Related papers: Lossless Prioritized Embeddings
It is shown that any bounded metric space can be isometrically embedded into the Gromov--Hausdorff metric class GH. This result is a consequence of local geometry description of the class GH in a sufficiently small neighborhood of a generic…
The basic tool for solving problems in metric geometry and isotonic regression is the metric projection onto closed convex cones. Isotonicity of these projections with respect to a given order relation can facilitate finding the solutions…
We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant…
The embedding problem is to decide, given an ordered pair of structures, whether or not there is an injective homomorphism from the first structure to the second. We study this problem using an established perspective in parameterized…
A major open problem in the field of metric embedding is the existence of dimension reduction for $n$-point subsets of Euclidean space, such that both distortion and dimension depend only on the {\em doubling constant} of the pointset, and…
The evaluation of recent embedding-based evaluation metrics for text generation is primarily based on measuring their correlation with human evaluations on standard benchmarks. However, these benchmarks are mostly from similar domains to…
The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in…
We introduce second-order vector representations of words, induced from nearest neighborhood topological features in pre-trained contextual word embeddings. We then analyze the effects of using second-order embeddings as input features in…
Consider an m by N matrix Phi with the Restricted Isometry Property of order k and level delta, that is, the norm of any k-sparse vector in R^N is preserved to within a multiplicative factor of 1 +- delta under application of Phi. We show…
The early development of a zygote can be mathematically described by a developmental tree. To compare developmental trees of different species, we need to define distances on trees. If children cells after a division are not…
Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental…
The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation…
An outstanding open question posed by Guha and Indyk in 2006 asks to characterize metric spaces in which distances can be estimated using efficient sketches. Specifically, we say that a sketching algorithm is efficient if it achieves…
Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric…
This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
We prove that every proper $n$-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space $\mathbb{R}^{3n+6,1}$. By an "approximate isometric embedding" we mean an embedding which preserves the energy…
We introduce a class of metrics on $\mathbb{R}^n$ generalizing the classical Grushin plane. These are length metrics defined by the line element $ds = d_E(\cdot,Y)^{-\beta}ds_E$ for a closed nonempty subset $Y \subset \mathbb{R}^n$ and…
Text embeddings from large language models (LLMs) have achieved excellent results in tasks such as information retrieval, semantic textual similarity, etc. In this work, we show an interesting finding: when feeding a text into the LLM-based…
Word-embeddings are vital components of Natural Language Processing (NLP) models and have been extensively explored. However, they consume a lot of memory which poses a challenge for edge deployment. Embedding matrices, typically, contain…