Let D = {(x, y),..., (In, Yn)} where xi R and y R be the...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let D = {(x₁, y₁),..., (In, Yn)} where xi Rª and y₁ € R be the training data that you are given. As you have to predict a continuous variable, one of the simplest possible models is linear regression, i.e. to predict y from wx for some parameter vector w Rd. We thus suggest minimizing the following loss n arg min Â(w) = arg min (y₁ - w ¹x;) ². WERD werd i=1 (LinearR) Let us introduce the nx d matrix XE Rxd with the x, as rows, and the vector y € Rr consisting of the scalars y. Then, (LinearR) can be equivalently re-written as arg min ly - Xw||²2. WERD We refer to any w* that attains the above minimum as a solution to the problem. 1. Show that if XX is invertible, then there is a unique w* that can be computed as w* = (XX)-¹Xw. 2. Show for n < d that (LinearR) does not admit a unique solution. Explain the reason. (Hint: Consider singular value decomposition (SVD) learned in linear algebra and null space of XTX.) 3. Consider the case n ≥d. Under what assumptions on X does (LinearR) admit a unique solution w*? Give an example with n = 3 and d = 2 where these assumptions do not hold. Activate W Let D = {(x₁, y₁),..., (In, Yn)} where xi Rª and y₁ € R be the training data that you are given. As you have to predict a continuous variable, one of the simplest possible models is linear regression, i.e. to predict y from wx for some parameter vector w Rd. We thus suggest minimizing the following loss n arg min Â(w) = arg min (y₁ - w ¹x;) ². WERD werd i=1 (LinearR) Let us introduce the nx d matrix XE Rxd with the x, as rows, and the vector y € Rr consisting of the scalars y. Then, (LinearR) can be equivalently re-written as arg min ly - Xw||²2. WERD We refer to any w* that attains the above minimum as a solution to the problem. 1. Show that if XX is invertible, then there is a unique w* that can be computed as w* = (XX)-¹Xw. 2. Show for n < d that (LinearR) does not admit a unique solution. Explain the reason. (Hint: Consider singular value decomposition (SVD) learned in linear algebra and null space of XTX.) 3. Consider the case n ≥d. Under what assumptions on X does (LinearR) admit a unique solution w*? Give an example with n = 3 and d = 2 where these assumptions do not hold. Activate W
Expert Answer:
Related Book For
Posted Date:
Students also viewed these programming questions
-
128 2. log2 2+4 Expand a logarithmic expression as much as possible.
-
"I'm not sure we should lay out $300,000 for that automated welding machine," said Jim Alder, president of the Superior Equipment Company. "That's a lot of money, and it would cost us $84,000 for...
-
:{"foster", "enthusiasm", "wagon", "ally", "lehigh", "programming", "dog", "cat", "Ally", "smile", "pet" }; a. Suppose you perform insertion sort in order to sort A in ascending order. How many...
-
What has motivated you most since accepting your Omega Nu Lambda nomination to grow in your career development? Give an example
-
Do you believe that international accounting harmonization/convergence will end investor concerns about cross-national differences in accounting practices?
-
Given: f ( x ) = ( 2 x + 3 ) sin ( x ) , find the first derivative of this function.
-
Describe or define the following terms: 1. Asset 2. Liability 3. Shareholders equity 4. Revenue 5. Expense Discuss how the five terms relate to one another.
-
Tri Designs, Inc. has the following data: Perform a vertical analysis of Tri Designss balance sheet for each year. TRI DESIGNS, INC. Comparative Balance Sheet December 31, 2016 and 2015 2016 2015...
-
3 . Contribution Margin Income Statement of a single product company Total Per unit Sales $ 1 , 2 0 0 , 0 0 0 $ 8 0 Less variable expenses $ 8 4 0 , 0 0 0 $ 5 6 Contribution Margin 3 6 0 , 0 0 0 $ 2...
-
You are the Owners project manager on an electrical upgrade project. Existing overhead power lines are being buried in a duct bank to hide them from view in an area of the town that is being...
-
The Slumber Store (TSS) is a national wholesaler of beds, mattresses, pillows, and sheets Key totals from TSS's most recent and forecasted financial statements are presented in the table below. From...
-
Computer Wholesalers restores and resells notebook computers. It originally acquires the notebook computers from corporations upgrading their computer systems, and it backs each notebook it sells...
-
You own a portfolio that is 20 percent invested in Stock X, 35percent in Stock Y, and 45 percent in Stock Z. The expected returnson these three stocks are 8 percent, 11 percent, and 13...
-
What is an alternative way to express your confidence in a measurement? Describe a time you had to make a precise and accurate measurement. Was there room for error? Would significant figures be...
-
Stone House Cafe has a 30 percent tax rate and total taxes of $35,280. What is the value of the interest tax shield if the interest expense is $16,700? SHOW WORK OR I WILL NOT RATE!
-
Factor completely. 5xy - 10xy + 5x
-
A 2-year term life insurance of 400 is issued to (x), with net annual premium determined by the equivalence principle. Calculate the probability that the loss at issue is less than 190, given the...
-
What are the two methods used to translate financial statements and how does the functional currency play a role in determining which method is used?
-
If in 10 flips of a coin we get the same ratio as in Prob. 1(3 heads and 7 tails), is the conclusion the same as in Prob. 1? First conjecture, then compute
-
A laterally insulated homogeneous bar with ends at x = 0 and x = 1 has initial temperature 0. Its left and is kept at 0, whereas the temperature at the right end varies sinusoidally according to u(t,...
-
Solve Prob. 11 by considering complements
-
The price at the intersection of the market demand curve and the market supply curve is called the _________ price, and the quantity is called the _________ quantity.
-
A situation where quantity supplied is greater than quantity demanded is called a(n) _________.
-
The supply of a good _________ if the price of one of its substitutes in production rises.
Study smarter with the SolutionInn App