20,282 research outputs found
Efficient Orthogonal Tensor Decomposition, with an Application to Latent Variable Model Learning
Decomposing tensors into orthogonal factors is a well-known task in
statistics, machine learning, and signal processing. We study orthogonal outer
product decompositions where the factors in the summands in the decomposition
are required to be orthogonal across summands, by relating this orthogonal
decomposition to the singular value decompositions of the flattenings. We show
that it is a non-trivial assumption for a tensor to have such an orthogonal
decomposition, and we show that it is unique (up to natural symmetries) in case
it exists, in which case we also demonstrate how it can be efficiently and
reliably obtained by a sequence of singular value decompositions. We
demonstrate how the factoring algorithm can be applied for parameter
identification in latent variable and mixture models
From exceptional collections to motivic decompositions via noncommutative motives
Making use of noncommutative motives we relate exceptional collections (and
more generally semi-orthogonal decompositions) to motivic decompositions. On
one hand we prove that the Chow motive M(X) of every smooth proper
Deligne-Mumford stack X, whose bounded derived category D(X) of coherent
schemes admits a full exceptional collection, decomposes into a direct sum of
tensor powers of the Lefschetz motive. Examples include projective spaces,
quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli
spaces. On the other hand we prove that if M(X) decomposes into a direct sum of
tensor powers of the Lefschetz motive and moreover D(X) admits a
semi-orthogonal decomposition, then the noncommutative motive of each one of
the pieces of the semi-orthogonal decomposition is a direct sum of the tensor
unit. As an application we obtain a simplification of Dubrovin's conjecture.Comment: 14 pages; revised versio
Empirical Evaluation of Four Tensor Decomposition Algorithms
Higher-order tensor decompositions are analogous to the familiar Singular Value Decomposition (SVD), but they transcend the limitations of matrices (second-order tensors). SVD is a powerful tool that has achieved impressive results in information retrieval, collaborative filtering, computational linguistics, computational vision, and other fields. However, SVD is limited to two-dimensional arrays of data (two modes), and many potential applications have three or more modes, which require higher-order tensor decompositions. This paper evaluates four algorithms for higher-order tensor decomposition: Higher-Order Singular Value Decomposition (HO-SVD), Higher-Order Orthogonal Iteration (HOOI), Slice Projection (SP), and Multislice Projection (MP). We measure the time (elapsed run time), space (RAM and disk space requirements), and fit (tensor reconstruction accuracy) of the four algorithms, under a variety of conditions. We find that standard implementations of HO-SVD and HOOI do not scale up to larger tensors, due to increasing RAM requirements. We recommend HOOI for tensors that are small enough for the available RAM and MP for larger tensors
Rank properties and computational methods for orthogonal tensor decompositions
The orthogonal decomposition factorizes a tensor into a sum of an orthogonal
list of rankone tensors. We present several properties of orthogonal rank. We
find that a subtensor may have a larger orthogonal rank than the whole tensor
and prove the lower semicontinuity of orthogonal rank. The lower semicontinuity
guarantees the existence of low orthogonal rank approximation. To fit the
orthogonal decomposition, we propose an algorithm based on the augmented
Lagrangian method and guarantee the orthogonality by a novel orthogonalization
procedure. Numerical experiments show that the proposed method has a great
advantage over the existing methods for strongly orthogonal decompositions in
terms of the approximation error.Comment: 19 pages, 2 figures, 3 table
From exceptional collections to motivic decompositions via noncommutative motives
Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X)_Q of every smooth and proper Deligne–Mumford stack X, whose bounded derived category D^b(X) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(X)_Q decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover D^b(X) admits a semiorthogonal decomposition, then the noncommutative motive of each one of the pieces of the
semi-orthogonal decomposition is a direct sum of ⊗-units. As an application we obtain a simplification of Dubrovin’s conjecture
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