Measure the covariance estimation rate

CodingIntermediate⏱ 1 hourHigh-Dim Statisticssimulationcovariancerateoperator norm

Problem

Empirically verify the rate $\|\hat\Sigma-\Sigma\| \asymp \sqrt{p/n}+p/n$ for Gaussian data with $\Sigma = I_p$ .

Fix $p$ and sweep $n$ ; plot the error on log–log axes and fit the slope.
Fix $n/p$ and sweep both; confirm the error depends only on the ratio.
Compare the measured error with the theoretical $(1+\sqrt{p/n})^2-1$ from the Marchenko–Pastur upper edge.

Implement it, then check the implementation against something you can verify independently.

Solve it here7 tests · Python runs in your browser

⌘/Ctrl + ↵ to run · limit 60s · draft saved in this browser

7 tests will run against your code. The first run downloads the Python runtime — several megabytes, once per browser.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Problem	Field	Type	Difficulty	Time
Simulate distance contrast run	Classical ML	Coding	Beginner	30 min
Balls into bins, with one choice and with two run	AI Systems	Coding	Intermediate	1 hour
Bounding the spectral norm by row and column sums	Foundations	Proof	Intermediate	30 min
Cache hit rates under Zipf popularity run	AI Systems	Coding	Intermediate	1 hour
Frobenius versus operator norm for random matrices	Random Matrices	Numerical	Intermediate	30 min