Suppose we want to classify a review as good ($1$) or bad ($0$). We have a training data set of $10,000$ reviews. Also, suppose we have a vocabulary of $100,000$ words $w_1, \dots, w_{100,000}$. So the data is a matrix of dimension $100,000 \times 10,000$. Let's represent each of the words in the reviews using a bag-of-words approach over tf-idf values. Also, we normalize the rows such that they sum to $1$.
In a logistic regression approach, would we have $10,000$ different logistic regression models as follows:
$$ \log \left(\frac{p}{1-p} \right)_{1} = \beta_{0_{1}} + \beta_{1_{1}}w_{11} + \dots + \beta_{100,000_{1}}w_{100,000} \\ \vdots \\ \log \left(\frac{p}{1-p} \right)_{10,000} = \beta_{0_{10,000}} + \beta_{1_{10,000}}w_{11} + \dots + \beta_{100,000_{10,000}}w_{100,000}$$
So are we estimating $100,000 \times 10,000$ coefficients?
