For an operator on a Banach space , let be the collection of all its invariant subspaces. We consider the index function on and we show, amongst others, that if is a bounded below operator and if , , then If in additi...For an operator on a Banach space , let be the collection of all its invariant subspaces. We consider the index function on and we show, amongst others, that if is a bounded below operator and if , , then If in addition are index 1 invariant subspaces of , with nonzero intersection, we show that . Furthermore, using the index function, we provide an example where for some , holds .展开更多
Various index structures have recently been proposed to facilitate high-dimensional KNN queries, among which the techniques of approximate vector presentation and one-dimensional (1D) transformation can break the curs...Various index structures have recently been proposed to facilitate high-dimensional KNN queries, among which the techniques of approximate vector presentation and one-dimensional (1D) transformation can break the curse of dimensionality. Based on the two techniques above, a novel high-dimensional index is proposed, called Bit-code and Distance based index (BD). BD is based on a special partitioning strategy which is optimized for high-dimensional data. By the definitions of bit code and transformation function, a high-dimensional vector can be first approximately represented and then transformed into a 1D vector, the key managed by a B+-tree. A new KNN search algorithm is also proposed that exploits the bit code and distance to prune the search space more effectively. Results of extensive experiments using both synthetic and real data demonstrated that BD out- performs the existing index structures for KNN search in high-dimensional spaces.展开更多
文摘For an operator on a Banach space , let be the collection of all its invariant subspaces. We consider the index function on and we show, amongst others, that if is a bounded below operator and if , , then If in addition are index 1 invariant subspaces of , with nonzero intersection, we show that . Furthermore, using the index function, we provide an example where for some , holds .
基金Project (No. [2005]555) supported by the Hi-Tech Research and De-velopment Program (863) of China
文摘Various index structures have recently been proposed to facilitate high-dimensional KNN queries, among which the techniques of approximate vector presentation and one-dimensional (1D) transformation can break the curse of dimensionality. Based on the two techniques above, a novel high-dimensional index is proposed, called Bit-code and Distance based index (BD). BD is based on a special partitioning strategy which is optimized for high-dimensional data. By the definitions of bit code and transformation function, a high-dimensional vector can be first approximately represented and then transformed into a 1D vector, the key managed by a B+-tree. A new KNN search algorithm is also proposed that exploits the bit code and distance to prune the search space more effectively. Results of extensive experiments using both synthetic and real data demonstrated that BD out- performs the existing index structures for KNN search in high-dimensional spaces.
