# 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.

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### Total variation is the sum of positive and negative variation

I've a doubt about the following proof: $V_f(a,b)=\\ sup\{V_f(P):P∈𝒫[a,b]\}=\\ sup\{p_f(P)+n_f(P):P∈𝒫[a,b]\}=\\ sup\{p_f(P):P∈𝒫[a,b]\}+sup\{n_f(P):P∈𝒫[a,b]\}= \\ p_f(a, b)+n_f(a,b)$ Here $f$ ...
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### Uses: Total Variation

I recently have come across the total variation metric on probability measures as a dual to a certain space of continuous functions. My question is, what is the use of this metric and what does it ...
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### Properties of the sequence

Given a sequence $\{a_i\}_{i\in \mathbb{Z}},$ consider the sequence defined by $b_i:=F(a_{i-1},a_{i},a_{i+1}),$ where $F:\mathbb{R}^3 \rightarrow \mathbb{R}$ is increasing in each of the variable, and ...
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### total variation of sinc function

The total variation of differentiable function $f$ is defined by $$||f||_V=\int_{-\infty}^{\infty} |f'(t)|\:dt$$ I must show $$||f||_V = \infty$$ if $$f(t)=\frac{\sin(\pi t)}{\pi t}$$ In other words ...
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### Total variance distance between high-dimension Gaussians with different means

I am trying to compute the total variation distance, also called total variance distance, and by abuse of notation sometimes even statistical distance (at least in the field of research of lattice-...
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### total variation distance and coupling

How can we show that $||\gamma M- \beta M||_{TV}\leq||\gamma -\beta||_{TV}$ (total variation distance) for a transition matrix of a markov chain $M$ ? $\gamma, \beta$ are two distributions on the ...
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### $L^{p}$ spaces for complex measures

Let $(\Omega,\mathscr{A})$ be a measurable space and $\mu\colon\mathscr{A}\to\mathbb{C}$ a complex measure (i.e. a $\sigma$-additive function from $\mathscr{A}$ to $\mathbb{C}$). Then $\mu$ can be ...
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### Total Variation Distance and Number of Samples

I don't know a distribution $P_{XY}$. However, I can randomly get some samples (X,Y) from this distribution. Based on $n$ samples, I approximated the distribution as $P'_{XY}$. How can I bound the ...
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### integration of bounded variation function is of bounded variation?

Let $g:X\times Y \rightarrow \mathbb{R}$, and define $f(x) = \int_{Y} g(x,y)\,dy$. Suppose that $g$ is of bounded variation, then can I say that $f(x)$ is also of bounded variation? I have shown \...
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### What is the performance difference between the Total Variation norm and Dirichlet energy norm in image processing

I have been looking at some image processing material lately, and see that two common norms are the Total Variation norm and the Dirichlet energy--also called the $H^1$ seminorm. Now these norms look ...
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### Bound an integral in terms of the total variation distance

Let $(E,\mathcal E,\lambda)$ be a measure space $\mu,\nu$ be measures on $\left(E^2,\lambda^{\otimes2}\right)$ with densities $f,g:E^2\to[0,\infty)$, respectively $\left\|\mu-\nu\right\|$ denote the ...
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### Riffle shuffling cards and cuts

In “Trailing a dovetail shuffle to its liar” by Diaconis and Bayers. In section 2, he says the following. Cutting a deck of cards respect the cyclic order of deck of cards, where card 1 follows card ...
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### If $(\pi_λ)_{λ\in\mathbb R}$ is a family of orthogonal projections, do $λ↦\left\|\pi_λx\right\|_H^2$ and $λ↦\pi_λx$ have the same variation?

Let $H$ be a $\mathbb R$-Hilbert space and $H_\lambda$ be a closed subspace of $H$ for $\lambda\in\mathbb R$. Assume $H_\lambda\subseteq H_\mu$ for all $\lambda,\mu\in\mathbb R$ with $\lambda\le\mu$ ...
It's from Brownian Motion and Stochastic Calculus by I. Karatzas chapter 1, problem 5.7 property (iv) trying to show an inequality wrt total variation and cross variation of martingales. $$\check{\xi}... 1answer 173 views ### total variation distance between 2 distributions decreases? Say we have 2 probability distributions \pi_0, \alpha_0 on the same state space S with some transition matrix P. Define \pi_1 = \pi_0 \cdot P, \alpha_1 = \alpha_0 \cdot P. The goal is to ... 1answer 47 views ### Can the total variation measure of a complex measure be related to its real and imaginary parts? If \mu is a signed measure with Jordan decomposition \mu=\mu^+-\mu^-, then the total variation measure of \mu is equal to \mu^++\mu^-. My question is, is it similarly possible to express the ... 0answers 95 views ### Total variation of a measure in a topological dual of the space of continuous functions I read the definition of total variation of a measure in the following link https://www.encyclopediaofmath.org/index.php/Signed_measure And then I have a question: If \mu is a vector measure in ... 1answer 42 views ### Existence minimizer for total variation over measures I want to prove that there exists a minimizer to the following problem$$ \min || \mu ||_{\text{TV}} \text{ such that } \mathcal{F} \mu = y $$where \mu \in \mathcal{M}([0,1]), the space of Radon ... 2answers 188 views ### Total variation distance of probaiblity measures Let E be a set and \mathcal E\subseteq 2^E with \emptyset\in\mathcal E. If \mu:\mathcal E\to\mathbb R with \mu(\emptyset)=0, then$$\operatorname{Var}_\mu(B):=\sup\left\{\sum_{i=1}^n\left|\...
Suppose $S$ is a smooth surface without boundary and $f:S\to\mathbb R$ is continuous. Then, we can define the total variation of $f$ as:  \mathrm{TV}[f]:=\sup_{\|\phi\|_\infty\leq1} \int_S f(x)\...