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Non-parametric B-spline decoupling of multivariate functions

Joppe De Jonghe, Mariya Ishteva

TL;DR

This work introduces CMTF-BSD, a decoupling framework that uses a non-parametric B-spline representation for the internal univariate functions in a multivariate decoupling model $\mathbf{f}(\mathbf{x}) = \mathbf{W}_1 \mathbf{g}(\mathbf{W}_0 \mathbf{x})$. By formulating a coupled matrix-tensor factorization that integrates zeroth-order information from function values and enforcing a B-spline structure on $\mathbf{g}$, the method achieves flexible, constraint-friendly internal function shaping. The approach preserves the computational efficiency of projection-based CPD methods while enabling nonnegativity and monotonicity constraints on the internal functions, demonstrated on synthetic examples including trigonometric and monotone cases. The results indicate stable, sub-1% Jacobian reconstruction errors across various spline configurations, and the monotonicity constraint provides guaranteed monotone internal functions, suggesting strong potential for applications in system identification and neural network compression.

Abstract

Many scientific fields and applications require compact representations of multivariate functions. For this problem, decoupling methods are powerful techniques for representing the multivariate functions as a combination of linear transformations and nonlinear univariate functions. This work introduces an efficient decoupling algorithm that leverages the use of B-splines to allow a non-parametric estimation of the decoupling's internal functions. The use of B-splines alleviates the problem of choosing an appropriate basis, as in parametric methods, but still allows an intuitive way to tweak the flexibility of the estimated functions. Besides the non-parametric property, the use of B-spline representations allows for easy integration of nonnegativity or monotonicity constraints on the function shapes, which is not possible for the currently available (non-)parametric decoupling methods. The proposed algorithm is illustrated on synthetic examples that highlight the flexibility of the B-spline representation and the ease with which a monotonicity constraint can be added. The examples also show that if monotonic functions are required, enforcing the constraint is necessary.

Non-parametric B-spline decoupling of multivariate functions

TL;DR

This work introduces CMTF-BSD, a decoupling framework that uses a non-parametric B-spline representation for the internal univariate functions in a multivariate decoupling model . By formulating a coupled matrix-tensor factorization that integrates zeroth-order information from function values and enforcing a B-spline structure on , the method achieves flexible, constraint-friendly internal function shaping. The approach preserves the computational efficiency of projection-based CPD methods while enabling nonnegativity and monotonicity constraints on the internal functions, demonstrated on synthetic examples including trigonometric and monotone cases. The results indicate stable, sub-1% Jacobian reconstruction errors across various spline configurations, and the monotonicity constraint provides guaranteed monotone internal functions, suggesting strong potential for applications in system identification and neural network compression.

Abstract

Many scientific fields and applications require compact representations of multivariate functions. For this problem, decoupling methods are powerful techniques for representing the multivariate functions as a combination of linear transformations and nonlinear univariate functions. This work introduces an efficient decoupling algorithm that leverages the use of B-splines to allow a non-parametric estimation of the decoupling's internal functions. The use of B-splines alleviates the problem of choosing an appropriate basis, as in parametric methods, but still allows an intuitive way to tweak the flexibility of the estimated functions. Besides the non-parametric property, the use of B-spline representations allows for easy integration of nonnegativity or monotonicity constraints on the function shapes, which is not possible for the currently available (non-)parametric decoupling methods. The proposed algorithm is illustrated on synthetic examples that highlight the flexibility of the B-spline representation and the ease with which a monotonicity constraint can be added. The examples also show that if monotonic functions are required, enforcing the constraint is necessary.

Paper Structure

This paper contains 14 sections, 19 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations and tensor background
  3. Notations
  4. Canonical polyadic decomposition
  5. Decoupling
  6. The decoupling problem
  7. Representation of the internal functions
  8. Optimization problem
  9. Algorithm
  10. Experiments and results
  11. Metrics
  12. Effect of DoF and degree of spline representation
  13. Monotonically increasing internal functions
  14. Conclusion

Figures (3)

  • Figure 1: Decoupling of a multivariate function $\mathbf{f}(\mathbf{x})$ (left) into the model $\mathbf{f}(\mathbf{x}) = \mathbf{W}_1 \mathbf{g}(\mathbf{W}_0\mathbf{x})$ (right) constitutes a linear transformation of the input by $\mathbf{W}_0$, followed by branches of univariate functions and a final linear transformation by $\mathbf{W}_1$.
  • Figure 2: Results for applying the CMTF-BSD(.) algorithm \ref{['alg:CMTF-BSD']} to the sytem $\mathbf{f}_{trig}$ of equation \ref{['eq:sys_trig']}, for $30$ executions per $(d,df)$ pair where $d \in \{1,2,3\}$ and $df \in \{4,6,\hdots,28\}$. Top figure: the relative reconstruction error of the Jacobian tensor $\mathcal{J}$; middle figure: the relative error of the computed system for the first output; bottom figure: the relative error for the second output. The red dotted line indicates an error of $1\%$.
  • Figure 3: Reconstruction errors for the Jacobian tensor $\mathcal{J}$, for $30$ executions of the CMTF-BSD algorithm \ref{['alg:CMTF-BSD']} and degrees of freedom $df \in \{8,10,12,\hdots,20\}$, with and without monotonicity constraint. The monotonic (+) results compute the coefficients on line $11$ of algorithm \ref{['alg:Bspline_projection']} with nonnegative least squares to retrieve monotonically increasing functions.