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Beyond Gaussian models: a mathematical framework for the analysis of real-world data structures

Lorenzo Bardone
Friday, February 10, 2023 - 14:00
Hybrid: in presence and online

Neural Network algorithms are very good at extracting information from real-world data. But what makes them perform so well where other algorithms fail? Recent works have linked the success of Neural Networks to their ability to easily access properties of the data distribution that are more complex than the mean vector and the covariance matrix (the high order cumulants-HOC).

To investigate the role of HOC for inference tasks, we consider a hypothesis testing problem that compares Gaussian noise with a non-Gaussian whitened distribution. The aim is to bound the sample complexity (SC), i.e. the number of samples necessary to understand which of the two distributions data was drawn from.

In this non-Gaussian model the first two cumulants of the data give no information, which is in fact concentrated on HOC. Hence random matrix results, usually employed for similar Gaussian models, cannot be readily applied and new ways to get SC bounds need to be found. In the talk, we will focus on the approach based on the Low-degree Likelihood Ratio.

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