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We explore two primary classes of approaches to dimensionality reduction(DR): Independent Dimensionality Reduction (IDR) and SimultaneousDimensionality Reduction (SDR). In IDR methods, of which Principal ComponentsAnalysis is a paradigmatic example, each modality is compressed independently,striving to retain as much variation within each modality as possible. Incontrast, in SDR, one simultaneously compresses the modalities to maximize thecovariation between the reduced descriptions while paying less attention to howmuch individual variation is preserved. Paradigmatic examples include PartialLeast Squares and Canonical Correlations Analysis. Even though these DR methodsare a staple of statistics, their relative accuracy and data set sizerequirements are poorly understood. We introduce a generative linear model tosynthesize multimodal data with known variance and covariance structures toexamine these questions. We assess the accuracy of the reconstruction of thecovariance structures as a function of the number of samples, signal-to-noiseratio, and the number of varying and covarying signals in the data. Usingnumerical experiments, we demonstrate that linear SDR methods consistentlyoutperform linear IDR methods and yield higher-quality, more succinctreduced-dimensional representations with smaller datasets. Remarkably,regularized CCA can identify low-dimensional weak covarying structures evenwhen the number of samples is much smaller than the dimensionality of the data,which is a regime challenging for all dimensionality reduction methods. Ourwork corroborates and explains previous observations in the literature that SDRcan be more effective in detecting covariation patterns in data. These findingssuggest that SDR should be preferred to IDR in real-world data analysis whendetecting covariation is more important than preserving variation.

Simultaneous Dimensionality Reduction: A Data Efficient Approach for Multimodal Representations Learning