Embedding and visualization techniques are essential for analyzing high-dimensional data, but they often struggle with complex data governed by multiple latent variables, potentially distorting key structural characteristics. This paper considers scenarios where the observed features can be partitioned into mutually exclusive subsets, each capturing a different smooth substructure. In such cases, visualizing the data based on each feature partition can better characterize the underlying processes and structures in the data, leading to improved interpretability. To partition the features, we propose solving an optimization problem that promotes graph Laplacian-based smoothness in each partition, thereby prioritizing partitions with simpler geometric structures. Our approach generalizes traditional embedding and visualization techniques, allowing them to learn multiple embeddings simultaneously. We establish that if several independent or partially dependent manifolds are embedded in distinct feature subsets in high-dimensional space, then our framework can reliably identify the correct subsets with theoretical guarantees. Finally, we demonstrate the effectiveness of our approach in extracting multiple low-dimensional structures and partially independent processes from both simulated and real data.

Scientific observations may consist of a large number of variables (features). Selecting a subset of meaningful features is often crucial for identifying patterns hidden in the ambient space. In this paper, we present a method for unsupervised feature selection, and we demonstrate its advantage in clustering, a common unsupervised task. We propose a differentiable loss that combines a graph Laplacian-based score that favors low-frequency features with a gating mechanism for removing nuisance features. Our method improves upon the naive graph Laplacian score by replacing it with a gated variant computed on a subset of low-frequency features. We identify this subset by learning the parameters of continuously relaxed Bernoulli variables, which gate the entire feature space. We mathematically motivate the proposed approach and demonstrate that it is crucial to compute the graph Laplacian on the gated inputs rather than on the full feature set in the high noise regime. Using several real-world examples, we demonstrate the efficacy and advantage of the proposed approach over leading baselines.