This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most "informative" projection direction in each iteration. We theoretically show the proposed dimension reduction method can consistently estimate the most "informative" projection direction in each iteration.

Dimension reduction (DR) is an important and widely studied technique in exploratory data analysis. However, traditional DR methods are not applicable to datasets with a contrastive structure, where data are split into a foreground group of interest (case or treatment group), and a background group (control group). This type of data, common in biomedical studies, necessitates contrastive dimension reduction (CDR) methods to effectively capture information unique to or enriched in the foreground group relative to the background group. Despite the development of various CDR methods, two critical questions remain underexplored: when should these methods be applied, and how can the information unique to the foreground group be quantified? In this work, we address these gaps by proposing a hypothesis test to determine the existence of contrastive information, and introducing a contrastive dimension estimator (CDE) to quantify the unique components in the foreground group. We provide theoretical support for our methods and validate their effectiveness through extensive simulated, semi-simulated, and real experiments involving images, gene expressions, protein expressions, and medical sensors, demonstrating their ability to identify the unique information in the foreground group.

Most popular dimension reduction (DR) methods like t-SNE and UMAP are based on minimizing a cost between input and latent pairwise similarities. Though widely used, these approaches lack clear probabilistic foundations to enable a full understanding of their properties and limitations. To that extent, we introduce a unifying statistical framework based on the coupling of hidden graphs using cross entropy. These graphs induce a Markov random field dependency structure among the observations in both input and latent spaces. We show that existing pairwise similarity DR methods can be retrieved from our framework with particular choices of priors for the graphs. Moreover this reveals that these methods relying on shift-invariant kernels su er from a statistical degeneracy that explains poor performances in conserving coarse-grain dependencies. New links are drawn with PCA which appears as a non-degenerate graph coupling model.

Unlike distance metric learning where the subsequent tasks utilizing the estimated distance metric is the usual focus, the proposal focuses on the estimated metric characterizing the geometry structure. Despite the illustrated taxi and MNIST examples, it is still open to finding more compelling applications that target the data space geometry. Interpreting mathematical concepts such as Riemannian metric and geodesic in the context of potential application (e.g., cognition and perception research where similarity measures are common) could be inspiring. Our proposal requires sufficiently dense data, which could be demanding, especially for high-dimensional data due to the curse of dimensionality. Dimensional reduction (e.g., manifold embedding as in the MNIST example) can substantially alleviate the curse of dimensionality, and the dense data requirement will more likely hold true.

X. Without loss of generality, we assume S(B) = S Hence, to prove Theorem 1, it is sufficient to show that S(B) = S(Σ) holds. To verify S(B) = S(Σ), we only need to show the following two results hold: (I). We now begin with the statement (I). This completes the proof for Statement I. We then turn to Statement II. This leads to a contradiction with (H.2) where the structure dimension is r.

Analyzing high-dimensional data presents challenges due to the "curse of dimensionality'', making computations intensive. Dimension reduction techniques, categorized as linear or non-linear, simplify such data. Non-linear methods are particularly essential for efficiently visualizing and processing complex data structures in interactive and graphical applications. This research develops an incremental non-linear dimension reduction method using the Geometric Multi-Resolution Analysis (GMRA) framework for streaming data. The proposed method enables real-time data analysis and visualization by incrementally updating the cluster map, PCA basis vectors, and wavelet coefficients. Numerical experiments show that the incremental GMRA accurately represents non-linear manifolds even with small initial samples and aligns closely with batch GMRA, demonstrating efficient updates and maintaining the multiscale structure. The findings highlight the potential of Incremental GMRA for real-time visualization and interactive graphics applications that require adaptive high-dimensional data representations.

Sufficient dimension reduction is a powerful tool to extract core information hidden in the high-dimensional data and has potentially many important applications in machine learning tasks. However, the existing nonlinear sufficient dimension reduction methods often lack the scalability necessary for dealing with large-scale data. We propose a new type of stochastic neural network under a rigorous probabilistic framework and show that it can be used for sufficient dimension reduction for large-scale data. The proposed stochastic neural network is trained using an adaptive stochastic gradient Markov chain Monte Carlo algorithm, whose convergence is rigorously studied in the paper as well. Through extensive experiments on real-world classification and regression problems, we show that the proposed method compares favorably with the existing state-of-the-art sufficient dimension reduction methods and is computationally more efficient for large-scale data.

This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most "informative" projection direction in each iteration. We theoretically show the proposed dimension reduction method can consistently estimate the most "informative" projection direction in each iteration.