analysis [@eeg], mechanical engineering [@mech], economics [@econ], network analysis [@network], blog spam detection [@blog], and decoding colored barcodes [@bar].

These applications are all based on two existing proposals for a distance between subspaces of different dimensions: The *containment gap* [@Kato pp. 197–199] and the *symmetric directional distance* [@SWF; @WWF]. They are however somewhat ad hoc and bear little relation to the natural geometry of subspaces. Also, it is not clear what they are suppose to measure 