Related papers: Topology-Preserving Dimensionality Reduction via I…
Dimensionality reduction is a fundamental task in modern data science. Several projection methods specifically tailored to take into account the non-linearity of the data via local embeddings have been proposed. Such methods are often based…
The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…
This paper presents a comprehensive overview of several multidimensional reduction methods focusing on Multidimensional Principal Component Analysis (MPCA), Multilinear Orthogonal Neighborhood Preserving Projection (MONPP), Multidimensional…
We introduce "TriMap"; a dimensionality reduction technique based on triplet constraints, which preserves the global structure of the data better than the other commonly used methods such as t-SNE, LargeVis, and UMAP. To quantify the global…
We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is…
Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially…
Real world re-identfication (ReID) algorithms aim to map new observations of an object to previously recorded instances. These systems are often constrained by quantity and size of the stored embeddings. To combat this scaling problem, we…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
For numerous reasons there raises a need for dimension reduction that preserves certain characteristics of data. In this work we focus on data coming from a mixture of Gaussian distributions and we propose a method that preserves…
This paper gives a review and synthesis of methods of evaluating dimensionality reduction techniques. Particular attention is paid to rank-order neighborhood evaluation metrics. A framework is created for exploring dimensionality reduction…
In traditional topology optimization, the computing time required to iteratively update the material distribution within a design domain strongly depends on the complexity or size of the problem, limiting its application in real engineering…
Dimensionality reduction (DR) plays a vital role in the visual analysis of high-dimensional data. One main aim of DR is to reveal hidden patterns that lie on intrinsic low-dimensional manifolds. However, DR often overlooks important…
This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can…
Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…
Multidimensional Scaling (MDS) is one of the most popular methods for dimensionality reduction and visualization of high dimensional data. Apart from these tasks, it also found applications in the field of geometry processing for the…
We demonstrate optimization of optical metasurfaces over $10^5$--$10^6$ degrees of freedom in two and three dimensions, 100--1000+ wavelengths ($\lambda$) in diameter, with 100+ parameters per $\lambda^2$. In particular, we show how…
Although transformer-based models have been dominating the field of deep learning, various studies of their embedding space have shown that they suffer from "representation degeneration problem": embeddings tend to be distributed in a…
Photonic topology optimization is a technique used to find the electric permittivity distribution of a device that optimizes an electromagnetic figure-of-merit. Two common techniques are used: continuous density-based optimizations that…
We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets…
We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth…