Add generalized repmat function as repeat

[johnmyleswhite]

This pull request provides a function called repeat that generalizes both R's rep function and Matlab repmat function. It allows the user to repeat the inner entries of a tensor as many times as desired and to also repeat outer slices of the tensor as many times as desired.

The major question I have is where the function fld1 should be placed: if it's not generally useful, I can hide it more.

[ViralBShah]

I believe there is an open discussion on meshgrid. In some cases, it is useful to have a RepeatMatrix type, which keeps the representation compact but overloads indexing operations and such. In some other cases, the cost of creating the repeated matrix (largely memory usage) may be small compared to overhead of accessing it. Clearly, one needs both.

One thought is that routines such as ndgrid, repmat, repeat, can create these RepeatedMatrix types, and you can then do something like [ a ] to force the conversion to a Matrix.

This is just food for thought - and should not hold up this PR in any way. This discussion should possibly be captured elsewhere.

[johnmyleswhite]

I'm not an experienced Matlab user, but my understanding is that meshgrid is effectively a way to compute Cartesian products. Is that right? If meshgrid behaves like R's expand.grid, then it has quite different behavior from what repeat is trying to do. repeat is just an attempt to have short notation for many recurring patterns in statistics -- like saying that the first 1000 rows of a matrix belong to Condition 1 and the second belong to Condition 2. This came up a lot while demoing out Vega.jl, which is why I finally focused on finishing this function. repeat would solve that problem by letting you write things like: repeat([1, 2], inner = [1000]).

Producing a RepeatMatrix type would be nice in many cases. I suspect I'd often want to get a standard editable Array back and not a repeated version, but having both sounds really helpful.

[ViralBShah]

Could you add repeat to the help as well?

[lindahua]

The computation of indices seem to be very expensive, and such computation is repeated for each element ..

[johnmyleswhite]

We can avoid that computation for vectors and matrices easily by writing special case functions, which will probably handle most use cases. I couldn't figure out how to express this computation for higher-order tensors using direct linear indexing.

[johnmyleswhite]

I'll write special case code to improve performance, then write up the function for help.

[johnmyleswhite]

@ViralBShah, I've added repeat to the help docs.

@lindahua, I've tried to make repeat faster by writing specialized ind2sub and sub2ind functions. We could try to write branches to handle special cases to make the code faster for matrices, but this revision of repeat is already only 2x slower than repmat on my machine while being substantially more general. With profiling I suspect we can make the gap smaller still.

[timholy]

I've been previously frustrated by our repmat's limitation to 2d, so I support merging this.

[johnmyleswhite]

I'll try to speed this up later today. Are you happy with this interface, @timholy?

I've come to want fld1 and rem1 more over time. I can make a separate pull request for those.

[timholy]

Yes indeed. I strongly approve of its generality :-).

And to clarify, speed improvements would be nice, but I'd rate them as optional because I frankly don't expect major improvements. (Dahua's suggestions were much more important; this is trying to get the last bit of juice out something that has already been squeezed pretty hard.) I would say removing the overly-restrictive type-qualifier is more important, and the optimizations are optional. Merge at will.

[johnmyleswhite]

Anybody else have comments on this? @StefanKarpinski, @JeffBezanson, @ViralBShah?

[johnmyleswhite]

I'm going to merge this in a bit unless someone says otherwise since it seems to not step on most people's toes and helps the rest of us a lot.

[StefanKarpinski]

Go for it, although I may tinker with the fld1 function.

[bjarthur]

given repeat, should repmat be deprecated? the latter cannot handle more than 2 dimensions, and does not have the flexibility provided by the former's inner argument.