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Geometrically robust least squares through manifold optimization

Jeremy Coulson, Alberto Padoan, Cyrus Mostajeran

TL;DR

The paper addresses robust least squares under geometric perturbations by formulating a minimax problem on a product manifold, with $x \in \mathbb{R}^n$ and $y \in \mathrm{Gr}(k,n)$, and the subspace variable constrained to a ball in the Grassmannian. It introduces an exact penalty for the ball constraint and a smoothing step to obtain a differentiable objective, solved via a first-order gradient descent-ascent method (TSRGDA) on the product manifold, using a chordal distance on Grassmannians and a Stiefel-based implementation. Key contributions include the manifold-based minimax formulation, the exact-penalty plus smoothing approach, and demonstration through a numerical example that converges to a local minimax point, with implications for subspace tracking and data-driven control. The framework provides a robust, geometry-aware method for least squares problems where the linear model is tied to subspace perturbations, offering practical impact in signal processing and control applications.

Abstract

This paper presents a methodology for solving a geometrically robust least squares problem, which arises in various applications where the model is subject to geometric constraints. The problem is formulated as a minimax optimization problem on a product manifold, where one variable is constrained to a ball describing uncertainty. To handle the constraint, an exact penalty method is applied. A first-order gradient descent ascent algorithm is proposed to solve the problem, and its convergence properties are illustrated by an example. The proposed method offers a robust approach to solving a wide range of problems arising in signal processing and data-driven control.

Geometrically robust least squares through manifold optimization

TL;DR

The paper addresses robust least squares under geometric perturbations by formulating a minimax problem on a product manifold, with and , and the subspace variable constrained to a ball in the Grassmannian. It introduces an exact penalty for the ball constraint and a smoothing step to obtain a differentiable objective, solved via a first-order gradient descent-ascent method (TSRGDA) on the product manifold, using a chordal distance on Grassmannians and a Stiefel-based implementation. Key contributions include the manifold-based minimax formulation, the exact-penalty plus smoothing approach, and demonstration through a numerical example that converges to a local minimax point, with implications for subspace tracking and data-driven control. The framework provides a robust, geometry-aware method for least squares problems where the linear model is tied to subspace perturbations, offering practical impact in signal processing and control applications.

Abstract

This paper presents a methodology for solving a geometrically robust least squares problem, which arises in various applications where the model is subject to geometric constraints. The problem is formulated as a minimax optimization problem on a product manifold, where one variable is constrained to a ball describing uncertainty. To handle the constraint, an exact penalty method is applied. A first-order gradient descent ascent algorithm is proposed to solve the problem, and its convergence properties are illustrated by an example. The proposed method offers a robust approach to solving a wide range of problems arising in signal processing and data-driven control.

Paper Structure

This paper contains 4 sections, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometrically robust least squares
  3. Example
  4. Conclusion

Figures (2)

  • Figure 1: The points $y_*$ and $P_{y_*}x_*$ obtained from iteratively applying update rule \ref{['eq:update']} are plotted as a green solid line and large black dot, respectively. The projected iterates $P_{y_k}x_k$ are plotted as small black dots. The boundaries of $\mathbb{B}^d_{\rho}(\hat{y})$ are depicted with dashed lines and its interior is shaded.
  • Figure 2: Frobenius norm of the gradient of the objective function in \ref{['eq:least_squares_robust_smooth']} at the iterates of update rule \ref{['eq:update']}.