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If you obtain a vector by taking $n$ discrete samples over some underlying function, then it's easy to compare that vector with another of the same size. With a bunch of $n$-dimensional vectors, you can do interesting analysis like PCA. However, if all your vectors have different dimensionality, such as a bunch of images taken of (potentially) the same object at different resolutions, the problem is trickier. In computer vision, they often use a feature transform (such as SIFT) that will provide a comparison metric for the two vectors.

Is there a name for the problem of finding a transformation of a subspace from one vector that is closely aligned with a different transformation of a different subspace from another vector? E.g., matching a single word between two speakers when the word is embedded within a sentence; or finding an image of a car at different resolutions embedded in two different images.

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You might want to ask this on astronomy.stackexchange.com. I think they do something similar to align sequences of images in order to get an enhanced composite image. –  Codie CodeMonkey Nov 1 '11 at 7:09

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