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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16 views

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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34 views

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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36 views

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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17 views

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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7 views

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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25 views

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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27 views

$L^{p}$ 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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18 views

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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77 views

$L_2$ /$L_1$ norm of Gradient

The question, briefly: I am interested in using "Total Variation Denoising" in order to recover a 2-dimensional signal (in particular, an image). In the existing literature, many authors use the $...
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27 views

Question concerning the decomposition of the Total Variation Measure

Let $\Omega \subseteq \mathbb{R}^n$ be open and $u \in BV(\Omega)$. Then there exists a finite signed Radon measure $Du$, which happens to be the weak derivative of $u$, with the property: $\...
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20 views

Total variation distance to stationary distribution decreases monotonically for Ergodic Markov processes

Proposition: Let $P$ be the transition matrix of a strongly connected aperiodic Markov chain. Let $\pi$ be the stationary distribution of $P$. Then for any distribution vector $x$ it holds that $$d_{...
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34 views

Equation of Variation y with x [closed]

$y-x= 1/x - 1/y$ where x and y are not equal to 0 then y varies ? (Complete) I am supposed to know if it is direct or inverse variation but I did not manage to get it
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57 views

Bounded Variation Function and its Total Variation

Suppose $f$ is a bounded variation function on $[a,b]$. $v(x) = TV([a,x])$ is the total variation of $f$ on $[a,x]$. From bounded variation of $f$, monotonicity of $v$, we know that $f$ and $v$ have ...
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22 views

Exterior measure of total variation measure

Let $(X,\Sigma,\mu)$ be a signed measure space. If $\mu^*$ denotes the exterior measure of $\mu$ and $\Lambda_{\mu^*}$ denotes the set of $\mu^*$ measurables sets, i'm trying to find a relation ...
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27 views

Definition of positive and negative variation

The total variation of a measure is defined as $$|\mu|(E)=\sup \sum_i |\mu(E_i)|$$ where the supremum is taken over all countable partitions of $E$. So the positive and negative variation of $\mu$ ...
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27 views

Total variation distance of a Markov kernel and its invariant distribution

Let $(E,\mathcal E,\lambda)$ be a measure space, $Q$ be a Markov kernel on $(E,\mathcal E)$ with $$Q(x,B)=\int_B\lambda({\rm d}y)q(x,y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$$ for some ...
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36 views

Convergence in Total Variation on a d-dimensional Torus

Let $\mathbb T_k$ be a d-dimensional torus with sides of length $k$, and let $S_n^{(k)}$ denote the lazy simple random walk on $\mathbb T_k$ (which jumps to each of its neighbors with equal ...
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24 views

Distance between a distribution and convex combination of distributions

I have a distribution $P$ and I have all the distributions $Q$s that are at most d distance from $P$. Let's assume the convex combination of all those $Q$s are $S$. Then what would be the maximum ...
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81 views

Limits of the total variation when $f(-\infty)=0$

Let $f\in\operatorname{BV}(\mathbb R)$ and $$V_f(x):= \displaystyle\sup_{x_1 < x_2 < x_3 < ... < x_n=x} \sum_{j=1}^n |f(x_{j+1})-f(x_j)|$$ be the total variation of $f$ on $]-\infty, x[$. ...
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42 views

Equality between expressions for total variation distance between two discrete probability distributions.

I found an excellent answer to a more general form of this question here: Two notions of total variation norms but it is a bit more sophisticated than what I am looking for. My question Let $X,Y$ ...
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43 views

Absolute continuity with respect to the product measure

Let $\mu$ be a measure over the product space $X\times Y$, $X$ is any topological space and $Y$ is either Lindelöf or compact, and let $\mu_y$ and $\mu_x$ denote the marginals of $\mu$ on $X$ and $Y$, ...
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13 views

total variation in reflected SDE, the Skorohod problem

I have been struggling to understand the Skorohod problem for reflected SDEs. My main confusion is the role of the total variation in the reflection term. I get that we need a term to basically freeze ...
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32 views

On the definition of bounded variation [duplicate]

In one dimension, a function $f: I:= [a,b] \to \mathbb{R}$ is said to be of bounded variation if $$\mathrm{Var}(f,I):= \sup_{P}\left \{ \sum_{i=1} ^n \|f(x_i) - f(x_{i-1})\| : \mathrm{for} \; P:=\{x_0,...
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32 views

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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20 views

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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31 views

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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59 views

Total variation between random variables or probability measures

Consider a die with probability space $(\Omega = \{one,two,three,four,five,six\}, \mu)$ such that $\forall \omega \in \Omega, \mu(\omega) = 1/6$. Consider also a coin with probability space $(\Lambda =...
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48 views

Toy dice problem and conditioning

Let $X_1$ and $X_2$ be the values of two fair dice (with 6 faces). For the sake of understanding (and quantifying) how conditioning on a particular sum of the dice modifies the joint distribution of $...
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27 views

What's the maximum total variation divergence between a gaussian with a gaussian mixture?

Define $\pi_\mu(z)=N(z; \mu, \sigma^2 I)$. Define $\phi(z)$ to be the pdf of a gaussian mixture where every gaussian also has covariance $\sigma^2 I$. I wonder what is the solution for $\hat\mu = \arg\...
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30 views

References for total variation distance between tensor products of two probability measures

Let $P$ and $Q$ be two probability measures on $\{1,\dots,n\}$. Are there references regarding $$d_{TV}(P\otimes Q,Q\otimes P)\,,$$ where $d_{TV}$ is the total variation distance? In other words, if ...
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77 views

Equivalent definitions of total variation norm

Let $E$ be a set, $\mathcal E\subseteq2^E$ with $\emptyset\in\mathcal E$ and $\eta:\mathcal E\to\mathbb R$ with $\eta(\emptyset)=0$. If $B\subseteq E$, let $$|\eta|(B):=\sup\sum_{i=1}^k|\eta(B_i)|,$$ ...
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382 views

Show that the total variation distance of probability measures $\mu,\nu$ is equal to $\frac{1}{2}\sup_f\left|\int f\:{\rm d}(\nu-\mu)\right|$

Let $(E,\mathcal E)$ be a measurable space, $\mu$ and $\nu$ be probability measures on $(E,\mathcal E)$ and $$|\nu-\mu|:=\sup_{B\in\mathcal E}|\nu(B)-\mu(B)|$$ denote the total variation distance of $\...
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34 views

Can we infer convergence in total variation distance from a Poincaré inequality?

Let $(E,\mathcal E,\mu)$ be a probability space, $\lambda>0$ and $\kappa_t$ be a Markov kernel on $(E,\mathcal E)$ with$^1$ $$\operatorname{Var}_\mu\left[\kappa_tf\right]\le\operatorname{Var}_\mu\...
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54 views

Total variation of $N_t - \lambda t$

How can I compute the total variation process of the martingale $ \{N_t - \lambda t \}$ under the assumption that $N_t$ is a Poisson process with parameter $\lambda$? Thank you very much !
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39 views

Prove $|\int_\mathbb{R} F(x)\varphi'(x)\,dx|\leq A$ where $F$ has bounded variation on $\mathbb{R}$ and $\varphi\in C^1$.

Let $F$ be of bounded variation on $\mathbb{R}$, i.e it is of bounded variation on any finite subinterval $[a,b]$, $\sup_{[a,b]}V_a^b(F)<\infty$. Given the fact that \begin{align*} \int_\...
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32 views

Intuition for the proof of optimal coupling

I am learning coupling from this text Markov Chains and Mixing time by David Levin et al.. My question is regarding the proof for the following proposition (pages 50-51 from the text). Proposition:...
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33 views

Relation between total variation and KS distance between measures on $[0,1]^d$

Let $P$ and $Q$ be two probability measures on the space $[0,1]^d$, $d \in \{1, 2, \ldots \}$, endowed with the $L_\infty$ norm and the corresponding Borel $\sigma$-field, $\mathcal{B}$. Let $$F_P(\...
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35 views

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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38 views

Difference in the total variation norm of measures is equal to two, implies measures are not absolutely continuous?

Given two probability measures $\mu, \nu$ on a probability space $X$, their total variation is defined by $$ ||\mu - \nu|| := 2 \sup_{A \subset X}|\mu(A)-\nu(A)|.$$ I would like to know: If $||\mu - \...
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144 views

Absolute continuity, implies bounded variation.

I am trying to directly prove from the definition of absolute continuity, that if a function $f:[a,b]\rightarrow\mathbb{R}$ is absolutely continuous, then it is of bounded variation, i.e $Vf$ is ...
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103 views

Intuition for Pinsker's first inequality

I'm learning about the Kullback-Leiber divergence from Introduction to Nonparametric Estimation by Alexandre B. Tsybakov. He proves Pinsker's first inequality on pages 88-89 (which I state below in ...
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25 views

Iterative least squares for TV-like regularizer

I am trying to implement a TV-like surface regularization in a least squares solver. The formulation is as follows: $E_{surface} = \sum |dA_\theta| $ Background $dA_{\theta} = \frac{\theta_1}{f_x ...
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60 views

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 $\lambda\le\mu$ ...
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128 views

total variation inequality involving cross variation

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}...
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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 ...
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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 ...
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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 $...
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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 ...
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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|\...
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193 views

Total variation on a creased surface

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)\...