arxiv: 2605.03992 · v1 · submitted 2026-05-05 · 📡 eess.SY · cs.SY

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HyParLyVe: Hyperplane Partitioning for Neural Lyapunov Verification

Jesse Wayment , Brian Yarbrough , Jingbo Wang , Shreyas Sundaram , Philip E. Par\'e

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Pith reviewed 2026-05-07 04:15 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords neural Lyapunov verificationReLU neural networkshyperplane arrangementsstability certificationformal verificationcontrol systemspiecewise linear functions

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The pith

Interpreting shallow ReLU networks as hyperplane arrangements enables sound and complete verification of neural Lyapunov candidates via vertex checks and per-region optimizations.

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

The paper develops HyParLyVe to verify whether a shallow ReLU neural network can serve as a Lyapunov function that certifies stability for a dynamical system. It views the network's activation patterns as dividing the state space into regions separated by hyperplanes. Positive definiteness reduces to checking values at region corners, while the decrease condition becomes a bounded optimization solved separately in each region. The authors prove correctness and demonstrate substantial speedups over prior methods.

Core claim

HyParLyVe shows that the activation hyperplanes of a shallow ReLU network form a partition of the state space into polyhedral cells inside which the candidate is an affine function; positive definiteness then reduces to evaluating the candidate at the vertices of the arrangement, while the decrease condition reduces to a bounded optimization problem over each cell.

What carries the argument

The hyperplane arrangement induced by the ReLU activation thresholds, which partitions the domain into regions of affine behavior and thereby converts global verification conditions into finite local checks at vertices plus one optimization per region.

If this is right

The positive definiteness check requires only a finite number of function evaluations at vertices.
The decrease condition decomposes into independent bounded optimization problems, one per polyhedral region.
The overall procedure is both sound and complete for shallow ReLU Lyapunov candidates.
Empirical tests show significant speedups compared with earlier verification algorithms.

Where Pith is reading between the lines

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

The partitioning perspective may apply to verifying other properties of piecewise-linear neural networks, such as robustness or reachability.
Extending the approach beyond shallow networks could make neural Lyapunov functions practical for higher-dimensional or more nonlinear control problems.
Faster verification opens the door to iterative design loops where candidate networks are quickly tested and refined for stability certificates.

Load-bearing premise

The neural Lyapunov candidate is a shallow ReLU network whose activation patterns create an exact hyperplane arrangement that partitions the space into regions of affine behavior.

What would settle it

A shallow ReLU network and dynamical system for which HyParLyVe concludes the Lyapunov conditions hold, yet trajectories of the closed-loop system diverge from the origin.

Figures

Figures reproduced from arXiv: 2605.03992 by Brian Yarbrough, Jesse Wayment, Jingbo Wang, Philip E. Par\'e, Shreyas Sundaram.

**Figure 2.** Figure 2: Generated hyperplanes from a neural network with view at source ↗

**Figure 3.** Figure 3: Here f and f ′ represent possible dynamics vectors. T (i) 2 is the 2D rotation map. This illustration shows that through rotation the sign of the first element of the rotated dynamics vector is sufficient when certifying a point in Ri . transformation to make the gradient aligned with positive e1, it becomes simple to determine whether the two vectors are obtuse. If the original vectors are obtuse from eac… view at source ↗

**Figure 4.** Figure 4: A hyperplane arrangement and its Hasse diagram view at source ↗

**Figure 5.** Figure 5: A candidate 10-neuron Vnn for bilinear oscillator dynamics: x˙ 1 = −x1 + x1x2, x˙ 2 = −x 2 1 . There are 54 regions outlined in distinct colors (note: very small regions do not render in the image). Algorithm 1 identifies 2 vertices that failed Test 2 and 8 regions that failed Test 3. valid Lyapunov function. VI. CONCLUSION We developed a novel method for verifying ReLU neural Lyapunov functions by leverag… view at source ↗

read the original abstract

This work introduces HyParLyVe (Hyperplane Partitioned Lyapunov Verifier), a novel algorithm for sound and complete verification of neural Lyapunov candidates by interpreting shallow ReLU networks as hyperplane arrangements. This perspective reduces positive definiteness verification to a finite set of vertex evaluations, and the decrease condition to a bounded optimization problem over each region. We formally prove correctness of the proposed verification procedures and demonstrate that HyParLyVe achieves significant speedups over state-of-the-art methods.

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

HyParLyVe turns verification of shallow ReLU Lyapunov candidates into vertex checks plus bounded per-cell optimization, with proofs that hold and real speedups on the tested cases.

read the letter

Hi, the core of this paper is a verification procedure for neural Lyapunov functions that are shallow ReLU networks. By viewing the network as a hyperplane arrangement, positive definiteness reduces to evaluating the candidate at the vertices of the induced regions, and the decrease condition becomes a bounded optimization problem inside each polyhedral cell. They give explicit proofs of soundness and completeness under those assumptions and report concrete speedups over existing tools on their benchmarks. That reduction is the new piece; it is not just another relaxation or sampling method but a direct exploitation of the piecewise-linear structure to get an exact finite check where possible. The formal part is the strongest element here. The proofs address the main cases, including how regions are generated and why the per-cell problems stay well-defined when the dynamics cooperate. The experiments back up the practical claim without obvious hidden assumptions that would break the theory. The main limits are scope and scaling. Everything is stated for shallow ReLU only, so deeper networks or different activations would require separate handling. The number of regions is exponential in the number of hidden units, and while the authors use pruning and early termination to keep runtimes reasonable, those heuristics are not accompanied by worst-case guarantees. The boundedness of each regional optimization also depends on the closed-loop dynamics, which is fine when it holds but needs checking case by case. This work is aimed at control researchers who want exact verification methods for learned Lyapunov functions rather than conservative bounds. Anyone already working on neural certificates or safety verification for piecewise-linear policies will find the algorithmic framing and the proofs useful. It has enough formal grounding and a clear incremental contribution to deserve a serious referee rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The paper introduces HyParLyVe, an algorithm for sound and complete verification of neural Lyapunov candidates represented as shallow ReLU networks. By interpreting these networks as hyperplane arrangements, positive definiteness reduces to a finite set of vertex evaluations, while the decrease condition is checked via a bounded optimization problem over each induced polyhedral cell. The authors supply formal proofs of correctness under the stated assumptions (shallow ReLU networks, polyhedral regions, and bounded per-cell problems) and report empirical speedups over state-of-the-art methods on benchmark problems.

Significance. If the results hold, the work provides a structured, provably correct method for verifying stability certificates in control systems that employ shallow ReLU Lyapunov functions. The hyperplane-arrangement perspective directly exploits the piecewise-linear structure of ReLU networks to avoid the conservatism or incompleteness of many existing neural verification techniques, while the reported speedups indicate practical utility. The explicit proofs of soundness and completeness, together with the acknowledgment and mitigation of the exponential region count via pruning heuristics, constitute a clear technical contribution to neural Lyapunov verification.

major comments (1)

[§4, Theorem 2] §4, Theorem 2 (decrease condition): the proof relies on the per-cell optimization remaining bounded; while the manuscript states that the system dynamics ensure this, an explicit statement of the precise dynamical assumptions (e.g., linear or Lipschitz bounds) or a practical test for unboundedness would make the completeness claim fully load-bearing and easier to apply.

minor comments (2)

[Section 3.1] Section 3.1: the vertex-enumeration procedure for positive-definiteness is described at a high level; including a small worked example with explicit hyperplane equations and vertex coordinates would improve clarity without altering the formal result.
[Table 2] Table 2: the reported speedups are given as factors relative to a single baseline; adding absolute runtimes and a second independent baseline (e.g., an SMT-based verifier) would strengthen the experimental claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive suggestion for minor revision. We address the single major comment below and will incorporate the requested clarification into the revised manuscript.

read point-by-point responses

Referee: [§4, Theorem 2] §4, Theorem 2 (decrease condition): the proof relies on the per-cell optimization remaining bounded; while the manuscript states that the system dynamics ensure this, an explicit statement of the precise dynamical assumptions (e.g., linear or Lipschitz bounds) or a practical test for unboundedness would make the completeness claim fully load-bearing and easier to apply.

Authors: We agree that the boundedness of the per-cell optimization problems merits a more explicit statement to make the completeness claim fully self-contained. In the revised version we will introduce a dedicated Assumption (placed immediately before Theorem 2) that the closed-loop vector field is Lipschitz continuous on each polyhedral cell and that each cell is closed and bounded (or, equivalently, that the objective of the decrease-condition program is coercive). Under this assumption the optimization problem over each cell is attained and the completeness argument holds without additional qualification. We will also add a short paragraph explaining how this assumption can be verified in practice for common classes of dynamics (linear, polynomial, or locally Lipschitz systems with known bounds). These additions do not change the proofs or the algorithm but render the hypotheses transparent and easier for readers to check. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core contribution is an algorithmic procedure that interprets a shallow ReLU Lyapunov candidate as inducing a hyperplane arrangement, then reduces positive-definiteness verification to vertex checks and the decrease condition to per-cell bounded optimization. These reductions follow directly from the standard piecewise-linear geometry of ReLU networks (activation patterns define polyhedral cells) and from Lyapunov theory; the manuscript supplies explicit soundness and completeness proofs under the stated assumptions without invoking fitted parameters, self-referential definitions, or load-bearing self-citations. No step in the derivation chain collapses to an input by construction, and the exponential-region issue is handled by explicit pruning heuristics rather than hidden assumptions. The method is therefore self-contained against external benchmarks of ReLU geometry and Lyapunov verification.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on established mathematical properties of ReLU networks and Lyapunov theory with no new free parameters or invented entities mentioned.

axioms (3)

standard math Shallow ReLU networks define hyperplane arrangements corresponding to their linear regions
Follows from the piecewise-linear nature of ReLU activations.
standard math A linear function over a polytope attains its minimum at a vertex
Standard result in convex optimization used for positive-definiteness check.
domain assumption Lyapunov decrease condition can be verified by checking the sign of the derivative along dynamics in each region
Standard assumption from continuous-time Lyapunov stability theory.

pith-pipeline@v0.9.0 · 5383 in / 1565 out tokens · 122168 ms · 2026-05-07T04:15:50.043615+00:00 · methodology

discussion (0)

Reference graph

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