Learning from higher-order statistics, efficiently: hypothesis tests, random features, and neural networks^*

Abstract Neural networks excel at discovering statistical patterns in high-dimensional data sets. In practice, higher-order cumulants, which quantify the non-Gaussian correlations between three or more variables, are particularly important for the performance of neural networks. But how efficient are neural networks at extracting features from higher-order cumulants? We study this question in the spiked cumulant model, where the statistician needs to recover a privileged direction or ‘spike’ from the order- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> cumulants of d -dimensional inputs. We first discuss the fundamental statistical and computational limits of recovering the spike by analysing the number of samples n required to strongly distinguish between inputs from the spiked cumulant model and isotropic Gaussian inputs. Existing literature established the presence of a wide statistical-to-computational gap in this problem. We deepen this line of work by finding an exact formula for the likelihood ratio norm which proves that statistical distinguishability requires <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≳</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> samples, while distinguishing the two distributions in polynomial time requires <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≳</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> samples for a wide class of algorithms, i.e. those covered by the low-degree conjecture. Numerical experiments show that neural networks do indeed learn to distinguish the two distributions with quadratic sample complexity, while ‘lazy’ methods like random features (RFs) are not better than random guessing in this regime. Our results show that neural networks extract information from higher-order correlations in the spiked cumulant model efficiently, and reveal a large gap in the amount of data required by neural networks and RFs to learn from higher-order cumulants.