All Questions
Tagged with geometric-probability linear-algebra
15
questions
5
votes
2
answers
391
views
A geometrical puzzle involving calculus
Some time ago I stumbled across a problem from the Putnam Mathematical Competition. I could not find it, but I remember the text quite well.
There are two vectors: a=(10, $y$) and b=($x$,10), where $0 ...
- 53
0
votes
0
answers
38
views
Expected norm-squared of one random vector projected onto others, all iid
For $k\leq n$, $x_1,\dots,x_k \in \mathbb{C}^n$ are independent identically distributed random vectors almost surely unit norm and with span dimension $k$. Call $X=[x_2,\dots,x_k]$. I am studying $\...
- 4,886
0
votes
0
answers
21
views
Expected Value of volume of any convex body excluding some points?
Consider the bounded region given by $A \in \mathbb{R}^n$. Let it be given that we have uniformly sampled $k$ i.i.d. points ${P_i} \sim U(A)$ where $k$ is some constant.
Now $S \subseteq A$ be any ...
- 2,479
1
vote
0
answers
78
views
Average number of hyperquadrants in a random subspace
Suppose I have a random $n$-dimensional linear subspace of $\mathbb{R}^m$. How many of the $2^m$ hyperquadrants does this space intersect, on average? Alternatively, what are the odds that this ...
- 1,629
3
votes
1
answer
93
views
Distribution of distances to a hyperplane
Suppose I have the unit sphere in $R^3$ and WLOG have a hyperplane be the $x-z$ plane. I uniformly at random choose points within the sphere. I know that the distribution of the coordinates of those ...
user1013993
2
votes
0
answers
65
views
consolidate codependent continuous probability distributions into one multivariable distribution: how?
Sorry if this has been answered satisfactorily elsewhere; if it has and eluded me, post the link and I shall close this.
This post is motivated by a couple of questions that I’ve asked recently (at ...
user867884
2
votes
0
answers
74
views
Finite set of vectors approximating a unit ball.
I am having difficulty proving that a unit ball can be approximated with a set of finite vectors. Specifically, I want to bound the error of the following approximation.
Let $D$ be a uniform ...
- 57
1
vote
1
answer
54
views
how close can you get to a random d-dimensional vector of +1 and -1 given k guesses?
Consider a uniformly randomly selected vector $v \in \lbrace +1,-1 \rbrace^d $ that is a vector of size d, consisting of +1 and -1 (there are 2^d such vectors)
I'm interested in understanding, how "...
- 17.5k
5
votes
0
answers
209
views
Singular vectors of random Gaussian matrix with non-isotropic rows
Suppose $G \in \mathbb{R}^{m \times n}$ has i.i.d. rows $g_i \sim \mathcal{N}(0, \Sigma)$ for some diagonal matrix $\Sigma = \text{diag}(\lambda_1,\dots,\lambda_n)$ where the diagonal entries satisfy $...
- 614
2
votes
1
answer
52
views
Probability that a Randomly Chosen Vector is a Positive Linear Combination
Choose a basis $\beta=\{v_i \in \mathbb{R}^n, 1\leq i\leq n\}$. What is the probability that a randomly chosen nonzero vector $x\in S^{n-1}\subset \mathbb{R}^n$ is a positive linear combination of ...
- 2,508
10
votes
1
answer
6k
views
Average area of the shadow of a convex shape [closed]
What is the average area of the shadow of a convex shape taken over all possible orientations?
If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be ...
- 365
0
votes
1
answer
149
views
Minimum number of random vectors needed to span a space
I am working with $\mathbb{R}^{nm}$ for some $n,m\geq3$.
What is the minimum number of random vectors I need for them to span $\mathbb{R}^{nm}$ (in terms of $n$ and $m$)?
I'm happy for this to ...
- 177
2
votes
1
answer
60
views
Expected distance from the span
I would like to pick at random $3$ vectors on the sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ and compute the expected distance of one vector to the span of the others. More precisely: let $a_1$, $a_2$, $...
- 333
7
votes
0
answers
234
views
Learning a point on the sphere using signs of dot products
An unknown point $x$ is fixed on the unit sphere in $\mathbb{R}^n$ and we iterate as follows: at each step, specify a unit vector $v\in \mathbb{R}^n$ and record $\mathrm{sign}(x\cdot v).$
The goal is ...
- 4,886
1
vote
1
answer
185
views
How does point A calculate position of a moving point B when only "direction" of point B is known?
I am currently designing a game. I have a "runner" and a "catcher" that sort of play tag with each other. The catcher always know in which direction the runner is, but he does not know where exactly ...
- 309