\min_A \|D - DA\|_F^2 + \lambda \|A\|_{2,1}Explain the admm optimization problem with the help of a toy diagram i need more visualization which is doing sparsitu for what how it is going to reconstruct and it is used for Extractive Summarization Using ℓ₂,₁ Regularization
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We introduce an ADMM optimization problem for extractive summarization. The objective function minimizes the reconstruction error using Frobenius norm, plus a sparsity regularization term with lambda parameter. Matrix D contains document sentences, while matrix A represents coefficients. The ℓ₂,₁ norm induces row-wise sparsity, effectively selecting representative sentences that can reconstruct the entire document.
We will explore ADMM optimization for extractive text summarization. The objective function minimizes reconstruction error while enforcing sparsity through L2,1 regularization. This approach identifies the most representative sentences for creating effective document summaries.
The document matrix D contains sentence feature vectors as rows. Self-representation means each sentence can be expressed as a linear combination of all sentences. Matrix A captures these relationships, where sparse entries in A identify the most representative sentences for summarization.
L2,1 regularization promotes row-wise sparsity in matrix A by summing the L2 norms of each row. This encourages entire rows to become zero, effectively selecting specific sentences for the summary. The regularization parameter lambda controls the trade-off between reconstruction accuracy and sparsity.
ADMM alternates between three update steps. First, update A by minimizing reconstruction error plus augmented Lagrangian terms. Second, apply the proximal operator for L2,1 regularization to update Z. Third, update the dual variable U. This process repeats until convergence, producing a sparse matrix A that identifies key sentences.
The extractive summarization process begins with feature extraction to build matrix D. ADMM optimization finds sparse matrix A that identifies the most representative sentences. Non-zero rows in A correspond to selected sentences, which are ranked by importance to generate the final summary. This approach ensures both diversity and representativeness in the extracted summary.
The ℓ₂,₁ norm is the sum of ℓ₂ norms of matrix rows, promoting row-wise sparsity. As lambda increases, more rows become zero, selecting fewer sentences. This visualization shows how different lambda values affect sentence selection - higher regularization leads to sparser solutions with fewer but more representative sentences for summarization.
ADMM decomposes the optimization problem by introducing auxiliary variables and constraints. The algorithm alternates between three steps: updating A to minimize reconstruction error, updating Z using proximal operators for sparsity, and updating dual variables U. This iterative process guarantees convergence to the optimal sparse solution for sentence selection.
The final extractive summarization system selects representative sentences based on sparse matrix A. Non-zero rows correspond to selected sentences, ranked by coefficient magnitude. Different lambda values create different trade-offs between sparsity and quality. The system achieves high ROUGE scores while maintaining coherence and computational efficiency for practical text summarization applications.