Identify Meaning: Identify the expression's meaning.The expression minw∥Xw−y∥22 represents the minimization of the squared Euclidean norm of the vector difference Xw−y, where X is a matrix, w is a vector of variables, and y is a vector. This is a common objective in least squares problems.
Rewrite Expression: Rewrite the expression in a more familiar form.The squared Euclidean norm ∣∣Xw−y∣∣22 can be expanded to (Xw−y)T(Xw−y), which is a standard form in optimization problems involving least squares.
Expand Expression: Expand the expression.Using matrix multiplication, expand (Xw−y)T(Xw−y) to wTXTXw−wTXTy−yTXw+yTy. Note that wTXTy and yTXw are scalars and equal, so this simplifies to wTXTXw−2yTXw+yTy.
Identify Objective: Identify the objective of minimization.To find the minimum of wTXTXw−2yTXw+yTy, we need to take the derivative with respect to w and set it to zero. This gives us the normal equations in the context of least squares: XTXw=XTy.
Solve for w: Solve for w. Assuming XTX is invertible, solve the equation XTXw=XTy by multiplying both sides by the inverse of XTX, leading to w=(XTX)−1XTy. This is the solution that minimizes the original expression.
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