[Paper Review] Splinets -- efficient orthonormalization of the B-splines

Xijia Liu, Hiba Nassar+1 more|arXiv (Cornell University)|Oct 16, 2019

Advanced Numerical Analysis Techniques7 references4 citations

TL;DR

This paper introduces splinets—efficiently orthonormalized B-spline bases that preserve locality and symmetry while achieving computational efficiency through a hierarchical, dyadic orthogonalization process. The method reduces inner product evaluations to O(n) for B-splines of order k, with total support size growing as O(log n), significantly outperforming prior orthogonalized bases that scale as O(n).

ABSTRACT

A new efficient orthogonalization of the B-spline basis is proposed and contrasted with some previous orthogonalized methods. The resulting orthogonal basis of splines is best visualized as a net of functions rather than a sequence of them. For this reason, the basis is referred to as a splinet. The splinets feature clear advantages over other spline bases. They efficiently exploit 'near-orthogonalization' featured by the B-splines and gains are achieved at two levels: locality that is exhibited through small size of the total support of a splinet and computational efficiency that follows from a small number of orthogonalization procedures needed to be performed on the B-splines to achieve orthogonality. These efficiencies are formally proven by showing the asymptotic rates with respect to the number of elements in a splinet. The natural symmetry of the B-splines in the case of the equally spaced knots is preserved in the splinets, while quasi-symmetrical features are also seen for the case of arbitrarily spaced knots.

Motivation & Objective

• To develop an efficient orthonormalization of B-spline bases that preserves their locality and structural symmetry.
• To reduce computational cost in functional signal decomposition by minimizing the number of inner product evaluations during orthogonalization.
• To construct a basis (splinets) with near-orthogonal support properties that scale logarithmically with the number of knots.
• To provide a generic algorithm for sparse diagonalization of band matrices, applicable beyond splines.
• To enable practical implementation via an R package leveraging the numerical and structural advantages of splinets.

Proposed method

• Proposes a hierarchical, one-sided orthogonalization process applied to B-splines in a dyadic knot structure, starting from the finest level.
• Uses a recursive procedure where each spline is orthogonalized only with respect to its two parent splines in the hierarchy, exploiting disjoint supports.
• Applies a modified Gram-Schmidt process with symmetry preservation, ensuring quasi-symmetrical features even for non-uniform knots.
• Derives a transformation matrix P such that P^T H P = I, where H is the Gram matrix of B-splines, enabling efficient spectral decomposition.
• Employs a stopping rule based on error norms (e.g., ‖ε‖² < 10⁻¹⁶ after 3 iterations), ensuring negligible numerical error.
• Introduces a computationally convenient representation of splines using the orthonormalized basis, facilitating numerical implementation.

Experimental results

Research questions

• RQ1 Can B-spline bases be orthogonalized in a way that preserves their locality and computational efficiency?
• RQ2 What is the asymptotic support size of an orthonormalized B-spline basis, and how does it compare to existing methods?
• RQ3 How many inner products are required to achieve orthonormality using the proposed method versus classical Gram-Schmidt?
• RQ4 Can the orthogonalization process be generalized to arbitrary band matrices with symmetric, banded structure?
• RQ5 To what extent do numerical errors in the iterative orthogonalization process affect the final orthonormality and stability?

Key findings

• The total support size of splinets grows as O(log n), compared to O(n) for previous orthogonalized bases, achieving near-locality.
• For B-splines of order k, the number of inner products required for orthogonalization is J²ₙ = (5k−1)/4 · n − (2k²/log 2) log(n+1) + ..., which is significantly less than the O(n²) of classical Gram-Schmidt.
• After three iterations of the orthogonalization loop, the error norm drops below 10⁻¹⁶, indicating numerical convergence to machine precision.
• The transformation matrix P is sparse and nearly banded, with most non-zero entries being negligible, enabling efficient storage and computation.
• The method preserves symmetry in the equispaced case and exhibits quasi-symmetry for irregular knots, maintaining structural fidelity.
• The approach generalizes to any positive definite band matrix with 2k−1 diagonals, enabling efficient diagonalization via the same algorithm.