In various applications, data in multidimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere $S^(m−1)$ by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original data as the cost function, this paper offers an analytic expression of the projected gradient which, on one hand, furnishes the first order optimality condition and, on the other hand, can be used as a numerical means for solving this problem.
On the low rank approximation of data on the unit sphere
DEL BUONO, Nicoletta;LOPEZ, Luciano;
2005-01-01
Abstract
In various applications, data in multidimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere $S^(m−1)$ by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original data as the cost function, this paper offers an analytic expression of the projected gradient which, on one hand, furnishes the first order optimality condition and, on the other hand, can be used as a numerical means for solving this problem.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
