Questions tagged [metric-embeddings]
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68 questions
9
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1
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Does every metric space of the form $X^2$ embed isometrically into some finitely iterated hyperspace of $X$?
This question occurred to me while I was writing my most recent answer.
Recall that given an extended metric space $(X,d)$ the hyperspace of $X$, which I'll write as $\def\Hc{\mathcal{H}}\Hc(X)$, is ...
- 19.7k
3
votes
0
answers
100
views
Finite-Dvoretzky theorem variant with codomain of prespecified dimension
Let $n,k\in \mathbb{N}_+$ and $F_k\subset \ell^n_2$ be a $k$-points subset of $\ell_n^2:=(\mathbb{R}^n,\|\cdot\|_2)$. How well can $F_k$ be bi-Lipschitzly embedded into $\ell^N_{\infty}$?
...
- 4,842
10
votes
2
answers
380
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Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
- 14.5k
1
vote
1
answer
173
views
Alexandrov's uniqueness theorem in Minkowski spacetime
Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$.
Each face of $P$ comes with induced metric tensor,
if the face is space-like, then it is euclidean metric;
every time-like face is isometric ...
- 47.1k
4
votes
1
answer
140
views
Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$
Suppose $(X,d)$ is a countably infinite set endowed with a metric $d$ that satisfies the following condition:
Every finite subset of $X$ with the induced metric is isometric to a subset of some ...
- 3,557
6
votes
1
answer
253
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Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
2
votes
2
answers
326
views
Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$
Question:
is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$?
I'm convinced it must be true, but can't remember having seen ...
- 14k
1
vote
0
answers
215
views
Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma
The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...
- 4,842
4
votes
1
answer
325
views
Bi-Lipschitz embeddings of compact doubling spaces
Suppose that $(X,\rho)$ is a compact doubling metric space. Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x_i\}_{i=1}^n\subseteq X$ such that the map
$$
\begin{...
- 4,842
1
vote
0
answers
161
views
Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
- 179
9
votes
0
answers
446
views
Embedding a graph into Euclidean space
I want to find a map $v\mapsto \tilde v$ from the vertex set of a connected infinite graph $\Gamma$ to a Euclidean space that meets the following two conditions:
there is $\varepsilon>0$ such that ...
- 47.1k
1
vote
1
answer
355
views
Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$
This is a follow-up to this question of mine.
It is well-known that the Banach space $\ell_1$ does not contain any isomorphic copies of $c_0$. One can even go further and show that $\ell_1$ does not ...
- 1,161
2
votes
1
answer
491
views
Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
- 521
3
votes
0
answers
274
views
Volume of sub-manifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations:
\begin{equation}
P_1(\vec x)=0, \\
\vdots \\
P_m(\vec x)=0,
\end{equation}
(where $\...
- 521
5
votes
1
answer
496
views
Isomorphic embedding of $l^n_{\infty}$ into $l_1^m$?
Given $n$, is there a $C(n)$-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large $m$ and $C(n)<<\log(n)$?
For $n=2$ this can done with $m=2$. There are some results about $...
- 755
4
votes
1
answer
241
views
Johnson-Lindenstrauss over torus?
The Johnson-Lindenstrauss lemma is a famous result in dimensionality reduction — Given $m$ points in $\mathbb{R}^N$, and $\varepsilon >0$, there exists a quasi-isometry $f : \mathbb{R}^N\to\mathbb{...
- 2,466
3
votes
0
answers
214
views
Constant in Naor and Neiman's Assouad Theorem
In Naor and Neiman's Assouad embedding theorem - "Assouad’s theorem with dimension independent of the snowflaking" Revisita Mathematica, the authors derive quantitative estimates on the ...
- 4,842
4
votes
0
answers
283
views
To what extent is the Nash embedding not unique?
Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique?
It is clear that the set of all such embeddings ...
- 521
7
votes
3
answers
422
views
Hyperbolic space embeds into Wasserstein space
Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
- 205
3
votes
1
answer
494
views
Explicit formula for embedding Cayley graph of free group into hyperbolic space
The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the ...
2
votes
0
answers
129
views
Finite approximations to the Kuratowski/Fréchet embedding
Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with
$$
\left\{B\left(x_k,\frac1{n}\right)...
- 205
18
votes
2
answers
2k
views
Are hyperbolic spaces actually better for embedding trees than Euclidean spaces?
There is a folklore in the empirical computer-science literature that, given a tree $(X,d)$, one can find a bi-Lipschitz embedding into a hyperbolic space $\mathbb{H}^n$ and that $n$ is "much ...
- 205
1
vote
0
answers
148
views
Best estimate on doubling constant of a finite metric space
Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant?
Probability based on its cardinality, diameter, and ...
- 205
5
votes
2
answers
263
views
Rozendorn's Article
I'm researching the isometric dips of the hyperbolic plane and in particular I'm interested in reading the results of Rozendorn who proved that the hyperbolic plane is isometrically immersed in $\...
- 143
2
votes
0
answers
119
views
Almost Lipschitz embedding of compact metric measure spaces into Euclidean spaces
Let $(X,d)$ be a compact metric space, $m$ be a metric outer-measure on $X$. Are there 'mild conditions' on $X$ ensuring the existence of a positive integer $N\geq 3$ such that there exist $x_1,\dots,...
- 109
7
votes
1
answer
269
views
Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?
Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? ...
- 73
1
vote
0
answers
104
views
intuition about Gaussian processes over a finite space
In a paper that I am reading the authors defines $\mathbb P(n,q)$ the space of covariance tensors for $\mathbb R^q$-valued Gaussian processes on an abstract finite space $K=\{x_1,\dots,x_n\}$. In his ...
- 5,255
2
votes
0
answers
129
views
Kernels with finite dimensional feature spaces
Suppose $x,y \in \mathbb{R}^n$ for some given fixed n.
Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
- 93
7
votes
1
answer
702
views
When is a metric space a snowflake?
Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.
...
9
votes
1
answer
569
views
Why $(\mathrm{Lip}([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?
In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation:
For a metric space X they write $\mathcal{P}_1(X)$ ...
- 1,088
4
votes
1
answer
330
views
Bi-Hölder embeddings of finite metric spaces
This is a reference request. There is a large body of work, I'm familiar with, that describes the existence of bi-Hölder embeddings of finite metric spaces into Euclidean space (such as this ...
- 4,842
5
votes
0
answers
204
views
Correspondence between Riemannian metrics and Euclidean embeddings
Given a sufficiently smooth manifold M,
a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely)
an embedding of M into ...
3
votes
1
answer
180
views
Banach embedding of finite dimensional spaces
Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).
I am wondering whether are any result for the case when $r>s>2$?
...
- 751
3
votes
0
answers
143
views
An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1
The following question is related to this previous question, Canonical immersion of the double torus:
Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ...
- 31
2
votes
0
answers
102
views
Dense embeddings into Euclidean space
The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
- 4,842
10
votes
0
answers
826
views
Two questions around the $abc$-conjecture
Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers.
The abc-conjecture can be formulated using these two metrics as:
For ...
user6671
3
votes
0
answers
144
views
Is the matrix $\mu_f(X_i \cap X_j)$ positive definite?
Let $X_1,\ldots, X_n$ be finite subsets of some larger finite set $Z$.
Let $f:Z \rightarrow \mathbb{R}_{>0}$ be any function, and define a (counting) measure $\mu_f(X) = \sum_{x \in X} f(x)$ for a ...
user6671
2
votes
1
answer
300
views
Partitions of unity with arbitrary Lip-constants
Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\...
- 29
2
votes
0
answers
154
views
Embedding a binary subspace to $l_2$ in a much lower dimension
I'm trying to find a way to embed a binary linear subspace of dimension $n$ (a linear code) to the Euclidian space while reducing the dimension significantly.
The subspace (or code) contains points ...
- 311
6
votes
0
answers
247
views
Is this function embeddable in Euclidean space?
Let $X = \{v_1,\ldots,v_n\}$ be a set of vectors non-zero vectors $v_i \ge 0$ and such that the vectors are pairwise linear independent. Define a function on this set $X$:
$$d(v,w) = 1-\frac{2 \...
user6671
35
votes
6
answers
2k
views
Trigonometry / Euclidean Geometry for natural numbers?
Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$.
The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ...
user6671
3
votes
0
answers
123
views
"Hoelder conjugate" version of the Johnson-Lindenstrauss transform
A variation of the well-known Johnson-Lindenstrauss transform (JLT) asserts that for $x_1,\ldots,x_m\in\mathbb{R}^n$ there exists a linear transformation $A:\mathbb{R}^n\to\mathbb{R}^k$ with $k=\...
- 111
5
votes
1
answer
685
views
Fast Bourgain embedding (or similar embeddings)?
Currently I am working on applications of Bourgain Embedding (or similar embeddings of finite metric spaces to $l_2$) to automatic feature engineering for machine learning/data science ( http://www....
user6671
0
votes
1
answer
119
views
Lower Estimate of A Lipschitz Map
Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function
$\rho:(0,\infty)\...
- 4,842
0
votes
0
answers
178
views
Green's Function for Fractional Laplacian on the Union of Two Balls
I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve:
$$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...
- 93
5
votes
2
answers
1k
views
Isometric embedding of a genus g surface
Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
4
votes
1
answer
303
views
Suppose that a metric space allows David–Semmes regular map to some Euclidean space. Does it allow bi-Lipschitz embedding into some Euclidean space?
I want to ask about the progress on Question 8 from "Thirty-three yes or no questions about mappings, measures, and metrics" by Juha Heinonen and Stephen Semmes. Is it still open? If yes, ...
- 1,088
2
votes
0
answers
153
views
Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$
I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball:
If you pick n points uniformly at random from the surface of a d
dimensional sphere of ...
- 133
8
votes
0
answers
252
views
Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?
QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...
- 1,088
6
votes
1
answer
254
views
Embedding Turing machine [closed]
I have some questions about Turing machines. Is there an embedding method where you embed Turing machines, finite automata into continuous space or graphs? Or are there geometrical approaches to ...
- 63