Tensor embedding preserves Hamming distance

Fri, 14 Oct 2022 00:00:00 +0200

Table of Contents

This is a proof that Tensor Embedding (Joudaki, Rätsch, and Kahles 2020) with $ℓ^2$-norm preserves the Hamming distance.

This is in collaboration with Amir Joudaki.

\begin{equation*} \newcommand{\I}{\mathcal I} \newcommand{\EE}{\mathbb E} \newcommand{\var}{\operatorname{Var}} \end{equation*}

Definitions Link to heading

Notation

The alphabet is $\Sigma$, of size $|\Sigma| = \sigma$.
The set of indices is $\I := \{(i_1, \dots, i_t) \in [n]^t: i_1 < \dots < i_t\}$.
Given a string $a_1\dots a_n = a\in \Sigma^n$, we define the $I$-index as $a_I = (a_{i_1}, \dots, a_{i_t})$.
We write $[ X ]$ for the indicator variable of event $X$, which is $1$ when $X$ holds and $0$ otherwise.

Definition 1: Tensor embedding

Given $a\in \Sigma^n$, the tensor embedding $T_a$ is the $\sigma^t$ tensor given by $T_a[s] = \sum_{I\in \I} [A_I = s]$ for each $s\in \Sigma^t$.

The normalized tensor embedding distance $d_{te}$ between two sequences $a$ and $b$ is defined as