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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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How this variational derivative is calculated?

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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
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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 ...
Michael Smith's user avatar
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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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
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Total variation convergence in context of stochastic processes

Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution $$X_t~\...
Oskar Vavtar's user avatar
1 vote
1 answer
38 views

Numerical Method for (Total Variation) TV Norm Minimization of Linear Combination of Matrices

I have a matrix $\mathbf{A} \in \mathbb{R}^{2000 \times 2000}$ represented in memory by an array of $2000 \times 2000$ float32 elements and I also have $10$ arrays $...
VojtaK's user avatar
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Questions about the proof of existence for maximal coupling

I'm trying to understand the proof of existence for the maximal coupling. That is: For any two probability measures $\mathbb{P},\mathbb{P}'$ on a measurable space $(E,\mathcal{E})$ there exists a ...
Almost-surely's user avatar
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On indistinguishability and addition/convolution

Happy new year. I'd like some help with this problem: Let $p,d \in {[0,1]}^{n}$ so that ${||p||}_{L1} = {||d||}_{L1}=1$ i.e p,d are probability distributions over Z/nZ. '$\star$' is the operator of ...
alon's user avatar
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2 votes
2 answers
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Distances in the space of probability measures and couplings of random variables

One-sentence summary: I want an intuitive explanation for why closeness of probability measures (in terms of, eg., TV or Lévy-Prokhorov) implies the existence of good couplings. Set up. Consider a ...
nootnoot's user avatar
3 votes
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73 views

Property of signed measure and total variation

Let $\nu$ be a signed measure on $(X,\mathcal{M})$. The total variation of $\nu$ is defined as $$ |\nu|= \nu^+ + \nu^-, $$ where $\nu = \nu^+ - \nu^-$ is the Jordan decomposition of $\nu$. For every $...
aaa's user avatar
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Variation norm and 1-Norm

In a sigmafinite measure space (Ω, S, µ) with a finite signed measure ν i should show I know what the definition of these two are but i have no idea how too prove it.
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Total variation $t\mapsto \|f\|_{t,\operatorname{var}}$ of a cadlag function $f$ with finite variation is cadlag

If $(X_t)_{t\in [0,T]}$ is cadlag and has finite variation, is it true that the total variation $t\mapsto \operatorname{Var}(X)_t$ is cadlag? ($\operatorname{Var}$ denotes the total variation) This ...
Analysis's user avatar
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Total variation in product

Let $a(x),b(x)$ be two density functions on $X$ and $c(y),d(y)$ be two density functions on $Y$, i.e. they are non-negative real functions such that $\int_X a=\int_X b=\int_Y c=\int_Y d=1$. Is it true ...
Cyrist's user avatar
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total variation distance: $\max_{A \subseteq \mathcal{A}} \left| P(A)-Q(A) \right| = \frac{1}{2} \sum_{x \in \mathcal{A}} \left| P(x) - Q(x)\right|$? [duplicate]

Could you please suggest a proof or a reference that shows a proof of the following equality (not "passing through the densities" as done in Definition of the total variation distance: $ V(P,...
Ommo's user avatar
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Total variation distance: a relationship between a Polish space $(\mathcal{X}, d)$ and a measurable space $\left(\mathcal{X},\mathcal{A}\right)$

Introduction (part 1). In the following excerpts of Villani (2008) Optimal transport, old and new, Villani (i) defines the Wasserstein distance among two probability measures $\mu$ and $\nu$ on a ${\...
Ommo's user avatar
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Equivalence between the "supremum formula" and the "maximum formula" of the "total variation distance"

Introduction. In a previous post, Definitions of the total variation distance, I asked about the equivalences among different definitions of the "total variation distance" $d_{TV}$: I found ...
Ommo's user avatar
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2 answers
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Definitions of the total variation distance

Introduction. I found the following definitions of the total variation distance $d_{TV}$ between two probability distributions (also called probability measures) $P$ and $Q$ on $\mathcal{A}$ (please ...
Ommo's user avatar
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An inequality between the total variation distance for one measure and the combination of projections of this measure

Let $p$ and $q$ be two probability measures on some measure space $(X,\mathcal X)$. Suppose that there is $\lambda>0$ such that $(1+\lambda) p-\lambda q\geqq 0$, in other words there is $\gamma\in]...
P. Quinton's user avatar
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Does the vector space of differences of quantile functions have a neat characterization?

Consider the convex cone of quantile functions of random variables on the real line (with finite second moment), that is $$ C := \{ Q_{\mu}: \mu \in \mathcal P_{(2)}(\mathbb R) \}, $$ where $Q_{\mu}(p)...
ViktorStein's user avatar
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Weighted sum of N images, which minimizes their TV norm

I have $K$ images $I_i, i\in{1 \ldots K}$ of the size $M \times N$. I wish to find weights $w_i$, s.t. $w_i \in [0,1]$ and $\sum_1^K w_i = 1$ so that $$|\sum_{i=1}^{K} w_i I_i|_{TV}$$ is minimal. I ...
VojtaK's user avatar
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1 answer
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What is the best regularity of a rough path?

Assume we have a one dimensional rough path $\mathbf{X} = (X, \mathbb{X})$ with finite $p$-variation, with $p \in (1,2]$. Assuming that $Y$ is controlled in the sense of Gubinelli with respect to the ...
qervert's user avatar
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What is the positive variation of a function?

I am having trouble understanding what the positive variation of a function $\frac{ 1}{2} ( T_F + F)$ represents. (Where, in this case, $T_F$ is the total variation of $F$ from $[0,x]$.) For consider $...
kam's user avatar
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1 answer
183 views

Space of finite signed measures

Given a measure space $(\Omega, \mathcal{A})$, let $\nu$ be a signed measure on that space and let $|\nu| := \nu^+ + \nu^-$ be the variation. Now consider the measurable functions $X:\Omega \...
guest1's user avatar
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1 answer
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Is $\mathcal{M}(\mathbb{R}^d)$ composed of finite measures?

Let $C_0(\mathbb{R}^d)$ be the space of real-valued continuous functions on $\mathbb{R}^d$ that vanish at infinity. By Riesz–Markov–Kakutani representation theorem, the dual of $C_0(\mathbb{R}^d)$ is $...
Lin2568's user avatar
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2 votes
1 answer
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How can I simulate on R the total variation distance between the Law of the number of fixed points and Pois(1)?

I am trying to make a graph on the Total Variation Distance between the Law of the number of Fixed Points of a Random Permutation and the Poisson($\lambda = 1$) distribution. I know that the Total ...
enrico didoli's user avatar
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1 answer
128 views

Prove the definitions of total variation for the case of real-valued signed measures are equivalent.

According to wikipedia, there are two definitions of total varation for the case of real-valued signed measure $\mu$: the first is classical total variation definition, where the total variation $|\...
wuxj's user avatar
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"Shady" witness for total variation distance

Intro: Hello! So I am wondering about the following problem. I have three discrete probability distributions (think $d$-dimensional $\ell_1$-normalized vectors with only non-negative entries), namely $...
alex.ander's user avatar
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1 answer
309 views

Decomposition of probability measures with a bounded total variation distance

Fix a probability space $(\Omega, \mathcal{F})$. Let $P$ and $Q$ be two probability measures on $(\Omega, \mathcal{F})$ such that there exist $\varepsilon$ and $\delta$ that for every $A \in \mathcal{...
MMH's user avatar
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Total variation of a complex Radon measure via continuous functions

The total variation norm $\mu$ of a complex measure can be defined as $$\lVert \mu \rVert = \sup_{\lVert f \rVert_u \le 1} \left| \int f\ \mathrm{d}\mu\right|$$ Here $\lVert \cdot \rVert_u$ is the ...
llf's user avatar
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TV Distance between Limiting Distribution and One Step Transition Probability Distribution of Markov Chains

Edit 05/01/2023: This question may seem obtrusive at first glance, but it is warranted by some reinforcement learning algorithms, such as TD algorithm. In fact, consider building the Bellman Equation ...
Sizhe Ding's user avatar
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1 answer
76 views

Total variation distance for sums of random variables

Consider two random variables $X_n \equiv X_n(\omega)$ and $Y_n \equiv Y_n(\omega)$ defined on the same probability space $(\Omega, \Sigma, P)$. Assume that $Y_n$ converges to zero in probability, i.e....
Jack London's user avatar
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1 answer
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Optimization Problem about total variation in the distribution parameter's neighborhood

Given two fixed distribution as $p_\theta,p_{\theta^*}$. In the neighborhood of $\theta$, there exists a small region $\epsilon$, which forms the distribution $p_{\theta+\epsilon}$. These three ...
1 1's user avatar
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1 answer
267 views

Equality relationship between Total Variation and KL divergence [closed]

I noticed that according to Pinsker's inequality, the following relationship can be got. $D_{TV}(p||q) \le \sqrt{{1\above{1pt}2}D_{KL}(p||q)} $. I'm curious whether, under some condition, the equality ...
1 1's user avatar
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1 vote
0 answers
28 views

Compute the arc length of the Cantor function using total variation [duplicate]

We're all familiar with the standard way to prove that the arc length of the Cantor function $C:[0,1] \rightarrow [0,1]$ is $2$. We also know that if a function $f:[a,b] \rightarrow \mathbb{R}$ is ...
LucasS's user avatar
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1 vote
1 answer
128 views

Need help understanding definition of total variation metric of two probability measures

Fix a measurable space $(X,\Sigma)$. Let $\mu:\Sigma\to\mathbb{R}$ be some signed measure. Then the total variation norm is defined by $$ \|\mu\|_{\textrm{TV}}=\mu_{+}(X)+\mu_{-}(X),\quad (*) $$ where ...
Miski123's user avatar
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2 votes
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Explicit form for the total variation measure of a symmetric matrix-valued measure

Let $X$ be a locally compact Polish space. Consider the set $M(X; S^d)$ of real symmetric matrix-valued measures on $X$, that is, the set of countably (with respect to the Frobenius norm $\| \cdot \|...
ViktorStein's user avatar
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1 vote
1 answer
361 views

Decomposition of Total Variation Distance

Let $X$, $Y$, and $Z$ be three random variables on a common measurable space. I am interested in the total variation distance between the two joint distributions $(X,Z)$ and $(Y,Z)$, i.e. \begin{...
Chris Harshaw's user avatar
2 votes
0 answers
39 views

Convergence in total variation implies convergence of some similarity criterion?

Let $(X,\Sigma)$ be a measure space, let $\mathcal P(X)=\{ p : (X,\Sigma,p) \text{ is a probability space} \}$. We use the total variation distance, i.e. for any $p, q\in\mathcal P(X)$, $\| p-q \| = \...
P. Quinton's user avatar
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1 vote
1 answer
193 views

Prove $\int_\mathbb{R}|F(x+h)-F(x)|dx\leq A |h|$ for bounded variation function F(x) [closed]

A bounded function $F$ is said to be of bounded variation on $\mathbb{R}$ if $F$ is of bounded variation on any finite sub-interval $[a, b]$, and $\sup_{a,b}T_F (a,b)<\infty$ $~~~~$(The $sup_{a,b}...
Gang men's user avatar
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0 answers
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Total variation distance for product measures

Consider two measurable spaces $X$ and $Y$ and form their Cartesian product $X\times Y$, with the product $\sigma$-algebra. In this question I asked whether, given probability measures $p$ and $q$ on ...
geodude's user avatar
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1 answer
35 views

Does exchanging the role data fit term and regularization term make sens?

For example the total variation regularization formulation is given as $$ \min_x \frac{1}{2} || Ax -y ||_2^2 + \alpha ||\nabla x||_1. $$ Now I am wondering, would it make sense to ask for a solution ...
stish's user avatar
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0 answers
102 views

How to determine the smoothness of a set of non-equidistant discrete values?

I have a set of points $(x_i, y_i)$ for which I want to determine the smoothness of the curve formed with them. The horizontal distance $(x_{i+1} - x_i)$ between these points is not identical. I have ...
user7468395's user avatar
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1 answer
327 views

How to find total variation of a signed measure??

Let $(X,F,\nu)$ be a signed measure on the sigma algebra F. now by Jordan-Hahn decomposition theorem $\nu = \nu_1 - \nu_2$, where $\nu_1$ and $\nu_2$ are positive and mutually singular, and such ...
Vishnudasa Srinivasan's user avatar
1 vote
1 answer
35 views

If $\mu_n \to \mu$ in the weak* topology does $|\mu_n - \mu|(X) \to 0$?

Let $X$ be a compact Hausdorff space and fix a net of complex Borel measures $(\mu_n)$ such that $\mu_n \to \mu$ in the weak* (aka vague) topology on $C(X)^*$. That is, for every continuous function $...
Zorngo's user avatar
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2 answers
300 views

Relationship between radon-nikodym derivatives and total variation distance

Let $(\mathcal{X},\mathcal{A})$ be a measure space on which we have defined two probability measures $P$ and $Q$. Let $f$ and $g$ denote Radon-Nikodym derivatives of $P$ and $Q$ with respect to a $\...
verygoodbloke's user avatar
1 vote
1 answer
143 views

Question on relation between radon-nikodym derivatives and the total variation distance

Let $(\mathcal{X},\mathcal{A})$ be a measure space on which we have defined two probability measures $P$ and $Q$. I am reading some notes online which makes the following jump without explanation in ...
verygoodbloke's user avatar
3 votes
0 answers
131 views

understanding why Wasserstein is weak

I am reading Wasserstein GAN paper and in Appendix A, it says Let $\mathcal{X} \subseteq \mathbb{R}^d$ be a compact set (such as $[0, 1]^d$ the space of images). We define Prob($\mathcal{X}$) to be ...
MoneyBall's user avatar
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1 answer
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Piecewise differentiable version: Total variation expressed as an integral

I have already successfully found a proof of the equality $$V_{a}^{b}(f) = \int_{a}^{b} |f'(x)| \, dx$$ where $f$ is a continuously differentiable function on $(a,b)$ and continuous on $[a,b]$ as well ...
TheOutZ's user avatar
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1 vote
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64 views

Relationship between limit and total variation

Consider a twice continuously differentiable and bounded function $f:\mathbb R \rightarrow \mathbb R$ has the following property: in any neighborhood of $\infty$, there are an infinite number of ...
Jong's user avatar
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1 vote
1 answer
185 views

Fréchet derivative of the total variation norm for measures on a manifold

Let $\Theta$ be a compact $d$-dimensional Riemannian manifold without boundary and $M(\Theta)$ (resp. $M_+(\Theta)$) denote the set of signed (resp. nonnegative) finite Borel measures on $\Theta$. ...
ViktorStein's user avatar
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3 votes
3 answers
214 views

Does the derivative of a vector-valued BV function $f(x)$ equal to the norm of $f'(x)$?

Let $f(x)$ be a real-valued function on $[a,b]$ of bounded variation. It is standard that $f(x)$ is almost everywhere differentiable, and that $\dfrac{{\rm d}}{{\rm d}x} V^x_a f = |f'(x)|$ for a.e. $x\...
Jianing Song's user avatar
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