Open dlfivefifty opened 1 year ago
By the way, expect progress with regard to:
Currently trying to adapt the method of recovering the measure from the Inverse Cauchy transform after knowing it's support
Awesome!
there’s something here I’m worried about: in this case we want a point cloud living on a Riemann surface but I don’t know how to tell which inverse lives on which sheet
ALTHOUGH… perhaps we don’t actually need this information as the “culling” process might still be valid.
Free Convolution of Semicircle and atoms $1/4 (2\delta{-2} + \delta{-1} + \delta_{-1})$
Turns out that the pruning method we discussed earlier (for every $j$, for each inverse $k$, $G{\mu}(G{\mu}^{-1}(y_j)_k) = y_j$ is enough to recover the measure here (this was the example someone else did with fixed point iteration)
This is awesome!!
I guess the main thing missing now is when both the input measures are multi-valued? That is, how can we tell which inverse goes with which inverse?
It's possible we can just try every combination and the pruning procedure will deduce whether the pair are admissible or not.
Is there an algorithm to find all roots within an interval?
It appears that Algorithm 5 is not applicable in the case of convolving a Square root measure with A point measure with 3 or more atoms i.e. there could be multiple roots of G^-1 in the interval [0, b_0]
A few approaches:
Oh I forgot an easy one: Newton + Deflation . Basically you divide by the first root you find. This can be numerically unstable so instead you multiply by something like λ + 1/(x - r)
I guess this would also work with bisection
Or even just use bisection multiple times?
I've been thinking about the convolution of a Sqrt measure with a point measure, and I have the following conjecture:
Let $Gb(w)$ be the Cauchy transform of the point measure $\mu{b}$ (For technical reasons, it must have at least 2 points, and finitely many points). Consider the function $1/Gb(w) - w$, which admits a Nevanlinna representation $-\int \textup{d} \mu{b} + \int \frac{1}{x-w}\textup{d} \mu'_{b}(x)$
This measure $\mu'_{b}$ has total measure equal to the variance of the point measure. And in this case, it happens to also be a point measure, with atoms whenever $G_b(w) = 0$.
If you represent the new measure as a sum of atoms $a1 \delta{\lambda_1} + a2 \delta{\lambda_2} + ...$
Let $\sigma^2$ be the variance of the Sqrt measure.
I conjecture that: A cubic singularity occurs in the convolution whenever $a_i = \sigma^2$, and such a singularity occurs at the point $\lambda_i$
Some experimental evidence (ignore the numerical artifacts lol)
This is the convolution of the measure $\frac{x^{2}+1}{4\pi}\sqrt{4-x^{2}}dx$ with $1/2 (\delta{-\sqrt{1.5}} + \delta{\sqrt{1.5}})$, which was predicted under the conjecture
Convolution of semicircle with $1/2 \delta{\approx -2.78} + 1/4 \delta{-1} + 1/4 \delta_{1}$ which was also predicted $\sqrt{5+\sqrt{7}}$ is the closed form
I also have some progress towards proving this, the two papers you sent me earlier have a lot of the machinery I think would be required to prove this, though neither paper really considered point measures on their own
Very cool.
Something I’m curious about is whether we can ever get higher order singularities such as quartic. This would probably have to correspond to a transition directly from one interval to 3
I realise you are taught in programming to always document your code. But in research code this is time consuming and a lot of times the function is self-explanatory. I feel like your documentation is too detailed and that you are wasting time on this.
For comparison: Julia itself has very little documentation, particularly for the internal machinery, see e.g.
https://github.com/JuliaLang/julia/blob/master/stdlib/LinearAlgebra/src/diagonal.jl
I can understand pretty much the whole code even though the documentation is so sparse.