Related papers: The Power of Recursive Embeddings for $\ell_p$ Met…
Metric embeddings are a widely used method in algorithm design, where generally a ``complex'' metric is embedded into a simpler, lower-dimensional one. Historically, the theoretical computer science community has focused on bi-Lipschitz…
Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion…
We present a framework for performing efficient regression in general metric spaces. Roughly speaking, our regressor predicts the value at a new point by computing a Lipschitz extension --- the smoothest function consistent with the…
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability…
It is known that if finite subsets of a locally finite metric space $M$ admit $C$-bilipschitz embeddings into $\ell_p$ $(1\le p\le \infty)$, then for every $\epsilon>0$, the space $M$ admits a $(C+\epsilon)$-bilipschitz embedding into…
Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p >…
Learning the embedding space, where semantically similar objects are located close together and dissimilar objects far apart, is a cornerstone of many computer vision applications. Existing approaches usually learn a single metric in the…
Metric learning aims to learn a highly discriminative model encouraging the embeddings of similar classes to be close in the chosen metrics and pushed apart for dissimilar ones. The common recipe is to use an encoder to extract embeddings…
Human memory is inherently prone to forgetting. To address this, multimodal embedding models have been introduced, which transform diverse real-world data into a unified embedding space. These embeddings can be retrieved efficiently, aiding…
Oblivious low-distortion subspace embeddings are a crucial building block for numerical linear algebra problems. We show for any real $p, 1 \leq p < \infty$, given a matrix $M \in \mathbb{R}^{n \times d}$ with $n \gg d$, with constant…
Bidirectional models are considered essential for strong text embeddings. Recent approaches to adapt autoregressive language models (LMs) into strong text embedding models have largely had the requirement to modify the LM architecture to be…
The distance metric plays an important role in nearest neighbor (NN) classification. Usually the Euclidean distance metric is assumed or a Mahalanobis distance metric is optimized to improve the NN performance. In this paper, we study the…
An embedding is a function that maps entities from one algebraic structure into another while preserving certain characteristics. Embeddings are being used successfully for mapping relational data or text into vector spaces where they can…
Modern deep learning-based recommendation systems exploit hundreds to thousands of different categorical features, each with millions of different categories ranging from clicks to posts. To respect the natural diversity within the…
We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range $0<p<\infty$ from a metric viewpoint and give a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with both their ad-hoc distances…
Tensor decomposition of high-dimensional data often struggles to capture semantically or physically meaningful structures, particularly when relying on reconstruction objectives and fixed-rank constraints. We introduce a no-rank tensor…
We give sufficient conditions for a metric space to bilipschitz embed in L_1. In particular, if X is a length space and there is a Lipschitz map u:X--->R such that for every interval I in R, the connected components of the inverse image…
Given an arbitrary set of high dimensional points in $\ell_1$, there are known negative results that preclude the possibility of always mapping them to a low dimensional $\ell_1$ space while preserving distances with small multiplicative…
Ordinal embedding aims at finding a low dimensional representation of objects from a set of constraints of the form "item $j$ is closer to item $i$ than item $k$". Typically, each object is mapped onto a point vector in a low dimensional…
For Euclidean space ($\ell_2$), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss, with a host of known applications. Here, we consider the problem of dimension reduction for all $\ell_p$ spaces $1 \le p…