I have a variant on linear regression. Can anyone tell me what it's called / point me to more info about it / tell me that it's (trivially reducible to / nothing like) standard linear regression?

Standard linear regression has a known matrix X = x(i,j) and a known target vector Y = y(j), and seeks to find weights W = w(i) to best approximate X * W = Y.

In my version, instead of knowing the values of the input variables (X), I know how much each contributes to the output. So I don't know x(i,j) but I kind of know x(i,j) * w(i), except that W isn't really a thing. And I know some structure on X: every value is either 0, or equal to every other value in its row. (I can tell those apart because the 0s contribute zero and the others contribute nonzero.) I want to find the best W to approximate X * W = Y, but that question will depend on what I want to do with the uncertainty in X, and I'm not sure about that.

I should probably avoid giving my specific scenario, so think widget sales. You can either sell a widget in a city or not. Sales of a widget will be well-correlated between cities: if widget sells well in New York, it will probably sell well in Detroit and in Austin and so on, with the caveat that selling well in New York means a lot more sales than selling well in Austin. I have a list of previous widgets, and how much they sold in each city. Received wisdom is that a widget will sell about twice as much in New York as in Detroit, and a third more than in Austin, but I want to improve on the received wisdom.

So I'm told that a widget will sell 10 million, and that it will be sold in (list of cities). I want to come up with the best estimate for its sales in New York, its sales in Austin, etc.

Hopefully this is clear?