subspace_distance.R

``` r
set.seed(27)

n <- 100
k <- 5

A <- matrix(rnorm(n * n), nrow = n, ncol = n)
s <- svd(A, k, k)

# generate two n x k orthonormal bases

u <- s$u
v <- s$v

# following vu and lei (2013) -- inefficient in space
frob_norm_sq1 <- norm(tcrossprod(u) - tcrossprod(v), type = "F")^2 / 2


# following rohe, chatterjee and yu (2011) -- efficient in space
s2 <- svd(crossprod(u, v))
frob_norm_sq2 <- k - sum(s2$d^2)

frob_norm_sq1
#> [1] 4.845742
frob_norm_sq2
#> [1] 4.845742
```

<sup>Created on 2021-03-18 by the [reprex package](https://reprex.tidyverse.org) (v1.0.0.9000)</sup>