Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study distributed low rank approximation in which the matrix to be approximated is only implicitly represented across the different servers. For example, each of s servers may have an n×d matrix At, and  may be interested in computing a low rank approximation to A = ƒ(Σt=1sAt), where ƒ is a function which is applied entrywise to the matrix Σt=1sAt.  show for a wide class of functions ƒ it is possible to efficiently compute a d × d rank-k projection matrix P for which ∥A − AP∥F2 ≤ ∥A − [A]k∥F2 + e ∥A∥F2, where AP denotes the projection of A onto the row span of P, and [A]k denotes the best rank-k approximation to A given by the singular value decomposition. The communication cost of our protocols is d·(sk=e)O(1), and they succeed with high probability. Our framework allows us to efficiently compute a low rank approximation to an entry-wise softmax, to a Gaussian kernel expansion, and to M-Estimators applied entrywise (i.e., forms of robust low rank approximation).  also show that our additive error approximation is best possible, in the sense that any protocol achieving relative error for these problems requires significantly more communication. Finally,  experimentally validate our algorithms on real datasets.
['David P. Woodruff', 'Peilin Zhong']
I am writing a research paper on a idea Distributed low rank approximation of implicit functions of a matrix
[{"from": "human", "value": "I am writing a research paper on a idea Distributed low rank approximation of implicit functions of a matrix"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study distributed low rank approximation in which the matrix to be approximated is only implicitly represented across the different servers. For example, each of s servers may have an n\u00d7d matrix At, and  may be interested in computing a low rank approximation to A = \u0192(\u03a3t=1sAt), where \u0192 is a function which is applied entrywise to the matrix \u03a3t=1sAt.  show for a wide class of functions \u0192 it is possible to efficiently compute a d \u00d7 d rank-k projection matrix P for which \u2225A \u2212 AP\u2225F2 \u2264 \u2225A \u2212 [A]k\u2225F2 + e \u2225A\u2225F2, where AP denotes the projection of A onto the row span of P, and [A]k denotes the best rank-k approximation to A given by the singular value decomposition. The communication cost of our protocols is d\u00b7(sk=e)O(1), and they succeed with high probability. Our framework allows us to efficiently compute a low rank approximation to an entry-wise softmax, to a Gaussian kernel expansion, and to M-Estimators applied entrywise (i.e., forms of robust low rank approximation).  also show that our additive error approximation is best possible, in the sense that any protocol achieving relative error for these problems requires significantly more communication. Finally,  experimentally validate our algorithms on real datasets."}]