Related papers: Intrinsic Dimension for Large-Scale Geometric Lear…

Fr\'echet Inception Distance (FID) is widely used to evaluate image generators, yet lower FID does not always correspond to better sample quality. We show that this mismatch depends in part on the geometry of the reference dataset. In a…

It is known that statistical model selection as well as identification of dynamical equations from available data are both very challenging tasks. Physical systems behave according to their underlying dynamical equations which, in turn, can…

In order to deal with multidimensional structure representations of real-world networks, as well as with their worst-case irreducible information content analysis, the demand for new graph abstractions increases. This article investigates…

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In…

The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid…

Outlier recognition is a fundamental problem in data analysis and has attracted a great deal of attention in the past decades. However, most existing methods still suffer from several issues such as high time and space complexities or…

Learning embeddings from large-scale networks is an open challenge. Despite the overwhelming number of existing methods, is is unclear how to exploit network structure in a way that generalizes easily to unseen nodes, edges or graphs. In…

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central…

In a human-robot collaborative task where a robot helps its partner by finding described objects, the depth dimension plays a critical role in successful task completion. Existing studies have mostly focused on comprehending the object…

Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…

Graphs are versatile tools for representing structured data. As a result, a variety of machine learning methods have been studied for graph data analysis. Although many such learning methods depend on the measurement of differences between…

The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…

Data are not only ubiquitous in society, but are increasingly complex both in size and dimensionality. Dimension reduction offers researchers and scholars the ability to make such complex, high dimensional data spaces simpler and more…

An additive autoencoder for dimension reduction, which is composed of a serially performed bias estimation, linear trend estimation, and nonlinear residual estimation, is proposed and analyzed. Computational experiments confirm that an…

Network embedding is a general-purpose machine learning technique that encodes network structure in vector spaces with tunable dimension. Choosing an appropriate embedding dimension -- small enough to be efficient and large enough to be…

In recent years, there has been interest in how geometric properties such as intrinsic dimension (ID) of a neural network's hidden representations change through its layers, and how such properties are predictive of important model behavior…

A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two…

Large scale image datasets are a growing trend in the field of machine learning. However, it is hard to quantitatively understand or specify how various datasets compare to each other - i.e., if one dataset is more complex or harder to…

How to evaluate the importance of nodes is essential in research of complex network. There are many methods proposed for solving this problem, but they still have room to be improved. In this paper, a new approach called local volume…

A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on…