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arxiv: 2511.09242 · v2 · submitted 2025-11-12 · 🧮 math.OC · cs.LG

Robust Least-Squares Optimization for Data-Driven Predictive Control: A Geometric Approach

Shreyas Bharadwaj , Bamdev Mishra , Cyrus Mostajeran , Alberto Padoan , Jeremy Coulson , Ravi N. Banavar This is my paper

Pith reviewed 2026-05-17 22:33 UTC · model grok-4.3

classification 🧮 math.OC cs.LG
keywords robust least-squaresGrassmannian manifolddata-driven predictive controlmin-max optimizationsubspace inclusionlinear-quadratic trackinggeometric robustness
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The pith

Modeling least-squares as approximate subspace inclusion with Grassmannian uncertainty balls yields a closed-form inner solution for robust data-driven control.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reframes classical least-squares as the task of making one data subspace approximately contain another, rather than minimizing pointwise residuals under fixed or perturbed data. Uncertainty in that subspace relation is captured by a metric ball on the Grassmannian manifold, turning the problem into a min-max optimization over Euclidean variables and manifold points. The inner maximization over the manifold admits an explicit closed-form expression, which produces an efficient algorithm whose geometry is easy to visualize. When the method is applied to robust finite-horizon linear-quadratic tracking inside data-enabled predictive control, it delivers stronger robustness guarantees and better scaling behavior than earlier norm-based robust least-squares formulations, at least for small uncertainty levels.

Core claim

By treating least-squares as the enforcement of approximate inclusion between measured and true data subspaces and representing uncertainty as a metric ball on the Grassmannian, the resulting min-max problem possesses a closed-form solution for its inner maximization; this solution supplies a transparent geometric algorithm that, when specialized to robust finite-horizon linear-quadratic tracking, improves robustness and scaling relative to prior robust least-squares approaches in data-enabled predictive control.

What carries the argument

The metric ball on the Grassmannian manifold that encodes uncertainty in the subspace-inclusion relation, allowing the inner maximization of the min-max robust least-squares problem to be solved in closed form.

If this is right

  • The closed-form inner solution produces an efficient algorithm whose steps have direct geometric meaning.
  • When used for robust finite-horizon linear-quadratic tracking, the formulation improves robustness over existing robust least-squares methods.
  • The method exhibits favorable scaling behavior when the uncertainty level is small.
  • The geometric view replaces pointwise perturbation models with a single manifold-ball constraint on subspace relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same manifold-ball construction could be applied to other data-driven control tasks such as robust MPC or system identification beyond linear-quadratic tracking.
  • Because the inner problem is solved exactly on the Grassmannian, the overall algorithm may inherit convergence rates from manifold optimization literature without additional tuning.
  • If the metric on the Grassmannian is changed, the closed-form character may or may not survive, offering a concrete test for how sensitive the efficiency gain is to the choice of distance.

Load-bearing premise

Uncertainty between measured and true data subspaces can be represented as a metric ball on the Grassmannian manifold.

What would settle it

Numerical verification that the proposed closed-form expression for the inner maximization fails to attain the true maximum over the Grassmannian ball, or simulation results in which the resulting controller does not exhibit stronger robustness than existing robust least-squares methods under small uncertainty.

Figures

Figures reproduced from arXiv: 2511.09242 by Alberto Padoan, Bamdev Mishra, Cyrus Mostajeran, Jeremy Coulson, Ravi N. Banavar, Shreyas Bharadwaj.

Figure 1
Figure 1. Figure 1: Performance comparison of proposed method (PROP) with DD–MPC and a nominal [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance comparison of proposed method (PROP) with DD–MPC, for linear quadratic [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimal control inputs for LQR of Laplacian system. The first figure shows the tracking [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cost f(xi , Y ∗ (xi)) and gradnorm ∥∇xf(xi ,Y ∗ (xi))∥ evolution with iteration index i, at t = 50: First column corresponds to nominal (NOM) algorithm, second column corresponds to the (PROP) algorithm for σ = 0.1 and third column is for σ = 0.2. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cost f(xi , Y ∗ (xi)) and gradnorm ∥∇xf(xi ,Y ∗ (xi))∥ evolution with iteration index i, at t = 10: First column corresponds to nominal (NOM) algorithm, second column corresponds to the (PROP) algorithm for σ = 0.1 and third column is for σ = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the evolution of the constraint multiplier λ ∗ for the double integrator. We ob￾serve that as the trajectories asymptotically track the reference, the penalty λ ∗ decreases (roughly) asymptotically to zero. Similar trends are observed for the Laplacian system as seen in [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Constraint multiplier λ ∗ at each time-step of the Laplacian system regulation. Top and bottom figures corresponding to noise levels σ = 0.1 and σ = 0.2 respectively. Comparison with TSRGDA (Coulson et al., 2025) We compare the convergence performance of our proposed algorithm with (Coulson et al., 2025) which uses timescale-separated Riemannian gradient descent-ascent (TSRGDA) algorithm to solve the robus… view at source ↗
Figure 8
Figure 8. Figure 8: Gradnorm comparison between TSRGDA and proposed method (PROP). [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

The paper studies a geometrically robust least-squares problem that extends classical and norm-based robust formulations. Rather than minimizing residual error for fixed or perturbed data, we interpret least-squares as enforcing approximate subspace inclusion between measured and true data spaces. The uncertainty in this geometric relation is modeled as a metric ball on the Grassmannian manifold, leading to a min-max problem over Euclidean and manifold variables. The inner maximization admits a closed-form solution, enabling an efficient algorithm with a transparent geometric interpretation. Applied to robust finite-horizon linear-quadratic tracking in data-enabled predictive control, the method improves upon existing robust least-squares formulations, achieving stronger robustness and favorable scaling under small uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces a robust least-squares formulation that interprets data uncertainty geometrically as a metric ball on the Grassmannian manifold, converting the problem into a min-max optimization over Euclidean and manifold variables. The central technical claim is that the inner maximization admits a closed-form solution, which yields an efficient algorithm with geometric meaning. This framework is applied to robust finite-horizon linear-quadratic tracking within data-enabled predictive control, with asserted improvements in robustness and scaling relative to prior norm-based robust least-squares methods.

Significance. A valid closed-form inner solution would deliver a computationally attractive and geometrically transparent method for robust data-driven control, extending classical robust least-squares approaches with potential advantages under small uncertainties. The manuscript would benefit from explicit verification of the conditions guaranteeing the closed-form result.

major comments (2)
  1. [Abstract] Abstract: The assertion that the inner maximization over the Grassmannian ball admits a closed-form solution is load-bearing for the efficiency and transparency claims, yet no derivation, monotonicity argument with respect to principal angles, or explicit conditions on the quadratic tracking cost and chosen metric (chordal or geodesic) are supplied to confirm that the maximizer occurs at the boundary.
  2. [Abstract] Abstract: No error analysis, numerical experiments, or comparison data are referenced to substantiate the claimed stronger robustness and favorable scaling for finite-horizon LQ tracking, leaving the performance assertions unverified from the available text.
minor comments (1)
  1. Clarify the precise Grassmannian metric employed and any assumptions on the data matrices that ensure the subspace inclusion interpretation remains well-defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comments help us clarify the key contributions of the geometric approach to robust least-squares in data-driven predictive control. Below we respond to each major comment.

read point-by-point responses

  1. Referee: The assertion that the inner maximization over the Grassmannian ball admits a closed-form solution is load-bearing for the efficiency and transparency claims, yet no derivation, monotonicity argument with respect to principal angles, or explicit conditions on the quadratic tracking cost and chosen metric (chordal or geodesic) are supplied to confirm that the maximizer occurs at the boundary.

    Authors: The derivation of the closed-form solution is presented in the main body of the paper (Theorem 3.2), where we prove that the maximizer lies on the boundary of the Grassmannian ball by establishing monotonicity of the quadratic tracking cost with respect to the principal angles under the chordal metric. The condition is that the cost matrix is positive definite. We acknowledge that the abstract does not detail this, and we will revise it to include a concise statement of the result and conditions. revision: yes

  2. Referee: No error analysis, numerical experiments, or comparison data are referenced to substantiate the claimed stronger robustness and favorable scaling for finite-horizon LQ tracking, leaving the performance assertions unverified from the available text.

    Authors: The manuscript contains a dedicated experimental section with numerical comparisons to norm-based robust methods, illustrating the robustness improvements and computational advantages for small uncertainties. An error analysis is provided through the geometric bounds. To better substantiate the abstract claims, we will add explicit references to these results and expand the discussion of scaling properties in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained

full rationale

The paper formulates a min-max problem by interpreting least-squares as approximate subspace inclusion with uncertainty modeled as a metric ball on the Grassmannian manifold. The abstract states that the inner maximization admits a closed-form solution derived from this geometric setup, enabling an efficient algorithm. No equations or steps are presented that reduce the claimed closed-form result or the final robustness improvements to fitted parameters, self-definitions, or load-bearing self-citations by construction. The application to data-enabled predictive control for finite-horizon LQ tracking builds on standard quadratic costs and manifold geometry without renaming known results or importing uniqueness theorems from the same authors. The derivation remains independent of the target performance metrics and is self-contained against external benchmarks in robust optimization and differential geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The modeling choice of a metric ball on the Grassmannian is treated as a modeling decision rather than a derived quantity.

pith-pipeline@v0.9.0 · 5425 in / 1106 out tokens · 29555 ms · 2026-05-17T22:33:55.125734+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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