代写Math 132A Assignment 6代做Python编程

Math 132A Assignment 6

1. Suppose we have a speech signal represented as a finite sequence of real numbers x1, x2, . . . , xn (at least one nonzero). If we record this signal on magnetic tape, suppose that xi is recorded as the real number yi for each i. Suppose that we believe that a good model of the relationship between the recorded signal and original signal is that they are scaled, i.e., yi = αxi for some constant α. Use the least squares method to find a formula for estimating α.

2. For this question, Consider the problem of finding the point x∗ ∈ R4 in the solution set of Ax = b with minimum norm, where

(a) Calculate x* exactly. (Feel free to use a computer for this!)

(b) Apply Kaczmarz’s Algorithm to generate the first 5 members of a sequence (x(0) , x(1) , . . .) converging to x*. Use µ = 1.

3. This question is about single linear neuron training. Suppose we have a training set T ⊂ R3 × R and we believe a good model for the training set is that of a function f : R3 → R defined by f(x) = wT x for some w ∈ R3.

(a) Suppose that

Apply Kaczmarz’s algorithm to find a good w. (Let’s say go up to w(3)).

(b) Now suppose

Use gradient descent to find a good candidate for w (again just do three iterations).