Covers theoretical and experimental aspects of information theory and coding.

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### How do you use the bits you get back from Bits Back Coding?

Bits Back coding is a scheme to transmit an observation x.
You can read about it here [1]. To my understanding, it works like this:
The encoder samples a message z from a distribution Q(z|x) that it ...

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29 views

### Information theoretic common sequence agreement (not secret key)

Supposing Alice and Bob share $\rho$-correlated sequences in $\{0,1\}^n$, what coding theory based schemes are available for Alice and Bod to extract sequences $A,B\in\{0,1\}^n$ respectively such that ...

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84 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
...

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42 views

### Rate-Distortion theory: What is the distribution of distortion on an optimal encoder?

If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, then decode it to create an estimate $\hat{X}$, rate-distortion theory tells us that the lowest mean-squared-error ...

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329 views

### Expected centered entropy of the binomial distribution

In short, the function I am interested in is the following:
$$I_n(p) = \sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} \left[h(p) - h\left(\tfrac{k}{n}\right)\right],$$
where $h(x) \triangleq -x \log x - ...

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34 views

### Channels of Practical Interest

Are there channels of practical interest whose capacity achieving probability distributions belong to a parametric family of probability measures?
Specifically, suppose $ \theta= (\theta_1 \theta_2 ...

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268 views

### Good covering of a sphere

Consider a sphere $S_r(0)$ with center at zero and radius $r$ in the Hamming space $\{0,1\}^n$.
We will be interested in covering this sphere with balls of radius $\rho < r$.
We know that there ...

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47 views

### Upper bound for chi-square divergence in terms of KL divergence

In my research I need an upper bound for chi-square divergence in terms KL divergence which works for general alphabets. To make this precise, note that for two probability measures $P$ and $Q$ ...

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217 views

### Coupon Collector Problem for Non-Uniform Coupons: Bound on the number of missed Coupons

Suppose $\mathcal B=\{1,2,..,b\}$ is the set of all possible coupons, with $\mathbf p = ( p_1,p_2,...,p_b)$ assigning the probability of occurrence for all coupons in $\mathcal B$.
The "traditional ...

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75 views

### On Rényi entropy/divergence

The Rényi entropy for a probability density function $f$ with dominating measure $\mu$ of order $\alpha>0$ is defined as
$$H_\alpha(f)={1 \over {\alpha-1}}\log\int f^\alpha d\mu.$$
If $f$ is ...

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40 views

### minimal entropy approximation of a truncated discrete measure

Consider a measure $\mu$ on $\mathbb{N}$ given by the sequence $(\mu(n))_{n \geq 0}$ with $\mu(0)>0$. For example $\mu(n)=n^2+1$ on the figure below.
For each $n$, let $X_n \sim \mu(\cdot \mid ...

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**1**answer

80 views

### Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...

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114 views

### What is the sum capacity of a scalar gaussian broadcast channel?

"On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing):
A transmitter with ...

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50 views

### minimal entropy approximation of a discrete random variable

Let $X$ be a $\mathbb{N}$-valued random variable. Define
$$
H^\epsilon_n(X) = \inf_f H(f(X))
$$
where $f$ runs over all functions $\mathbb{N} \to \mathbb{N}$ such that $\Pr(f(X)\neq X)<\epsilon$ ...

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63 views

### Continuous self-information

Let $I(X,Y)$ be the mutual information between two continuous random variables $X$ and $Y$.
We have $I(X,Y) = H(X)-H(X|Y)$, and setting $X=Y$ leads to $I(X,X) = H(X)-H(X|X)$. If $X$ was discrete, ...

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78 views

### On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote
$$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$
collection of pairs with Hamming distance $1$ from $[uv]$ string ...

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102 views

### Convex Conjugate of Relative Entropy

The convex conjugate of a function, say, $f:X\mapsto \mathbb{R}$ is a function $f^*:X^*\mapsto \mathbb{R}$ defined as
$$f^*(x^*):=\sup_{x\in X} ~\langle x, x^*\rangle-f(x),$$ where $X^*$ is the ...

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141 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by ...

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32 views

### Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then ...

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191 views

### Higher moments of information and Renyi entropy

For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: What is entropy, really?, What is the role of the logarithm ...

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38 views

### Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...

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141 views

### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...

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97 views

### Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and ...

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134 views

### On the inverse problem of Dobrushin

Dobrushin, in this paper, looked into the following problem. Suppose We are given a Markov kernel (conditional distribution) $P_{Y|X}$. Information theorist usually call $W$ a channel. It is known ...

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115 views

### Form of the Shannon capacity for Heptagon?

Is the $0$-error capacity of $7$-cycle:
$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?

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96 views

### Coding for channels with concentrated error

Can we implement a reduced-error transmission over a channel with error frequency having $\liminf<\frac{1}{2}$ and $\limsup>\frac{1}{2}$?
We know that if a channel with error flips (in the ...

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381 views

### Asking for an English version of a paper

I have been looking for the paper
"almost independence and secrecy capacity" by Csiszar. But all I could find was a Russian version published in Problems of Information Transmission.
I am wondering ...

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95 views

### Noise reduction in capacity-0 channels

Suppose we have a binary symmetric channel with $p=\frac{1}{3}$; that is, a communications channel in which each bit is flipped with independent probability $\frac{1}{3}$. I know that there is a code ...

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160 views

### Total conditional complexity

By $C(|)$ denote conditional complexity.
By $CT(|)$ denote total conditional complexity.
For every n there exist two strings $x$ and $y$ of length $n$ such that $C(x|y) = O(1)$
but $CT(x|y) \ge n $.
...

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98 views

### Entropy on a draw from a random distribution.

Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate
...

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104 views

### Applications of list decoding

This is citation from http://en.wikipedia.org/wiki/List_decoding:
Algorithms developed for list decoding of several interesting code families have found interesting applications in computational ...

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80 views

### $q$-deformations of fundamental equation of information and entropies

Classical information theory: fundamental equation of information
In classical information theory, the information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$ of a ...

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73 views

### Assumption of equal prior message probabilities in the standard proofs of the converse of Shannon's theorem

One of the first steps in the standard proofs for the (weak) converse of the Shannon's theorem (a.k.a. noisy-channel coding theorem) for the discrete memoryless sources is the assumption that messages ...

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339 views

### decomposition of Hilbert space into tensor product $L^2([0,\tfrac{1}{2}]) \otimes L^2([\tfrac{1}{2},1]) \simeq L^2([0,1])$

The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$.
As an example, is it ...

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100 views

### Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution.
The mutual information between the input and the output of ...

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728 views

### A conjecture about the entropy of matrix vector products

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...

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88 views

### Invariance of mutual information

Let $I(X,Y):=H(X)+H(Y)-H(X,Y)$ be the mutual information of the joint probability distribution $p_{XY}$ (here $H(\cdot)$ is the Shannon entropy of its argument). I know that the mutual information is ...

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133 views

### Is this graph of reciprocal power means always convex?

Let
$$
p = (p_1, \ldots, p_n)
$$
be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$.
Is the function
...

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105 views

### Lower convex envelope of a function involving entropy

Suppose two discrete random variables $X$ and $Y$ defined on finite sets $\mathcal{X}$ and $\mathcal{Y}$ are given and also suppose the conditional distribution $P_{Y|X}$ (i.e, channel) is fixed. We ...

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### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...

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### Variance of continuous stochastic process

In the paper "Directed Information, Causal Estimation, and Communication in Continuous Time" the author show an example of continuous Gaussian Channel:
Let $\{B_t\}$ be a standard Brownian motion and ...

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144 views

### Another formulation of error-correcting coding problem

Consider classic error-correcting problem:
there is finite set $A$ and string $a_1...a_n$, $a_i \in A$ in the begin.
in the end we have $b_1...b_n$. Set places of errors $E = \{i| a_i\not= b_i \}$, ...

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108 views

### Will relative entropy increase with majorization?

Suppose that three probability distributions have the relation $P\succ Q\succ R$, then do the following relations between their relative entropy valid? (assume that $p_1\geq p_2\geq\cdots \geq p_n$ ...

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261 views

### Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$.
Is it true that:
If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...

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422 views

### Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$,
i.e., each $p=(p(x_0),\dots,p(x_n))\in ...

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124 views

### Why private randomization does not help the Shannon's source coding

I am wondering why stochastic encoder and decoder can not help the Shannon source coding? I know the achievability scheme of source coding, which is based on typicality, is deterministic, and hence we ...

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347 views

### Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...

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135 views

### How to estimate the entropy of a distribution on a power set?

Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum_{x\in X} p(x)\log p(x)$.
Given a sample of observations $x_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x_i=x\}}{N}$ and ...

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79 views

### Connection between inf-entropy rate and min-entropy

I am reading the paper "Generating random bits from an arbitrary source: fundamental limits" by Vembu and Verdu. This paper is written in the language of information theory, however, I need to ...

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114 views

### Is there an universal (dis)similarity measure between two structures?

I'm always wondering is there an universal (dis)similarity measure
between two structures (let's say between two undirected simple
graphs)? I mean, not "the measure with universal parameter that we
...