\min_{X, Z} \|D - DX\|_F^2 + \|Z\|_{2,1} \quad \text{s.t. } X = Z used for ADMM Reformulation for Efficient Solving the video should be 2 min each and every term for what reason why we use that methods or that term why minus OK.Real-World Analogy (Chef's Teamwork) A group of chefs must recreate a grand buffet but use only a few special recipes. One chef focuses on matching the flavors exactly (reconstruction), while the other ensures they only use a limited variety of dishes (sparsity). A head chef makes sure both stick to the same list. They work together until both goals are met. Exact Use Case (Paper-Style) > Use: Decompose the problem into subproblems solvable via Alternating Direction Method of Multipliers (ADMM) for faster, modular optimization. Why: To split reconstruction and sparsity into two convex problems that are easier to solve and converge effectively.So explain it very clear visually

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Welcome to the buffet challenge! Imagine a grand buffet with many dishes representing our data matrix D. We have two chefs working together: the head chef focuses on reconstruction, trying to recreate the buffet exactly using a master recipe list X. The sparsity chef ensures we use only a few special recipes, maintaining a short ingredient list Z. The constraint X equals Z means both chefs must work with the same list. This ADMM formulation splits our optimization into two manageable parts: minimizing reconstruction error and enforcing group sparsity. The original optimization problem is hard to solve directly because it combines reconstruction and sparsity terms in a non-separable way. ADMM provides an elegant solution by splitting this into two separate convex subproblems. Think of it as two gears working in sync: one gear handles reconstruction error, the other manages sparsity. Each gear can turn independently, making the problem much easier. Since both subproblems are convex, we get guaranteed convergence and can even solve them in parallel. This divide-and-conquer approach is what makes ADMM so powerful for complex optimization. In ADMM step one, we focus on reconstruction. The head chef works alone while the ingredient list Z is fixed and pinned to the wall. The chef adjusts the master recipe list X to minimize the gap between the original buffet and the recreated buffet. The Frobenius norm measures this reconstruction error - it's like measuring how different the recreated buffet looks from the original. The minus sign in D minus DX is crucial because we want D to approximately equal DX, so we minimize their difference. This gives us a convex quadratic problem with a beautiful closed-form solution involving the data term, regularization, consensus with Z, and the dual variable Y. In ADMM step two, we focus on sparsity. The sparsity chef works alone while the master recipe list X is now fixed and pinned. The chef shortens the ingredient list Z by discarding entire recipe groups whose combined weight is below the threshold lambda over rho. The ℓ₂,₁ norm promotes group sparsity, meaning entire rows of Z can become zero simultaneously. This is different from regular sparsity - we're removing whole groups of ingredients, not just individual items. The soft-thresholding operator shrinks rows that are above the threshold and completely removes rows that fall below it. This creates the sparse structure we need while maintaining the most important recipe groups. Finally, we need synchronization. Both chefs work side-by-side, but they must use identical lists. The conveyor belt represents the dual variable Y, which carries the mismatch between X and Z back to the head chef as feedback. When X doesn't equal Z, this mismatch gets amplified by rho and added to Y, creating pressure for the chefs to align their lists. The convergence plot shows how beautifully ADMM works: both the objective value and constraint residual drop smoothly to near zero in just five iterations. This feedback loop ensures that ADMM converges to the optimal solution where the buffet is perfectly recreated using minimal recipes.