• Publications
  • Influence
Singular values and eigenvalues of tensors: a variational approach
  • Lek-Heng Lim
  • Mathematics
    1st IEEE International Workshop on Computational…
  • 13 December 2005
We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix
Most Tensor Problems Are NP-Hard
TLDR
It is proved that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
TLDR
It is argued that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations, and a natural way of overcoming the ill-posedness of the low-rank approximation problem is proposed by using weak solutions when true solutions do not exist.
Statistical ranking and combinatorial Hodge theory
TLDR
Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistencies.
Symmetric Tensors and Symmetric Tensor Rank
TLDR
The notion of the generic symmetric rank is discussed, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order.
Multilinear PageRank
TLDR
This paper first extends the celebrated PageRank modification to a higher-order Markov chain, then develops convergence theory for a simple fixed-point method, a shifted fixed- point method, and a Newton iteration in a particular parameter regime of multilinear PageRank.
Schubert Varieties and Distances between Subspaces of Different Dimensions
  • Ke Ye, Lek-Heng Lim
  • Mathematics, Computer Science
    SIAM J. Matrix Anal. Appl.
  • 3 July 2014
TLDR
It is shown that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian, which reduces to the Grassman distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space.
Hodge Laplacians on graphs
TLDR
This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph LaPLacian, requiring only knowledge of linear algebra and graph theory.
Nuclear norm of higher-order tensors
TLDR
An analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for Tensor rank is established --- for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm.
Rank aggregation via nuclear norm minimization
TLDR
An algorithm for matrix completion is extended to handle skew-symmetric data and use that to extract ranks for each item and shows a formal recovery result for the noiseless case and presents a detailed study of the algorithm on synthetic data and Netflix ratings.
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