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Abstract: We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and th

by andreyf 11y ago

Abstract: We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors.

ZenoArrow 11y ago
Thanks andreyf. I'm not familiar with most of those terms, but I'm guessing it's a big deal if it's got a decent number of upvotes. Do these findings have any benefits that a layperson might be able to understand (you can assume some programming knowledge if that's relevant)?
- cinquemb 11y ago
  I was wondering about how someone would extend LSA techniques into tensor rank-k spaces where k > 2, so one could add as many datapoints/features collected other than terms and frequency of such (like parts of speech, sound clips, etc). I also want to use something like this for spacial filtering of eeg data. Ideally, i would use the Jacobi-Davidson Method[0], but armadillo only has n x m partial singular value decomposition. [0] http://netlib.org/utk/people/JackDongarra/etemplates/node340.html http://netlib.org/utk/people/JackDongarra/etemplates/node340...
- lsorber 11y ago
  Attempt at a short summary: A matrix M has an SVD U S V', where U and V have orthonormal columns and S is a diagonal matrix. How does this generalize to multidimensional arrays? A third-order tensor T has a multilinear SVD of the form S x (U, V, W), where S is a core tensor and U, V, and W are again orthonormal matrices. In the matrix SVD, U transforms the columns of S, while V transforms its rows (hence it appears on the right of S). The product S x (U, V, W) is similar: U transforms the columns of the core tensor S, V its rows and W the mode-3 vectors of S. The authors show that slices of S must be orthogonal under the inner product, e.g., <S(:,i,:), S(:,j,:)> = c * delta(i,j). In the case of a matrix SVD, this reduces to S being diagonal. An example application is predicting how a user would rate a movie at a given time by approximating (the known entries of) a user x movie x time tensor with a truncated multilinear SVD.

by andreyf 11y ago

ZenoArrow 11y ago

Thanks andreyf. I'm not familiar with most of those terms, but I'm guessing it's a big deal if it's got a decent number of upvotes. Do these findings have any benefits that a layperson might be able to understand (you can assume some programming knowledge if that's relevant)?

cinquemb 11y ago
I was wondering about how someone would extend LSA techniques into tensor rank-k spaces where k > 2, so one could add as many datapoints/features collected other than terms and frequency of such (like parts of speech, sound clips, etc). I also want to use something like this for spacial filtering of eeg data. Ideally, i would use the Jacobi-Davidson Method[0], but armadillo only has n x m partial singular value decomposition. [0] http://netlib.org/utk/people/JackDongarra/etemplates/node340.html http://netlib.org/utk/people/JackDongarra/etemplates/node340...
lsorber 11y ago
Attempt at a short summary: A matrix M has an SVD U S V', where U and V have orthonormal columns and S is a diagonal matrix. How does this generalize to multidimensional arrays? A third-order tensor T has a multilinear SVD of the form S x (U, V, W), where S is a core tensor and U, V, and W are again orthonormal matrices. In the matrix SVD, U transforms the columns of S, while V transforms its rows (hence it appears on the right of S). The product S x (U, V, W) is similar: U transforms the columns of the core tensor S, V its rows and W the mode-3 vectors of S. The authors show that slices of S must be orthogonal under the inner product, e.g., <S(:,i,:), S(:,j,:)> = c * delta(i,j). In the case of a matrix SVD, this reduces to S being diagonal. An example application is predicting how a user would rate a movie at a given time by approximating (the known entries of) a user x movie x time tensor with a truncated multilinear SVD.