Marion REVOLLE

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Algorithmic information theory for automatic classification Marion Revolle, [email protected] Nicolas le Bihan, [email protected] Fran¸cois Cayre, [email protected] Main objective Files : any byte string in a computer (text, music, ..) & Similarity measure in a non-probabilist context : Similarity metric Algorithmic information theory : Kolmogorov complexity % 1.1 Complexity 1.2 GZIP 1.3 Examples GZIP : compression algorithm = DEFLATE + Huffman. Given x a file string of size jxj define on the alphabet Ax A- x = ABCA BCCB CABC of size αx. A- DEFLATE L(x) = 6 Z(-1! 1) A- Simple complexity Dictionary compression based on LZ77 : make ref- x : L(A) L(B) L(C) Z(-3!3) L(C) Z(-6!5) K(x) : Kolmogorov complexity : the length of a erences from the past. ABC ABCC BCABC shortest binary program to compute x. B- y = ABCA BCAB CABC DEFLATE(x) generate two kinds of symbol : L(x) : Lempel-Ziv complexity : the minimal number L(y) = 5 of operations making insert/copy from x's past L(a) : insert the element a in Ax = Literal. y : L(A) L(B) L(C) Z(-3!3) Z(-6!6) to generate x. Z(-i ! j) : paste j elements, i elements before = Refer- ABC ABC ABCABC ence of length j. L(yjx) = 2 yjx : Z(-12!6) Z(-12!6) B- Conditional complexity ABCABC ABCABC K(xjy) : conditional Kolmogorov complexity : the B- Complexity C- z = MNOM NOMN OMNO length of a shortest binary program to compute Number of symbols to compress x with DEFLATE L(z) = 5 x is y is furnished as an auxiliary input. ≈ number of symbols to compress with LZ77 y : L(M) L(N) L(O) Z(-3!3) Z(-6!6) L(xjy) : Ziv-Merhav complexity : the minimal num- = Lempel-Ziv complexity MNO MNO MNOMNO ber of operation making insert/copy from y to L(zjx) = 12 Random byte string : lim L(x) = jxj zjx : L(M) L(N) L(O) L(M) L(N) L(O) L(M) L(N) L(O) ... generate x. jxj!1 logαjxj MNOMNOMNO ... 2.1 Metrics 2.2 Classification A- Vitanyi A.1- Normalized Information Distance The length of a shortest binary program that compute x from y as well as x from y. maxfK(xjy);K(yjx)g NID(x; y) = (1) maxfK(x);K(y)g A.2- Normalized Compression Distance Approximations : • K(x) = C(x) = x's compression size • C(xjy) = C(xy) { C(y) C(xy) − minfC(x);C(y)g NCD(x; y) = (2) maxfC(x);C(y)g B- Our proposals B.1- Normalized Lempel-Ziv Distance Approximation : K(x) = L(x) maxfL(xjy) − 1;L(yjx) − 1g NLD(x; y) = (3) maxfL(x);L(y)g B.2- Salza • explicitude : keep only references which give information. X ex(xjy) = li (4) s2S li>ly0 • weighted complexity : the shorter the references, the greater the complexity. X1 LP (xjy) = (5) li s2S LP (xjy) 2 LP (yjx) 2 S(x; y) = max ly0; lx0 (6) ex(xjy) ex(yjx) Fig. 1: Markov chain classification with a)NCD, b)NLD and c)Salza Fig. 2: Language classification with a)NCD, b)NLD and c)Salza Grenoble Images Parole Signal Automatique UMR CNRS 5216 - Grenoble Campus 38400 Saint Martin d'Hères - FRANCE.

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Lecture 24 1 Kolmogorov Complexity
6.441 Spring 2006 24-1 6.441 Transmission of Information May 18, 2006 Lecture 24 Lecturer: Madhu Sudan Scribe: Chung Chan 1 Kolmogorov complexity Shannon’s notion of compressibility is closely tied to a probability distribution. However, the probability distribution of the source is often unknown to the encoder. Sometimes, we interested in compressibility of specific sequence. e.g. How compressible is the Bible? We have the Lempel-Ziv universal compression algorithm that can compress any string without knowledge of the underlying probability except the assumption that the strings comes from a stochastic source. But is it the best compression possible for each deterministic bit string? This notion of compressibility/complexity of a deterministic bit string has been studied by Solomonoff in 1964, Kolmogorov in 1966, Chaitin in 1967 and Levin. Consider the following n-sequence 0100011011000001010011100101110111 ··· Although it may appear random, it is the enumeration of all binary strings. A natural way to compress it is to represent it by the procedure that generates it: enumerate all strings in binary, and stop at the n-th bit. The compression achieved in bits is |compression length| ≤ 2 log n + O(1) More generally, with the universal Turing machine, we can encode data to a computer program that generates it. The length of the smallest program that produces the bit string x is called the Kolmogorov complexity of x. Definition 1 (Kolmogorov complexity) For every language L, the Kolmogorov complexity of the bit string x with respect to L is KL (x) = min l(p) p:L(p)=x where p is a program represented as a bit string, L(p) is the output of the program with respect to the language L, and l(p) is the length of the program, or more precisely, the point at which the execution halts.
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