# Questions tagged [sampling]

The sampling tag has no usage guidance.

74
questions

0
votes

0
answers

28
views

### Alternative to the Sampling Theorem / Invertible transform with sampling criteria

I seek a transform $T$ that operates on real-valued $x(t)$, that
Is perfectly invertible
Has discrete counterpart with continuous reconstructor
Provides conditional reconstruction guarantees
...

3
votes

0
answers

33
views

### Invertibility of the sampling matrix

Given a function $f: \mathbb{R}^2\rightarrow\mathbb{C}$ sampled as a matrix $F_{ij}$ on some ractangle $[a,b]\times[c,d]\subset\mathbb{R}^2$ with steps $\Delta x$ and $\Delta y$ as the stepsizes so ...

1
vote

1
answer

77
views

### Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$?

Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
We don't assume $X$ and $Y$ are ...

4
votes

1
answer

167
views

### Integral of $\ln(1/|f|)$ for $f$ bandlimited

I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...

2
votes

1
answer

130
views

### Shift-invariant spaces

We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi(\cdot-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...

5
votes

3
answers

187
views

### Probability of an edge in a random graph

Consider a vertex set $V$ and a degree sequence $(d_v)_{v\in V}$. I want to know the probability that an edge exists between two given vertices $u$ and $v$ in a random graph with this degree sequence.
...

1
vote

0
answers

27
views

### Correlating two matrices $A,B$ with stochastic dependency structure imposed by cross-validation

Consider a labelled data set
$$D = \{(x_1, y_1),...,(x_n, y_n)\} $$
on which we want to evaluate a machine learning algorithm using $k$-fold cross validation with $m$ different random seeds. This ...

1
vote

1
answer

118
views

### Intuition of the "work" done by random variables in Monte Carlo methods (incl. MCL)

(I've tried Math SE, but have so far come up empty handed, so I'm trying my luck here.)
I would like to get a better intuitive understanding of why Monte Carlo works so well in approximating a ...

3
votes

0
answers

172
views

### Injectivity of the covolution operator $f \mapsto (k*f(s))_{s \in S}$ via sampling at $S=\alpha \mathbb Z$

Let $S \subset \mathbb R$ be a uniform grid, say $S = \alpha \mathbb Z, \alpha >0$.
Let $k$ be a square integrable function and $A$ the operator which maps $f \in L^2(\mathbb R)$ to the sequence
$$
...

5
votes

0
answers

108
views

### Which operations commute with fractional translation?

Let $\mathbf{v}$ be a real signal (i.e., an infinitely long vector).
A translation operation $T_{s}\mathbf{v}$ with integer $s$ is trivially
defined by $\forall i,s:\left(T_{s}\mathbf{v}\right)_{i}=v_{...

0
votes

0
answers

28
views

### Sampling after sorting vs sorting without sampling

Given a probability distribution $(p_1,\cdots,p_d)$ and integer $k$.
We can sample the distribution $k$ times. For any obtained sample $(X_1,\cdots,X_d)$ with $X_i$ denoting the occurrence of $i$, we ...

0
votes

0
answers

14
views

### Subset estimation from positive and negative samples

I am looking for the name of the following (family of) problems:
given a known set $K$ and an unknown subset $G \subset K$, and the values of the characteristic function $\mathbb 1_{G}(x_i)$ for some ...

1
vote

1
answer

54
views

### Distribution of a two-part sampling process

I have a known distribution $f(x)$ (in fact, I can safely assume that $f(x)$ is the Maxwell-Boltzmann distribution, i.e. $f(x)\propto x^2 \exp(-x^2)$). I take $N$ samples from the distribution, but am ...

0
votes

1
answer

134
views

### Random sampling from modified Erlang distribution [closed]

I am tasked with randomly sampling from the following probability density function, which is a modified Erlang Function:
$$f(k,q,\nu)=\frac{(k q)^{k-1}}{[(k-1) !]^{v}} \quad \text { with } \quad q \...

1
vote

1
answer

59
views

### Column subset selection with least angle optimization

Given a matrix $A$ I need to select a "representative" subset of its columns so that the each non-selected column is as close as possible to a selected one.
Formally:
Given $A \in \mathbb{R}...

2
votes

1
answer

116
views

### Fast sampling of matroids

In his classic paper, Donald E. Knuth described how random matroids of fixed rank can be generated.
What is the currently the fastest (in terms of mixing behaviour) known way to sample matroids of ...

2
votes

0
answers

57
views

### Are two degree sequences compatible, for random simple graph generation?

Consider a set $V$ of $n$ vertices, and three degree sequences $a_i$, $b_i$ and $c_i$ such that $c_i = a_i+b_i$, $i=1..n$.
Assume these degree sequences are graphical: there exist simple graphs (no ...

3
votes

1
answer

315
views

### Relation between signal derivative and frequency spectrum

I want to sample a signal whose derivative I know to be bounded by physical constraints. The sampling is disturbed by gaussian noise, hence I need to filter the sample with a lowpass filter.
Since I ...

1
vote

1
answer

119
views

### Sampling i.i.d. variables with restrictions

General Problem: Suppose $X_1,\ldots,X_n \sim \mathbb{P}_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "...

6
votes

0
answers

235
views

### Probability that a random multigraph is simple

Question.
Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence ...

1
vote

1
answer

415
views

### Interior point of a convex polytope

Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to ...

0
votes

0
answers

31
views

### Condition on the point cloud matrix making the points "generic" in the uniform sense

For a matrix $X\in\mathbb{R}^{d\times n}$, what condition can I impose on $X$ to make the collection of its columns generic in the sense that they look like the result of uniformly sampling a convex ...

0
votes

0
answers

43
views

### Yates-Grundy draw-by-draw sampling inclusion probabilities

Is there an efficient algorithm to calculate the inclusion probabilities (the probability that an item will be included in a sample) in the Yates-Grundy draw-by-draw sampling?
Sampling description: ...

3
votes

2
answers

503
views

### Invariant measure vs Riemannian measure on Stiefel manifold

I'm interested in sampling uniformly from the Stiefel manifold $V(k, n)$, but while researching how to do this I came to wonder the following.
In Edelman et al. [1] there are presented two ...

1
vote

1
answer

121
views

### Random optimization problem

Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution ...

5
votes

1
answer

186
views

### What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?

Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.
Let
$$p(x) \propto \exp(-\|x\|_1/\sigma)$$
be the pdf of the Laplace ...

-2
votes

1
answer

45
views

### Using common samples to numerically estimate pairwise equality of three random variables

Let $X,Y,Z$ be three discrete random variables which I can numerically sample. I need to numerically estimate the probability that $X=Y$ and the probability that $X=Z$. I would like to know whether ...

2
votes

0
answers

105
views

### Can computers take uniform samples from a polytope?

For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$.
Suppose the plane $P \subset \mathbb R^N$ is ...

0
votes

2
answers

261
views

### Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?

I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...

2
votes

2
answers

87
views

### Draw samples from distribitions in the neighborhood of a fixed distribution

Disclaimer
Sorry in advance for vagueness. I'm still trying to get my ideas right on this one.
Setup
So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...

8
votes

1
answer

330
views

### Expectation inequality for sampling without replacement

Is the following proposition correct?
$X_1, X_2, X_3$ are uniformly at random sampled from a finite set $\mathcal X$ without replacement.
$f : \mathcal X^2 \rightarrow \mathbb R_{\ge0}$ is symmetric:...

4
votes

1
answer

352
views

### Minimize the variance of a Boltzmann distribution

N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here).
...

1
vote

1
answer

461
views

### Uniform sampling of random connected graph with given number of vertices/edges

I am looking for algorithms for the exact uniform sampling of connected labelled graphs with $n$ vertices and $m$ edges. By "exact" I mean that every such graph should be generated with precisely (not ...

2
votes

1
answer

66
views

### Sampling with non-uniform probabilities

Let $p_1,p_2,...,p_n$ are given probabilities. ($\sum_{i=1}^n p_i =1, p_i \geq 0 $). Is there any distribution, which picks $k\leq n$ distinct elements from $1,2,...,n$ such that $P(i \in S) = k p_i$ ...

1
vote

1
answer

117
views

### Plancharel-Pólya inequality for functions of exponential type

If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$
then it can be proven that $$\int_{-\infty}^{+\...

4
votes

1
answer

248
views

### Combinatorial computational problem about 0-1 vectors and sampling algorithms

Let $M \in \{0,1\}^{m\times n}$, where $n\gg 1$ and $m\le n$. A procedure consisting of the following three steps is repeated $t\gg 1$ times:
A row $\require{amsmath} \boldsymbol{r}$ of $M$ picked in ...

1
vote

1
answer

133
views

### Sampling set: relatively dense and uniformly discrete

The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\...

7
votes

1
answer

585
views

### Randomly covering a sphere

Let $S$ be the $n$-dimensional unit sphere in the Euclidean space. Further,
let $X_1,\ldots,X_k$ and $Y_1,\ldots,Y_m$ be iid $S$-valued random variables with common (unknown) distribution $\mu$. With $...

4
votes

1
answer

1k
views

### How to measure distribution of high-dimensional data

I have to methods of projecting random samples in $\mathbb{R}^n$ onto a manifold defined by $C(q)=0$, which is a lower-dimensional subset. Now, samples in $\mathbb{R}^n$ are uniformly distributed. ...

1
vote

0
answers

42
views

### RMHMC sampling in non-parametric setup

The aim is to sample distributions using Fisher information (as mass matrix in Hamiltonian MCMC sampling). Details can be found in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.580&...

0
votes

0
answers

179
views

### Expected value of parametrized Gibbs distribution w.r.t another probability distribution

Let $\mu$ be a compactly supported probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ has density. For a random point $x \sim \mu$,
consider ...

1
vote

1
answer

189
views

### How to sample a path between 2 states in a Markov chain

Given an ergodic Markov chain and 2 states $x$ and $y$, how may one algorithmically sample a path (of finite length) $x =: s_0 \rightarrow s_1 \rightarrow \ldots \rightarrow s_T = y$ between $x$ and $...

1
vote

1
answer

103
views

### Bounding function by random sampling

Let $f(x): \mathbf{C} \to \mathbf{C}$ be a complex valued polynomial of degree $n$, and suppose that for a random point $z$ sampled uniformly from the unit disk $|z|\leq 1$ we have
$$\mathbf{P}_{z}( |...

0
votes

1
answer

139
views

### Efficiently Sampling of Multivariate Distributions in the Vicinity of a Manifold

I am given a multivariate distribution, that maps each point of $\mathbb{R}^n$ to its probability of being drawn as sample and, the convolution $\mathcal{S_r}\times\mathcal{M}$ of a manifold $\mathcal{...

3
votes

1
answer

770
views

### Sampling from a particular multivariate probability distribution

Given $3$ real variables $x_1, x_2, x_3 \equiv \bf{x}$, consider their probability density function (PDF)
\begin{equation}
P({\bf x}) = C \, p(x_1) \cdots p(x_3) \exp[f({\bf x})],
\end{equation}
where ...

1
vote

0
answers

59
views

### 2-step sampling from a conditional density

The setting is as follows:
We are given two random variables $X : \Omega \to \mathbb{R}$ and $\Theta : \Omega \to T$ for some 'parameter space' $T \subset \mathbb{R}$, and
1) we know the density of $...

3
votes

2
answers

225
views

### Sampling Theorem for non-bandlimited Functions

The classical Shannon sampling theorem states that
a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol $...

1
vote

0
answers

62
views

### Simulate a graph from a certain distribution

I am wondering if anyone can indicate whether the following is a solved problem. I don't care about time of the algorithm currently.
Consider a general probability distribution F on simple graphs ...

6
votes

1
answer

185
views

### Sample integer points of cross-polytope uniformly

For $r,d\in\mathbb{N}$, let
$$C_{r,d}=\{x\in\mathbb{Z}^d: \|x\|_1\le r\}\subset\mathbb{Z}^d$$
be the set of integer points of the $d$-dimensional cross-polytope with radius $r$.
What is (...

0
votes

0
answers

48
views

### Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...