In this paper we propose and study an optimization problem over a matrix group orbit that we call (GOO). We prove that GOO can be used to induce matrix decomposition techniques such as singular value decomposition (SVD), LU decomposition, QR decomposition, Schur decomposition and Cholesky decomposition, etc. This gives rise to a unified framework for matrix decomposition and allows us to bridge these matrix decomposition methods. Moreover, we generalize GOO for tensor decomposition. As a concrete application of GOO, we devise a new data decomposition method over a special linear group to normalize point cloud data. Experiment results show that our normalization method is able to obtain recovery well from distortions like shearing, rotation and squeezing.
In @cite_23 the following formulation is used to get the Ky-Fan @math -norm @cite_25 of a matrix @math when @math and @math : [ ^ m k , ^ = , ^ n k , ^ = ( ^ ) . ]
@cite_25 @cite_23
191339423 2952509110
A method of manufacturing a semiconductor device, in particular a monolithic integrated circuit, in which highly doped zones are provided according to a given pattern on one side of a monocrystalline silicon substrate body by local diffusion of at least one impurity in a substantially flat surface of the substrate body and the substrate surface on said side is given a profile in a pattern which corresponds to the pattern of the highly doped zones, after which an epitaxial silicon layer is provided on said side and one or more semiconductor circuit elements are then formed while using at least one photoresist step, characterized in that the substantially flat substrate surface is given a crystal orientation lying between a 001 face and an adjacent 111 face, which orientation deviates at least 10 DEG from the said 001 face and at least 15 DEG from said 111 face and is present in a strip within 10 DEG from the crystallographic zone formed by the said two faces. We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results hold even when the row space is highly coherent, in contrast with previous analyses. In the absence of noise, we show that one can exactly recover a @math matrix of rank @math from merely @math matrix entries. We also show that one can recover an order @math tensor using @math entries. For noisy recovery, our algorithm consistently estimates a low rank matrix corrupted with noise using @math entries. We complement our study with simulations that verify our theory and demonstrate the scalability of our algorithms.