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"This problem is mainly of sparsity & data compression. The global optimization principle is min_x 1/2 parallel Ax - b parallel_2^2 + lambda parallel x parallel_p^p with iterative decomposition v_i = b - A x: Spare representation rightarrow lambda rightarrow lambda 0 for Ax = b Spare approximation rightarrow lambda > lambda 0 for Ax almostequalto b The LP ""norms'' level sets are bullet strictly convex when p > 1 bullet convex p = 1 bullet non convex p < 1 Hence it satisfies real case value only when A = an m times n real valued matrix, where m < n b = an m-dimensional real valued vector X = set of all minimum L1 norm solutions to Ax = b x bar elementof X leftrightdoublearrow parallel x bar parallel_1 = min_x parallel x parallel_1 s.t. Ax = b \r\n Let we introduce an auxiliary variable X almostequalto Yq: min_q, x S(q, x) = 1/2 parallel y_q - x parallel_2^2 + lambda parallel x parallel_1 such that parallel q parallel_2 = 1 "
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