Related papers: Isotropy, Clusters, and Classifiers

Constrained clustering has been well-studied for algorithms such as $K$-means and hierarchical clustering. However, how to satisfy many constraints in these algorithmic settings has been shown to be intractable. One alternative to encode…

Two structures are said to be equimorphic if each embeds in the other. Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case of linear orders, here also called chains. In particular we…

Motivated by recent progress on the scaling behavior of entanglement entropy, we study the scaling behavior of the number of clusters crossing the boundary between two subsystems for several classical statistical models in two dimension.…

Multidimensional scaling is an important dimension reduction tool in statistics and machine learning. Yet few theoretical results characterizing its statistical performance exist, not to mention any in high dimensions. By considering a…

Embedding models trained separately on similar data often produce representations that encode stable information but are not directly interchangeable. This lack of interoperability raises challenges in several practical applications, such…

We present a clustering-based language model using word embeddings for text readability prediction. Presumably, an Euclidean semantic space hypothesis holds true for word embeddings whose training is done by observing word co-occurrences.…

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the…

A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.

Modern AI is opening the door to collective decision-making in which participants express their views as free-form text rather than voting on a fixed set of candidates. A natural idea is to embed these opinions in a vector space so that the…

Clustering high-dimensional datasets is hard because interpoint distances become less informative in high-dimensional spaces. We present a clustering algorithm that performs nonlinear dimensionality reduction and clustering jointly. The…

A fundamental question in adversarial machine learning is whether a robust classifier exists for a given task. A line of research has made some progress towards this goal by studying the concentration of measure, but we argue standard…

We are given a set of elements in a metric space. The distribution of the elements is arbitrary, possibly adversarial. Can we weigh the elements in a way that is resistant to such (adversarial) manipulations? This problem arises in various…

It has been hypothesized that some form of "modular" structure in artificial neural networks should be useful for learning, compositionality, and generalization. However, defining and quantifying modularity remains an open problem. We cast…

During the last decade, entity embeddings have become ubiquitous in Artificial Intelligence. Such embeddings essentially serve as compact but semantically meaningful representations of the entities of interest. In most approaches, vectors…

Contextualized embeddings vary by context, even for the same token, and form a distribution in the embedding space. To analyze this distribution, we focus on the norm of the mean embedding and the variance of the embeddings. In this study,…

Representation learning aims to extract meaningful lower-dimensional embeddings from data, known as representations. Despite its widespread application, there is no established definition of a ``good'' representation. Typically, the…

Embedding spaces contain interpretable dimensions indicating gender, formality in style, or even object properties. This has been observed multiple times. Such interpretable dimensions are becoming valuable tools in different areas of…

The increasing needs of clustering massive datasets and the high cost of running clustering algorithms poses difficult problems for users. In this context it is important to determine if a data set is clusterable, that is, it may be…

Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the…

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…