Blessing of dimensionality: mathematical foundations of the statistical physics of data (1801.03421v1)

Abstract: The concentration of measure phenomena were discovered as the mathematical background of statistical mechanics at the end of the XIX - beginning of the XX century and were then explored in mathematics of the XX-XXI centuries. At the beginning of the XXI century, it became clear that the proper utilisation of these phenomena in machine learning might transform the curse of dimensionality into the blessing of dimensionality. This paper summarises recently discovered phenomena of measure concentration which drastically simplify some machine learning problems in high dimension, and allow us to correct legacy artificial intelligence systems. The classical concentration of measure theorems state that i.i.d. random points are concentrated in a thin layer near a surface (a sphere or equators of a sphere, an average or median level set of energy or another Lipschitz function, etc.). The new stochastic separation theorems describe the thin structure of these thin layers: the random points are not only concentrated in a thin layer but are all linearly separable from the rest of the set, even for exponentially large random sets. The linear functionals for separation of points can be selected in the form of the linear Fisher's discriminant. All artificial intelligence systems make errors. Non-destructive correction requires separation of the situations (samples) with errors from the samples corresponding to correct behaviour by a simple and robust classifier. The stochastic separation theorems provide us by such classifiers and a non-iterative (one-shot) procedure for learning.

• The paper proposes that the 'curse of dimensionality' can become a 'blessing' for machine learning by leveraging measure concentration theorems from statistical physics to simplify high-dimensional data geometry.
• The stochastic separation theorem demonstrates that individual points in high-dimensional datasets can be linearly separated from others with high probability, making linear classifiers highly effective.
• This theoretical framework offers practical implications for correcting legacy AI systems, enabling knowledge transfer between AI models, and explaining rapid learning in neuroscience.

Blessing of Dimensionality: Mathematical Foundations of the Statistical Physics of Data

In the paper "Blessing of Dimensionality: Mathematical Foundations of the Statistical Physics of Data," authors A.N. Gorban and I.Y. Tyukin explore the transformation of high-dimensional complexities into simplified mathematical frameworks, facilitating the resolution of machine learning challenges. Delving into the phenomenon where measure concentration significantly enhances machine learning applications, the authors propose that the commonly perceived "curse of dimensionality" can be reversed to a "blessing."

High-Dimensional Concentration Phenomena

The essence of this paper lies in the application of measure concentration theorems, originally rooted in statistical mechanics, to modern machine learning datasets. Classical measure concentration results show that i.i.d. random points in high-dimensional spaces tend to localize around surfaces like spheres or equators. This localization simplifies the geometry and energy distribution of the data, making it tractable despite the high dimensionality.

A novel aspect introduced is the stochastic separation theorem, which extends the concentration concept to assert that individual points in these large sets can be separated linearly from the rest with high probability. These points can be discriminated using linear Fisher's discriminant, emphasizing the practicality of linear classifiers even as data dimensionality increases. In many practical scenarios, even exponentially large sets remain linearly separable, meaning computations remain efficient and scalable.

Implications and Applications

This theoretical advancement offers several practical implications:

1. Correction of Legacy AI Systems: Emphasizing that all AI systems can make errors, the paper suggests that stochastic separation theorems afford non-destructive correction mechanisms. Legacy systems can benefit from these non-iterative, linear classifiers to distinguish correct samples from errors swiftly.
2. Knowledge Transfer Between AI Systems: The stochastic separation framework provides an avenue for efficient knowledge transfer among AI systems without extensive retraining. It facilitates the rapid adaptation of one system to mimic the decision patterns preferred by another more accurately.
3. Neuroscience Applications: The concepts apply beyond AI to explain rapid learning and selective behavior in biological neural systems, positing that individual neurons could acquire precise stimulus representations rapidly based on the high-dimensional nature of their inputs.

Future Directions

In theoretical terms, the implications point towards exploring new dimensions of AI robustness and adaptability. Such findings encourage a reassessment of machine learning algorithm designs that leverage dimensionality rather than diminishing it. The paper suggests further investigations into the intrinsic dimensionality of data to focus computational resources efficiently and advocates an extension of measure concentration principles to diverse probabilistic distributions beyond those examined.

Conclusion

By leveraging phenomena originating in statistical physics, the paper delivers a robust methodology likely applicable across various domains involving high-dimensional data. It exemplifies the potential to transform ostensibly complex settings into simplified tasks, enhancing the performance and adaptability of machine learning systems. The simplicity and efficiency of linear methods in high-dimensional regimes—as demonstrated—stand as pivotal insights into future applications of AI and data analytics.