Efficient Multivariate Polynomial Interpolation on Unisolvent Nodes to Overcome the Curse of Dimensionality

핵심 개념

This work presents a generalized notion of unisolvent nodes that enables efficient and numerically stable multivariate polynomial interpolation, reaching optimal exponential convergence rates while requiring only a sub-exponential number of nodes. This overcomes the curse of dimensionality that plagues previous approaches.

초록

The key insights and highlights of this work are:

The authors introduce a generalized concept of unisolvent nodes that extends beyond tensorial grids. This allows constructing non-tensorial grids of nodes that are unisolvent for the corresponding polynomial spaces.

They develop multivariate extensions of Newton and Lagrange interpolation schemes that can operate on these generalized unisolvent nodes. The algorithms have at most quadratic runtime and linear memory requirements.

The authors show that for a class of analytic "Trefethen functions", the proposed interpolation schemes can reach the optimal exponential convergence rates, while requiring only a sub-exponential number of nodes. This overcomes the curse of dimensionality that affects previous approaches.

Numerical experiments validate the theoretical predictions, demonstrating that the Runge function can be interpolated to machine precision up to 5 dimensions using the proposed methods.

The authors also introduce the dual notion of unisolvence, which allows determining the polynomial space for which a given set of nodes is unisolvent. This enables polynomial regression on scattered data, including manifolds like the torus.

In summary, the generalized unisolvent nodes and the associated multivariate interpolation schemes presented in this work provide an efficient solution to the long-standing problem of high-dimensional polynomial approximation, lifting the curse of dimensionality.

통계

The Runge function f(x) = 1/(1 + 10||x||₂) can be interpolated to machine precision up to 5 dimensions using the proposed methods.

인용구

"Combining sub-exponential node numbers with exponential approximation rates, non-tensorial unisolvent nodes are thus able to lift the curse of dimensionality for multivariate interpolation tasks."
"Theorem 2.6 and Corollary 2.9 allow establishing a polynomial regression scheme for scattered data on planar and curved manifolds."

핵심 통찰 요약

Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

by Michael Hech... 게시일 arxiv.org 04-17-2024

https://arxiv.org/pdf/2010.10824.pdf

더 깊은 질문

How can the proposed interpolation schemes be extended to handle non-analytic or non-smooth functions?

The proposed interpolation schemes can be extended to handle non-analytic or non-smooth functions by incorporating appropriate modifications to the interpolation process. One approach could involve adapting the interpolation nodes and polynomial spaces to accommodate the characteristics of non-analytic functions. For non-smooth functions, techniques such as piecewise interpolation or spline interpolation could be employed to ensure continuity and differentiability at the interpolation nodes. Additionally, regularization techniques could be applied to handle noisy or irregular data points in the interpolation process. By adjusting the interpolation methodology to suit the specific properties of non-analytic or non-smooth functions, the proposed schemes can effectively handle a broader range of functions.

What are the limitations of the current approach, and how could it be further generalized to handle an even broader class of functions?

One limitation of the current approach is its reliance on polynomial interpolation, which may not be suitable for functions with complex or oscillatory behavior. To overcome this limitation and handle an even broader class of functions, the approach could be extended to include other types of basis functions such as radial basis functions or wavelets. By incorporating a more diverse set of basis functions, the interpolation scheme can better capture the characteristics of a wider range of functions, including those that are non-polynomial or exhibit rapid variations. Additionally, the concept of unisolvence could be generalized to include more flexible node arrangements, allowing for adaptive node placement based on the function's behavior. This adaptability would enhance the approach's ability to interpolate diverse functions effectively.

Can the concepts of generalized unisolvence and sub-exponential node grids be applied to other areas of high-dimensional approximation beyond polynomial interpolation, such as sparse grid methods or deep learning?

Yes, the concepts of generalized unisolvence and sub-exponential node grids can be applied to other areas of high-dimensional approximation beyond polynomial interpolation. In sparse grid methods, the idea of unisolvence can be utilized to optimize the selection of sparse grid points, leading to more efficient and accurate approximations. By strategically placing nodes based on the unisolvence criteria, sparse grid methods can achieve better interpolation results with reduced computational complexity.
In the context of deep learning, the concept of sub-exponential node grids can be beneficial for tasks involving high-dimensional data. By leveraging the principles of unisolvence to determine optimal node configurations, deep learning models can effectively handle complex and multi-dimensional data sets. This approach can enhance the efficiency and scalability of deep learning algorithms, particularly in scenarios where data dimensionality poses a challenge. By incorporating these concepts into sparse grid methods and deep learning frameworks, researchers can improve the accuracy and performance of high-dimensional approximation tasks across various domains.