Chebyshev-Polynomial-Based Algorithm Overview

• Chebyshev-polynomial-based algorithms exploit exponential approximation and interpolation to efficiently invert operators and approximate analytic functions.
• They enable distributed implementations with local computations, leveraging sparse polynomial filters for rapid convergence in graph filtering.
• Empirical studies confirm that these methods achieve lower per-iteration costs and faster error decay compared to gradient descent and classical polynomial approximations.

A Chebyshev-polynomial-based algorithm is any algorithm that exploits properties of Chebyshev polynomials—particularly their exponential approximation rates for analytic functions and their structured recurrences—to achieve computational or analytic tasks with enhanced efficiency, accuracy, or scalability. Such algorithms arise in approximation theory, numerical linear algebra, optimization, spectral graph theory, cryptography, and scientific computing. The common foundation is the utilization of Chebyshev polynomials' powerful approximation, interpolation, and spectral properties, typically in settings where such properties can yield exponential convergence, robust stability, or computational acceleration.

1. Exponential Approximation and Chebyshev Interpolation

Chebyshev interpolation polynomials exhibit spectral (i.e., exponential) convergence when approximating analytic functions over intervals or multidimensional cubes. For a function analytic in a neighborhood of a compact interval or box, Chebyshev interpolation achieves an error bound

$$\sup_{t\in[\mu,\,\nu]} |1 - h(t)\,C_M(t)| \leq D\,r^M$$

for some constants $D>0$ and $0 < r < 1$, where $C_M$ is the Chebyshev interpolation polynomial, $h$ is the function under consideration, and $M$ is the maximum degree used in the polynomial (Cheng et al., 19 Apr 2025). This property underpins the construction of highly efficient polynomial approximations for both univariate and multivariate analytic functions, crucial for the iterative inversion of polynomial operators on graphs or similar settings.

2. Algorithmic Realization: Iterative Polynomial Approximation for Graph Inverse Filtering

A prominent concrete instance is the "Chebyshev-interpolation-polynomial-based algorithm" (CIPA) for the distributed inversion of graph polynomial filters. Given a signal $y$ and a polynomial graph operator $H=h(S_1,\dots,S_d)$, with $S_i$ commutative graph shift matrices (such as adjacency or Laplacian operators), the objective is to recover $x = H^{-1}y$. Direct inversion is often dense and not distributed; however, approximating $H^{-1}$ by a low-degree polynomial $G = C_M(S_1,\ldots,S_d)$ (where $C_M$ is a multivariate Chebyshev interpolation polynomial to $f=1/h$) yields an iterative scheme: $$\begin{cases} e^{(m)} = Hx^{(m-1)} - y \ x^{(m)} = x^{(m-1)} - G e^{(m)} \end{cases}$$ where $x^{(0)}$ is arbitrary and $G$ ensures the operator $I-GH$ is a contraction on the relevant spectrum (Cheng et al., 19 Apr 2025). This iteration converges exponentially: $$\|x^{(m)} - H^{-1}y\|_2 \leq C r^m \|x^{(0)} - H^{-1}y\|_2$$ with $r < 1$, as guaranteed by the exponential approximation property of $C_M$.

3. Structural Features and Distributed Implementation

The CIPA structure leverages three essential features:

• Commutativity of graph shifts: $S_iS_j = S_jS_i$, enabling joint spectral analysis and diagonalization.
• Locality of polynomial filters: $G$ and $H$ act via sparse polynomials in $S_i$, so each iteration involves only local computations and one-hop communication—the communication per node is $O(\sum_{k} L_k)$, and computation per node is $O(\prod_{k}(L_k+1))$.
• Storage and computational efficiency: Each node stores local polynomial coefficients and neighbor connections, requiring only $O(D_H + D_G)$ memory and arithmetic operations per iteration, where $D_H = \prod_k(L_k+1)$, $D_G = \prod_k(M+1)$.

These properties permit large-scale, fully distributed implementations where each agent/node processes and exchanges only local data (Cheng et al., 19 Apr 2025).

4. Comparative Analysis: Convergence and Complexity

Chebyshev-polynomial-based iterative algorithms such as CIPA offer key advantages compared to classical approaches:

• Classical Chebyshev polynomial approximation (CPA): Applies a single high-degree polynomial filter to approximate $H^{-1}y$ in one step; to reach an accuracy $\epsilon$ typically requires $M = O(\log(1/\epsilon))$.
• Gradient Descent (GD): Requires $O(D_H)$ operations and converges linearly with rate $(\mathrm{cond}(H)-1)/(\mathrm{cond}(H)+1)$.
• CIPA: Distributes the approximation across multiple low-degree polynomial steps: a moderate M yields a robust contraction factor $b_M<1$ such that $\epsilon$ accuracy is achieved after $O(\log(1/\epsilon))$ iterations—a much lower per-iteration cost and better empirical convergence (few iterations with modest $M$ suffice in practice).

Empirical simulations confirm that for typical problem sizes ($N=1000$), CIPA with $M=3$ achieves $10^{-3}$ relative error in four iterations, whereas GD requires over $20$ iterations, and CPA needs polynomials of degree $10$ for comparable accuracy (Cheng et al., 19 Apr 2025).

5. Simulation and Application Results

Practical implications are highlighted by applications in signal processing on graphs:

• On circulant graphs, CIPA achieves rapid error decay with minimal iterations and low polynomial degree.
• In Tikhonov denoising of complex datasets (e.g., the "walking dog" motion with a product graph structure), CIPA achieves higher output SNR with fewer iterations than GD, with ARMA filters underperforming unless heavily regularized.
• These findings demonstrate that Chebyshev-polynomial-based iterative schemes combine robust convergence, low per-node computational cost, and straightforward decomposability for large, resource-constrained networks (Cheng et al., 19 Apr 2025).

6. Broader Contexts and Theoretical Implications

Chebyshev-polynomial-based algorithms are not exclusive to graph filtering:

• In numerical linear algebra, Chebyshev-based recurrences underlie efficient simulation of Gaussian Markov random fields, sparse matrix functions, and preconditioned iterative solvers.
• Chebyshev interpolation and spectral properties are the foundation for high-accuracy polynomial approximation in scientific computing.
• Algorithms based on Chebyshev polynomials are used for root-finding, validated numerics, and optimization, exploiting their minimal Lebesgue constants and optimality in the uniform norm.
• In distributed systems and signal processing, these algorithms achieve parallelism and communication efficiency not available to classical direct or Newton-type methods.

The unifying principle is leveraging the exponential approximation power of Chebyshev polynomials, their stable recurrences, and their natural fit for structured, distributed, or recursive computational frameworks.

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References:

• "Iterative Polynomial Approximation Algorithms for Inverse Graph Filters" (Cheng et al., 19 Apr 2025)