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min_(w)||Xw-y||_(2)^(2)

minwXwy22 \min _{w}\|X w-y\|_{2}^{2}

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Q. minwXwy22 \min _{w}\|X w-y\|_{2}^{2}
  1. Identify Meaning: Identify the expression's meaning.\newlineThe expression minwXwy22\min_{\mathbf{w}}\|\mathbf{Xw}-\mathbf{y}\|_{2}^{2} represents the minimization of the squared Euclidean norm of the vector difference Xwy\mathbf{Xw} - \mathbf{y}, where X\mathbf{X} is a matrix, w\mathbf{w} is a vector of variables, and y\mathbf{y} is a vector. This is a common objective in least squares problems.
  2. Rewrite Expression: Rewrite the expression in a more familiar form.\newlineThe squared Euclidean norm Xwy22||\mathbf{Xw} - \mathbf{y}||_2^2 can be expanded to (Xwy)T(Xwy)(\mathbf{Xw} - \mathbf{y})^T(\mathbf{Xw} - \mathbf{y}), which is a standard form in optimization problems involving least squares.
  3. Expand Expression: Expand the expression.\newlineUsing matrix multiplication, expand (Xwy)T(Xwy)(Xw - y)^T(Xw - y) to wTXTXwwTXTyyTXw+yTyw^T X^T X w - w^T X^T y - y^T X w + y^T y. Note that wTXTyw^T X^T y and yTXwy^T X w are scalars and equal, so this simplifies to wTXTXw2yTXw+yTyw^T X^T X w - 2y^T X w + y^T y.
  4. Identify Objective: Identify the objective of minimization.\newlineTo find the minimum of wTXTXw2yTXw+yTyw^T X^T X w - 2y^T X w + y^T y, we need to take the derivative with respect to ww and set it to zero. This gives us the normal equations in the context of least squares: XTXw=XTyX^T X w = X^T y.
  5. Solve for ww: Solve for ww. Assuming XTXX^T X is invertible, solve the equation XTXw=XTyX^T X w = X^T y by multiplying both sides by the inverse of XTXX^T X, leading to w=(XTX)1XTyw = (X^T X)^{-1} X^T y. This is the solution that minimizes the original expression.

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