Linear regression on sparse matrix?

I have a matrix with sparse data. A small extract from it is seen below. The columns represent years and the rows represent different race tracks. The feature values are velocities on that specific track a specific year. Generally the velocity increases with the year but that is not necessarily true. As seen below the matrix is sparse and for some tracks I only have values for a single year. How can one most accurately predict the missing values? I suspect one can use some kind of algorithm that resembles a recommender system like the ones used for recommending movies. Those are also based on matrices with lots of missing values but I don't really know how to adapt this to my problem.

I'd be very grateful for any hints on how to solve this.

/Hanson