Affine Polynomial Projection

• Affine Polynomial Projection is a method for determining parameter vectors that yield low-rank affine matrix maps using structured low-rank approximation techniques.
• The approach leverages variable projection to reduce optimization dimensionality and efficiently satisfy rank constraints within polynomial projection models.
• Algorithmic advancements include fast evaluations of cost functions, gradients, and Hessians by exploiting block structures in Hankel and mosaic Hankel matrices.

Definition and Overview

The Affine Polynomial Projection Problem is centered around identifying a parameter vector such that a matrix, defined by a linear combination of parameter-dependent terms, achieves low rank while remaining close to given data in a specified norm. This involves techniques like structured low-rank approximation (SLRA) and variable projection methods which reduce problem dimensionality, especially in applications involving Hankel and mosaic Hankel matrices.

Problem Formulation

The SLRA problem addresses general affine structures by minimizing the distance between a parameter vector $p$ and given data $p_D$ under a weighted 2-norm, subject to rank constraints on an affine matrix map $S(p)$. The affine polynomial projection involves embedding constraints as matrix structures relevant to data modeling tasks, particularly for polynomial projections.

§ $$\min_{\widehat{p} \in \mathbb{R}^p}\|p_D - \widehat{p}\|_W \quad \text{subject to} \quad \operatorname{rank} S(\widehat{p}) \le r$$ § $S(p) = S_0 + \sum_{k=1}^p p_k S_k$ Here, $W$ is a weighting matrix allowing fixed elements by setting weights to infinity.

Variable Projection Principle

The variable projection method leverages the quadratic relationship in $p$ but non-linearity in rank constraints expressed through kernel matrices. This reduction in dimensionality from $p$ to $dm$ where $d = m-r$ allows optimization on a low manifold and supports manifold-based nonlinear optimization routines facilitating efficient solutions.

§ $R S(\widehat{p}) = 0$ § $f(R) := \min_{\widehat{p} \|p_D - \widehat{p}\|_W^2 \quad \text{subject to}\quad R S(\widehat{p}) = 0$

Algorithmic Development

Algorithms for evaluating cost functions, gradients, and Hessian approximations are constructed to exploit matrix structures, enabling efficient computation. Mosaic Hankel matrices exhibit block structures, allowing fast evaluations relevant to system identification or signal processing tasks.

Complexity Results

• General case: $O(m^2 n)$ for $m \times n$ mosaic Hankel matrices.
• Block/Toeplitz structure: $O(m n)$ per evaluation using block-wise weights.

Application to Affine Polynomial Projection Problems

The SLRA method can directly apply to polynomial projection problems via embedding constraints into matrix structures such as Hankel and Toeplitz matrices. Fixed elements are treated using infinite weighting schemes. Block structures in system identification demonstrate algorithmic efficiency improvements.

Extensions and Implications

The methodology spans beyond mosaic Hankel matrices, extending application across affine structures with efficient block computations within SLRA problems. Significant computing savings are achieved compared to previous naive methods through dimensional reduction and structural exploitation.

Projected Algorithms in Related Research

Relevant theoretical foundation covers the application of dual methods, interior-point methods, and alternating direction methods across problem constraints reflecting polynomial optimizations, ensuring scalability and efficiency (Henrion et al., 2011). Examples include conic projections onto intersecting affine subspaces supporting polynomial optimization formulations.

Conclusion

The Affine Polynomial Projection Problem explores minimizing the discrepancy between polynomial projections and actual values under constraint systems, emphasizing computational efficiency through matrix structure exploitation, reductions in problem dimensionality, and advancements in applicable optimization algorithms. This framework aids robust polynomial approximations in expansive data modeling fields.