The basic condition under which the ideal principle should be. On data mining, compression, and kolmogorov complexity article in data mining and knowledge discovery 151. It is hoped that this approach could inform novel cloudbased data compression methods that more. Notably the kolmogorov complexity of a data object is invariant to how we encode it into a string. Kolmogorov complexity an overview sciencedirect topics. In chapter 2, we described the notion of kolmogorov complexity, noting that the kolmogorov complexity of a sequence was not. Informally, cx measures the information content, degree of redundancy, degree of structure, of x neil conway cisc 876. While data compression and kolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. It is hoped that this approach could inform novel cloudbased data compression. Any better understanding of the data can therefore be used to find struc ture in the data, and consequently to compress it better. Causal inference by compression exploratory data analysis.
Quantum data compression entropy kolmogorov complexity what is information. Therefore, data mining will always be an art, where our goal will be to find better models patterns that fit our datasets as best as possible. Yes, it should be the case with a practical compression algorithm, and is theoretically possible with kolmogorov complexity. Empirical results indicate that grammarbased data compression is competitive with other techniques in practice 8, 12. Let lm be the complexity required to describe the model itself. Kolmogorov s theory, and which lies at the basis of most practical data compression methods. The size of the smallest contextfree grammar generating a given string is a natural,but more tractable variant of kolmogorov complexity. Is there a limit to data compression without mentioning. All structured data from the file and property namespaces is available under the creative commons cc0 license. The smallest grammar is known to be hard to approxi.
While data compression and kolmogorov complexity are both about e ective coding of words, the two settings di er in the following respect. Khalid sayood, in introduction to data compression fourth edition, 2012. Although all are 24bit binary strings and therefore equally likely to represent the result of 24 ips of a. The relationship between the bayesian approach and the minimum description length approach is established. More generally, with the universal turing machine, we can encode data to a computer program that. Kolmogorov complexity wikimili, the best wikipedia reader. On data mining, compression, and kolmogorov complexity 7 attribute, the class label e. Files are available under licenses specified on their description page. Shannon entropy versus kolmogorov complexity, the relation of both to.
Course notes for data compression kolmogorov complexity archived 20090909 at the wayback machine, 2005, p. Our algorithm for approximating the smallest grammar employs a variant of the lz77 compression scheme 15. Information theory has applications in many fields. The kolmogorov complexity of a string is the length in bits of the shortest program that produces that string. Pdf minimum description length induction, bayesianism, and. An intuitive reason for the former fact is that to nd the kolmogorov complexity of x, we should run all possible programs in parallel, and. To this end we propose an inference framework based on solid information theoretic foundations, i. Causal inference via conditional kolmogorov complexity using. Kolmogorov complexity, return, efficiency, compression. T he kolmogorov theory of complexity kolmogorov l leads to the notion of a universal minimal sufficient statistic for the optimal compression of data as discussed in v yugin 2, cover 3, 4, and cover, gacs, and gray 5. Unlike kolmogorov complexity, information theory is only concerned with the average information of a random source.
Simply put, we propose causal inference by compression. On data mining, compression, and kolmogorov complexity core. Megalooikonomou data mining and knowledge discovery, 2007 problems. In general, we show that data compression is almost always the best strategy, both in model selection and prediction. Adding the complexity approximation of each array in the partition matrix of a spacetime. Pdf reconciling data compression and kolmogorov complexity.
The general idea is that any regularity, or structure, detected in the data can be used to compress it. A data object can be a random variable, sequence of events, a temporal graph, etc. If decidable, we could design an algorithm that prints the shortest incompressible string of length n but such a string could then be succinctly described. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of kolmogorov complexity a word has many di erent. A string x is random if it cannot be compressed, i. Kolmogorov complexity formalizes this intuitive notion of complexity. Reconciling data compression and kolmogorov complexity core. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Kolmogorov complexity which is based on the knowledge of the length of x, denoted as lx. Kolmogorov complexity, data compression, and inference. This inequality gives the fundamental relationship between probability density functions and pre. Here too there is a tradeoff between complexity and inference. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of kolmogorov complexity a word. If we restrict the model class to finite sets then application of the ideal principle turns into kolmogorov s minimal sufficient statistic.
Compression based data mining of sequential data 2. Estimating the algorithmic complexity of stock markets. We look at instance complexity that allows us to look at the complexity of a string in relation to a set and how it compares to timebounded traditional and distinguishing kolmogorov complexity. The elementary theories of shannon information and kolmogorov complexity are cmpared, the extent to which they have a common purpose, and where they are fundamentally di. When the source distribution is not known, kolmogorov complexity estimation and analysis. The kolmogorov complexity of an object is a form of absolute information of the individual object. Im just asking is there or could there possibly be a limit to data compression without mr. This idea, put forth by kolmogorov, chaitin, and solomonoff, says that the intrinsic complexity of a sequence is the length of its shortest description. The kolmogorov complexity of a string x is the length of the smallest program that outputs x, relative to some model of computation.
Kolmogorov complexity and computational complexity. Kieffer and yang show that a good solution to the grammar problem leads to a good universal compression algorithm for. Beyond kolmogorov complexity two of the above metrics, ncd and clm, were. Even worse, it is never possible to prove that the kolmogorov complexity of an object is large. In the last few years, the internet use is growing and web pages are being populated by multimedia objects. Although all are 24bit binary strings and therefore equally likely to represent the result of 24 ips of a fair coin, there. Compress is undecidable can an algorithm perform optimal compression. Instead, each compression algorithm operates within a concrete feature space, and compressionbased measures calculate similarity between vectors in that space. Algorithmic information theory and kolmogorov complexity. In this work, we focus on causal inference for binary data.
Causal inference by compression max planck society. Bounds on practical compression and kolmogorov complexity. Short strings, however, are di cult to compress in practice, and the theory does not provide a satisfactory. The kolmogorov complexity of a data set was the length of the shortest program that generates this data on a computer. Video coding quantum data compression entropy kolmogorov complexity information representation show example from jpeg webpage choosing how to. Although the entropy of such a sequence is not defined, there is a notion of its intrinsic descriptive complexity. To quantify information one needs to look at the data compression and transmission rate. The best model minimizes ld, m, resulting in optimal compression.
On the other hand, for kolmogorov complexity, the common approach to evaluating the algorithmic complexity of a string has been by using lossless compression algorithms because the length of lossless compression is an upper bound of kolmogorov complexity. Can data mining be formalized into a logical system like the relational algebra. Im looking for a algorithm that can compute an approximation of the kolmogorov complexity of given input string. In this section we study the relation between k m and approaches to kolmogorov complexity based on compression. Information theory is a branch of mathematics that attempts to quantify information. Malicious url detection based on kolmogorov complexity. The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms pdf. We show that the answer to all these questions is negative, because data mining is closely related to compression and kolmogorov complexity. Choose the simplest hypothesis that is consistent with the data neil conway cisc 876. Keywords data mining compression kolmogorov complexity clustering.
We further consider polynomialtime bounded versions of compression complexity whose results depend on unproven assumptions in computational complexity. Reconciling data compression and kolmogorov complexity 2007. Reconciling data compression and kolmogorov complexity. Malicious url detection based on kolmogorov complexity estimation hsingkuo pao, yanlin chou,yuhjye lee. Katie moenkhaus kolmogorov complexity a major result is that the kolmogorov complexity of a random sequence on average is close to the. According to kolmogorov, the complexity of an object is the length of the shortest computer program that can reproduce. A complexity theoretic approach to the classical data compression problem is presented. Also known as kolmogorov chaitin complexity, descriptional complexity, programsize. The test file enwik8 is a bulk extract from wikipedia, and i dont think that mr kolmogorov can help us here. The kolmogorov theory is applicable to arbitrary, possibly nonrandom, data. Introduction 010101010101010101010101 100111011101011100100110 110100110010110100101100 consider the three strings shown above. It was proposed by kolmogorov in 1965 to quantify the randomness of strings and other objects in an objective and absolute manner. Kolmogorov complexity in natural models moses charikar princeton university. Pdf comparing data compression in webbased animation.
This will involve concatenating strings for which our method establishes a kolmogorov complexity, which then are given to a lossless compression algorithm in order to determine whether it provides consistent estimations, that is, to determine whether strings are less compressible where our method says that they have greater kolmogorov. However, because kolmogorov complexity is uncomputable li and vitanyi, 1997, compression algorithms are employed to approximate an upper bound on kx. In algorithmic information theory a subfield of computer science and mathematics, the kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program in a predetermined programming language that produces the object as output. A compression algorithm or compressor, for short, has.
It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity. Empirical entropy entropy is calculated from the source distribution producing a given string 10. We design and build a virtual one instruction set computer that we use to empirically compute the kolomogorov complexity of short strings within this computing model. The kolmogorov complexity of x, denoted cx, is the length of the shortest program that generates x. The kolmogorov function has a beautiful theory that we will for the most part ignore in this survey though we list a few results that prove very useful in complexity. In fact, kx is the best compression that one could possibly achieve for the text string x. Ee5585 data compression lecture 19 kolmogorov complexity. Shannon entropy versus kolmogorov complexity, the relation of both to universal coding, shannon mu.
Twodimensional kolmogorov complexity and an empirical. We sharpen and clarify the general modeling principles minimum description length mdl and minimum message length mml, abstracted as the ideal mdl principle and defined from bayess rule by means of kolmogorov complexity. We will map feature data and their corresponding probability distributions to binary strings to determine causal features in data by estimating kolmogorov complexity and conditional kolmogorov complexity of these strings. Andrei kolmogorov was the pioneering mathematician. Ee5585 data compression april 4, 20 lecture 19 instructor. A notion of language compressibility is defined, and it is shown that essentially all strings in a sufficiently sparse easy e.
In a sense, strings with high kolmogorov complexity cant be described in any shorter language. A complexitytheoretic approach to the classical data compression problem is presented. Short strings, however, are difficult to compress in practice, and the theory does not pro. Algorithms for estimating information distance with. On data mining, compression, and kolmogorov complexity. The absolute shortest program to produce a string gets the highest possible compression ratio. Apr 21, 2012 second, kolmogorov complexity is essentially the ideal compression technique. At the same time, traffic over the internet is growing and, conseque ntly, the importance of data compression. Algorithmic information theory and kolmogorov complexity alexander shen. That is, can we remove the need for a human to decide upon a data mining technique. Kolmogorov complexity provides a reasonable justi cation for \inductive. However, kolmogorov complexity is not computable, and hence we propose a practical and computable instantiation based on the minimum description length mdl principle.
The purpose of this paper is comprehensively present a subset of information theory as it applies to computer science. Correlation by compression exploratory data analysis. We catalog many of the timebounded variants of kolmogorov complexity. Compression and ranking siam journal on computing vol. Can algorithms tell us if a given string is compressible. Given a data compression algorithm, we define cx as the size of the compressed size of x and cxy as the compression achieved by first training the compression on y, and then compressing x. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of kolmogorov complexity a word has many different codes and the minimum code for a given word cannot be found effectively.
894 210 794 905 866 1447 44 1065 275 896 1035 1265 356 1448 740 783 537 1496 1100 862 252 1508 702 647 45 392 1334 1050 173 1100 302 151 1162 650