I've seen references to the "unit torus" in papers such as this (Start of Sec 3.3, page 5). So, what exactly is a unit torus? Is it just a square or cube in d-dimensions with periodic boundary conditions? I've seen this picture and this mathematical definition but neither seem to provide much clarity.
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2$\begingroup$ Does the second answer on the math stack exchange link not answer your question? $\endgroup$– user3209427Commented Mar 20 at 8:44
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2$\begingroup$ It does, thank you very much. Didn't see it. $\endgroup$– NNNCommented Mar 20 at 8:47
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1$\begingroup$ Looking at the context of that paper's use of "$d$-dimensional unit torus", it sets out function spaces $L^2(\mathbb T^d;\mathbb R^m)$ of varying dimensions $m$. What is needed is a measure on $\mathbb T^d$, presumably the product measure induced by Lebesgue measures on each unit circle factor of $\mathbb T^d$. Other contexts could well give rise to different definitions based on needs of the authors. $\endgroup$– hardmath ♦Commented Mar 20 at 14:43
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2$\begingroup$ Yes you can think of it as just $[0,1)^d$ with periodic boundary conditions. Basically if I give you a point $(x_1, \ldots, x_d)$ then for $0\leq x_i$ you would map that to $f_i = x_i - \lfloor x_i\rfloor$, and for $x_i<0$ you can use $f_i = \lceil -x_i \rceil + x_i$. Also note that your distance is also typically toroidal, i.e. the distance between two points $p,q$ is not just $\|p-q\|$ but rather the minimum out of all distances $\min_{e\in\{-1,0,1\}^d}\|p - (e + q)\|$. $\endgroup$– lightxbulbCommented Mar 20 at 23:55
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$\begingroup$ They seem to be using the discrete fourier transform in that paper which assumes that a signal/image/data is periodically extended in space (it's a discrete analogue of Fourier series which also assumes your signal is periodic), or equivalently that it is on a torus. $\endgroup$– lightxbulbCommented Mar 21 at 0:03
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