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 $m$, ...
- 14.2k
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 ...
- 50.5k
6
votes
Largest cell in an arrangement
Somehow doing better than $O(n^d)$ looks hard. If the cell is significantly larger than its average expected size, one can use sampling, to find it. Formally, assume the bounded cells (in the plane) ...
- 9,586
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 ...
- 36.3k
3
votes
Far point queries in high dimensions
See the following paper. The two problems are equivalent more or less. To see that, assume that the points are on the unit sphere centered at the origin, and observe that if your NN query $q$ is on ...
- 9,586
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 $R^d$ ...
- 10.2k
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 ...
- 46
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/...
- 9,380
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 $z$.
- 9,380
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 ...
- 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, ...
- 551
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....
- 11.1k
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|...
- 10.2k
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)...
- 266
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 ...
- 629
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 ...
- 1,085
1
vote
Batch membership testing for convex polyhedron specified in vertex representation
Instead of testing each point individually whether it is contained in the convex polyhedron, you should search for a supporting hyperplane of the polyhedron which separates the point from the ...
- 111
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+...
- 46
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