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Questions and Answers
What is the primary focus of the notes provided?
What is the primary focus of the notes provided?
Convex Optimization
Which of the following is a type of approximation mentioned?
Which of the following is a type of approximation mentioned?
Penalty function approximation involves minimizing the sum of a convex penalty function and the residuals.
Penalty function approximation involves minimizing the sum of a convex penalty function and the residuals.
True
What does the variable 'x⋆' represent in norm approximation?
What does the variable 'x⋆' represent in norm approximation?
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What equation is satisfied by the solution of the least-squares approximation?
What equation is satisfied by the solution of the least-squares approximation?
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Which penalty function leads to linear growth for large values of 'u'?
Which penalty function leads to linear growth for large values of 'u'?
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The term '_____ approximation' involves minimizing the sum of absolute residuals.
The term '_____ approximation' involves minimizing the sum of absolute residuals.
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What type of equations must be solved for least-squares approximation?
What type of equations must be solved for least-squares approximation?
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Study Notes
Approximation and Fitting
- Norm approximation aims to minimize the difference between a linear transformation of a variable and a target vector, represented as ( \min kAx - bk ).
- Solution interpretation for ( x^* = \arg\min_x kAx - bk ):
- Geometric: ( Ax^* ) is the point in the range of ( A ) closest to ( b ).
- Estimation: In a linear measurement model ( y = Ax + v ), ( y ) represents observed measurements with ( v ) as measurement error; ( x^* ) provides the best estimate of ( x ).
- Optimal Design: Design variables ( x ) yield outputs ( Ax ); ( x^* ) represents the best design to approximate the desired output ( b ).
Examples of Norm Approximation
-
Least-Squares Approximation ( (k · k_2) ):
- Satisfies normal equations ( A^T A x = A^T b ).
- If ( A ) has full rank ( n ), solution is ( x^* = (A^T A)^{-1} A^T b ).
-
Chebyshev Approximation ( (k · k_\infty) ):
- Solved through linear programming to minimize ( t ) subject to ( -t \leq Ax - b \leq t ).
-
Sum of Absolute Residuals Approximation ( (k · k_1) ):
- Also solvable via linear programming, minimizing ( 1^T y ) with constraints ( -y \leq Ax - b \leq y ).
Penalty Function Approximation
- Formulated as ( \min \phi(r_1) + · · · + \phi(r_m) ) with ( r = Ax - b ), where ( \phi ) is a convex penalty function.
Types of Penalty Functions
- Quadratic Penalty: ( \phi(u) = u^2 )
- Deadzone-Linear Penalty: ( \phi(u) = \max{0, |u| - a} ) creates a linear growth after a certain threshold.
- Log-Barrier Penalty: ( \phi(u) = -a^2 \log(1 - (u/a)^2) ) is applicable within limits and infinite otherwise.
Characteristics of Penalty Functions
- The shape of the penalty function significantly influences the distribution of residuals encountered in approximations.
Huber Penalty Function
- Defined with parameter ( M ):
- ( \phi_{hub}(u) = \begin{cases} u^2 & |u| \leq M \ M(2|u| - M) & |u| > M \end{cases} )
- Offers linear growth for large ( |u| ), making the approximation robust to outliers compared to simple quadratic penalties.
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Description
This quiz covers key concepts from Chapter 6 of 'Convex Optimization' by Stephen Boyd and Lieven Vandenberghe, focusing on approximation and fitting techniques. Topics include norm approximation, least-norm problems, and regularized approximation. Test your understanding of these essential optimization strategies.
