Thread by @ccanonne_

Test your intuition (or, in my case, lack thereof): hypothesis testing in high dimensions. You are given n i.i.d. samples from an unknown multivariate Gaussian p = N(μ,Σ) in ℝ^d. You want to know if p is *the* standard Gaussian N(0,I). 1/8

Now, as a first reminder, the total variation distance b/w N(μ,Σ) and N(0,I) is "basically" captured by
max(||μ||₂, ||Σ-I||_F)
So if we know that Σ=I (the unknown p is spherical with unit variance), then TV distance between p and N(0,I) is equivalent to ||μ||₂. 2/8

As a second reminder, remember we are considering testing here. Learning gives an upper bound, of course, but learning μ (resp. p) to ℓ₂ (resp. TV) distance ε may not be the optimal thing. OK, now here we go. 3/8

Question 1: How many samples are necessary and sufficient (how big n has to be) to test p=N(0,I) v. TV(p,N(0,I)) > ε, *assuming p has identity covariance matrix*?

(Up to constants, and thinking of ε as small) 4/8

Question 2: How many samples are necessary and sufficient (how big n has to be) to test p=N(0,I) v. ||μ||₂ > ε, *assuming p has identity covariance matrix*?

(Up to constants, and thinking of ε as small) 5/8

Question 3: How many samples are necessary and sufficient (how big n has to be) to test p=N(0,I) v. TV(p,N(0,I)) > ε, *assuming nothing on the covariance matrix Σ of p*?

(Up to constants, and thinking of ε as small) 6/8

Question 4 (and last): How many samples are necessary and sufficient (how big n has to be) to test p=N(0,I) v. ||μ||₂ > ε, *assuming nothing on the covariance matrix Σ of p*?

(Up to constants, and thinking of ε as small) 7/8

I'll give the answers tomorrow: in the meantime, feel free to chime in and comment below (note: as far as I know, the answer the Q4 was unknown—or at least not publicly available—until a few weeks ago. (cc/ @thegautamkamath))

tl;dr: what is normal may surprise you. 8/8

@thegautamkamath .@threadreaderapp please, unroll.