کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
8898204 1631324 2018 11 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Similarity matrix framework for data from union of subspaces
ترجمه فارسی عنوان
چارچوب ماتریس شباهت برای داده ها از اتحاد از زیر فضای
کلمات کلیدی
تقسیم بندی زیر فضای، اتحاد زیرمجموعه ها، خوشه بندی داده ها، ماتریس مشابهی، شکل ماتریس تعامل، تجزیه اسکلت،
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
چکیده انگلیسی
This paper presents a framework for finding similarity matrices for the segmentation of data W=[w1⋯wN]⊂RD drawn from a union U=⋃i=1MSi of independent subspaces {Si}i=1M of dimensions {di}i=1M. It is shown that any factorization of W=BP, where columns of B form a basis for data W and they also come from U, can be used to produce a similarity matrix ΞW. In other words, ΞW(i,j)≠0, when the columns wi and wj of W come from the same subspace, and ΞW(i,j)=0, when the columns wi and wj of W come from different subspaces. Furthermore, ΞW=Qdmax, where dmax=max⁡{di}i=1M and Q∈RN×N with Q(i,j)=|PTP(i,j)|. It is shown that a similarity matrix obtained from the reduced row echelon form of W is a special case of the theory. It is also proven that the Shape Interaction Matrix defined as VVT, where W=UΣVT is the skinny singular value decomposition of W, is not necessarily a similarity matrix. But, taking powers of its absolute value always generates a similarity matrix. An interesting finding of this research is that a similarity matrix can be obtained using a skeleton decomposition of W. First, a square sub-matrix A∈Rr×r of W with the same rank r as W is found. Then, the matrix R corresponding to the rows of W that contain A is constructed. Finally, a power of the matrix PTP where P=A−1R provides a similarity matrix ΞW. Since most of the data matrices are low-rank in many subspace segmentation problems, this is computationally efficient compared to other constructions of similarity matrices.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Applied and Computational Harmonic Analysis - Volume 45, Issue 2, September 2018, Pages 425-435
نویسندگان
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