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Questions tagged [total-variation]

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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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TVD of numerical scheme and proper CFL

I was reading the paper Total Variation Diminishing Runge-kutta Schemes and I came across the following line It is easy to prove, by using Harten's Lemma, that the Euler forward time ...
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46 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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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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24 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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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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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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69 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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27 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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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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Lower bound of the total variation

Let $f\in C^1(\Omega)$ be a continuously differentiable function on a domain $\Omega \subset \mathbb{R}^n$. It is known that the total variation of $f$ is equal to $$V(f,\Omega)=\int_\Omega |\nabla f(...
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upper bound for KL divergence of two sub-gaussians with different mean and variance

I was wondering if there is any upper bound for KL divergence of two subgaussian with means $\mu_1$ and $\mu_2$ and variances $\sigma_1$ and $\sigma_2$? The definition of KL divergence is $$D_{KL}(...
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1answer
29 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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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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Multi-Variable CoArea formula for Total Variation

The total variation of a scalar function over a 2-manifold can be defined using the coarea formula. If I now use a N-d function, and I want to integrate the frobenius norm of its gradient over the 2-...
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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)\...
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Hahn-Vitali-Saks theorem: do we need to ask for finite total variations?

In Yosida's functional analysis (p70), we encounter the Hahn-Vitali-Saks theorem, as also stated on wikipedia: https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem In particular, ...
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How to obtain this equation of hyperspectral image restoration using total variation in the following equation?

While reading the paper "Total-Variation-Regularized Low-Rank Matrix Factorization for Hyperspectral Image Restoration" I came through the following equation. Can someone help me how the last equation ...
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Absolute value of difference between expected values of two distributions and total variation

I'm trying to prove following inequality. For any probability distribution $P$ и ${Q}$ on $U$ and function $f(x): U \rightarrow [0; B]$ it is true that: $$ |E_{\tilde P}[f(x)] − E_{\tilde Q}[f(x)]| \...
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1answer
90 views

Showing $\int |f_n - g_n| dx \to 0$ when normal $X_n \overset{p}{\to} \mu, Y_n \overset{p}{\to} \mu $

I'd like to show the following. Let $X_n, Y_n$ be sequences of normal random variables. If $$ X_n \overset{p}{\to} \mu, Y_n \overset{p}{\to} \mu $$ then $\int |f_n - g_n| dx \to 0$, with $f_n, ...
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31 views

Riemann-Stieltjies integral. Find total variation.

I need to find the total variation of $V_g([0,2\pi])$, when $g(x)=cos(x)$ According to the formula: $V_g([a,b])=g(b)-g(a)$ $V_g([0,2\pi])=cos(2\pi)-cos(0)=1-1=0$ But this answer is shown as wrong. ...
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112 views

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 ...
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1answer
45 views

Total Generalized Variation [closed]

While reading the paper Total Generalized Variation, I came across the following formula (Equation 1.5), which describes the regularization term: Total Generalized Variation formula While there are ...
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100 views

$d_{TV}$ 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 ...
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Fraction of Variation Explained by a Variable

I have perhaps a basic question, but I cannot seem to find an answer online. Essentially I would like to explain how much of the variation a change in one independent variable contributes to a ...
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3answers
87 views

Minimizing total variation with small norm in a compact set

Take $\Omega\subseteq\mathbb R^n$ to be compact. Suppose for $t>0$ we define $f$ via the variational problem $$ \begin{array}{rl} \inf_{f\in\mathrm{BV}(\mathbb R^n)} & \mathrm{TV}[f]\\ \textrm{...
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1answer
39 views

Convergence to zero in total variation: Seeking a contradiction

Suppose I have a compact measurable set $\Omega\subseteq\mathbb R^n$ and a sequence of functions $\{f_k\}_{k=1}^\infty\subset C^\infty(\mathbb R^n)$ with the following set of properties: $f_k\geq0$ $\...
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Proving from definitions that total variation of an absolutely continuous function is absolutely continuous

I know this can also be done with integrals, but I'm doing it from definitions as an exercise. Say we have a collection of open intervals $\{(a_{k}, b_{k})\}_{k =1}^{n}$ in the interval $(a, b)$. Then ...
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53 views

Questions about Newman's simplified proof of Ramanujan's partition formula

Recently I started to go through Newman's proof of Ramanujan's asymptotic formula for the number of partitions $p(n)$. I got stuck right in the beginning, where we have $f(z) = \prod_{n=1}^\infty \...
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51 views

Finding the gradient of a variant of a total variation regularized least squares cost function

As the question states, given a sum-function : $$f(x) = \sum_{ij}\left({\sqrt{(x_{ij} - y_{ij})^2+1}}+\frac{1}{2}\sqrt{(x_{ij}-x_{i+1j})^2+(x_{ij}-x_{ij+1})^2 +1}\right)$$ where $x_{ij} $ describes ...
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1answer
77 views

Sum of total variation over countable partition of interval equals variation over interval? [duplicate]

Let $I=[a,b]$, and consider some $f\in BV(I)$ (the space of all functions with bounded total variation on $I$). Let $$\mathcal Q=\{[a_k,b_k]:k\in\mathbb N,~\operatorname{cl}\left(\bigcup_{k=1}^\infty[...
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174 views

The total $p$-variation of a standard Brownian motion is infinite almost surely for any $p > 1/2$

How can I show that the $p$-variation of a standard Brownian motion is infinite almost surely for any $p > 1/2$. By this I mean the total variation, $lim_{\delta\to0} (sup_{\pi:\delta(\pi)=\delta}...
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Are $L^2$ and total variation-linked norms equivalent on this function space?

Let $K > 0$. Let $\mathcal{C} = \mathcal{C}^1([0,K], [-1,1] )$ be the space of $C^1$ functions $f : [0, K] \rightarrow [-1,1]$, such that $f(0)=0$. Let's use the norm: $$||f||_2 = \sqrt{\int_0^K ...
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1answer
111 views

Given a family of probability distributions that are “close to each other” and have expected values 1, 2, 3…, find a lower bound on their variance

Let $\left(\mathcal{D}_{n}\right)_{n\in\mathbb{N}}$ be a family of probability distributions such as: $\forall n\in\mathbb{N}$, $\mathbb{E}\left[\mathcal{D}_{n}\right]=n$: the average value of the n-...
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200 views

The Dual Norm of Total Variation Norm (Form of $ \left \langle \cdot, \cdot \right \rangle $) By Smoothing

Recently, I am following paper Becker S, Bobin J, Candès E J. NESTA: A fast and accurate first-order method for sparse recovery[J]. SIAM Journal on Imaging Sciences, 2011, 4(1): 1-39. In section 6.3 ...
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29 views

Total Variation sign

I read a form of total variation like this: $\sum\sqrt{||\nabla f||_2^{2}+^2}$ What does the plus and squred sign mean? Shouldn't it just be $\sum\sqrt{||\nabla f||_2^{2}}$ ?
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Total variation of two distributions

If we are given two distributions, $P_1$ and $P_2$ over the same measure set $(X,E)$, where $X$ is the domain and $E$ is a collection of subsets of $X$. Then total variance between $P_1$ and $P_2$ ...
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123 views

Bounded variation along given sequence of subdivisions

Does there exist a continuous function $f:[0,1]\to\mathbb{R}$, along with a sequence $(\pi_n)_{n\geq 0}$ of subdivisions of $[0,1]$ $$\pi_n\equiv\Big(0=t_0^n<t_1^n<\cdots <t_{p_n}^n=1\Big)$$...
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$|\mu|(E) = \sup\{|\int_E f\,d\mu | ; |f|\leq 1 \}$, total variation of a measure.

The definition of total variation of a complex measure $\mu $, where $ (X, \mathfrak M, \mu ) $ is a measure space is $$ |\mu|(E) := \sup\left\{ \sum_{i=1}^\infty |\mu (E_i)| \right\} \text{ for ...
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On searching conditions, other than the continuity of $f,$ under which $ \lim_{\mu (P) \rightarrow 0} V(f,P) = V_a^b f $ is also valid.

Let $ f [a,b] \rightarrow \mathbb{R}, [a,b] \subset \mathbb{R} $ and a partition $ P = \{ a=t_0 < t_1 < \cdots<t_n = b \} $ of $ [a,b]$. If: $$ V(f,P)=\sum_{i=1}^n |f(t_i) - f(t_{i-1})| $$ ...
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1answer
114 views

When the total variation of a function equals the integral of its gradient

As you probably know, for any function $u\in L^1_{loc} (\Omega)$ the total variation is defined as $$\text{TV}(u,\Omega)= \sup \, \bigg\{ -\int_{\Omega} u\, div \phi \, dx : \phi \in C_c^{\infty} (\...
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The old and modern definitions of total variation are actually equivalent?

According to wikipedia, the total variation of the real-valued function $f$, defined on an interval $[a,b]\subset \mathbb{R}$, is the quantity $$V_b^a=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}\left | f(...
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232 views

Derivative of total variation

Problem: Let $\alpha \in BV[a,b]$ (bounded variation on the interval $[a,b]$) and let $\beta(x) = V_a^x \alpha$ be the total variation of $\alpha(x)$ from $a$ to $x \in [a,b]$. Prove or disprove: If ...
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A reference on Total Variation and its applications in Image Processing

The title is almost self-explanatory; I need a beginners' readable reference (book or article) on total variation and its applications in image processing, know any? Thanks bunches.
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1answer
167 views

If $ \Omega $ is a bounded domain, is a $ BV(\Omega) $ function also $ L^\infty(\Omega) $?

Let $ \Omega $ be a (non-empty) bounded domain. The space of functions of bounded variation is defined by $$ BV(\Omega) = \{ u\in L^1(\Omega) \mid \|u\|_{TV} < \infty \} $$ where $$ |u|_{TV} = \...
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3answers
376 views

Least Squares with Total Variation Regularization - How to Set the Lambda ($ \lambda $) Parameter?

I am trying to use total-variation minimization for an image reconstruction problem. Essentially, I am trying to penalize different in the intensity of the two pixels in the reconstructed image. For ...