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Register a new userUnderstanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization
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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.
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A first line of attack in exploratory data analysis is data visualization, i.e., generating a 2-dimensional representation of data that makes clusters of similar points visually identifiable. Standard Johnson-Lindenstrauss dimensionality reduction does not produce data visualizations. The t-SNE heuristic of van der Maaten and Hinton, which is based on non-convex optimization, has become the de facto standard for visualization in a wide range of applications. This work gives a formal framework for the problem of data visualization - finding a 2-dimensional embedding of clusterable data that correctly separates individual clusters to make them visually identifiable. We then give a rigorous analysis of the performance of t-SNE under a natural, deterministic condition on the ground-truth clusters (similar to conditions assumed in earlier analyses of clustering) in the underlying data. These are the first provable guarantees on t-SNE for constructing good data visualizations. We show that our deterministic condition is satisfied by considerably general probabilistic generative models for clusterable data such as mixtures of well-separated log-concave distributions. Finally, we give theoretical evidence that t-SNE provably succeeds in partially recovering cluster structure even when the above deterministic condition is not met.
Modern datasets and models are notoriously difficult to explore and analyze due to their inherent high dimensionality and massive numbers of samples. Existing visualization methods which employ dimensionality reduction to two or three dimensions are often inefficient and/or ineffective for these datasets. This paper introduces t-SNE-CUDA, a GPU-accelerated implementation of t-distributed Symmetric Neighbor Embedding (t-SNE) for visualizing datasets and models. t-SNE-CUDA significantly outperforms current implementations with 50-700x speedups on the CIFAR-10 and MNIST datasets. These speedups enable, for the first time, visualization of the neural network activations on the entire ImageNet dataset - a feat that was previously computationally intractable. We also demonstrate visualization performance in the NLP domain by visualizing the GloVe embedding vectors. From these visualizations, we can draw interesting conclusions about using the L2 metric in these embedding spaces. t-SNE-CUDA is publicly available athttps://github.com/CannyLab/tsne-cuda
We introduce an improved unsupervised clustering protocol specially suited for large-scale structured data. The protocol follows three steps: a dimensionality reduction of the data, a density estimation over the low dimensional representation of the data, and a final segmentation of the density landscape. For the dimensionality reduction step we introduce a parallelized implementation of the well-known t-Stochastic Neighbouring Embedding (t-SNE) algorithm that significantly alleviates some inherent limitations, while improving its suitability for large datasets. We also introduce a new adaptive Kernel Density Estimation particularly coupled with the t-SNE framework in order to get accurate density estimates out of the embedded data, and a variant of the rainfalling watershed algorithm to identify clusters within the density landscape. The whole mapping protocol is wrapped in the bigMap R package, together with visualization and analysis tools to ease the qualitative and quantitative assessment of the clustering.
Based on the theory of reproducing kernel Hilbert space (RKHS) and semiparametric method, we propose a new approach to nonlinear dimension reduction. The method extends the semiparametric method into a more generalized domain where both the interested parameters and nuisance parameters to be infinite dimensional. By casting the nonlinear dimensional reduction problem in a generalized semiparametric framework, we calculate the orthogonal complement space of generalized nuisance tangent space to derive the estimating equation. Solving the estimating equation by the theory of RKHS and regularization, we obtain the estimation of dimension reduction directions of the sufficient dimension reduction (SDR) subspace and also show the asymptotic property of estimator. Furthermore, the proposed method does not rely on the linearity condition and constant variance condition. Simulation and real data studies are conducted to demonstrate the finite sample performance of our method in comparison with several existing methods.
Inference with population genetic data usually treats the population pedigree as a nuisance parameter, the unobserved product of a past history of random mating. However, the history of genetic relationships in a given population is a fixed, unobserved object, and so an alternative approach is to treat this network of relationships as a complex object we wish to learn about, by observing how genomes have been noisily passed down through it. This paper explores this point of view, showing how to translate questions about population genetic data into calculations with a Poisson process of mutations on all ancestral genomes. This method is applied to give a robust interpretation to the $f_4$ statistic used to identify admixture, and to design a new statistic that measures covariances in mean times to most recent common ancestor between two pairs of sequences. The method more generally interprets population genetic statistics in terms of sums of specific functions over ancestral genomes, thereby providing concrete, broadly interpretable interpretations for these statistics. This provides a method for describing demographic history without simplified demographic models. More generally, it brings into focus the population pedigree, which is averaged over in model-based demographic inference.
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