is twice differentiable this is the same as the Hessian matrix H x 2 f x 2 1 2

# Is twice differentiable this is the same as the

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is twice differentiable, this is the same as the Hessian matrix H x = 2 f ∂x 2 1 2 f ∂x 1 ∂x 2 · · · 2 f ∂x 1 ∂x N 2 f ∂x 2 ∂x 1 2 f ∂x 2 2 · · · 2 f ∂x 2 ∂x N . . . . . . . . . 2 f ∂x N ∂x 1 2 f ∂x 1 ∂x 2 · · · 2 f ∂x 2 N being symmetric positive semi-definite. Show that this is indeed the case for f ( x ) = k y - Ax k 2 2 . 1 Last updated 14:22, November 7, 2019

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(f) Show that the minimizer ˆ x of min x k y - Ax k 2 2 + δ k x k 2 2 is always ˆ x = ( A T A + δ I ) - 1 A T y , no matter what A is. Make sure to include an explanation of why A T A + δ I is always invertible when δ > 0. 3. Download the file blocksdeconv.mat . This file contains the vectors: x : the 512 × 1 “blocks” signal h : a 30 × 1 boxcar filter y : a 541 × 1 vector of observations of h convolved with x yn : a noisy observation of y . The noise is iid Gaussian with standard deviation . 01. (a) Write a function which takes an vector h of length L and a number N , and returns the M × N (with M = N + L - 1) matrix A such that for any x R N , Ax is the vector of non-zero values of h convolved with x . (b) Use MATLAB’s svd() command to calculate the SVD of A . What is the largest singular value? What is the smallest singular value? Calculate A y and plot it ( y is the noise-free data). (c) Apply A to the noisy yn . Plot the result. Calculate the mean-sqaure error k x - ˆ x k 2 2 and compare to the measurement error k y - yn k 2 2 .
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