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Comment by thaumasiotes | Hacker News Reader

> But map X into Y of different dimensions and you get a non-square matrix

This is true, but it's also something that is arguably impossible on the one hand and "unwise" on the other.

You can easily map a higher-dimensional space into a lower-dimensional space, but you will irretrievably lose a lot of information when you do so.

And in the other direction, you can't actually map a lower-dimensional space into a higher-dimensional space with a matrix. The image of X can never have more dimensions than X does -- the choice to represent it with Y > X dimensions is just that, a representational choice. This idea is only meaningful in terms of the semantics behind the representation.

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> This is true, but it's also something that is arguably impossible on the one hand and "unwise" on the other.

If I understand "unwise" to refer to a linear map into a lower-dimensional space: of course that's something you'll often want to do! Suppose, for example, that most of the interesting structure of your data is close to lying on a n-dimensional subspace of R^m, with n<m. Constructing a clever linear map from R^m to R^n can be very wise and useful!

> And in the other direction, you can't actually map a lower-dimensional space into a higher-dimensional space with a matrix. The image of X can never have more dimensions than X does -- the choice to represent it with Y > X dimensions is just that, a representational choice.

Any matrix involves a choice (of basis), so that complaint is moot. For sure you can have a linear map whose codomain is higher-dimensional. You are correct that the image can't be higher-dimensional, but that doesn't prevent the existence of such a map.

Sometimes losing that information is very useful however, to arrive at a representation that has a specific meaning.

For example, an x-ray computed tomography (CT) image volume in 3D may be projected into various 2D synthetic planar x-ray projections (digitally reconstructed radiographs).

There are countless situations where projections are very useful.

I don't disagree, but I think my comment helps to illustrate a reason why square matrices really are more special than other matrices.

In fact, the 3D CT image itself is reconstructed from projections! See https://en.wikipedia.org/wiki/Tomographic_reconstruction

Yes, but this is a case where you have three dimensions in the input and also three dimensions in the output. (You're assembling the 3D image from different 2D slices with different depth coordinates.)

The third dimension is discretized while the other two are continuous; the reconstruction consists of smoothing out the third dimension.

In the real world, all of the dimensions are discretized. The input dimensions are theta (gantry angle), n (detector channel), and z (table position). The output dimensions are x, y, z. For a fixed z, the plot of projection intensity as a function of theta and n is called a "sinogram" (which is indeed a 2D space) which gets reconstructed to form an image (also 2D). It is true that there are three dimensions of raw data and three dimensions of reconstructed image. However, due to various tricks with reconstruction models, the total number of samples does not have to be the same in the raw data and in the output. As a result you can see recon models employing nonsquare matrices. For more information, you can read about methods like iterative reconstruction, compressed sensing, and differentiated backprojection. This description is adequate for axial tomography, whereas other geometries like cone beam tomography are more complicated (see FDK method).

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