Sharp kernel clustering algorithms and their associated Grothendieck inequalities

In the kernel clustering problem we are given a (large) $n\times n$ symmetric positive semidefinite matrix $A=(a_{ij})$ with $\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0$ and a (small) $k\times k$ symmetric positive semidefinite matrix $B=(b_{ij})$. The goal is to find a partition ${S_1,…,S_k}$ of ${1,… n}$ which maximizes $ \sum_{i=1}^k\sum_{j=1}^k (\sum_{(p,q)\in S_i\times S_j}a_{pq})b_{ij}$. We design a polynomial time approximation algorithm that achieves an approximation ratio of $\frac{R(B)^2}{C(B)}$, where $R(B)$ and $C(B)$ are geometric parameters that depend only on the matrix $B$, defined as follows: if $b_{ij} = < v_i, v_j>$ is the Gram matrix representation of $B$ for some $v_1,…,v_k\in \R^k$ then $R(B)$ is the minimum radius of a Euclidean ball containing the points ${v_1, …, v_k}$. The parameter $C(B)$ is defined as the maximum over all measurable partitions ${A_1,…,A_k}$ of $\R^{k-1}$ of the quantity $\sum_{i=1}^k\sum_{j=1}^k b_{ij}< z_i,z_j>$, where for $i\in {1,…,k}$ the vector $z_i\in \R^{k-1}$ is the Gaussian moment of $A_i$, i.e., $z_i=\frac{1}{(2\pi)^{(k-1)/2}}\int_{A_i}xe^{-|x|_2^2/2}dx$. We also show that for every $\eps > 0$, achieving an approximation guarantee of $(1-\e)\frac{R(B)^2}{C(B)}$ is Unique Games hard.

💡 Research Summary

The paper studies a generalized clustering problem called kernel clustering. Given a large symmetric positive semidefinite matrix A∈ℝ^{n×n} with zero total sum (∑{i,j}a{ij}=0) and a small symmetric positive semidefinite matrix B∈ℝ^{k×k}, the task is to partition the index set {1,…,n} into k disjoint subsets S₁,…,S_k so as to maximize

 Φ(S₁,…,S_k)=∑{i=1}^k∑{j=1}^k b_{ij}·∑{(p,q)∈S_i×S_j} a{pq}.

Matrix A encodes a kernel (similarity) among data points, while B encodes the desired interaction between clusters. The authors introduce two purely geometric parameters that depend only on B:

• R(B) – the radius of the smallest Euclidean ball that contains the Gram vectors v₁,…,v_k satisfying B_{ij}=⟨v_i,v_j⟩. This measures how “spread out’’ the vectors representing the clusters are.

• C(B) – the optimum of a continuous Gaussian partition problem. Consider a (k‑1)-dimensional Gaussian space with density (2π)^{-(k‑1)/2}e^{−‖x‖²/2}. For any measurable partition {A₁,…,A_k} of ℝ^{k‑1}, define the Gaussian moment z_i = ∫{A_i} x γ{k‑1}(dx). Then

 C(B)=max_{partitions} ∑{i,j} b{ij}⟨z_i,z_j⟩.

C(B) can be interpreted as the best value achievable by a “soft’’ partition of Gaussian space, and it is always ≤ k because the moments are bounded by the unit ball.

Algorithmic contribution.
The authors design a polynomial‑time approximation algorithm based on a semidefinite programming (SDP) relaxation. The SDP replaces each data point p by a unit vector u_p∈ℝ^n and maximizes

 ∑{p,q} a{pq}⟨u_p,u_q⟩·⟨v_{σ(p)},v_{σ(q)}⟩

over all possible labelings σ: {1,…,n}→{1,…,k}. This SDP can be solved efficiently and yields an upper bound on the optimal discrete objective.

The rounding step draws a random Gaussian vector g∈ℝ^n and projects the SDP vectors onto g. The signs and magnitudes of the projections induce a partition of ℝ^{k‑1} that approximates the optimal continuous partition defining C(B). By carefully analyzing the joint distribution of (u_p·g, u_q·g) and using the geometry of the vectors v_i, the authors prove that the expected value of the rounded solution satisfies

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