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Tensor Representation of Non-linear Models Using Cross Approximations

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Abstract

Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed.

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Notes

  1. We shall use the term tensorization throughout the entire paper for the sake of conciseness.

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Authors and Affiliations

  1. High Performance Computing Institute, Ecole Centrale de Nantes, 1 rue de la Noë, 44321, Nantes, France

    José V. Aguado, Domenico Borzacchiello & Kiran S. Kollepara

  2. ESI Group Chair at PIMM, ENSAM ParisTech, 151 boulevard de l’Hôpital, 75013, Paris, France

    Francisco Chinesta

  3. Laboratori de Càlcul Numèric (LaCaN), Universitat Politècnica de Catalunya, BarcelonaTech, 08034, Barcelona, Spain

    Antonio Huerta

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  1. José V. Aguado
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Appendices

Appendix A: Coefficients Definition for Alternating Directions Optimization

Coefficients \(\beta _*^r\in \mathbb {R}\), \(\tilde{\varvec{\alpha }}_*^r\in \mathbb {R}^{M}\) and \(\tilde{\varvec{\gamma }}_*\in \mathbb {R}^{S}\) in Eq. (10) are defined as follows:

$$\begin{aligned} \begin{aligned} \beta _*^r&:= \prod _{d=0,d\ne *}^{D}{ \langle \mathbf {A}_{d}^r{\varvec{v}}_{d},{\varvec{v}}_{d} \rangle _{{d}}},\\ \tilde{\varvec{\alpha }}_*^r&:= \varvec{\alpha } \,\circ \left( \bigodot _{d=0,d\ne *}^{D}{ \langle \mathbf {A}_{d}^r\mathbf {W}_{d},{\varvec{v}}_{d} \rangle _{{d}} }\right) ^T \quad \text {and}\\ \tilde{\varvec{\gamma }}_*&:= \varvec{\gamma } \,\circ \left( \bigodot _{d=0,d\ne *}^{D}{ \langle \mathbf {V}_{d},{\varvec{v}}_{d} \rangle _{{d}} }\right) ^T, \end{aligned} \end{aligned}$$

where “\(\bullet ^T\)” denotes the transpose and “\(\circ \)” stands for the Hadamard (component-wise) product. Recall that \(\varvec{\alpha }\) contains the representation coefficients of the rank-M approximation of the solution, while \(\varvec{\gamma }\) are the representation coefficients of the rank-S representation of the non-linear term. See Eqs. (6) and (8), respectively.

Using the coefficients defined above, we can define the following quantities:

$$\begin{aligned} \begin{aligned} \tilde{\mathbf {A}}_*:= \sum _{r=1}^{R}{\beta ^r_*\mathbf {A}_*^r} \quad \text {and} \quad \tilde{{\varvec{b}}}_*:= - \sum _{r=1}^{R}{\mathbf {A}_*^r\mathbf {W}_*\tilde{\varvec{\alpha }}_*^r} - \mathbf {V}_*\tilde{\varvec{\gamma }}_*, \end{aligned} \end{aligned}$$

that can be introduced in Eq. (10), leading to the following minimization problem:

$$\begin{aligned} {\varvec{w}}_*= \arg \min _{{\varvec{v}}_*\in \mathbb {R}^{N_*}} \frac{1}{2} \langle \tilde{\mathbf {A}}_*{\varvec{v}}_*,{\varvec{v}}_* \rangle _*+ \langle \tilde{{\varvec{b}}}_*,{\varvec{v}}_* \rangle _*, \end{aligned}$$
(38)

whose solution is \({\varvec{w}}_*= \tilde{\mathbf {A}}_*^{-1}\,\tilde{{\varvec{b}}}_*\).

Appendix B: Fiber Search Algorithm in Detail

Let \(\mathbf {P}:=\bigodot _{d=0}^{D}{\mathbf {P}_d}\in \mathbb {R}^{N\times 1}\) be an extractor matrix, with \(\mathbf {P}_*:= \mathbf {I}\in \mathbb {R}^{N_*\times N_*}\). Additionally, \(\mathbf {P}_d\in \mathbb {R}^{N_d\times 1}\), for \(d\ne *\), are extractor matrices full of zeros, except the entry \(\rho _d\) (see Sect. 4.1), which is equal to one.

The restriction of the approximation residual onto direction “\(*\)” can be written as follows:

$$\begin{aligned} {\varvec{r}}_*:= \mathbf {P}^T({\varvec{f}}({\varvec{u}}_M^\ell ) - {\varvec{f}}_{S}) \equiv {\varvec{f}}(\mathbf {P}^T{\varvec{u}}_M^\ell ) - \mathbf {P}^T{\varvec{f}}_{S}. \end{aligned}$$
(39)

Recalling the definition of both \({\varvec{u}}_M^\ell \) and \({\varvec{f}}_S\), we arrive to:

$$\begin{aligned} {\varvec{r}}_*= {\varvec{f}}(\mathbf {W}_*\,\tilde{\varvec{\alpha }}_*) - \mathbf {V}_*\,\tilde{\varvec{\gamma }}_*, \end{aligned}$$
(40)

where the coefficients \(\tilde{\varvec{\alpha }}_*\in \mathbb {R}^M\) and \(\tilde{\varvec{\gamma }}_*\in \mathbb {R}^{S}\) are defined as follows:

$$\begin{aligned} \begin{aligned} \tilde{\varvec{\alpha }}_*&:= \varvec{\alpha } \,\circ \left( \bigodot _{d=0,d\ne *}^{D}{\mathbf {P}_d^T\mathbf {W}_d} \right) ^T,\quad \text {and}\\ \tilde{\varvec{\gamma }}_*&:= \varvec{\gamma } \,\circ \left( \bigodot _{d=0,d\ne *}^{D}{\mathbf {P}_d^T\mathbf {V}_d} \right) ^T. \end{aligned} \end{aligned}$$

Here “\(\circ \)” stands for the Hadamard (component-wise) product.

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Aguado, J.V., Borzacchiello, D., Kollepara, K.S. et al. Tensor Representation of Non-linear Models Using Cross Approximations. J Sci Comput 81, 22–47 (2019). https://doi.org/10.1007/s10915-019-00917-2

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