Exploring relative advantages of dual vs single dimensionality reduction

The current experimental advancements in studying complex systems enable the recording of the activity of a large number of constituents of the system, while simultaneously recording the resultant system behavior. Usually, the modeling process begins with dimensionality reduction. Here we explore the dual dimensionality reduction, aiming to identify simultaneously which collective dynamics from the individual constituents are responsible for which reduced dimensional response features. Using linear modeling, we show that the dual dimensionality reduction approach requires significantly fewer data points than the regular methods (i.e., reducing each feature independently, and then identifying relations between the reduced descriptions). However, increasing the number of recorded dimensions is reflected as more noise in the measurements (quantified by the Signal to Noise Ratio - SNR). We numerically investigate the effect of SNR and the number of samples or observations N on single versus dual dimensionality reduction, showing in which regimes and conditions dual reduction is better and when both are the same. Then we outline how we study these results analytically.