Related papers: Clustering with UMAP: Why and How Connectivity Mat…
Seeking effective neural networks is a critical and practical field in deep learning. Besides designing the depth, type of convolution, normalization, and nonlinearities, the topological connectivity of neural networks is also important.…
Standard deep learning models for image segmentation cannot guarantee topology accuracy, failing to preserve the correct number of connected components or structures. This, in turn, affects the quality of the segmentations and compromises…
Advanced deep neural networks (DNNs), designed by either human or AutoML algorithms, are growing increasingly complex. Diverse operations are connected by complicated connectivity patterns, e.g., various types of skip connections. Those…
UMAP has supplanted t-SNE as state-of-the-art for visualizing high-dimensional datasets in many disciplines, but the reason for its success is not well understood. In this work, we investigate UMAP's sampling based optimization scheme in…
Multidimensional scaling is a statistical process that aims to embed high dimensional data into a lower-dimensional space; this process is often used for the purpose of data visualisation. Common multidimensional scaling algorithms tend to…
Nonlinear dimension reduction (NLDR) techniques such as tSNE, and UMAP provide a low-dimensional representation of high-dimensional data ($p\text{-}D$) by applying a nonlinear transformation. NLDR often exaggerates random patterns. But NLDR…
Subspace clustering algorithms are notorious for their scalability issues because building and processing large affinity matrices are demanding. In this paper, we introduce a method that simultaneously learns an embedding space along…
The topology of city street networks (SNs) is constrained by spatial embedding, requiring non-crossing links and preventing random node placement or overlap. Here, we analyzed SNs of $33$ Indian cities to explore how the spatial embedding…
In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the…
We first show that the commonly used dimensionality reduction (DR) methods such as t-SNE and LargeVis poorly capture the global structure of the data in the low dimensional embedding. We show this via a number of tests for the DR methods…
Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, whose number, orientations, and dimensions are all unknown. In practice one may have access to…
We discuss topological aspects of cluster analysis and show that inferring the topological structure of a dataset before clustering it can considerably enhance cluster detection: theoretical arguments and empirical evidence show that…
Deep clustering has increasingly been demonstrating superiority over conventional shallow clustering algorithms. Deep clustering algorithms usually combine representation learning with deep neural networks to achieve this performance,…
With the recent surge in big data analytics for hyper-dimensional data there is a renewed interest in dimensionality reduction techniques for machine learning applications. In order for these methods to improve performance gains and…
SpectralNet is a graph clustering method that uses neural network to find an embedding that separates the data. So far it was only used with $k$-nn graphs, which are usually constructed using a distance metric (e.g., Euclidean distance).…
Stochastic Neighbor Embedding and its variants are widely used dimensionality reduction techniques -- despite their popularity, no theoretical results are known. We prove that the optimal SNE embedding of well-separated clusters from high…
Real-world data usually have high dimensionality and it is important to mitigate the curse of dimensionality. High-dimensional data are usually in a coherent structure and make the data in relatively small true degrees of freedom. There are…
UMAP is a non-parametric graph-based dimensionality reduction algorithm using applied Riemannian geometry and algebraic topology to find low-dimensional embeddings of structured data. The UMAP algorithm consists of two steps: (1) Compute a…
We present a new nonlinear dimensionality reduction method, MAPLE, that enhances UMAP by improving manifold modeling. MAPLE employs a self-supervised learning approach to more efficiently encode low-dimensional manifold geometry. Central to…
It has become standard to use gradient-based dimensionality reduction (DR) methods like tSNE and UMAP when explaining what AI models have learned. This makes sense: these methods are fast, robust, and have an uncanny ability to find…