Journal of Combinatorial Theory Series A
Modeling the Shape of the Scene: A Holistic Representation of the Spatial Envelope
International Journal of Computer Vision
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
A tutorial on spectral clustering
Statistics and Computing
Graph Laplacians and their Convergence on Random Neighborhood Graphs
The Journal of Machine Learning Research
Towards a theoretical foundation for Laplacian-based manifold methods
Journal of Computer and System Sciences
Optimal construction of k-nearest-neighbor graphs for identifying noisy clusters
Theoretical Computer Science
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
SHREC'12 track: generic 3D shape retrieval
EG 3DOR'12 Proceedings of the 5th Eurographics conference on 3D Object Retrieval
ACM Transactions on Knowledge Discovery from Data (TKDD) - Special Issue on ACM SIGKDD 2012
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Ranking is one of the key problems in information retrieval. Recently, there has been significant interest in a class of ranking algorithms based on the assumption that data is sampled from a low dimensional manifold embedded in a higher dimensional Euclidean space. In this paper, we study a popular graph Laplacian based ranking algorithm [23] using an analytical method, which provides theoretical insights into the ranking algorithm going beyond the intuitive idea of "diffusion." Our analysis shows that the algorithm is sensitive to a commonly used parameter due to the use of symmetric normalized graph Laplacian. We also show that the ranking function may diverge to infinity at the query point in the limit of infinite samples. To address these issues, we propose an improved ranking algorithm on manifolds using Green's function of an iterated unnormalized graph Laplacian, which is more robust and density adaptive, as well as pointwise continuous in the limit of infinite samples. We also for the first time in the ranking literature empirically explore two variants from a family of twice normalized graph Laplacians. Experimental results on text and image data support our analysis, which also suggest the potential value of twice normalized graph Laplacians in practice.