graphs – Similarity measures for (geometric) triangulations

In a project I am working on, we are looking at multiple different (optimal with respect to some cost measure) triangulations of a fixed pointset $S$. I would like to cluster similar triangulations.

My first Idea was something like the edge flip distance, i.e. the minimal number of edge-flips in a quadrilateral needed to get from one triangulation to the other. Calculating this number is Np-Hard and since my datasets are not small, there are no algorithms that can calculate the flip distance in reasonable time.

Next I had a look at similarity measures for geometric graphs, but they emphasize the translation and insertion/deletion of points to get from one graph to the other one. These kinds of modifications are not interesting in my case, since the positions of the vertices are fixed for both triangulations.

In the figure two different triangulations on the same pointset are given. Note that we already have a one to one correspondence between the vertices of the two Triangulations, since they are generated with respect to the same set of points.

All in all I would like to know if there are any known similarity measures in the context of graphs/geometric graphs/triangulations, that may be interesting and could possibly be adapted to triangulations.

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real analysis – Proof that product measure is $sigma$-finite if the single measures are $sigma$-finite

I have two measures $mu$ on $(X,mathcal{M})$ and $nu$ on $(Y,mathcal{N})$ and they are both $sigma$-finite, the product measure ($mu otimes nu$) is $sigma$-finite.
Let’s take the family {$E_i$} in $(X,mathcal{M})$ and {$F_j$} in $(Y,mathcal{N})$.
I understand that if $mu(E_i)< infty$ and $nu(F_j)< infty$ and $cup E_i=X$, $cup F_j=Y$, this all is the requirement of $sigma$-finiteness, then measurable rectangle $R=E_i times F_j quad forall i,j quad$ is such that $(mu otimes nu)(R)=mu(E_i)nu(F_j) < infty$ because we multiply two non-infinity numbers.
My problem is to show that $cup_i cup_j E_i times F_j$ necessarily gives the whole $X times Y$, so, in the end to show that the ($mu otimes nu$) is $sigma$-finite.
For $ mathbb{R} times mathbb{R}$ I can imagine it visually, by imaging cartesian coordinate grid covered, but for more general cases I don’t know how to prove it rigourously.

Apple Relents on New Surveillance Measures

Apple Privacy J/KApple announced today that it is delaying rollout of its controversial new surveillance technology that sought to identify child pornography.

They’ve updated their previous press release with this disclaimer at the top:

“Update as of September 3, 2021: Previously we announced plans for features intended to help protect children from predators who use communication tools to recruit and exploit them and to help limit the spread of Child Sexual Abuse Material. Based on feedback from customers, advocacy groups, researchers, and others, we have decided to take additional time over the coming months to collect input and make improvements before releasing these critically important child safety features.”

Initial concerns were wide-ranging:

  • How accurate would this scanning be?  This was a particular concern when the consequences of a false match are so extreme.  A misclassification could potentially mean a phone owner would be reported to law enforecement.
  • Many people didn’t like Apple accessing their photo libraries or scanning their media.
  • Apple could review your searches (potentially in real time) and block them or report them.
  • A number of privacy groups were concerned that once this “thin wedge” was installed, it could be misused in the future.

Note that Apple has only announced they are taking “additional time” not necessarily canceling the initiative.  What are your thoughts on this?


I’m Andrew, techno polymath and long-time LowEndTalk community Moderator. My technical interests include all things Unix, perl, python, shell scripting, and relational database systems. I enjoy writing technical articles here on LowEndBox to help people get more out of their VPSes.

set theory – Reference request: large generalized probability measures

I’m interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size?

I’m thinking of something like the following: let $F$ be an ordered field, let $F_+$ be the nonnegative elements of $F$, and let $phantom{}^omega F$ be the set of infinite sequences of elements of $F$. We say that a partial function $sigma: phantom{}^omega F rightharpoonup F$ is a summability notion for $F$ if (1) it extends finite sums in the obvious way: whenever only finitely many members of $vec{a} in phantom{}^omega F$ are nonzero, $sigma(vec{a})= sum_{{ i in omega: a_i neq 0 }} a_i$ and (2) it has an appropriate order-independence property for positive partial sums: if $a_i in F_+$ for all $i in omega$, then if $sigma(vec{a})$ exists and $vec{b}$ is a permutation of $vec{a}$, then $sigma(vec{a}) = sigma(vec{b})$. (There are probably other properties we would want, but I leave this open for the moment.)

We can then come up with a generalized version of the Kolmogorov axioms: given a set $Omega$, a $sigma$-algebra $A$ over $Omega$, and an ordered field $F$ equipped with a summability notion (which we can just write $sum$), we say that an $f: A times A rightarrow F_+$ is a generalized probability measure if $f(Omega) = 1$ and $f(bigcup X)=sum { f(x): x in X }$ for $X$ a countable pairwise disjoint collection of elements of $A$. The advantage of using a formal sum, rather than a standard notion of convergence, is that it can make sense even when there are no nontrivial converging sequences in the natural topology on $F$ (as will be the case, e.g., when $F$ is larger than continuum size, as there will generally be no countable dense subsets).

I can’t help thinking that this must be well-trodden ground, but I can’t find any references that are exactly on point. The closest thing is the generalization of metric spaces by Ralph Kopperman (“All Topologies Come From Generalized Metrics”, American Mathematical Monthly 95 (1988): 89–97, doi:10.1080/00029890.1988.11971974); I suppose I’m seeking something similar for measure theory. Any pointers to sources would be much appreciated.

reference request – Pushforward of invariant measures (equivariant Moser theorem)

There is a well-known theorem that between any two absolutely continuous Borel probability measures $mu$ and $nu$ on $mathbb{R}^n$ there is an increasing triangular
transformation $T : mathbb{R}^n to mathbb{R}^n$, such that $nu = T_∗mu$ (see, for example, “Triangular
transformations of measures
” by Bogachev, Kolesnikov, and Medvedev).

There is another version of this theorem for smooth manifolds (non-compact Moser theorem).
If $M$ is a noncompact connected oriented manifold and if $mu$ and $nu$ are volume forms on M with $int_M mu = int_M nu le infty$ and if each end of the manifold $M$ has finite $mu$ volume if it has finite $nu$ volume and infinite $mu$ volume if it has infinite $nu$ volume, then there is a diffeomorphism $phi: M to M$, such that $phi^* mu = nu$. (See “Diffeomorphisms and volume-preserving embeddings for noncompact manifolds” by Greene and Shiohama.)

I am curious about the equivariant analog of either of these results.

More precisely, assume that a (compact?) group $G$ acts on $mathbb{R}^n$ and $mu$ and $nu$ are two $G$-invariant (smooth) densities. Does there always exist a $G$-equivariant diffeomorphism $T : mathbb{R}^n to mathbb{R}^n$, such that $nu = T_∗mu$?

Could you point out if this result is proved somewhere in the literature? Does the group $G$ have to be compact for it to hold?

reference request – Fourier transform of measures on $mathbb{T}$

I’m currently working with Fourier transforms of measures on the $mathbb{T}^n$ (more specifically in dimension two), i.e.
hat{mu}(k) = int_{mathbb{T}^n} e^{i k cdot x} dmu(x)

or something of that form. I am unfamiliar with this theory and would really appreciate a good reference on this topic.

Would anyone be able to point me to a good reference on the Fourier transform of measures over some unit cell? I have found literature for when $mathbb{T}$ is replaced with $mathbb{R}$, but am struggling to find a good reference for the requested case.

In case it is relevant, I am interested in the case when $k$ takes values on some lattice.

graph theory – Centrality measures in a network with negative correlations

I have a bidirectional network where the weights of edges are based on partial correlation matrix. I have both positive and negative values as weights. Now, I want to compute centrality measures as degree, closeness, betweenness and eigenvector. How can I handle the negative values? Would I get correct values for these measures, if I keep the negatives? Should I use absolute value or take (1-absolute value)?

Basically, I am confused about if these values would affect the outcome in any way. I have not found any resources that would discuss this. Please recommend, if you know any.

fa.functional analysis – Reference request: sequential weak* topology on the space of signed Radon measures

Consider the space $mathcal{M}_{loc} (mathbb{R}^d)$ of locally finite signed Radon measures, equipped with the weak* topology in duality with $C_b (mathbb{R}^d)$. It is known that this is space is not metrizable, nor first countable (although I believe it is a Souslin space?).

On the other hand, in practice one often works not with the weak* topology directly, but with weak* convergence. So it would be interesting to look at the sequential weak* topology on $mathcal{M}_{loc} (mathbb{R}^d)$ (that is, a set is declared to be closed provided it is sequentially weak* closed).

My question is, is this topology something that has already been studied in the literature? The only thing I can find is this MO question from several months ago.

pr.probability – Existence of measures with given 1d marginals

This is a question about marginals of probability measures, which seems unrelated to previous questions.

Let $mathbb{S}^{d-1}subset mathbb{R}^d$ be the unit sphere. Assume that for each $thetain mathbb{S}^{d-1}$ there is an associated probability measure $mu_theta$ on $mathbb{R}$.

Question: Under what conditions does there exist a probability measure $mu$ on $mathbb{R}^d$ such that

$$mbox{if }Xsim mu,mbox{ then }langle theta,Xrangle sim mu_thetambox{ for all }thetain mathbb{S}^{d-1}.$$

(Here $langlecdot,cdotcdotrangle$ is the Euclidean inner product and $Asim nu$ means object $A$ has prob. law $nu$.)

A necessary condition is that the map $thetamapsto mu_theta$ be continuous under the weak topology on probability measures on $mathbb{R}$. Is this condition sufficient?

(Motivation comes from the study of the Sliced Wasserstein distance. If the answer to the above question is “yes”, then in principle one can “easily compute” barycenters for SW.)

dg.differential geometry – Orthogonality in Wasserstein tangent space for discrete measures with equal mass

Let say I have $N$ discrete probability measures $(mu_1,…,mu_N)$ where each of them has $n$ points in $mathbb{R}^2$ of equal mass.

Let $P(mathcal{X})$ be the space of these probability measures on a Hilbert space $mathcal{X}$ (which is $mathbb{R}^2$ in our case if I’m not mistaken).

We can define the Wasserstein barycenter:

$overline{mu} = argmin_{nu in P(mathcal{X})} sum_i^N alpha_i W_2^2(mu_i, nu)$.

Under the 2-Wasserstein metric $W_2$ and with $alpha_i = frac{1}{N},_{i=1..N}$ (in our case).

I’m interested to grap the intuition with the tangent space at $overline{mu}$. Especially in the notion of orthogonality in this tangent space.

Let say I have a vector (or velocity) field $v_1 in L^2(overline{mu}, mathcal{X})$ (the tangent space at $overline{mu}$), containing $n$ vectors, that move all the $n$ masses of $overline{mu}$ in a direction $v_1^i,_{i=1..n}$ respectively.

What would it mean to have a velocity field $v_2 in L^2(overline{mu}, mathcal{X})$ orthogonal to $v_1$ in the $L^2(overline{mu}, mathcal{X})$ sense? It is just that we need to have $v_2^i perp v_1^i,_{i=1..n}$ in the $mathbb{R}^2$ sense?

I’m interested in any reference that could help (especially because I could need to cite something) and any explications.

Thank you very much.