• You are encouraged to discuss ideas with each other; but
• you must acknowledge your collaborator, and
• you must compose your own writeup and/or code independently.
• We strongly encourage answers to theory questions in LaTeX, and answers to coding questions
in Python (Jupyter notebooks)
- (5 points) Let {x1, x2, . . . , xn} be a set of points in d-dimensional space. Suppose we wish to
produce a single point estimate µ ∈ R
d
that minimizes the mean squared-error:
1
n
kx1 − µk
2
2 + kx2 − µk
2
2 + . . . + kxn − µk
2
2
Find a closed form expression for µ and prove that your answer is correct.
- (10 points) Not all norms behave the same; for instance, the
1-norm of a vector can be dramatically different from the2-norm, especially in high dimensions. Prove the following
norm inequalities for d-dimensional vectors, starting from the definitions provided in class and
lecture notes. (Use any algebraic technique/result you like, as long as you cite it.)
a. kxk∞ ≤ kxk2 ≤
√
dkxk∞
b. kxk∞ ≤ kxk1 ≤ dkxk∞ - (10 points) In this problem, you will practice using Python for exploratory data analysis.
• Open the Colab notebook from the following URL: click here
• Use File > Save a Copy in Drive to create an editable copy in your own Google Drive
• As you work through the notebook, look for the text and code cells marked with a TODO
at the top. In these cells, answer the questions or fill in code as indicated.
• When you are finished, run the entire notebook from beginning to end, using Runtime >
Run All. Check the output and make sure everything looks OK.
• Then, you can save your notebook as a PDF (to upload along with the rest of your
submission) using File > Print.