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Lecture 2 Linear Inverse Problems and Introduction to Least Squares
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Surprise Quiz! Lemma (you can assume) Given the Lemma, prove by induction the following inequality: Once you have proven the aforesaid inequality, derive the Cauchy Schwartz Inequality from it.
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Solution
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How many solutions? Case 1 Case 2 Case 3
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How many solutions? Case 1 Infinite Case 2 One Case 3 None
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Under-determined System Matrix-Vector Notation: The system is ‘Fat’ YΨ = θ
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Perfectly Determined System The system is ‘Square’ YΨ = θ
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Over-Determined System The system is ‘Tall’ YΨ = θ
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Curve Fitting in Noisy Observation
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Least Squares: Line The observation is ‘noisy’ Fit a line that minimizes the sum of least squared error, i.e. Like any other minimization, set gradient = 0
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Least Squares: Line contd.
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We have the solution:
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Homework Find the least squares fit for a second degree polynomial of the form,
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General Form The least squares problem is expressed as:
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Solution
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Taking the gradient
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Solution Taking the gradient The ‘Normal Equations’
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Properties of Least Squares θ LS is a linear function of Y Pseudo Inverse: This is also the left inverse of a ‘tall’ matrix Ψ, i.e.
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