I do not know what is the notation, $\ell$, means in the below formula about negative sampling and noise contrastive estimation (NCE) because I don't have a strong math background. Does it stands for log? Is it the ell symbol for sequence space; if so, what does sequence space means here?
Additionally, what does $\mathcal{N}$ means and the bold upper $\mathsf{T}$ means? I always wonder if the authors made mistakes or not.
$$ \sum_{t=1}^{\mathsf{T}}\sum_{c\in c_{t}}\ell(s(w_{t}, w_{c})) + \sum_{n\in \mathcal{N}_{t,c}}\ell(-s(w_{t}, n)) $$
This from the book, Deep Learning for NLP and Speech Recognition (Kamath et al., 2019) in the 5th Chapter for Distribution Representations, the authors go over word2vec and negative sampling for skip-gram. In particular, they explain the differences between negative sampling and NCE.
Attaching Page for More Context
Along with the following articles:
- NCE https://arxiv.org/pdf/1809.01812.pdf
- Negative sampling https://arxiv.org/pdf/1310.4546.pdf