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Measuring Difficulty

27.12.2020

An algorithm for quantifying the complexity of structures has been developed. It is able to work not only with physical systems up to quantum ones, but also with images, video and even music.

An intuitive understanding of the complexity of systems and processes is inherent in any person. For example, one can easily distinguish the complexity of two drawings based on the dissimilarity of their elements to each other and the number of differing details. This is key information for the human brain to distinguish between objects of approximately the same size and shape. But how to digitize an idea of the complexity of an object and express it mathematically? After all, the need for a mathematical description that properly reflects the complexity of hierarchical non-random structures exists in many fields of science, from physics and geology to the social sciences.

An international team of scientists has developed a universal machine algorithm that can quantify, with a single number, the complexity of any two-dimensional or three-dimensional system. They successfully applied the method to accurately detect the phase transitions of magnetic materials by varying the complexity of the initial and final states. However, the algorithm is also capable of working with images, video, music, quantum systems, neural network algorithms for pattern recognition, and other systems. The authors argue that the proposed scheme is much simpler and cheaper than standard methods based on the calculation of correlation functions or using machine learning methods. They reported this in the journal of the US National Academy of Sciences (PNAS).

The method of calculating the complexity is based on the step-by-step division of the structure into blocks and the subsequent averaging of a certain characteristic within them. At each step, the algorithm compares the averaged ("fuzzy") structure with the original one and fixes the degree of change in the form of a numerical coefficient. For example, if the system analyzes an image, then the pixels in it are divided into blocks, in each of which they are averaged by color. Thus, if the image consists of many small details, then they will disappear, which will increase the difference between the "blurred" and the original structure. At the same time, the numerical coefficient expressing the complexity of the image also increases. The same operation is repeated with the "blurred" image. As a result, the algorithm calculates a numerical coefficient characterizing the degree of complexity of the image.

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