Exploring Bit-Difference for Approximate KNN Search in High-dimensional Databases

In this paper, we develop a novel index structure to support efficient approximate k-nearest neighbor (KNN) query in high-dimensional databases. In high-dimensional spaces, the computational cost of the distance (e.g., Euclidean distance) between two points contributes a dominant portion of the overall query response time for memory processing. To reduce the distance computation, we first propose a structure (BID) using BIt-Difference to answer approximate KNN query. The BID employs one bit to represent each feature vector of point and the number of bit-difference is used to prune the further points. To facilitate real dataset which is typically skewed, we enhance the BID mechanism with clustering, cluster adapted bitcoder and dimensional weight, named the BID⁺. Extensive experiments are conducted to show that our proposed method yields significant performance advantages over the existing index structures on both real life and synthetic high-dimensional datasets.

Coordinate-Independent Computations on Differential Equations

This project investigates the computational representation of differentiable manifolds, with the primary goal of solving partial differential equations using multiple coordinate systems on general n- dimensional spaces. In the process, this abstraction is used to perform accurate integrations of ordinary differential equations using multiple coordinate systems. In the case of linear partial differential equations, however, unexpected difficulties arise even with the simplest equations.

Formalizing Triggers: A Learning Model for Finite Spaces

In a recent seminal paper, Gibson and Wexler (1993) take important steps to formalizing the notion of language learning in a (finite) space whose grammars are characterized by a finite number of parameters. They introduce the Triggering Learning Algorithm (TLA) and show that even in finite space convergence may be a problem due to local maxima. In this paper we explicitly formalize learning in finite parameter space as a Markov structure whose states are parameter settings. We show that this captures the dynamics of TLA completely and allows us to explicitly compute the rates of convergence for TLA and other variants of TLA e.g. random walk. Also included in the paper are a corrected version of GW's central convergence proof, a list of "problem states" in addition to local maxima, and batch and PAC-style learning bounds for the model.