Before revision:  Our work extends the problem coverage to optimal sketched LRD, least-squares regression (LS), and ] k-means clustering .
After revision:  Our work extends the problem coverage to optimal sketched LRD, least-squares regression (LS), and ], the sketch matrix is found by choosing a random sparse matrix, e.g., the CountSketch, and then updating the values of the non-zero entries by running gradient descent on a training data set. Despite the growing body of work on this paradigm, a noticeable omission is that the locations of the non-zero entries of previous algorithms were fixed, and only their values were learned. In this work we propose the first learning algorithm that also optimizes the locations of the non-zero entries. We show this algorithm gives better accuracy for low rank approximation than previous work, and apply it to other problems such as k-means clustering .