Approximation of High-Dimensional Rank One Tensors


Many real world problems are high-dimensional in that their solution is a function which depends on many variables or parameters. This presents a computational challenge since traditional numerical techniques are built on model classes for functions based solely on smoothness. It is known that the approximation of smoothness classes of functions suffers from the so-called ‘curse of dimensionality’. Avoiding this curse requires new model classes for real world functions that match applications. This has led to the introduction of notions such as sparsity, variable reduction, and reduced modeling. One theme that is particularly common is to assume a tensor structure for the target function. This paper investigates how well a rank one function f(x 1,…,x d )=f 1(x 1)⋯f d (x d ), defined on Ω=[0,1]d can be captured through point queries. It is shown that such a rank one function with component functions f j in \(W^{r}_{\infty}([0,1])\) can be captured (in L ) to accuracy O(C(d,r)N r) from N well-chosen point evaluations. The constant C(d,r) scales like d dr. The queries in our algorithms have two ingredients, a set of points built on the results from discrepancy theory and a second adaptive set of queries dependent on the information drawn from the first set. Under the assumption that a point zΩ with nonvanishing f(z) is known, the accuracy improves to O(dN r).

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This research was supported by the Office of Naval Research Contracts ONR N00014-08-1-1113, ONR N00014-09-1-0107, and ONR N00014-11-1-0712; the AFOSR Contract FA95500910500; the NSF Grants DMS-0810869, and DMS 0915231; and the DFG Special Priority Program SPP-1324. This research was done when R.D. was a visiting professor at RWTH and the AICES Graduate Program. This publication is based in part on work supported by Award No. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).

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Correspondence to Wolfgang Dahmen.

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Communicated by Vladimir N. Temlyakov.

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Bachmayr, M., Dahmen, W., DeVore, R. et al. Approximation of High-Dimensional Rank One Tensors. Constr Approx 39, 385–395 (2014).

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  • Query algorithms
  • High-dimensional approximation
  • Separable functions
  • Rate of approximation

Mathematics Subject Classification (2000)

  • 41A25
  • 65D15