Open sethaxen opened 1 year ago
made a comment and removed it (just me mixing up notation again, which makes the need to make it clear even more important). i keep thinking in terms of $\Delta^N$ and forgetting we're actually using $\Delta^{N-1}$ but defining $\Delta^{N}$ in the first section
@Raoul-Kima if you got a github email, ignore that!
We want the standard definition of unit simplex,
$\Delta^N = \{ \theta \in \mathbb{R}^{N+1} : \theta_n \geq 0, \textrm{sum}(\theta) = 1 \}.$
Then we just have to keep the dimensions consistent elsewhere. I'm about to take a notational pass through the doc.
Did everyone decide to just skip over doing the simplex version and go straight to full generality? The intro no longer mentions simplexes and Section 3 currently begins,
In this section, we consider transforms from unconstrained space to constrained vectors, including unit-simplex, sum-to-zero, and unit-length constraints
Given the change in focus, I didn't want to get in an edit war of trying to change it back, so I just left it.
P.S. I didn't know where to comment with so many issues, so just chose this one at random.
P.P.S. I like the newly framed transforms intro section. That's about as clear as I think we'll be able to be.
Did everyone decide to just skip over doing the simplex version and go straight to full generality? The intro no longer mentions simplexes and Section 3 currently begins,
In this section, we consider transforms from unconstrained space to constrained vectors, including unit-simplex, sum-to-zero, and unit-length constraints
Given the change in focus, I didn't want to get in an edit war of trying to change it back, so I just left it.
P.S. I didn't know where to comment with so many issues, so just chose this one at random.
P.P.S. I like the newly framed transforms intro section. That's about as clear as I think we'll be able to be.
i thought we agreed on simplex. and not general. i was making rough patches of things about aitchison geometry in general to add, @spinkney adding the ilr as well makes it seem like a good prospect to me - so we get a chance to introduce the transforms idea for hmc in general as well as the compositional data analysis utility point of view is there
yeah i think it looks good enough!
PS - sorry for not responding a lot on the issues, missed 3 weeks of classes so had some catching up to do, will add my changes in a day or two, the cluster had some rotation shift going on, so will check check and then put the other transforms to sample as well.
We want the standard definition of unit simplex,
ΔN={θ∈RN+1:θn≥0,sum(θ)=1}.
Then we just have to keep the dimensions consistent elsewhere. I'm about to take a notational pass through the doc.
yeah i continued this over the email thread - unit and standard simplex have similar definitions and some places use them interchangeably which isnt correct and then there's aitchison's definition too
The "unit" in "unit simplex" refers to the sum being unity (i.e., 1). In the Wikipedia entry for simplex, "standard N-simplex" and "unit N-simplex" are synonyms that both refer to $\Delta^{N}$. I suggest we just stick to the "unit simplex" terminology so as not to cause confusion.
In the paper, we define the $N$-simplex as being an $N+1$-length vector. In our model implementations, we use $N$ to refer to the number of elements in the vector on the simplex. We should harmonize these notations.