Skip to main content
Share Your Experience: Take the 2024 Developer Survey
12 votes

Johnson and Lindenstrauss lemma for hamming space

Consider the $n$ vectors $e_1,\ldots,e_n$ of weight $1$, and the zero vector $e_0$. The Hamming distance between $e_0$ to any $e_i$ is $1$. Let $\varphi$ be a map into a Hamming cube of dimension $m$, ...
Yuval Filmus's user avatar
  • 14.5k
11 votes
Accepted

Maximum Polyhedron Volume in Given $n$ Points

This was shown to be hard (more precisely $\mathsf{NP}$-hard to approximate to better than exponential in $k$) by Marco Di Summa, Friedrich Eisenbrand, Yuri Faenza, Carsten Moldenhauer, "On largest ...
David Eppstein's user avatar
5 votes

Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

I think this is open. Note that if instead of testing equivalence under rotations you ask for equivalence under the general linear group, then already testing equivalence of degree three polynomials ...
Joshua Grochow's user avatar
3 votes
Accepted

How to find the size of an ϵ-net of a vector space?

Note that $\mathcal{W}_{\epsilon}$ is an epsilon-net in the parametrization space, which is just $p$-dimensional Euclidean space. (So there is no need to think about covering numbers in e.g. spaces of ...
Mark Sellke's user avatar
3 votes

Lower bound on probability of getting two close points in a sample of $n$ points

Here's a counter-example showing your desired bound is not possible, unless I am mistaken. It's a simple variant of the example in Roei's comment. Fix any $n$ and $N\ge 4n$. Take $D$ to contain $N/...
Neal Young's user avatar
  • 10.8k
3 votes

VC dimension of intersection of half-spaces

It has been recently shown by Csikos, Kupavskii, Mustafa in "Optimal Bounds on the VC-dimension" that the VC dimension of $k$-fold unions (or intersections or XORs) of half-spaces in $R^d$ ...
Aryeh's user avatar
  • 10.6k
1 vote

An inequality about median of points in higher dimensions

Yes. By the triangle inequality, $\|x-z\| \le \|x-m\| + \|m -z\|$, which implies the desired inequality (with $K=1$) for any $m$ and $z$.
Neal Young's user avatar
  • 10.8k
1 vote

How to calculate complexity in a high dimensional space?

As far as I have understood, you aim to develop a framework to capture the hardness of combinatorial problems in 3D. However, there are major problems in your question. Your first sentence lacks a ...
padawan's user avatar
  • 93
1 vote

Multivariable concave function $(n - 1) f(x) >= \sum_{i=1}^{n} f(x_{-i})$

Consider the function $f(x, y) = 1 - e^{-(x + y)}$. Now $f(0, 0) = 0$, $f$ is increasing and concave, since $g(t) = -e^{-t}$ is concave. But $f(1, 0) + f(0, 1) = 2(1 - e^{-1}) > 1 - e^{-2} = f(1, ...
Antti Röyskö's user avatar
1 vote

Lattice generation inside d-dimensional unit ball

In general, even telling whether any such point exists is hard; it is equivalent to the Shortest Vector Problem (SVP), and it is conjectured that there is no polynomial-time algorithm for this problem....
D.W.'s user avatar
  • 12.1k
1 vote
Accepted

Application of the inequality with expectations

I don't think the claim is correct. Take $n=1$ (so the norm is just absolute value), $N=1$, and $u_1=1$. Let $X$ be any random variable with a finite first moment and infinite second moment (i.e., $E|...
Aryeh's user avatar
  • 10.6k
1 vote

Johnson and Lindenstrauss lemma for hamming space

While other answers are correct, I want to mention one result from Polynomial Time Approximation Schemes for Geometric k-Clustering, which is weaker, which roughly says that there exists a (randomized)...
Sudipta Roy's user avatar
1 vote
Accepted

Johnson and Lindenstrauss lemma for hamming space

Let the matrix consist of $n$ points in $d$-dimensional space. We first generate a projection matrix $d\times K$ whose each entry is sampled from the Cauchy distribution. Then the sketch matrix is ...
Ram's user avatar
  • 639
1 vote
Accepted

Data structure for storing points and finding a predecessor of a point

What about first computing the skyline (a.k.a. maximal vectors, etc.) of all points, then maintain a data structure for orthogonal range reporting? The range you are interested in is the orthant ...
Joseph Stack's user avatar
  • 1,095
1 vote
Accepted

VC dimension of Voronoi cells in R^d?

Please check Theorem 21.5, Section 21 in the book "A probabilistic Theory of Pattern Recognition (1996)" from Devroye, Gyorfi, and Lugosi. I think the following upper bound is valid: VC $\leq$ $k + (d+...
AlexGj's user avatar
  • 46

Only top scored, non community-wiki answers of a minimum length are eligible