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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.

In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the co-domain of a function or a measure. For a real-valued continuous function $~f~$, defined on an interval $~[a, b] ⊂ ℝ~$, its total variation on the interval of definition is a measure of the one-dimensional arc length of the curve with parametric equation $~x ↦ f(x)~$, for $~x ∈ [a, b]~$.

In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance or variational distance.

Applications:

Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like optimal control, numerical analysis, and calculus of variations, where the solution to a certain problem has to minimize its value. As an example, use of the total variation functional is common in the following two kind of problems:

  • Numerical analysis of differential equations: it is the science of finding approximate solutions to differential equations. Applications of total variation to these problems are detailed in the article "total variation diminishing"

  • Image denoising: in image processing, denoising is a collection of methods used to reduce the noise in an image reconstructed from data obtained by electronic means, for example data transmission or sensing. "Total variation denoising" is the name for the application of total variation to image noise reduction; further details can be found in the papers of (Rudin, Osher & Fatemi 1992) and (Caselles, Chambolle & Novaga 2007). A sensible extension of this model to colour images, called Colour TV, can be found in (Blomgren & Chan 1998).

References:

https://en.wikipedia.org/wiki/Total_variation

http://mathworld.wolfram.com/TotalVariation.html

https://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures

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Computing the total variation for a multivariable function

I am trying to write an example computation with multivariable total variation to include in my functional analysis notes using the following definition from Wikipedia: Let $\Omega$ be an open subset of $\mathbb{R}^n$. Given a function $f$…
WDR
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Comparison of definitions for Functions of Bounded Variation

I have been trying to understand the functions of bounded variation and I came across the following definitions Defintion 1: A function $f:\mathbb{R^d} \rightarrow \mathbb{R}$ is of bounded variation…
Celestina
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Relation between Shannon Entropy and Total Variation distance

Let $p_1(\cdot), p_2(\cdot)$ be two discrete distributions on $\mathbb{Z}.$ Total variation distance is defined as $d_{TV}(p_1,p_2)= \frac{1}{2} \displaystyle \sum_{k \in \mathbb{Z}}|p_1(k)-p_2(k)|$ and Shannon entropy is defined the usual way,…
pikachuchameleon
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Variational distance of product of distributions

Let $F(\bar{x})=\prod_{i=1}^{n}f(x_i)$ and $G(\bar{x})=\prod_{i=1}^{n}g(x_i)$, where $f(x)$ and $g(x)$ are probability density functions, and $\bar{x}=(x_1,\ldots,x_n)$. The variational distance between $F$ and $G$ is: $$V(F,G)=\int…
Albert
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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 $$f(x)=\sum_{y\leq x}f(y^+)-f(y^-).$$ These functions occur as the…
Chris Culter
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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 $\mu$ and $\nu$. How can we show that…
0xbadf00d
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$d_{TV}$ VS correlation coefficient.

Consider two RV $X,Y$. If $d_{TV}(X,Y)=0$ you may couple them in such a way that $$\rho_{XY}=\frac {\operatorname{COV}(X,Y) }{\sigma_X\sigma_Y}=1.$$ So, is there any formula to bound $d_{TV}$ in terms of $\rho$?
MR_BD
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Total variation of stochastic process and total variation of a signed measure.

In this definition of total variation of a stochastic process $A$, I don't understand what the author means by "the measure $dD_t(\omega)$ is the total variation of the signed measure signed measure $dA_t(\omega)$." The definition 1.7.7 is the…
An old man in the sea.
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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(\mathbf{u})=P([\mathbf{0},\mathbf{u}]), \, \quad…
Jack London
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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’m not getting it. Can someone please help me,…
user1286545
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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. $x\in [a,b]$, where $V^x_a f$ is the total…
Jianing Song
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Signed Borel Measures and Functions of Bounded Variation

Let $\nu$ be a finite signed Borel measure on the closed interval $[a,b]$. We can define a function $F_\nu : [a,b]$ by $$F_\nu(x) = \nu([a,x]).$$It can be shown that $F_\nu$ has bounded variation and is right-continuous. An exercise in my class…
Nick A.
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Understanding total, quadratic, and $\Phi$ variation of functions

I've started to study stochastic calculus on my own recently (I'll read Fima's book for a simpler introduction and then Steele's for a maybe more formal approach). I've come across the definition of the total variation, quadratic variation and other…
YetAnotherUsr
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Compact subset of infinite-dimensional space has empty interior

My question is related to this question. My space is the set of all Borel probability measures on $\Theta=[0,1]$, which is a compact metric space under the Prokhorov metric. Call this space $\Delta \Theta$. This is a compact subset of the space of…
Canine360
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Fourier series of function with bounded variation is bounded by its norm

We have $\mathbb{T} = \mathbb{R}/\mathbb {Z}.$ We define a space $BV(\mathbb{T}) = \{f| f:\mathbb{T} \to \mathbb{C}, V(f) <\infty \},$ where $V(f)$ denotes the total variation. We also equip this space with norm $\lVert f \rVert_{BV(\mathbb{T})} =…
Bombadil
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