Visualizing Neural Networks with the Grand Tour

The implied semantics of direct manipulation is that when a user drags an UI element (in this case, an axis handle), they are signaling to the system that they wished that the corresponding data point had been projected to the location where the UI element was dropped, rather than where it was dragged from. In our case the overall projection is a rotation (originally determined by the Grand Tour), and an arbitrary user manipulation might not necessarily generate a new projection that is also a rotation. Our goal, then, is to find a new rotation which satisfies the user request and is close to the previous state of the Grand Tour projection, so that the resulting state satisfies the user request. In a nutshell, when user drags the ithi {th}ith axis handle by (dx,dy)(dx, dy)(dx,dy), we add them to the first two entries of the ithi {th}ith row of the Grand Tour matrix, and then perform Gram-Schmidt orthonormalization on the rows of the new matrix. Rows have to be reordered such that the ithi {th}ith row is considered first in the Gram-Schmidt procedure.