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Questions tagged [it.information-theory]

Questions in Information Theory

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78 views

Is it there any cellular automata model of the universe where uncomputable things could happen?

There are several papers that propose that the universe itself is analogous to a cellular automata. Obviously, these papers assume that the universe is computable. But is it there any cosmological ...
6
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1answer
204 views

Is a binary sequence computable iff the Kolmogorov complexity of its initial segments is bounded?

Disclaimer: I am mostly unfamiliar with theoretical computer science, making it hard for me to navigate literature in the field. I ask the following out of curiosity. Background/Motivation: Coming ...
5
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1answer
150 views

Minimal information needed for determine some function

From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of ...
10
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1answer
229 views

Is algorithmic information theory still evolving?

I am currently looking for a subject for a thesis and encountered the field of algorithmic information theory. The field seems very interesting for me, but it seems everything is the field was done ...
3
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2answers
171 views

How can AIC converge in the limit when even 2 parameter models can have infinite VC dimension?

AIC-based model-selection converges to zero error in the limit, and also has finite-sample convergence that is rate-optimal with respect to worst case minimax error [1]. (Note that AIC refers to ...
1
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0answers
45 views

Is there a theoretical guarantee that an autoencoder $g$ has $I(x;g(x)) \approx H(x)$?

I know that in general, a function $g$ can be a good auto-encoder (i.e., $g(x) \approx x$ for $x \sim D$) and on the same time $I(g(x);x)$ is small. This is the case when $g$ forms a good correlation ...
0
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0answers
110 views

Specific probability bound on minimal Kolmogorov sufficient statistic?

Using the notation from this question and defining the additional notation: $K^*(B) = \ell(P^*_B)$ is there some universal Turing machine such that $\forall c \exists n,k\Pr\{K^*(B) \geq c+n\} \geq ...
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1answer
74 views

Reducing disjoint or indexing or inner-product problem to s-t connectivity problem in directed graph

I am asked to prove that an O(1)-pass randomized streaming algorithm that solves s-t connectivity problem in a simple directed graph $G=(V,E)$ with $|V|=n$ vertices, with sucess possibility $>\frac{...
6
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1answer
228 views

Algorithmic mutual information between random string and minimal Kolmogorov sufficient statistic

Regarding notation in the following, the function $\ell(B)$ returns the length of bitstring $B$, and the cardinality of set $S$ is denoted by $|S|$. A bitstring $B$ is generated by drawing 0s and 1s ...
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1answer
37 views

Relationship between worst case length of transcript and entropy of transcript

Consider the two party model of communication complexity where Alice and Bob are given inputs $X$ and $Y$ sampled from some distribution $\mu$, and their goal is to solve some problem $P$ (the details ...
4
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1answer
2k views

Improving Bloom filter - can we distinguish elements of a database using less than 2.33275 bits/element?

While we usually use large e.g. 64 bit hashes, there are many techniques to reduce this size, e.g. for savings in storage and transmission. Popular Bloom filter instead of marking just 1 hash ...
12
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1answer
1k views

Entropy and computational complexity

There are researcher showing that erasing bit has to consume energy, now is there any research done on the average consumption of energy of algorithm with computational complexity $F(n)$? I guess, ...
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0answers
65 views

Using idea of entropy (maybe Shannon entropy or other continuous entropy) to the topic of functional analysis

I am an electrical engineer without a detailed background in theoretical computer science. I am posting here since I hypothesize that the concept of entropy or other branches of information theory (as ...
4
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1answer
63 views

Can entropicly secure encryption algorithms be used on low-entropy messages by adding noise

There exist information-theoretic notions of security like Shannon's "perfect security" that one-time pads exhibit. All methods which achieve perfect security will require long keys, however. If we ...
4
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1answer
223 views

The Maxwell's Demon and Computer Science

What is the best source -in terms of quality- that would explain the argument that uses computations concepts to demonstrate that the Maxwell's Demon does not break the second law of thermodynamics? I ...
0
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1answer
78 views

Error correction that adapts to different error rates

Say we have $N$ bits that we'd like to store in an $M$-bit error correcting code, where $M > N$. Given $\epsilon > 0$, as long as $N > N_0(\epsilon)$ we can recover the original bits as long ...
4
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1answer
482 views

Autoencoders and information compression

Disclaimer: I know very (very) little about deep nets, besides what an introductory course on machine learning would teach on neural networks, and skimming some paper abstracts and introductions. If ...
1
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0answers
31 views

Problem dependent lower bound for stochastic bandits with full information

Suppose you have a $K$ armed stochastic bandit problem but with full information. There are $K$ arms with mean rewards $\mu_1,...,\mu_K$. At each step we have to select an arm, collect the reward from ...
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0answers
46 views

Parametrically-relaxed Kolmogorov complexity

Consider the following problem: Input: An integer $n$ and a subset $S \subseteq \{0...n-1\}$ in some representation. Output: The encoding of some kind of automaton (say, a Turing machine) which ...
4
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1answer
208 views

Relation between variance and mutual information

Given two discrete random variables $X,Y$ such that $X,Y \in \mathbb{R}$ and $0 \leq X,Y \leq 1$, is it true that $$|\text{Cov}[X,Y] \leq \sqrt{\frac{1}{2} \text{I}[X,Y]}|. $$ This bound may be ...
1
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0answers
111 views

Connection between diamond norm and output purity norm

Setting of the problem: Given a quantum channel $\mathcal{E}: \mathcal{H}_A\rightarrow \mathcal{H}_B$ (where $\mathcal{H}$ refers to a Hilbert space and subscript refers to the quantum register ...
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1answer
96 views

Does a code need at least two symbols to be defined as a code? [closed]

I am wondering whether you could still call a code something that, if transmitting, only transmits one symbol. Or does the formal definition of code require 2 or more symbols? (and would the answer ...
2
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1answer
230 views

Entropy of a byte in a compression algorithm?

I have a (fixed, long) string of bytes that I want to compress, $C$. I use a typical (good) lossless compression algorithm on it, to generate a compressed string of bytes, $C^*$. Then I define a ...
1
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0answers
111 views

Information theoretic lower-bound on object graph serialization

This might be a daft quesstion, but here comes. I became intriqued about data serialization formats and tried to look for research on what could be the information theoric lower bound on encoding ...
1
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1answer
51 views

Families of LDPC codes with constant error fraction corrected

I am looking for families of error-correcting LDPC codes with a constant error fraction corrected by a decoding algorithm. For example, I know that Sipser and Spielman proved that there is an ...
4
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1answer
207 views

Word length using entropy : Maximum entropy criteria

The question is based on research paper titled, Markovian language model of the DNA and its information content In the supplementary document, the Authors show how they determine the word length of ...
0
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0answers
198 views

Submodularity of KL divergence

Entropy on a set of random variables is known to be submodular. Now given a set of random variables $P_1, P_2, P_3, \dots, P_N$ and $Q_1, Q_2, Q_3, \dots, Q_N$, where every $P_i$ is a true ...
6
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1answer
1k views

What are some standard books/papers on Information Theory?

I have started Information Theory classes just recently and was wondering what would be a standard book to purchase. I know I can go for basic introductory books but I also like to purchase standard ...
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1answer
90 views

Lower bound on the number of objects in the universe [closed]

From Cover & Thomas' Elements of Information Theory: Player A chooses some object in the universe, and player B attempts to identify the object with a series of yes–no questions. Suppose ...
3
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1answer
162 views

The average number of compressible strings in a random set of random strings

In the book Elements of Information Theory (p.446), it is stated: ...although there are some simple sequences, most sequences do not have simple descriptions. Similarly, most integers are not simple. ...
4
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0answers
95 views

Why primitive rotation is $53.13^\circ$ in the quantum Turing machine used by Vitanyi for Quantum Kolmogrov Complexity?

Right now I am going through Quantum Kolmogorov Complexity Based on Classical Descriptions by Vitanyi. In the introduction, the author assumed the primitive rotation $\theta = 53.13^\circ$ to have ...
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1answer
166 views

Why don't we transmit at rates higher than the Shannon capacity if we are going to get a nonzero probability of error anyways ?

Shannon capacity $C$ is the upper limit on a rate $R$ defined as the number of information symbols $k$ divided by the number of transmitted symbols $n$, that can be transmitted over a channel such ...
4
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2answers
497 views

Relation between group theory and information theory

Motivation: I am interested about the application of group theory to information theory. To be precise, I am interested in data compression (source coding theory). Question: Is there any paper/survey ...
4
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0answers
61 views

What is the shortest description of a universal computational structure that includes a meta-circular evaluator?

I am wondering whether there is a minimal (or the shortest known) way of specifying a universal computational structure that includes a specification of a meta-circular evaluator within that structure....
6
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1answer
175 views

Lower bound on prefix code lengths

For a prefix code $C:\{0,1\}^*\to\{0,1\}^*$, define $f(n)$ as the length of the longest encoding of a number with up to $n$ bits: $$ f(n)=\max_{|k|\le n}\left|C(k)\right|. $$ (Note that by taking ...
8
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3answers
217 views

Showing that interval-sum queries on a binary array can not be done using linear space and constant time

You are given a $n$-sized binary array. I want to show that no algorithm can do the following (or to be surprised and find out that such algorithms exist after all): 1) Pre-process the input array ...
1
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1answer
232 views

Does Huffman coding always produce shorter codes than the Shannon code?

Let $X\in\{1,2,\ldots,m\}$ be a discrete random variable with $X\sim p$. Let $C$ be a code for $X$ with $l_i$ being the length $i$-th codeword and let $L(C)$ be the expected length of the code. The ...
2
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2answers
324 views

What is empirical mutual information?

The topic says it all - I've been seeing this referenced a few times in information theory literature (Feedback in the non-asymptotic regime, Y. Polyanskiy et. al. among others), oftentimes making ...
1
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0answers
99 views

Maximal correlation vs correlation coefficient when one RV is Gaussian

Last week I asked a question on MOF (see here), but I got no reply. So I am asking my question here. Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have ...
3
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0answers
62 views

Estimate smooth vector, from dot-product queries

I have a secret $n$-dimensional vector $\mathbb{s} \in \mathbb{Z}^n$. I don't know $\mathbb{s}$; my goal is to estimate $\mathbb{s}$. I do have an oracle for the function $f_\mathbb{s} : \mathbb{Z}^...
5
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1answer
291 views

Turbo codes and message passing

I'm self-studying turbo codes for a graduate course in coding theory. I understood how turbo codes works by directly reading Berrou' paper and some of the following works on this topic. Given that, ...
0
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1answer
102 views

Kraft Macmillan inequality explanation [closed]

I am going through some questions and answers regarding Information Theory and I found this question and its solution. Can some one explain this solution to me. We would like to encode a sequence of ...
3
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0answers
111 views

What is the problem of finding a largest subset of smallest Kolmogorov complexity?

What do you call the problem of finding a largest possible subset of strings with smallest possible information content? I'm studying a particular instantiation of this problem in a different setting ...
1
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2answers
195 views

Information-theoretic Diffie-Hellman

The following non-standard description of Diffie-Hellman is entirely my own, by which I mean that I came up with it having not read about it anywhere else beforehand. In Diffie-Hellman Alice and Bob ...
3
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1answer
179 views

Is joint Kolmogorov Complexity order invariant?

Due to the symmetry of information, it follows up to an additive constant that K(X,Y) = K(Y,X) Does this hold for more than two data objects as well?
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112 views

“Elements of Information Theory”: Some (basic) help needed here

I was following the textbook by Cover & Thomas (2006): Elements of Information Theory. (hyperlink is not owned by me) I have one question that has been irking for me some time. It is regarding ...
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0answers
123 views

Initialization of errata evaluator polynomial for simultaneous computation in Berlekamp-Massey for Reed-Solomon

This is a continuation of this post on SO. I am trying to implement an errata (errors-and-erasures) decoder for Reed-Solomon. My current approach is to use Berlekamp-Massey (because it's the most ...
8
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1answer
474 views

Information theory and convex optimization

I'm taking a graduate level course in information theory and I'm constantly struck by how much convex optimization there is in this subject. However, the proofs seem to shy away from using the full ...
14
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5answers
893 views

The utility of Renyi entropies?

Most of us are familiar with — or at least have heard of — the Shannon entropy of a random variable, $H(X) = -\mathbb{E} \bigl[ \log p(X)\bigr]$, and all the related information-theoretic ...
15
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1answer
373 views

Information complexity of query algorithms?

Information complexity has been a very useful tool in communication complexity, mainly used to lower bound the communication complexity of distributed problems. Is there an analogue of information ...