The Random Projection Method
Santosh S. Vempala
αα»αααα 2005 Β· DIMACS Series ααααα
ααΈ 65 Β· American Mathematical Soc.
ααααα
βα’αα‘α·α
ααααΌαα·α
105
ααααα
reportααΆαααΆαααααα αα·αααα·ααΆαααααααα·αααααΌαααΆααααααααααΆαααα αααααααααααααα
α’αααΈααααα βα’αα‘α·α ααααΌαα·αααα
Random projection is a simple geometric technique for reducing the dimensionality of a set of points in Euclidean space while preserving pairwise distances approximately. The technique plays a key role in several breakthrough developments in the field of algorithms. In other cases, it provides elegant alternative proofs. The book begins with an elementary description of the technique and its basic properties. Then it develops the method in the context of applications, which are divided into three groups. The first group consists of combinatorial optimization problems such as maxcut, graph coloring, minimum multicut, graph bandwidth and VLSI layout. Presented in this context is the theory of Euclidean embeddings of graphs. The next group is machine learning problems, specifically, learning intersections of halfspaces and learning large margin hypotheses. The projection method is further refined for the latter application. The last set consists of problems inspired by information retrieval, namely, nearest neighbor search, geometric clustering and efficient low-rank approximation. Motivated by the first two applications, an extension of random projection to the hypercube is developed here. Throughout the book, random projection is used as a way to understand, simplify and connect progress on these important and seemingly unrelated problems. The book is suitable for graduate students and research mathematicians interested in computational geometry.
ααΆααααααααααα βα’αα‘α·α ααααΌαα·αααα
ααααΆααααΎαα’αααΈααΆααααααΎαααααα’αααα
α’αΆαβααααααΆα
ααΌαααααααααΆααα αα·αβααααααα
ααα‘αΎααααααα·ααΈ Google Play Books αααααΆαα Android αα·α iPad/iPhone α ααΆβααααΎααααΆαααααβαααααααααααααααα·ααΆαα½αβααααΈβααααα’αααβ αα·αβα’αα»ααααΆαα±ααβα’αααα’αΆααααβααΆαα’ααΈαααΊαα·α α¬ααααΆαβα’ααΈαααΊαα·αβαα
αααααααΈααααααα
αα»αααααΌαααβαα½ααα αα·ααα»αααααΌααα
α’αααα’αΆα
ααααΆααααααα
ααΆααα‘αααααααΆααα·ααα
αααα»α Google Play αααααααΎαααααα·ααΈαα»αααααΆαα’ααΈαααΊαα·ααααα»ααα»αααααΌαααααααα’αααα
eReaders αα·αβα§αααααβαααααβααα
ααΎααααΈα’αΆααα
ααΎβα§ααααα e-ink ααΌα
ααΆβα§αααααα’αΆαβααααα
α’αα‘α·α
ααααΌαα·α Kobo α’αααααΉαααααΌαβααΆαααβα―αααΆα α αΎαβαααααααΆαα
βα§αααααβααααα’αααα ααΌαα’αα»ααααααΆαβααΆαααααΆααααα’α·αααααααααααααααααα½α ααΎααααΈαααααα―αααΆαβαα
α§αααααα’αΆαααααα
βα’αα‘α·α
ααααΌαα·ααααααααΆααα
