arxiv: 2604.05374 · v1 · submitted 2026-04-07 · 💻 cs.LG

Recognition: no theorem link

LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers

Sunbochen Tang , Andrea Goertzen , Navid Azizan

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:40 UTC · model grok-4.3

classification 💻 cs.LG

keywords linear matrix inequalitiesdifferentiable projection layersneural networksDouglas-Rachford splittingcontrol certificatessemidefinite programminginvariant sets

0 comments

The pith

LMI-Net transforms generic neural networks into models that satisfy linear matrix inequality constraints exactly by inserting a differentiable projection layer.

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

The paper develops LMI-Net to enforce hard LMI constraints inside neural networks for control and certification tasks. It lifts the constraint set to an affine equality intersected with the positive semidefinite cone, solves the projection with Douglas-Rachford splitting during the forward pass, and uses implicit differentiation for the backward pass. Theoretical guarantees establish that the layer converges to a feasible point. This approach bridges learning-based methods with formal verification by ensuring constraint satisfaction by construction rather than through soft penalties that may break under distribution shift.

Core claim

LMI-Net establishes that a generic neural network can be turned into a reliable model satisfying LMI constraints through a modular differentiable projection layer that converges to a feasible point, with the projection performed via Douglas-Rachford splitting on the lifted affine-PSD formulation and implicit differentiation for training.

What carries the argument

The differentiable projection layer that lifts LMI constraints to the intersection of an affine equality constraint and the positive semidefinite cone, then applies Douglas-Rachford splitting for the forward pass with implicit differentiation for backpropagation.

If this is right

Trained models satisfy the LMI constraints exactly by construction rather than approximately.
Feasibility rates improve substantially over soft-constrained baselines when inputs shift from the training distribution.
Inference speed remains comparable to an unconstrained network because the projection is efficient and avoids runtime SDP solves.
The method supports joint learning of controllers and certificates for families of disturbed linear systems.

Where Pith is reading between the lines

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

The lifting-plus-splitting construction could be reused to enforce other convex matrix constraints inside learned models.
LMI-Net layers might be stacked or composed with existing neural modules to maintain invariance properties in learned dynamical systems.
Hybrid pipelines could alternate between LMI-Net training steps and occasional full SDP verification for added rigor.

Load-bearing premise

The LMI feasible set can be reliably lifted to an affine equality intersected with the PSD cone such that Douglas-Rachford splitting converges efficiently and differentiably without excessive cost or loss of expressivity.

What would settle it

A case in which the projection layer output violates the original LMI for a valid input or fails to converge to feasibility within a fixed iteration budget on a standard LMI problem.

Figures

Figures reproduced from arXiv: 2604.05374 by Andrea Goertzen, Navid Azizan, Sunbochen Tang.

read the original abstract

Linear matrix inequalities (LMIs) have played a central role in certifying stability, robustness, and forward invariance of dynamical systems. Despite rapid development in learning-based methods for control design and certificate synthesis, existing approaches often fail to preserve the hard matrix inequality constraints required for formal guarantees. We propose LMI-Net, an efficient and modular differentiable projection layer that enforces LMI constraints by construction. Our approach lifts the set defined by LMI constraints into the intersection of an affine equality constraint and the positive semidefinite cone, performs the forward pass via Douglas-Rachford splitting, and supports efficient backward propagation through implicit differentiation. We establish theoretical guarantees that the projection layer converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints. Evaluated on experiments including invariant ellipsoid synthesis and joint controller-and-certificate design for a family of disturbed linear systems, LMI-Net substantially improves feasibility over soft-constrained models under distribution shift while retaining fast inference speed, bridging semidefinite-program-based certification and modern learning techniques.

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.

Desk Editor's Note private letter to a colleague

LMI-Net adds a DR-based differentiable projection layer to enforce LMIs in NNs, but the convergence claim needs checking when the feasible set is empty for some inputs.

read the letter

The main thing here is a modular projection layer that lifts an LMI to an affine-plus-PSD intersection, runs Douglas-Rachford splitting in the forward pass, and uses implicit differentiation for the backward pass. This turns a generic network into one that satisfies the LMI by construction, at least when a solution exists. The experiments on invariant ellipsoid synthesis and joint controller-certificate design for disturbed linear systems report higher feasibility rates than soft-constrained baselines, especially under distribution shift, while keeping inference fast. That is the practical bridge the abstract advertises, and it builds on standard convex primitives without obvious circularity. The construction itself looks new enough in its combination for this setting. A clear soft spot is the case when the LMI set is empty for a given input. Douglas-Rachford only converges to a point in the intersection when that intersection is nonempty; the abstract states convergence to a feasible point without saying what the layer outputs or whether the result still meets the original LMI when no certificate exists. Under distribution shift or strong disturbances this can happen, so the “by construction” certification is not automatic for arbitrary inputs. The full derivations and any fallback behavior need to be verified. This work is aimed at people doing learning-based control who want hard LMI certificates rather than penalties. A reader already familiar with SDP methods and implicit layers will get the most out of the experiments and the projection details. It deserves a serious referee to check the proofs, the edge-case handling, and the experimental protocols, because the core idea connects two areas that have stayed separate.

Referee Report

1 major / 1 minor

Summary. The paper introduces LMI-Net, a modular architecture that augments generic neural networks with a differentiable projection layer enforcing linear matrix inequality (LMI) constraints by construction. The layer lifts the LMI set {y | A(y) ≽ 0} (A affine) to the intersection of an affine subspace and the PSD cone, applies Douglas-Rachford splitting in the forward pass, and uses implicit differentiation for backpropagation. Theoretical guarantees are claimed for convergence of the projection to a feasible point; experiments on invariant ellipsoid synthesis and joint controller-certificate design for disturbed linear systems report substantially improved feasibility over soft-constrained baselines under distribution shift while preserving fast inference.

Significance. If the convergence guarantees are complete and the method remains efficient and expressive, LMI-Net would provide a practical bridge between SDP-based certification and end-to-end learning for control and dynamical systems, enabling neural models with hard formal guarantees on stability, robustness, or invariance.

major comments (1)

[Abstract] Abstract and theoretical-guarantee statement: the claim that the projection layer 'converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints' is not supported when the lifted intersection is empty. Douglas-Rachford splitting converges to a point in the intersection only if that intersection is non-empty; the manuscript does not specify the layer's output or any fallback when the original LMI is infeasible for a given input (possible under distribution shift). This directly undermines the 'by construction' certification for arbitrary inputs.

minor comments (1)

[Methods] The abstract states that the approach 'supports efficient backward propagation through implicit differentiation' but provides no concrete statement of the implicit-function theorem assumptions or the conditioning of the Jacobian; this should be clarified in the methods section with reference to the specific DR iteration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point regarding the scope of our theoretical guarantees. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and theoretical-guarantee statement: the claim that the projection layer 'converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints' is not supported when the lifted intersection is empty. Douglas-Rachford splitting converges to a point in the intersection only if that intersection is non-empty; the manuscript does not specify the layer's output or any fallback when the original LMI is infeasible for a given input (possible under distribution shift). This directly undermines the 'by construction' certification for arbitrary inputs.

Authors: We agree that the abstract statement requires qualification. The convergence result for Douglas-Rachford splitting holds only when the intersection of the affine subspace and the PSD cone is non-empty; our analysis in the manuscript establishes this conditional guarantee. The current text does not explicitly describe the layer's behavior or any fallback mechanism when the original LMI is infeasible for a given input. We will revise the abstract to state the guarantee under the feasibility assumption and add a dedicated paragraph in Section 3 discussing the infeasible case. In practice, non-convergence can be detected by monitoring the residual, after which the layer can return the last iterate together with a feasibility flag or fall back to a soft-constrained approximation. This change will make the 'by construction' claim precise without altering the method or the reported experiments. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on standard external convex optimization results

full rationale

The central claim rests on lifting LMIs to an affine-PSD intersection and applying Douglas-Rachford splitting plus implicit differentiation. These are established, externally verifiable algorithms whose convergence properties (when the intersection is nonempty) are imported from the convex optimization literature rather than derived from the paper's own fitted quantities or self-citations. No step equates a prediction to a fitted input by construction, renames a known result, or loads the argument on an unverified self-citation chain. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard results from convex optimization; no new free parameters or invented entities are introduced in the abstract description.

axioms (2)

standard math Douglas-Rachford splitting converges to the projection onto the intersection of an affine set and the positive semidefinite cone.
Invoked for the forward pass of the projection layer.
standard math Implicit differentiation applies to the fixed-point iteration of the splitting method.
Used to obtain gradients for the backward pass.

pith-pipeline@v0.9.0 · 5490 in / 1260 out tokens · 60435 ms · 2026-05-10T19:40:54.734056+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

HardNet++: Nonlinear Constraint Enforcement in Neural Networks
cs.LG 2026-04 unverdicted novelty 5.0

HardNet++ enforces general nonlinear equality and inequality constraints on neural network outputs via an end-to-end trainable iterative process using damped local linearizations.

Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

S. Boyd, L. El Ghaoui, E. Feron, and V . Balakrishnan,Linear matrix inequalities in system and control theory. SIAM, 1994

1994
[2]

A survey on explicit model predictive control,

A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” inNonlinear model predictive control: towards new chal- lenging applications, pp. 345–369, Springer, 2009

2009
[3]

Parameterized LMIs in control theory,

P. Apkarian and H. D. Tuan, “Parameterized LMIs in control theory,” SIAM journal on control and optimization, vol. 38, no. 4, pp. 1241– 1264, 2000

2000
[4]

End-to-end learning to warm-start for real-time quadratic optimization,

R. Sambharya, G. Hall, B. Amos, and B. Stellato, “End-to-end learning to warm-start for real-time quadratic optimization,” inLearning for dynamics and control conference, pp. 220–234, PMLR, 2023

2023
[5]

Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers,

P. D. Grontas, A. Terpin, E. C. Balta, R. D’Andrea, and J. Lygeros, “Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers,” inInternational Conference on Learning Represen- tations (ICLR), 2026

2026
[6]

Parametric semidefinite programming: geometry of the trajectory of solutions,

A. Bellon, D. Henrion, V . Kungurtsev, and J. Mare ˇcek, “Parametric semidefinite programming: geometry of the trajectory of solutions,” Mathematics of Operations Research, vol. 50, no. 1, pp. 410–430, 2025

2025
[7]

Safe control with learned certificates: A survey of neural lyapunov, barrier, and contraction methods for robotics and control,

C. Dawson, S. Gao, and C. Fan, “Safe control with learned certificates: A survey of neural lyapunov, barrier, and contraction methods for robotics and control,”IEEE Transactions on Robotics, vol. 39, no. 3, pp. 1749–1767, 2023

2023
[8]

Learning stable deep dynamics models,

J. Z. Kolter and G. Manek, “Learning stable deep dynamics models,” Advances in neural information processing systems, vol. 32, 2019

2019
[9]

Learning stability certificates from data,

N. Boffi, S. Tu, N. Matni, J.-J. Slotine, and V . Sindhwani, “Learning stability certificates from data,” inConference on Robot Learning, pp. 1341–1350, PMLR, 2021

2021
[10]

Learning dissipative chaotic dynamics with boundedness guarantees,

S. Tang, T. Sapsis, and N. Azizan, “Learning dissipative chaotic dynamics with boundedness guarantees,”arXiv preprint arXiv:2410.00976, 2024

work page arXiv 2024
[11]

Eco: Energy-constrained operator learning for chaotic dynamics with boundedness guarantees,

A. Goertzen, S. Tang, and N. Azizan, “ECO: Energy-constrained operator learning for chaotic dynamics with boundedness guarantees,” arXiv preprint arXiv:2512.01984, 2025

work page arXiv 2025
[12]

Learning hybrid control barrier functions from data,

L. Lindemann, H. Hu, A. Robey, H. Zhang, D. Dimarogonas, S. Tu, and N. Matni, “Learning hybrid control barrier functions from data,” inConference on robot learning, pp. 1351–1370, PMLR, 2021

2021
[13]

Data-driven control with in- herent lyapunov stability,

Y . Min, S. M. Richards, and N. Azizan, “Data-driven control with in- herent lyapunov stability,” in2023 62nd IEEE Conference on Decision and Control (CDC), pp. 6032–6037, IEEE, 2023

2023
[14]

Learning contraction policies from offline data,

N. Rezazadeh, M. Kolarich, S. S. Kia, and N. Mehr, “Learning contraction policies from offline data,”IEEE Robotics and Automation Letters, vol. 7, no. 2, pp. 2905–2912, 2022

2022
[15]

Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,

H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine, “Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,”Annual Reviews in Control, vol. 52, pp. 135–169, 2021

2021
[16]

Hardnet: Hard-constrained neural networks with universal approximation guarantees.arXiv preprint arXiv:2410.10807,

Y . Min and N. Azizan, “HardNet: Hard-constrained neural net- works with universal approximation guarantees,”arXiv preprint arXiv:2410.10807, 2024

work page arXiv 2024
[17]

RAYEN: Imposition of Hard Convex Constraints on Neural Networks

J. Tordesillas, J. P. How, and M. Hutter, “RAYEN: Imposition of hard convex constraints on neural networks,”arXiv preprint arXiv:2307.08336, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
[18]

Splitting algorithms for the sum of two nonlinear operators,

P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,”SIAM Journal on Numerical Analysis, vol. 16, no. 6, pp. 964–979, 1979

1979
[19]

S-procedure in nonlinear control theory,

V . A. Yakubovich, “S-procedure in nonlinear control theory,”V estnik Leningrad University Mathematics, vol. 4, pp. 73–93, 1977. English translation; original Russian publication inV estnik Leningradskogo Universiteta, Seriya Matematika(1971), pp. 62–77

1977
[20]

Convex analysis and monotone operator theory in hilbert spaces,

H. H. Bauschke and P. L. Combettes, “Convex analysis and monotone operator theory in hilbert spaces,” Springer
[21]

Convergence rate analysis of several splitting schemes,

D. Davis and W. Yin, “Convergence rate analysis of several splitting schemes,” inSplitting methods in communication, imaging, science, and engineering, pp. 115–163, Springer, 2017

2017
[22]

CVXPY: A Python-embedded modeling language for convex optimization,

S. Diamond and S. Boyd, “CVXPY: A Python-embedded modeling language for convex optimization,”Journal of Machine Learning Research, vol. 17, no. 83, pp. 1–5, 2016

2016