Questions tagged [total-variation]
This tag is for questions relating to Total Variation.The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper (Jordan 1881). He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.
211 questions
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Finding sequences of Gaussian measures for different modes of convergence (norm, narrow, weak*)
I am working on a problem about modes of convergence for measures and would like to find sequences of Gaussian measures that satisfy specific criteria.
Let $\mu_{m,s}$ be the Gaussian probability ...
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Is conditional mutual information $I(X;Y\mid Z)$ lower semicontinuous for general $X,Y,Z$?
Assume $(X_n,Y_n,Z_n)\Rightarrow (X,Y,Z)$ weakly on standard Borel spaces. Is it always true that
$$I(X;Y\mid Z)\ \le\ \liminf_{n\to\infty} I(X_n;Y_n\mid Z_n)?$$
It is classical that relative entropy $...
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Perimeter of $E \cup B_{\tau\rho}$ in $B_\rho$
Let $E \subset \mathbb{R}^n$ be a set of finite perimeter. Fix $\rho>0$ and a center $x_0\in\mathbb{R}^n$, and write
$
B_r := \{x\in\mathbb{R}^n:\ |x-x_0|<r\}, \qquad \partial B_r := \{x:\ |x-...
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Variable splitting on helping with non-differentiability
I am working on an implementation of deconvolution and Wang et al.'s paper$^\color{magenta}{\dagger}$ mentions something I do not quite understand. The objective function is, in essence,
$$\min_u\...
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When is a sum of jump functions still a jump function?
Setup
Let's adopt the definition at https://encyclopediaofmath.org/wiki/Jump_function:
A right-continuous function of bounded variation $f:[0,1]\to\mathbb R$ is a jump function if $$f(x)=\sum_{y\leq ...
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V-uniform ergodicity
Let's say $P$ is a $V$-uniformly ergodic Markov process on a general state space with some function $V \geq 1$, that is there exist a probability measure $\pi$ and constants $C>0$ and $\rho \in (0,...
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A lower bound for total variation distance of product
Let $X$ and $Y$ be independent random variables. Suppose their Total variation distance is "big":
$$
tv(X, Y) = \frac{1}{2}\sum_{\alpha}|\mathbb{P}[X = \alpha] - \mathbb{P}[Y = \alpha]| \ge \...
user824478
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Equivalent condition for a function f on R having the form f(x) = ∫ [-∞, x] φ(t)dt for some φ ∈ L1(R), i.e. φ is Lebesgue-integrable and ∫φ(t)dt<∞
Hewitt-Stromberg (Real and Abstract Analysis, Springer Verlag 1969) claim in theorem (18.18): A function $f$ on $R$ has the form $$f(x)=\int^x_{-∞}φ(t)dt$$ for some $φ∈L_1(R)$ if and only if $f$ is ...
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Prove $\sup_{P=(t_0,\dots,t_k)}\sum_{j=1}^k |f(t_j)-f(t_{j-1})|=\sup_{n\in\mathbb N}\sum_{j=1}^{k_n} |f(t_j^n)-f(t_{j-1}^n)|$
Let $f:[0,1]\rightarrow \mathbb R$ be right continuous $P_n=(0=t_0^n<t_1^n<\dots<t_{k_n}^n=1)$ a sequence of grids with $\max_{1\leq j\leq k_n} |t_j^n-t_{j-1}^n|\to 0 (n\to \infty)$. Show ...
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Total Variation between Conditional Probabilities
Let $\mu_1$ and $\mu_2$ be two probability measures on a measurable space $(S,\mathcal{S})$, where $\mathcal{S}$ is a $\sigma$-algebra over $S$. Take $E\in\mathcal{S}$ such that $\mu_1(E),\mu_2(E)>...
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Total variation of integral function
Consider a function $f\in L^1([a,b])$ and define $F(x):=\int_a^x f(y)dy$. I should prove that $V_a^bF=||f||_{L^1([a,b])}$.
My attempt was to proceed by approximation with a test function $\phi$, but I’...
user1286545
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Example of a function with infinite (total) variation and zero quadratic variation
In the first chapter of "Introduction to Stochastic Calculus with applications" by Klebaner, there is a very brief mention of the existence of functions with zero quadratic variation but ...
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Weak convergence of measure and their total variation measures.
Does weak convergence of Radon measures implies weak convergence of their total variation measures? Precisely speaking, I want to know whether the following proposition is correct.
Let $X$ be a ...
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Total Variation Distance of Normal Approximation for a Poisson 100.
Ross and Pekoz's book A Second Course in Probability gives the following question in their chapter on Stein-Chen Method:
Compute a bound on the accuracy of a normal approximation for a Poisson random ...
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How can we calculate the Euler-lagrange equations?
In this paper https://arxiv.org/pdf/1907.09605.pdf \
let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
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