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### Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization

Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMAP have demonstrated impressive visualization performance on many real world datasets. One tension that has always faced these methods is the trade-off between preservation of global structure and preservation of local structure: these methods can either handle one or the other, but not both. In this work, our main goal is to understand what aspects of DR methods are important for preserving both local and global structure: it is difficult to design a better method without a true understanding of the choices we make in our algorithms and their empirical impact on the lower-dimensional embeddings they produce. Towards the goal of local structure preservation, we provide several useful design principles for DR loss functions based on our new understanding of the mechanisms behind successful DR methods. Towards the goal of global structure preservation, our analysis illuminates that the choice of which components to preserve is important. We leverage these insights to design a new algorithm for DR, called Pairwise Controlled Manifold Approximation Projection (PaCMAP), which preserves both local and global structure. Our work provides several unexpected insights into what design choices both to make and avoid when constructing DR algorithms.

### Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality

Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine learning, existing performance guarantees for generic loss functions are either overly conservative due to the curse of dimensionality, or plausible only in large sample asymptotics. In this paper, we develop a non-asymptotic framework for analyzing the out-of-sample performance for Wasserstein robust learning and the generalization bound for its related Lipschitz and gradient regularization problems. To the best of our knowledge, this gives the first finite-sample guarantee for generic Wasserstein DRO problems without suffering from the curse of dimensionality. Our results highlight the bias-variation trade-off intrinsic in the Wasserstein DRO, which balances between the empirical mean of the loss and the variation of the loss, measured by the Lipschitz norm or the gradient norm of the loss. Our analysis is based on two novel methodological developments that are of independent interest: 1) a new concentration inequality controlling the decay rate of large deviation probabilities by the variation of the loss and, 2) a localized Rademacher complexity theory based on the variation of the loss.

### Two-sample Test using Projected Wasserstein Distance: Breaking the Curse of Dimensionality

We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. In particular, we aim to circumvent the curse of dimensionality in Wasserstein distance: when the dimension is high, it has diminishing testing power, which is inherently due to the slow concentration property of Wasserstein metrics in the high dimension space. A key contribution is to couple optimal projection to find the low dimensional linear mapping to maximize the Wasserstein distance between projected probability distributions. We characterize the theoretical property of the finite-sample convergence rate on IPMs and present practical algorithms for computing this metric. Numerical examples validate our theoretical results.

### Sufficient dimension reduction for classification using principal optimal transport direction

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.

### Curse of Dimensionality

Let us start with a question. Dimension in very simple terms means the attributes or features of a given dataset. So why are we using such a negative word curse associated with dimensions? What is the curse here? If anybody asks me What is a Machine learning Model? Speaking in layman terms, when we give the dataset for training and the output of the training phase is a Model.

### Dimension Reduction in Contextual Online Learning via Nonparametric Variable Selection

We consider a contextual online learning (multi-armed bandit) problem with high-dimensional covariate $\mathbf{x}$ and decision $\mathbf{y}$. The reward function to learn, $f(\mathbf{x},\mathbf{y})$, does not have a particular parametric form. The literature has shown that the optimal regret is $\tilde{O}(T^{(d_x+d_y+1)/(d_x+d_y+2)})$, where $d_x$ and $d_y$ are the dimensions of $\mathbf x$ and $\mathbf y$, and thus it suffers from the curse of dimensionality. In many applications, only a small subset of variables in the covariate affect the value of $f$, which is referred to as \textit{sparsity} in statistics. To take advantage of the sparsity structure of the covariate, we propose a variable selection algorithm called \textit{BV-LASSO}, which incorporates novel ideas such as binning and voting to apply LASSO to nonparametric settings. Our algorithm achieves the regret $\tilde{O}(T^{(d_x^*+d_y+1)/(d_x^*+d_y+2)})$, where $d_x^*$ is the effective covariate dimension. The regret matches the optimal regret when the covariate is $d^*_x$-dimensional and thus cannot be improved. Our algorithm may serve as a general recipe to achieve dimension reduction via variable selection in nonparametric settings.

### Dynamic Batch Learning in High-Dimensional Sparse Linear Contextual Bandits

We study the problem of dynamic batch learning in high-dimensional sparse linear contextual bandits, where a decision maker, under a given maximum-number-of-batch constraint and only able to observe rewards at the end of each batch, can dynamically decide how many individuals to include in the next batch (at the end of the current batch) and what personalized action-selection scheme to adopt within each batch. Such batch constraints are ubiquitous in a variety of practical contexts, including personalized product offerings in marketing and medical treatment selection in clinical trials. We characterize the fundamental learning limit in this problem via a regret lower bound and provide a matching upper bound (up to log factors), thus prescribing an optimal scheme for this problem. To the best of our knowledge, our work provides the first inroad into a theoretical understanding of dynamic batch learning in high-dimensional sparse linear contextual bandits. Notably, even a special case of our result (when no batch constraint is present) yields the first minimax optimal $\tilde{O}(\sqrt{s_0T})$ regret bound for standard online learning in high-dimensional linear contextual bandits (for the no-margin case), where $s_0$ is the sparsity parameter (or an upper bound thereof) and $T$ is the learning horizon. This result (both that $\tilde{O}(\sqrt{s_0 T})$ is achievable and that $\Omega(\sqrt{s_0 T})$ is a lower bound) appears to be unknown in the emerging literature of high-dimensional contextual bandits.

### What is the relationship between Curse of Dimensionality and isotropic neighborhoods?

The problem that Hastie, Tibshirani and Friedman are talking about here is that the number of fixed-size neighborhoods goes up exponentially with the dimension. If you're trying to get some intuition for how isotropic neighborhoods are affected by the curse of dimensionality, think about approximating ball-shaped (isotropic) neighborhoods with cube-shaped neighborhoods. Suppose we have an $d$-dimensional unit cube $[0, 1] d$ that we want to divide up into cube-shaped neighborhoods. If I want a neighborhood of side length $\delta 0.1$, in one dimension this requires $10 1 10$ neighborhoods. In two dimensions, this requires $10 2 100$ neighborhoods.

### The Lasso with general Gaussian designs with applications to hypothesis testing

The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates $p$ is of the same order or larger than the number of observations $n$. Classical asymptotic normality theory is not applicable for this model due to two fundamental reasons: $(1)$ The regularized risk is non-smooth; $(2)$ The distance between the estimator $\bf \widehat{\theta}$ and the true parameters vector $\bf \theta^\star$ cannot be neglected. As a consequence, standard perturbative arguments that are the traditional basis for asymptotic normality fail. On the other hand, the Lasso estimator can be precisely characterized in the regime in which both $n$ and $p$ are large, while $n/p$ is of order one. This characterization was first obtained in the case of standard Gaussian designs, and subsequently generalized to other high-dimensional estimation procedures. Here we extend the same characterization to Gaussian correlated designs with non-singular covariance structure. This characterization is expressed in terms of a simpler fixed design'' model. We establish non-asymptotic bounds on the distance between distributions of various quantities in the two models, which hold uniformly over signals $\bf \theta^\star$ in a suitable sparsity class, and values of the regularization parameter. As applications, we study the distribution of the debiased Lasso, and show that a degrees-of-freedom correction is necessary for computing valid confidence intervals.

### Overcoming the Curse of Dimensionality in Density Estimation with Mixed Sobolev GANs

We propose a novel GAN framework for non-parametric density estimation with high-dimensional data. This framework is based on a novel density estimator, called the hyperbolic cross density estimator, which enjoys nice convergence properties in the mixed Sobolev spaces. As modifications of the usual Sobolev spaces, the mixed Sobolev spaces are more suitable for describing high-dimensional density functions. We prove that, unlike other existing approaches, the proposed GAN framework does not suffer the curse of dimensionality and can achieve the optimal convergence rate of $O_p(n^{-1/2})$, with $n$ data points in an arbitrary fixed dimension. We also study the universality of GANs in terms of the existence of ReLU networks which can approximate the density functions in the mixed Sobolev spaces up to any accuracy level.