# Comparing minimas of two different functions

The goal is to find vectors $$x_u$$ and $$y_i$$, both of the same length $$f=64$$, and to do this the following loss function is minimized:

$$\sum_{u, i} (1 + \alpha \cdot r_{ui})(p_{ui} - x_{u}^{T}y_i)^2$$

The vectors obtained are user-factors and item-factors, used for the purpose of rating calculation in a Recommender System. My goal is to find the optimum value of $$\alpha$$, and the initial thought was to try different values for $$\alpha$$, for which I would calculate the min of the above function; in the end, I would pick $$\alpha$$ for which the function has the lowest value. Is this a good idea? How could one find an optimal $$\alpha$$ value?

As for the $$r_{ui}$$ and $$p_{ui}$$ values, these would be the same for each user-item pair in each run of the minimization. Given that the function is used for a Recommender System, one way to identify good $$\alpha$$ would be to withhold a test set, and then check how the recommendations (top $$x_u^{T}y_i$$ values) are recovered.

• Since x and y make a dot product and aren't associated with any other constraints why can't you simplify to using a single variable? Aug 24 at 14:43
• @Richard I was too quick when writing the loss function. Please have a look at (3) in this paper: yifanhu.net/PUB/cf.pdf I took the first part only Aug 24 at 14:53
• A good question should be self-contained. The paper you link might disappear in a year or two rendering your question unintelligible. Please copy all relevant information into the question via the Edit button. Aug 24 at 19:05
• @Richard Ok, I think I've provided all the necessary info now. Aug 25 at 7:10
• It still looks as though either x or y is redundant. Aug 25 at 15:04