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Efficient Stochastic Complexity Calculation for Clustered Data.

Daniel Nicorici, Olli Yli-Harja, Jaakko Astola· September 2, 2026 View original

Key takeaways

  • A new recursive formula efficiently calculates stochastic complexity for clustered data.
  • The method reduces computation time from polynomial to linear.
  • This improves data clustering based on the Minimum Description Length principle.
  • It enables faster and more accurate estimation of optimal cluster numbers.

Who benefits

Data ScienceMachine LearningBioinformaticsMarket ResearchCybersecurity

Summary

This paper introduces an efficient recursive formula for calculating the stochastic probability (shortest code length) of vectors with cluster structures using the Normalized Maximum Likelihood (NML) model, reducing computation time from polynomial to linear. This advancement is crucial for data clustering based on the Minimum Description Length (MDL) principle.

Understanding the inherent complexity of data is fundamental in machine learning, especially for tasks like clustering. The Minimum Description Length (MDL) principle is a powerful framework for this, where the "shortest code length" or stochastic complexity of data indicates its underlying structure. For data exhibiting cluster structures, calculating this complexity using the Normalized Maximum Likelihood (NML) model has traditionally been computationally intensive, requiring polynomial time relative to the data size and number of clusters. This research presents a significant breakthrough by introducing a recursive formula that dramatically speeds up this calculation. The new method efficiently computes the normalizing constant within the NML model, reducing the time complexity from polynomial to linear. This improvement makes the problem of determining the stochastic complexity of clustered vectors tractable, which has profound implications for data clustering. It allows for more efficient estimation of the optimal number of clusters and the best cluster structure, enhancing the practical application of MDL-based clustering algorithms.

Why it matters

Professionals in data science and machine learning can now more efficiently apply MDL-based clustering techniques, leading to faster and more accurate determination of optimal cluster numbers and structures in large datasets.

How to implement this in your domain

  1. 1Integrate the new recursive formula into existing MDL-based clustering algorithms for improved efficiency.
  2. 2Apply the optimized stochastic complexity calculation to determine the optimal number of clusters in large datasets.
  3. 3Develop new data analysis tools that leverage this faster computation for real-time clustering insights.
  4. 4Benchmark the performance gains of the new method against traditional polynomial-time approaches.

Original post by Daniel Nicorici, Olli Yli-Harja, Jaakko Astola

"arXiv:2609.00084v1 Announce Type: new Abstract: This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and prac…"

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Originally posted by Daniel Nicorici, Olli Yli-Harja, Jaakko Astola on X · view source

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