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Replacing K-infinity Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems

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Pith's one-line read A Leaky ReLU with two slopes can replace arbitrary K∞ comparison functions in barrier-function safety certificates, and unions of invariant sets can be certified by taking pointwise maxima.

desk verdict UIS union argument is valid, but Theorem 1 is false as stated and the paper's main equivalence claim needs major repair. read the letter →

arxiv 2502.03765 v2 pith:SXZU2OAB submitted 2025-02-06 eess.SY cs.SY

Pouya Samanipour , Hasan Poonawala This is my paper · ORCID
classification eess.SYcs.SY MSC 93C1093D30
keywords barrierfunctionsLeakyReLUclassK-infinitypiecewiseaffinesystemsneuralnetworksinvariantsetsforwardinvariancesafetyverification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that in barrier-function safety analysis for ReLU-neural-network and piecewise-affine systems, the extended class K∞ comparison function in the barrier condition can be replaced by a fixed-shape Leaky ReLU without losing any certified invariant set. If true, safety analysts can stop searching over arbitrary nonlinear comparison functions and instead tune two slopes, one for the positive side of the barrier and one for the negative. The paper also proposes the Union of Invariant Sets (UIS) method, which merges invariant sets obtained from several linear comparison functions into a single piecewise-affine barrier function whose superlevel set is their union. The authors demonstrate on inverted-pendulum and barrier-certificate examples that the union can be larger than any individual set.

What carries the argument

The load-bearing identity is the ratio bound α(h(x))/h(x) at equation (8) and its negative-side analogue at equation (10), which converts any class K∞ comparison function into a two-slope Leaky ReLU. The second piece is the max-of-barrier-functions construction h(x)=max_i $h^{{(i)}}$(x) over a product partition, with α(x)=α_max σ(α_min/α_max)(x), which turns individual invariant sets into a single certified union.

What would settle it

Construct a bounded, continuously differentiable h and a PWA dynamics where condition (6) holds with α(h)=sign(h)√|h|, an extended class K∞ function, but for every pair 0<α_1≤α_m the inequality L_f h ≥ −α_m σ(α_1/α_m)(h) is violated at some point arbitrarily close to the boundary h=0; if such a case exists, Theorem 1's equivalence is false as stated.

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Extended reading notes

Core claim

The central claim is Theorem 1: for a bounded, continuously differentiable h, the existence of any extended class K∞ function α satisfying L_f h ≥ −α(h) is equivalent to the existence of a Leaky ReLU α(x)=α_m σ(α_1/α_m)(x) — with slopes 0<α_1≤α_m and σ(α_1/α_m)(x)=x for x≥0, (α_1/α_m)x for x<0 — satisfying the same inequality. The proof argues that finiteness of h and α(h) forces the ratio α(h)/h to be bounded above for h>0 and bounded below for h<0, so one can pick a dominating positive slope α_m and a smaller slope α_1. The paper then uses this equivalence to justify combining barrier functions from multiple linear α values by taking their pointwise maximum, yielding a PWA barrier function with Leaky ReLU α that certifies the union of the individual invariant sets.

Load-bearing premise

The proof of Theorem 1 assumes that for the given class K∞ function α, the ratio α(h)/h stays bounded as h approaches zero from either side; this fails for functions like sign(h)√|h|, where the ratio blows up.

Editorial extensions

If this is right

  • If Theorem 1 holds, barrier-function synthesis (17) can be run with a fixed Leaky ReLU shape instead of searching over K∞ functions, reducing the search to two slopes.
  • The UIS method yields a certified invariant set that contains every individual set from the chosen α values (Corollary 1), so it never performs worse than the previous best single-α approach and can be strictly larger.
  • The resulting PWA barrier function is compatible with the same partition-based optimization framework, so no new machinery beyond a product partition and a pointwise max is needed.
  • Because the Leaky ReLU is non-smooth, validity at non-differentiable points rests on the nonsmooth barrier-function framework, extending the method to systems where the active barrier switches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Theorem 1's equivalence is local in shape near h=0; for comparison functions that vanish faster than linearly, such as sign(h)√|h|, the ratio bound fails and no finite-slope Leaky ReLU can dominate, so the equivalence as stated appears to require an additional regularity assumption.
  • The UIS max construction suggests a general recipe: any finite collection of valid barrier functions with comparable slopes can be merged by pointwise max, which might extend to non-ReLU nonlinear systems on compact sets, as the paper hints in Remark 2.
  • A testable extension is to treat the Leaky ReLU slopes α_1 and α_m as decision variables in the optimization (17) rather than choosing them by bisection, using the UIS certification to validate the result and potentially recovering larger invariant sets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a framework for constructing piecewise-affine (PWA) barrier functions and invariant sets for dynamical systems identified by ReLU neural networks. The main theoretical contribution is Theorem 1, which claims that validity of a barrier condition (6) with some extended class-K∞ function α is equivalent to validity with a Leaky ReLU α of the form α(h)=α_m σ(α_1/α_m)(h). Building on this, the authors introduce a Union of Invariant Sets (UIS) method: multiple invariant sets S^(i), each certified by a linear α_i h via the earlier optimization (17), are combined into one set S = ∪_i S^(i), with a max-of-barriers PWA function as the certificate and a Leaky ReLU class-K∞ function. The paper includes two numerical examples and makes the code publicly available.

Significance. If the theoretical claims were correct, the paper would offer a practical way to avoid searching over class-K∞ functions and to enlarge invariant sets without additional computational cost, and the UIS idea is conceptually appealing. Credit should be given for making the code available and for the clear algorithmic presentation of the union construction. However, the central theorem is false as stated, and the proof of Lemma 1 has gaps concerning the PWA representation and the behavior at non-differentiable points and outside the set. These issues are load-bearing for the paper's main message, and the numerical examples are demonstrations rather than substitutes for proof.

major comments (3)
  1. [Theorem 1, Eq. (8)-(10)] The implication (a)⇒(b) is false. The proof asserts that α(h(x))/h(x) is bounded above on the set 0 < h(x) ≤ h_max merely because both α(h(x)) and h(x) are finite. This fails for valid extended class-K∞ functions whose ratio α(h)/h diverges near h=0, e.g., α(h)=sign(h)√|h|. A concrete counterexample within all the stated smoothness assumptions is: D=[0,1], h(x)=x^2 (so 0≤h≤1), f(x)=-1/2, and α(h)=sign(h)√|h|. Then L_f h = h'(x)f(x) = 2x·(-1/2) = -x = -√h(x), so (6) holds with equality. Yet for any Leaky ReLU with slopes 0<α_1≤α_m, condition (6) on h>0 reads -√h + α_m h ≥ 0, i.e., -x + α_m x^2 ≥ 0 for all x∈[0,1]. For 0 < x < 1/α_m this quantity is negative, so no Leaky ReLU parameters satisfy (6). Thus boundedness of h and finiteness of α do not imply the bounded-ratio condition used in (8) and (10); a Lipschitz-type condition on α near h=0 is needed but is neither stated nor proved.
  2. [Lemma 1, Eq. (21) and its proof] The product partition P = P_1 × ⋯ × P_m defined in (21) does not necessarily refine the active regions of the max function h(x,P,α)=max_i h^(i)(x). On a cell of P, each h^(i) is affine, but the pointwise maximum of several affine functions is generally not affine on the cell; the active index changes at the hyperplanes where h^(i)=h^(j), and those hyperplanes need not be part of the product partition. Consequently, the proof's assertion that 'we defined P to ensure that in each cell, we have only one affine function h_i(x,P,α)' is not justified, and the resulting h is not shown to be PWA in the sense of (1). A correct construction would require a common refinement of P_1,…,P_m together with the comparison hyperplanes {x : h^(i)(x)=h^(j)(x)}, and the proof does not provide or analyze such a refinement.
  3. [Lemma 1, proof of part 2] The proof explicitly omits the demonstration that h(x,P,α)<0 for all x∉S ('Due to brevity, the proof of h(x,P,α)<0 for x∉S is eliminated'), and it defers all non-differentiable points to Proposition 2 of [15]. Both omissions are load-bearing: the negative-side argument around Eq. (26) uses h(x,P,α)<0, and condition (6) must hold for every x∈D, including the non-differentiable boundaries of the max function. The paper neither states the hypotheses required for Proposition 2 of [15] nor verifies them for the particular h in (19). As written, the proof establishes (6) only at differentiable points and under the extra assumption that S is exactly the superlevel set of h, which is part of what is left unproved.
minor comments (4)
  1. [Definition 3] The sentence 'Equation (6) makes set S an asymptotically set in D' appears to be a typo; it should read 'asymptotically stable set' or 'asymptotically attractive set'.
  2. [Theorem 1] In the statement of Theorem 1, the Leaky ReLU is written as α(x)=α_m σ(α_1/α_m)(x), but α should be a function of h(x), namely α(h)=α_m σ(α_1/α_m)(h). The current notation makes α a function of the state x, which is inconsistent with the barrier condition (6) in which α is evaluated at h(x).
  3. [Algorithm 1] The while condition 'P_N i=1 τbi ≠ 0' is a typo; it should be '∑_i τ_bi ≠ 0'.
  4. [Example 1] The description of the ReLU neural network approximation of the inverted pendulum is incomplete: no training data, network training procedure, or approximation-error bound is provided, so the reader cannot verify that the computed invariant set is valid for the true nonlinear dynamics rather than only for the identified PWA model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the UIS construction and Leaky ReLU barrier claim are derived from the paper's own optimization and external nonsmooth-BF results; the theorem's proof flaw is a correctness issue, not a circular fit.

full rationale

The paper's central derivation chain is not circular in the identification sense. Theorem 1 attempts to show equivalence between an arbitrary class-K∞ barrier condition and a Leaky ReLU condition; although equation (8)'s bounded-ratio assertion is mathematically false (e.g., α(h)=sqrt(h) makes α(h)/h unbounded as h→0+), this is an unsupported step in the proof, not a reduction of the conclusion to the hypothesis. Lemma 1's union argument is direct: each S(i) is obtained by solving the LP (17) with a linear α_i, and the union of forward-invariant sets is forward-invariant; the max-of-barriers certificate is then argued using the Leaky ReLU choice and [15]'s nonsmooth BF proposition. No parameter is fitted to a target invariant set and then reported as a prediction; the examples are demonstrations, not fits. The repeated references to the authors' prior [14] for the LP formulation and feasibility are normal building on earlier published work, not a self-citation chain that forces the paper's new claims. The omitted proof that h<0 outside S and the reliance on [15] for nondifferentiable points are gaps, but they are not circularity. Score 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central construction depends on the optimization from the authors' prior work [14], on the nonsmooth barrier-function theorem of [15], and on an unstated bounded-ratio property of class-K∞ functions that is false in general. The only hand-tuned continuous parameters are the α list and tolerances; the BF coefficients are decision variables of the optimization. No new physical or dynamical entities are introduced.

free parameters (3)
  • α parameter set for UIS = Example 1: [0.025, 0.05, 0.06]; Example 2: [0.1, 0.5]
    Users must supply the list of linear slopes; UIS unions the resulting sets. The final Leaky ReLU slopes α_min and α_max are taken from this list. The paper gives no rule for choosing it.
  • Barrier function coefficients s_i, t_i = not reported
    These are decision variables of the linear program (17); the resulting invariant-set shape depends on them. They are optimized, not derived from first principles.
  • Tolerances ε1, ε2, ε3 and numerical tolerance = 10^-4 for ε; 10^-6 for nonzero detection
    Hand-set constants in the optimization and refinement steps; they affect feasibility and the size of the certified set.
assumptions (4)
  • ad hoc to paper The ratio α(h)/h is bounded above for h>0 and bounded below for h<0 for any extended class K∞ function appearing in a valid barrier condition.
    Needed for the proof of Theorem 1, equations (8) and (10), but not true for non-Lipschitz K∞ functions such as sign(h)√|h|.
  • domain assumption All partition cells are bounded polytopes, and the affine dynamics inside each cell allow derivative evaluation at vertices.
    Assumption 1 and Assumption 2 in Section II-A and Section III-C restrict the system class and justify the vertex-based optimization.
  • standard math The nonsmooth barrier-function result of Glotfelter et al. [15], Proposition 2, applies to the maximum of PWA barrier functions at non-differentiable points.
    Invoked in the proof of Lemma 1 to certify the max barrier function on non-differentiable points; the argument is not reproduced.
  • domain assumption The dynamics have an equilibrium at the origin and the domain D is compact.
    Used in Lemma 1 and in Theorem 1's boundedness requirement on h.

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Cite this review

Pith. "Pith review of Replacing K-infinity Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems." pith.science (2026). https://pith.science/paper/SXZU2OAB

@misc{pith2026250203765,
  author       = {Pith},
  title        = {Pith review of: Replacing K-infinity Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXZU2OAB}},
  note         = {Machine review of arXiv:2502.03765}
}
read the original abstract

In this paper, a systematic framework is presented for determining piecewise affine PWA barrier functions and their corresponding invariant sets for dynamical systems identified via Rectified Linear Unit (ReLU) neural networks or their equivalent PWA representations. A common approach to determining the invariant set is to use Nagumo's condition, or to utilize the barrier function with a class K-infinity function. It may be challenging to find a suitable class K-infinity function in some cases. We propose leaky ReLU as an efficient substitute for the complex nonlinear K-infinity function in our formulation. Moreover, we propose the Union of Invariant Sets (UIS) method, which combines information from multiple invariant sets in order to compute the largest possible PWA invariant set. The proposed framework is validated through multiple examples, showcasing its potential to enhance the analysis of invariant sets in ReLU-based dynamical systems. Our code is available at: https://github.com/PouyaSamanipour/UIS.git.

Figures

Figures reproduced from arXiv: 2502.03765 by the authors.

Figure 1
Figure 1. The new invariant set S shown with a red dashed line is the union of four invariant sets obtained from optimization problem (17) with different α. As can be seen S(P4, α4) does not contribute to the S [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The final invariant set will be the Union of invariant [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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  1. SEROAISE: Advancing ROA Estimation for ReLU and PWA Dynamics through Estimating Certified Invariant Sets

    eess.SY 2025-04 reject novelty 6.0 of 10

    SEROAISE estimates larger certified regions of attraction for PWA and ReLU dynamical systems by growing a certified invariant set via the NUGIS procedure and then solving a linear program for a Lyapunov-like function ...

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