arXiv2023
Signal extraction is difficult when the number of variables $N$ is much larger than the number of observations $M$. We address this problem under the working hypothesis that an empirical dataset consists of states sampled from underlying multivariate dynamics. Instead of treating the $M$ observations as points in an $N$-dimensional variable space, we treat the $N$ variables as points in an $M$-dimensional sample-coordinate space, interpreted as an effective time-delay coordinate space of the latent dynamics. This representation allows a large $N$ to provide many points for estimating the variable distribution even when $M$ is small. Transposed singular value decomposition (SVD) and the unsupervised feature-selection method of Taguchi are used to extract variable-side deviations from an estimated Gaussian background as signal candidates. As $M$ is reduced, the effective separation between sampled states increases; contributions from finite-correlation components are expected to decay, whereas sufficiently long-correlation components can persist toward the small-sample limit. We define these persistent components as intrinsic signals and estimate them by extrapolation toward $M=0$. We first tested the method on high-dimensional, small-sample data explicitly generated by a randomised coupling strength globally coupled map (RCS-GCM), for which the long- and short-correlation components were known. The extracted intrinsic signals corresponded to the known long-correlation variables. We then applied the method to The Cancer Genome Atlas (TCGA) pan-kidney gene-expression data, which are not ordinarily treated as data explicitly generated by a dynamical system. Using an SVD component associated with the pathologic-M category, variable-side signal components were extracted from the 20,531-dimensional data even under small-sample conditions.