Questions tagged [it.information-theory]
Questions in Information Theory
163
questions
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1answer
103 views
Explicit Bits-back Coding (a.k.a. Free Energy Coding) applied to Gaussian mixtures
I've been trying to understand Bits-back coding (Frey, B. J., and G. E. Hinton. 1997.) a bit more (pun intended), which can be used to encode data with latent variable models. This tutorial by Pieter ...
1
vote
1answer
94 views
Data processing inequality for interaction information
The interaction information is defined as $I(X;Y)-I(X;Y|Z)$. Let $Z-(X, Y) -(X', Y')$ be a Markov chain. Is there an inequality similar to the data processing inequality, relating $I(X';Y')-I(X';Y'|Z)...
2
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1answer
85 views
Why Asymptotic Equipartition Property theorem proofs assume the source is memoryless?
I do not understand the assumption $X_1, X_2, \cdots$ are i.i.d. ~p(x) in the AEP proofs I have seen. I have read some different sources for understanding the Asymptotic Equipartition Property. Using ...
5
votes
1answer
138 views
Kolmogorov Complexity of a Decidable Language
The Kolmogorov Complexity (KC) of a string $y$ is the size of the smallest program $f$ and input $x$ that: $y = f(x)$. Let's define a variation of Kolmogorov's complexity$^1$. Suppose a decidable ...
-1
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1answer
48 views
Notation in proof for Asymptotic Equipartition Property
In the following lecture notes chapter 3, page 12-13, they state the following
We begin by introducting some important notation:
- For a set $\mathcal{S},|\mathcal{S}|$ denotes its cardinality (...
0
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1answer
90 views
Volume of elements mapped to the same codeword is $2^{H(X|\hat{X})}$
In this paper by Tishby, Pereira and Bialek they mention on page 4 in the Relevant quantization chapter the setting is the following; Given some signal space $X \sim p(x)$ and a quantized codebook $\...
0
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1answer
94 views
Notation of sequences in rate distortion theory
I have been reading whatever sources I could get my hands on today, regarding this problem.
Most notes online about rate distortion theory come from the book Elements of Information Theory by Thomas ...
7
votes
1answer
186 views
Expected vs worst-case communication complexity
In the set disjointness problem of 2-party communication complexity, Alice and Bob are both given an $n$-bit string as input; denoted by $X$ for Alice's input, and $Y$ for Bob's input. They need to ...
4
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0answers
76 views
Strong data-processing inequality: bound $TV(T_{\#}P_0,T_{\#}P_1)$ if $\|T(x)-x\|_\infty \le \varepsilon;\forall x \in \mathbb R^p$
Disclaimer. I've moved this question from MO hoping that here is the right venue. Also, this is my first post on this channel, so please have some patience.
So, Iet $X = (X,d)$ be a Polish space, ...
2
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0answers
97 views
Representing data with Shannon entropy predicted bits
Let us assume a file based on a character set where each character has equal probability of occurance. This will result in the maximum entropy for that character set. On calculating the entropy, let ...
4
votes
1answer
380 views
Maximization of Mutual Information
Let $X\in\{0,1\}^d$ be a Boolean vector and $Y, Z\in\{0,1\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y, Z$ and we'd like to find a joint distribution $\...
1
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0answers
131 views
Minimum number of hours of speech needed to train a neural net to recognize speech [closed]
From a theoretical computer science point of view, is there a lower limit on the number of hours of speech needed to train a neural net to translate speech to text? An estimate from CMU is 3000-5000 ...
1
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1answer
315 views
Chain rule for KL divergence
Is there an inequality to relate the KL divergence of two joint distribution and the sum of the KL divergence of their marginals? Or in particular, is there a proof or a counter example for the ...
2
votes
1answer
129 views
Why not include private randomness in internal communication information definition?
I am using https://www.cs.toronto.edu/~toni/Courses/CommComplexity2014/Lectures/lecture12.pdf as a reference.
This isn't exactly a research question but I can't find a good place to ask it.
Suppose ...
4
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0answers
103 views
Expected vs actual amount of information leaked by an $l$-bits message
Say we have a random variable $X$ that contains $k$ bits of information, and a message $M = f(X)$ ($M$ is deterministic given $X$) that is $l$ bits long, where $l<k$. This implies $H(X) = k$ and $...
1
vote
1answer
124 views
Uniqueness of the distribution maximizing the channel capacity
Setting: We look at a discrete memoryless channel which takes an input probability distribution acting over symbols in $\mathcal{X}$ to an output probability distribution over symbols in $\mathcal{Y}$....
6
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1answer
307 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
votes
1answer
177 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
318 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
votes
2answers
216 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
58 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 ...
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1answer
148 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
votes
1answer
242 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 ...
-1
votes
1answer
70 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 ...
6
votes
2answers
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 ...
13
votes
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, ...
4
votes
1answer
97 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
270 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
votes
1answer
80 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
votes
1answer
655 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
51 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 ...
1
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0answers
56 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
votes
1answer
480 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
128 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 ...
-1
votes
1answer
100 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
votes
1answer
257 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
131 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
vote
1answer
58 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
votes
1answer
282 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 ...
9
votes
1answer
6k 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 ...
-2
votes
1answer
116 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
votes
1answer
193 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
votes
0answers
99 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 ...
-2
votes
1answer
172 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 ...
5
votes
2answers
756 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 ...
3
votes
0answers
65 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....
5
votes
1answer
201 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
votes
3answers
234 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
vote
1answer
307 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 ...
3
votes
2answers
651 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 ...
