[Paper Review] Embedding the diamond graph in $L_p$ and dimension reduction in $L_1$

J. R. Lee, Assaf Naor|ArXiv.org|Jul 30, 2004

Computational Geometry and Mesh Generation5 references46 citations

TL;DR

This paper establishes a lower bound on the distortion required to embed the level-$k$ diamond graph into $L_p$ spaces for $1 < p \leq 2$, showing it is at least $\sqrt{1 + (p-1)k}$. This result implies that any $D$-embedding of such graphs into $\ell_1^d$ requires dimension $d \geq n^{\Omega(1/D^2)}$, proving that no $L_1$ analogue of the Johnson-Lindenstrauss dimension reduction lemma exists.

ABSTRACT

We show that any embedding of the level-k diamond graph of Newman and Rabinovich into $L_p$, $1 &lt; p \le 2$, requires distortion at least $\sqrt{k(p-1) + 1}$. An immediate consequence is that there exist arbitrarily large n-point sets $X \subseteq L_1$ such that any D-embedding of X into $\ell_1^d$ requires $d \geq n^{Ω(1/D^2)}$. This gives a simple proof of the recent result of Brinkman and Charikar which settles the long standing question of whether there is an $L_1$ analogue of the Johnson-Lindenstrauss dimension reduction lemma.

Motivation & Objective

• To establish a lower bound on the distortion required to embed the diamond graph $G_k$ into $L_p$ for $1 < p \leq 2$.
• To demonstrate that $L_1$ does not admit dimension reduction analogous to the Johnson-Lindenstrauss lemma.
• To provide a geometric, non-linear programming-based proof of the impossibility of $L_1$ dimension reduction.
• To show that arbitrarily large $n$-point subsets of $L_1$ require super-polynomial dimension for constant-distortion embeddings into $\ell_1^d$.

Proposed method

• Uses a generalized short diagonals inequality in $L_p$ for $1 < p \leq 2$, extending the classical $p=2$ case.
• Applies the inequality to anti-edges and edges in the diamond graph construction at each level $i$.
• Derives a recursive inequality relating the $L_p$-distances of anti-edges and edges across levels.
• Employs convexity and averaging to bound the sum of squared distances over anti-edges in terms of edge distances.
• Combines the inequality with the non-expansive $D$-embedding assumption to derive a distortion lower bound.
• Translates the $L_p$ distortion bound into a lower bound on the dimension $d$ required for embedding into $\ell_1^d$.

Experimental results

Research questions

• RQ1 What is the minimum distortion required to embed the level-$k$ diamond graph into $L_p$ for $1 < p \leq 2$?
• RQ2 Can the Johnson-Lindenstrauss dimension reduction lemma be extended to $L_1$ spaces?
• RQ3 What is the relationship between the distortion of $L_p$ embeddings and the dimension of the target $\ell_1^d$ space?
• RQ4 How does the rate of decay of distortion as $p \to 1$ affect the required embedding dimension in $\ell_1$?
• RQ5 Can geometric intuition replace linear programming in proving lower bounds for $L_1$ dimension reduction?

Key findings

• The distortion of any embedding of the level-$k$ diamond graph into $L_p$ for $1 < p \leq 2$ is at least $\sqrt{1 + (p-1)k}$.
• For every $n \in \mathbb{N}$, there exists an $n$-point subset $X \subseteq L_1$ such that any $D$-embedding into $\ell_1^d$ requires $d \geq n^{\Omega(1/D^2)}$.
• The lower bound on distortion in $L_p$ directly implies the impossibility of $L_1$ dimension reduction analogous to the Johnson-Lindenstrauss lemma.
• The proof relies on a novel geometric inequality in $L_p$ spaces, avoiding linear programming techniques used in prior work.
• The result confirms that $L_1$ does not support efficient dimension reduction, even for sets with $L_1$-embeddable metrics.
• The bound $d \geq n^{\Omega(1/D^2)}$ is tight up to the constant in the exponent, as shown by the construction and asymptotic analysis.