Loss functions in Instance Representation Learning [R]

In Wu et. al, the MLE objective is computationally infeasible due to the high number of images in the dataset.

Non-parametric Softmax

Negative Log-Likelihood

With large n, the denominator in (2) is hard to compute. Therefore, they use NCE (Noise-Contrastive Estimation).

The NCE Objective

Essentially, they approximate the difficult loss in (3) with the easier to compute loss in (7). However, we end up estimating the denominator anyways in (8). Why not just approximate the denominator in (2) with (8)?

I asked Claude about this and it said something about it being a biased estimator, but I didn't really get that. I'm also a little confused on the connection of the original NCE formulation as being a way to estimate density and the way it is used here; do we do this because NCE loss is easier to compute and as m (the number of noise samples) increases, we get the gradients of NCE loss and gradients of NLL loss to match?