Related papers: DeepAtlas: a tool for effective manifold learning
Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and…
This work proposes an algorithm for explicitly constructing a pair of neural networks that linearize and reconstruct an embedded submanifold, from finite samples of this manifold. Our such-generated neural networks, called Flattening…
Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…
Transformer models have consistently achieved remarkable results in various domains such as natural language processing and computer vision. However, despite ongoing research efforts to better understand these models, the field still lacks…
A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent)…
Likelihood-based, or explicit, deep generative models use neural networks to construct flexible high-dimensional densities. This formulation directly contradicts the manifold hypothesis, which states that observed data lies on a…
A core challenge in the interpretation of deep neural networks is identifying commonalities between the underlying algorithms implemented by distinct networks trained for the same task. Motivated by this problem, we introduce DYNAMO, an…
Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…
Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of…
The manifold hypothesis is a core mechanism behind the success of deep learning, so understanding the intrinsic manifold structure of image data is central to studying how neural networks learn from the data. Intrinsic dataset manifolds and…
Understanding low-dimensional structures within high-dimensional data is crucial for visualization, interpretation, and denoising in complex datasets. Despite the advancements in manifold learning techniques, key challenges-such as limited…
In recent years there has been increased interest in understanding the interplay between deep generative models (DGMs) and the manifold hypothesis. Research in this area focuses on understanding the reasons why commonly-used DGMs succeed or…
This work develops the global equations of neural networks through stacked piecewise manifolds, fixed-point theory, and boundary-conditioned iteration. Once fixed coordinates and operators are removed, a neural network appears as a…
In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets.…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
Manifold learning (ML), known also as non-linear dimension reduction, is a set of methods to find the low dimensional structure of data. Dimension reduction for large, high dimensional data is not merely a way to reduce the data; the new…
For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric…
Manifold learning methods are an invaluable tool in today's world of increasingly huge datasets. Manifold learning algorithms can discover a much lower-dimensional representation (embedding) of a high-dimensional dataset through non-linear…
Consider a set of $n$ data points in the Euclidean space $\mathbb{R}^d$. This set is called dataset in machine learning and data science. Manifold hypothesis states that the dataset lies on a low-dimensional submanifold with high…
Deep Metric Learning (DML) methods have been proven relevant for visual similarity learning. However, they sometimes lack generalization properties because they are trained often using an inappropriate sample selection strategy or due to…