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Regularized Projection Matrix Approximation with Applications to Community Detection

Zheng Zhai, Jialu Xu, Mingxin Wu, Xiaohui Li

TL;DR

Numerical experiments reveal that the regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.

Abstract

This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. The model is formulated as a projection approximation problem, incorporating an entry-wise penalty function. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. To solve this problem, we propose direct optimization on the Stiefel manifold, utilizing the Cayley transformation along with the Alternating Direction Method of Multipliers (ADMM) algorithm. Additionally, we provide a theoretical analysis that establishes the convergence properties of ADMM, demonstrating that the convergence point satisfies the KKT conditions of the original problem. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.

Regularized Projection Matrix Approximation with Applications to Community Detection

TL;DR

Numerical experiments reveal that the regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.

Abstract

This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. The model is formulated as a projection approximation problem, incorporating an entry-wise penalty function. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. To solve this problem, we propose direct optimization on the Stiefel manifold, utilizing the Cayley transformation along with the Alternating Direction Method of Multipliers (ADMM) algorithm. Additionally, we provide a theoretical analysis that establishes the convergence properties of ADMM, demonstrating that the convergence point satisfies the KKT conditions of the original problem. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.
Paper Structure (27 sections, 4 theorems, 56 equations, 6 figures, 5 tables, 3 algorithms)

This paper contains 27 sections, 4 theorems, 56 equations, 6 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Constraints and Penalties
  4. Bounded Penalty
  5. Non-negativity Penalty
  6. Sparsity Penalty
  7. First-order optimal condition
  8. KKT Condition
  9. Geometric Interpretation
  10. Algorithm
  11. Optimization on Stiefel Manifold
  12. A Faster Curvilinear Search Approach
  13. Perturbed Curvilinear Search
  14. Derivatives for different scenarios
  15. Optimization on Projection Matrix Manifold
  16. ...and 12 more sections

Key Result

Proposition 1

Denote the the canonical metric $\langle \cdot,\cdot \rangle_c$ by $\langle A,B \rangle_c := {\rm trace} (A^T (I-\frac{1}{2}UU^T)B), \forall A,B\in T_U.$ Let $P_{T_U}(\nabla F(U))$ denote the projection of $\nabla F(U)$ onto $T_U$. Then, for any $V\in T_U$, we have: where the projection operator is defined as $P_{\rm T_U}(W) = U \frac{(U^T W - W^T U)}{2} + U^\perp ({U^\perp})^T W$.

Figures (6)

  • Figure 1: Illustration demonstrating why minimizing the $\ell_1$ norm leads to a sparse solution. The left diagram shows the curve $r(x_1,x_2)$ with $r = |x_1|+|x_2|$ and $x_1^2+x_2^2=1$, while the right diagram depicts the surface $r(x_1,x_2,x_3)$ with $r=|x_1|+|x_2|+|x_3|$ and $x_1^2+x_2^2+x_3^2=1$.
  • Figure 2: The comparison of the convergence trajectories when applying the unperturbed (left diagram) and perturbed (right diagram) curvilnear search method for solving $\min_{\|x\|_2=1} \|x\|_1$. Both of the two methods start from $(-0.5,-0.5,0.4)$. While the unperturbed method converges to $(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},0)$ while the perturbed method converges to $(-1,0,0)$
  • Figure 3: Illustration of why the perturbed curvilinear search accelerates convergence compared to the unperturbed version: The first and third diagrams display the changes in the function value $F(X_k)$ across iterations $k$, while the second and fourth diagrams illustrate the convergence trajectory of $\phi(X_k)\in {\mathbb R}^2$. The first two diagrams correspond to a perturbation parameter $\lambda=0.7$, and the last two represent $\lambda=1.0$.
  • Figure 4: The function value $F(X_k),k=1,2,\dots$ ($g$ select as the Huber loss penalty function) varies with the iterations $k$ and the convergence trajectory $\phi(X_k),k=1,2,\cdots$ with different set of $\lambda \in \{0.1,0.4,0.7,1.0\}$ for the curvilinear search and ADMM method.
  • Figure 5: Illustration of solutions for the BPMA, PPMA and SPMA with varying $\lambda$ for a synthetic dataset where $P(A_{i,j}=1)=0.65$ when $c(i)=c(j)$ and $P(A_{i,j}=1)=0.40$ when $c(i)\neq c(j)$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1