Answer: Least Squares Optimization

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Show that 𝐱̃ = 𝐀 𝐓 (𝐀𝐀 𝐓 ) −𝟏 𝐬 + 𝐏null𝐚̃ Given that: 1) 𝐱̃ = argmin 𝐱 ‖𝐱‖1 s.t. 𝐬 = 𝐀𝐱 𝐀: 𝑚 × 𝑛 , 𝑛 > 𝑚 dim(ker(𝐀)) = 𝑛 − 𝑚 𝐱 ∈ 𝑅 𝑛 , 𝐚𝐧𝐮𝐥𝐥 ∈ 𝑅 𝑛−𝑚 2) 𝐚̃ = arg min 𝐚null ‖𝐱LS + 𝐏null𝐚null‖1  The columns of 𝐏null spans ker𝐀 and 𝐏null: 𝑛 × 𝑛 − 𝑚  𝐱LS is the least squares solution of 𝐬 = 𝐀𝐱 𝐱LS = argmin 𝐱 ‖𝐱‖2 2 s. t. 𝐬 = 𝐀𝐱 → 𝐱LS = 𝐀 𝐓 (𝐀𝐀 𝐓 ) −𝟏 𝐬

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Least Squares Optimization

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