October 30, 2022
Similar papers 2
October 21, 2021
A new dimension reduction (DR) method for data sets is proposed by autonomous deforming of data manifolds. The deformation is guided by the proposed deforming vector field, which is defined by two kinds of virtual interactions between data points. The flattening of data manifold is achieved as an emergent behavior under the elastic and repelling interactions between data points, meanwhile the topological structure of the manifold is preserved. To overcome the uneven sampling ...
February 21, 2019
Dimensionality reduction methods, also known as projections, are frequently used for exploring multidimensional data in machine learning, data science, and information visualization. Among these, t-SNE and its variants have become very popular for their ability to visually separate distinct data clusters. However, such methods are computationally expensive for large datasets, suffer from stability problems, and cannot directly handle out-of-sample data. We propose a learning ...
June 20, 2022
TSNE and UMAP are two of the most popular dimensionality reduction algorithms due to their speed and interpretable low-dimensional embeddings. However, while attempts have been made to improve on TSNE's computational complexity, no existing method can obtain TSNE embeddings at the speed of UMAP. In this work, we show that this is indeed possible by combining the two approaches into a single method. We theoretically and experimentally evaluate the full space of parameters in t...
September 14, 2023
Einstein field equations are notoriously challenging to solve due to their complex mathematical form, with few analytical solutions available in the absence of highly symmetric systems or ideal matter distribution. However, accurate solutions are crucial, particularly in systems with strong gravitational field such as black holes or neutron stars. In this work, we use neural networks and auto differentiation to solve the Einstein field equations numerically inspired by the id...
May 30, 2013
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifo ld learnin...
October 10, 2022
ParaDime is a framework for parametric dimensionality reduction (DR). In parametric DR, neural networks are trained to embed high-dimensional data items in a low-dimensional space while minimizing an objective function. ParaDime builds on the idea that the objective functions of several modern DR techniques result from transformed inter-item relationships. It provides a common interface for specifying these relations and transformations and for defining how they are used with...
August 8, 2020
When performing classification tasks, raw high dimensional features often contain redundant information, and lead to increased computational complexity and overfitting. In this paper, we assume the data samples lie on a single underlying smooth manifold, and define intra-class and inter-class similarities using pairwise local kernel distances. We aim to find a linear projection to maximize the intra-class similarities and minimize the inter-class similarities simultaneously, ...
June 7, 2021
This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality ...
August 10, 2022
In this paper, we present a method of embedding physics data manifolds with metric structure into lower dimensional spaces with simpler metrics, such as Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful step in the data analysis pipeline for many applications. Using progressively more realistic simulated collisions at the Large Hadron Collider, we show that this embedding approach learns the underlying latent structure. With the notion of volume i...
June 18, 2017
The vast majority of Dimensionality Reduction (DR) techniques rely on second-order statistics to define their optimization objective. Even though this provides adequate results in most cases, it comes with several shortcomings. The methods require carefully designed regularizers and they are usually prone to outliers. In this work, a new DR framework, that can directly model the target distribution using the notion of similarity instead of distance, is introduced. The propose...