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      Questions tagged [it.information-theory]

      Theoretical and experimental aspects of information theory and coding theory. This tag covers but is not limited to following branches: information theory, information geometry, optimal transportation theory, coding theory.

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      183 views

      What is the status of the Born Rule in axiomatic QM?

      While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
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      36 views

      Entropy density equals limit of conditional entropy

      Consider a probability distribution of m-length sequences $P_m(x_1,x_2......,x_m)$ where $x_i\in\{1,-1\}$. We then have that $h=\lim_{m\to \infty}\frac{H_m}{m}$ where $H_m$ is defined as $$H_m :\, =\...
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      53 views

      Convexity of conditional relative entropy for Markov distributions

      Consider two Markov processes $p$ and $q$. The conditional relative entropy between them is \begin{align} D(p\parallel q)& =\sum_a p(a)\sum_b p(b\mid a)\log\frac{p(b\mid a)}{q(b\mid a)}\\ & =\...
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      1answer
      43 views

      Entropy of distribution with block matrix support

      Let $P(X_1,X_2)$ be a discrete bivariate distribution that has the form shown in the figure below, i.e. its support can be split into blocks that do not overlap on either dimensions. Let's build $P'(...
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      1answer
      69 views

      Relationship between $\alpha$-divergences?

      I am working with $\alpha$-divergences and was wondering how understand the relationship between the definitions of Renyi and Amari? Renyi: $D_{\alpha}[p||q] = \frac{1}{\alpha - 1} \log \int p^{\...
      4
      votes
      1answer
      84 views

      Convexity of exponential family

      It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the ...
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      147 views

      Non-negative interaction information for special trivariate case

      Consider a discrete trivariate distribution $P(X_1, X_2, Y)$, which satisfies $$ p(x_1, x_2, y) = \min( p(x_1,y), p(x_2,y) ), $$ for all $x_1$ and $x_2$ for which $p(x_1, x_2) > 0$ and for all ...
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      1k views

      Is there a Kolmogorov complexity proof of the prime number theorem?

      Lance Fortnow uses Kolmorogov complexity to prove an Almost Prime Number Theorem (https://lance.fortnow.com/papers/files/kaikoura.pdf, after theorem $2.1$): the $i$th prime is at most $i(\log i)^2$. ...
      2
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      1answer
      97 views

      Mutual information inequality

      I am trying to prove three inequalities that would help me solve the proof of a larger theorem. Let $P(X,Y)$ be a discrete bivariate distribution and $$ I(X;Y) = \sum_{i,j} p(x_i, y_j) \log \frac{p(...
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      47 views

      Information theoretic lower bounds for sparse recovery

      For the well-known problem of sparse recovery using $\ell_1$ minimization, it was shown in this paper that for any random measurement matrix, a recovery procedure that succeeds with constant ...
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      51 views

      How to derive formula (10) norm to obtain formula (11) in Uncorrelated Group LASSO?

      In Uncorrelated Group LASSO, Eq. (10) and Eq. (11) are as follows: $J_2(W)=f(W)+\alpha Tr(W^TFW)$. (10) $F_{ii}=\sum_{g}\frac{(I_{G_{g}})_i||W_{G_g}||_{2,1}}{||W_{G_g}^i||_2}$. (11) where $w_{...
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      0answers
      76 views

      Shannon-McMillan-Breiman theorem for expander graphs: rate of convergence?

      Is the following uniform SMB theorem for random walks on expander graphs true? For simplicity, I will state it for a finite group $G=\langle S \rangle$ and a uniform probability measure $\mu$ on the ...
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      0answers
      166 views

      On the difference of conditional differential entropy of two correlated random variables

      Problem Definition Let $\mathbf{G}$ and $\mathbf{S}$ be jointly distributed random variables where $\mathbf{S}$ is continuous and is related to $\mathbf{G}$ through a conditional pdf $f(s|g)$ defined ...
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      74 views

      Binary search extension for determining a hyperplane splitting a set of points in $\mathbb{R}^d$

      We are given a set $S$ of $n$ points in $\mathbb{R}^d$ and a (hidden) vector $\mathbf{w}\in\mathbb{R}^d$, where each point $\mathbf{v}\in S$ is associated with a binary label equal to the sign of $\...
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      1answer
      201 views

      Conceptual explanation for the appearance of entropy in $\frac{d}{dp}\|x\|_p$

      For $x\in \mathbb{R}^d$, an elementary computation yields that $$\frac{d}{dp}\log \|x\|_p =\frac{1}{p^2}\sum_{i=1}^d \frac{|x_i|^p}{\|x\|_p^p}\log \frac{|x_i|^p}{\|x\|_p^p}=-\frac{1}{p^2}\operatorname{...

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