On the (Non) Injectivity of Piecewise Linear Janossy Pooling

Ilai Reshef; Nadav Dym (Technion - Israel Institute of Technology)

REVIEW 1 major objections 4 minor 31 references

On the (Non) Injectivity of Piecewise Linear Janossy Pooling

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read No piecewise-linear Janossy pooling is injective

desk verdict Settles the k>=2 Janossy pooling expressivity question with a genuinely new polytope-partition lemma; a solid paper with only minor presentation gaps. read the letter →

arxiv 2505.20150 v1 pith:LMSITJW6 submitted 2025-05-26 cs.LG cs.AI

Ilai Reshef , Nadav Dym (Technion - Israel Institute of Technology) This is my paper · ORCID

classification cs.LGcs.AI MSC 68T07

keywords multisetfunctionsJanossypoolinginjectivitybi-LipschitzembeddingspiecewiselineardeepsetssettransformersWassersteindistance

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

This paper answers a question about the expressiveness of standard neural network layers that process unordered sets of vectors. It proves that $k$-ary Janossy pooling—the family that includes deep sets ($k=1$) and set transformers ($k=2$)—cannot be injective when the pooled function is continuous piecewise linear (the activation class of ReLU networks) and the multiset size $n$ is larger than $k$. The only exception is the degenerate $k=n$ case, where pooling over all permutations can be injective by using a sorting-based map. On the positive side, the paper shows that if the domain is restricted to compact sets of multisets with no repeated points, even 1-ary deep sets can be injective and bi-Lipschitz with respect to the Wasserstein distance. The embedding dimension of this construction grows like $(1/R)^d$, where $R$ is the minimal separation between points in the domain.

What carries the argument

The central object is the polytope-covering lemma (Theorem 3.2) that drives the negative result: for any finite partition of $\mathbb{R}^k$ into convex polytopes, there exists a strictly decreasing vector $w$ of length $n$ whose every $k$-dimensional order-preserving subvector falls inside a single polytope $P_0$. This reduces the non-injectivity of Janossy pooling to a counting argument: around such a $w$ the pooling map is affine, and the system of $k$ equations expressing the sum of perturbations over all $k$-subtuples has a nonzero solution in $\mathbb{R}^n$ because $n > k$. For the positive result, the key mechanism is a hypercube tessellation of $\mathbb{R}^d$ with side length $s = R(D)/2$, where $R(D)$ is the minimal pairwise distance among points of any multiset in the compact domain $D$; each hypercube carries a CPwL indicator of membership plus a CPwL coordinate feature that equals the point's location inside the cube and interpolates to zero at the margin boundary, so the pooled sum reveals each point uniquely.

What would settle it

Try to construct a CPwL function $f: \mathbb{R}^2 \to \mathbb{R}$ on a partition of the unit square into four quadrants and show that its 2-ary Janossy pooling on triples is injective; the theorem predicts such injectivity is impossible, so any explicit injective example would refute it. More directly, a computational search over polytope partitions of $\mathbb{R}^2$ for one that defeats Theorem 3.2—no strictly decreasing triple $w$ whose three ordered pairs share a polytope—would test the proof's engine.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.1: let $C$ be any subset of $\mathbb{R}^d$ containing a line segment, let $f:(\mathbb{R}^d)^k \to \mathbb{R}^m$ be continuous piecewise linear, and let $n > k$. Then the $k$-ary Janossy pooling of $f$, defined by averaging $f$ over all ordered $k$-tuples of the input, is not injective on multisets of size $n$ from $C$. The proof hinges on a new geometric lemma (Theorem 3.2): for every finite partition of $\mathbb{R}^k$ into polytopes, there is a strictly decreasing vector $w=(w_1,\ldots,w_n)$ in $(0,1)^n$ and a single polytope $P_0$ such that every order-preserving $k$-subvector of $w$ lies in the interior of $P_0$. This makes the pooled output affine on a neighborhood of $w$, and since summing over $k$-element subsets leaves $k$ linear equations in $n > k$ unknowns, a nonzero perturbation preserves the pooled vector while changing the multiset. The companion positive result, Theorem 4.3, constructs, on a compact domain of multisets with $n$ distinct points, a CPwL function $f$ such that 1-ary pooling is injective and bi-Lipschitz; the construction tessellates $\mathbb{R}^d$ into hypercubes of side length $R(D)/2$ and attaches to each hypercube an indicator feature and a coordinate feature, so that the pooled output reveals every point of the multiset.

Load-bearing premise

The non-injectivity theorem depends on $f$ being continuous piecewise linear, so that its domain splits into finitely many polytopes on each of which $f$ is affine; if $f$ were smooth, injective Janossy pooling would be possible, and the theorem would no longer hold.

Editorial extensions

If this is right

Set transformers and other $k$-ary pooling models with ReLU activations cannot faithfully represent multisets of size $n > k$ in general: for every such model there exist distinct multisets that its output cannot tell apart.
The negative result provides a theoretical justification for the higher cost of sorting-based and quantile-based bi-Lipschitz encoders: no simple piecewise-linear pooling can match their injectivity guarantee.
For datasets with well-separated points, the positive result shows that ordinary deep sets are sufficient, both injective and bi-Lipschitz, with the embedding dimension determined by the minimal separation $R(D)$.
The construction's dimension scales as $(1/R(D))^d$, so the mathematical guarantee degrades as point clouds approach each other; near-duplicate points are exactly where specialized bi-Lipschitz layers are needed.
The paper's conjecture that $k$-ary pooling is injective on multisets with multiplicity at most $k$, if true, would establish a precise expressivity hierarchy across pooling orders.

Reading between the lines

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

A concrete, testable consequence of Theorem 3.1: for any ReLU set transformer with $k=2$, one can run an adversarial search over pairs of size-$n$ point sets ($n>2$) and expect to find identical embeddings; such an experiment would empirically demonstrate the collision guaranteed by the theorem.
The geometric lemma may generalize beyond polytopal partitions: the argument uses only finiteness and convexity, so a similar non-injectivity could hold for any function that is affine on finitely many convex regions, e.g., hinge-type or other CPwL-like activations.
The QM9 analysis in the paper (minimal within-molecule distances above 0.1 of the diameter for all sampled molecules) suggests a rule of thumb for practitioners: small-molecule point clouds are likely in the 'well-separated' regime where deep sets suffice, while continuous surfaces or dense clouds are not.
One could implement the positive construction explicitly and measure, on real point clouds, whether the guaranteed injectivity translates into useful Lipschitz constants for downstream tasks such as learning Wasserstein distances.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, this settles the open k>=2 case: CPwL k-ary Janossy pooling is never injective for multisets of size n>k, so sort-based bi-Lipschitz models are not just an option but necessary if you want injectivity in general. Second, the positive result is real and explicit: on compact domains with a positive minimal separation, even 1-ary CPwL deep sets can be injective and bi-Lipschitz, with embedding dimension scaling like (1/R(D))^d.

The core new piece is Theorem 3.2, a combinatorial lemma about polytope partitions of R^k. I checked the appendix proof. The induction building v_0,...,v_k, the nest property of POLY(v_i), and the convex-combination argument are all sound. The proof of Theorem 3.1 then reduces to k linear homogeneous equations in n>k unknowns, which is correct. The d>1 reduction by an affine segment is fine. The positive construction is a standard grid tessellation with local indicator and coordinate features, and the separation bound R(D)>0 via compactness is proven properly.

Soft spots are minor. Appendix A.3 has a small presentation gap: the claim that v_k lies in int(P0) does not follow just from v_k being in one polytope and inside [0,1]^k; you need to use that the other polytopes are closed, finite in number, and do not contain v_k. The stress-test note caught this and it is readily repaired. Also, the text calls P a 'covering' where the theorem needs a 'partition'—clearly a wording slip. And the QM9 sentence says the ratio was 'not larger than 1/10' when the histogram and the argument need 'not smaller than 1/10'; an obvious typo that does not touch the math.

I do not see a load-bearing flaw. The negative result is proven from first principles, the positive result is limited to separated domains, and the authors say so themselves in the limitations section. Citation practice is normal; the only self-citation (Sverdlov et al.) is used for the bi-Lipschitz upgrade, not for the main theorems.

Who is this for: people working on expressivity of permutation-invariant networks, set transformers, and geometric GNNs. It will get cited. It deserves a serious referee. I would accept it with minor revisions.

Referee Report

1 major / 4 minor

Summary. The paper studies the injectivity and bi-Lipschitzness of k-ary Janossy pooling when the inner function f is continuous piecewise linear (CPwL). The main negative result (Theorem 3.1) states that if C contains a line segment and n > k, no CPwL k-ary Janossy pooling is injective on multisets of size n in M_n(C); the proof reduces to a combinatorial statement (Theorem 3.2) that every polytope partition of R^k contains a polytope P0 and a strictly ordered vector w in (0,1)^n all of whose ordered k-subvectors lie in the interior of P0, after which a homogeneous linear system in n > k unknowns yields a nontrivial perturbation with equal pooling value. The positive result (Theorem 4.3) constructs, for compact domains D of multisets with distinct elements, an explicit CPwL 1-ary pooling that is injective and bi-Lipschitz with respect to the Wasserstein distance, using a hypercube tessellation of scale R(D)/2. The paper also discusses the degenerate k=n case, the dependence of the embedding dimension on R(D), and a conjecture for bounded-multiplicity domains.

Significance. If correct, the negative result resolves the open question for k >= 2 and shows that injective CPwL multiset representations require either sorting-based maps or the degenerate k = n construction; this strengthens the case for bi-Lipschitz models built on sorting rather than standard Janossy pooling. The proofs are self-contained and rely on no fitted parameters: Theorem 3.1 is proven from the explicit combinatorial lemma in the appendix, and the construction in Theorem 4.3 is explicit. The compact-domain separation assumption is stated and justified in Proposition 4.4, and the limitations and conjecture are clearly identified. The main theorems give falsifiable predictions about expressivity of standard architectures, which is a useful contribution to the theory of permutation-invariant networks.

major comments (1)

[Section 4, proof of Theorem 4.3, final paragraph] The bi-Lipschitz conclusion is justified by an unspecified auxiliary set D-hat, said to be a finite union of polytopes containing D and containing no multisets with repeated elements. This claim is not established. A finite union of grid hypercubes in the point space R^d is not a set of multisets, and the set of all multisets over such a union contains repeated points; if D-hat is intended as a set of multisets, its structure as a finite union of polytopes requires proof. Please give an explicit construction (for example, in sorted-coordinate space, fix an assignment of the n points to distinct grid cells and verify that each such region is a polytope) and prove that the constructed F is injective on that D-hat before applying [Sverdlov et al., 2024, Lemma 3.4]. As written, the bi-Lipschitz claim in Theorem 4.3 is not fully supported.

minor comments (4)

[Appendix A.1, Proposition A.3] The sentence concluding that v_k lies in int(P0) because it lies in the interior of [0,1]^k is not a valid inference. The conclusion is nonetheless correct: since |POLY(v_k)| = 1 and the finitely many other closed polytopes do not contain v_k, there is a positive distance from v_k to their union, so a small ball around v_k is contained in P0. Please replace the argument.
[Section 3, proof of Theorem 3.1] The proof refers to a finite polytope covering of [0,1]^k, while Theorem 3.2 is stated for a partition. Please clarify that the collection is the partition into linear regions of the symmetrized function (or take a common refinement) so that Theorem 3.2 applies directly.
[Section 4.1] The sentence 'the ratio was not larger than 1/10' contradicts the histogram and the preceding paragraph; it should read 'not smaller than 1/10'.
[Throughout] There are numerous typos and misspellings, including 'Currenlty', 'inejctivity', 'Lipshitz', 'partion', 'elments', 'Wasserstien', 'montonely', 'represetative', and 'a a finite union'. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Theorems 3.1 and 4.3 are proven constructively from first principles.

full rationale

The derivation chain is self-contained. Theorem 3.1 is reduced to the new combinatorial Theorem 3.2, whose appendix proof uses only polytope-convexity arguments and Rockafellar's Accessibility Lemma; the final perturbation solves k homogeneous linear equations in n > k unknowns, with no fitted parameter. The d > 1 case reduces explicitly to d = 1 via an affine segment, and the compatibility of sorted and unsorted vectors is argued directly. Theorem 4.3 is a constructive construction: f is built from a cube tesselation of side s = R(D)/2 and a triangulated delta-margin, with injectivity demonstrated by the explicit recovery algorithm in Lemmas 4.5 and 4.6; the compactness/separation Proposition 4.4 is proven, not assumed. The only self-citation is [Sverdlov et al. 2024, Lemma 3.4], used solely to upgrade the constructively injective map to bi-Lipschitz; it is a general CPwL lemma whose assumptions do not include the target function, so it does not make the argument circular. There is a presentation gap in Appendix A.3 (membership in int(P0) needs the positive-distance argument from the finite family of other closed polytopes), and a harmless factor-n! slip in the k=n remark of Section 3.1, but these are correctness issues, not circularities. No 'prediction' in the paper is a fitted parameter renamed, and the QM9 separation histogram is illustrative motivation, not a fitted input to the theorems.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims are proven with standard convex analysis and polytope geometry. No new physical entities are introduced. The construction's parameters s and δ are chosen from R(D), not fitted to data, and any admissible values work for the existence result.

free parameters (2)

hypercube side length s = R(D)/2
Chosen in Theorem 4.3 so that s + 2δ < R(D) for δ < R(D)/4, ensuring the support of each feature function contains at most one point of any multiset in D.
margin width δ = 0 < δ < R(D)/4
Chosen to satisfy (s + 2δ) < R(D). Any value in this range works; the theorem only needs existence.

assumptions (4)

standard math Rockafellar's Theorem 6.1 (Accessibility Lemma): if v is in the interior of a convex set P and α>0, then a convex combination with positive weight on v is in the interior of P.
Used in Proposition A.4 to show the constructed point lies in the interior of P0.
standard math Goodman-Pach cell decomposition theorem: the δ-margin of a hypercube can be triangulated without new vertices so each simplex has vertices on both the inner boundary and the outer border.
Used in Theorem 4.3 to construct the CPwL coordinate component f_Q,coords by piecewise affine interpolation.
domain assumption Sverdlov et al. Lemma 3.4: a CPwL injective function on a finite union of compact polytopes is bi-Lipschitz with respect to a CPwL distance with the same zero set.
Used in Theorem 4.3 to upgrade injectivity of F to bi-Lipschitzness against the Wasserstein distance; cited from the authors' prior work and not proved in this paper.
standard math Two convex polytopes with empty-interior intersection lie in a common hyperplane (Boyd & Vandenberghe, Convex Optimization, §2.5.2).
Used in Proposition A.3 to show that if the final point v_k belonged to two distinct polytopes, the affine-independence of the constructed basis would be contradicted.

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Cite this review

Pith. "Pith review of On the (Non) Injectivity of Piecewise Linear Janossy Pooling." pith.science (2026). https://pith.science/paper/LMSITJW6

@misc{pith2026250520150,
  author       = {Pith},
  title        = {Pith review of: On the (Non) Injectivity of Piecewise Linear Janossy Pooling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMSITJW6}},
  note         = {Machine review of arXiv:2505.20150}
}

read the original abstract

Multiset functions, which are functions that map multisets to vectors, are a fundamental tool in the construction of neural networks for multisets and graphs. To guarantee that the vector representation of the multiset is faithful, it is often desirable to have multiset mappings that are both injective and bi-Lipschitz. Currently, there are several constructions of multiset functions achieving both these guarantees, leading to improved performance in some tasks but often also to higher compute time than standard constructions. Accordingly, it is natural to inquire whether simpler multiset functions achieving the same guarantees are available. In this paper, we make a large step towards giving a negative answer to this question. We consider the family of k-ary Janossy pooling, which includes many of the most popular multiset models, and prove that no piecewise linear Janossy pooling function can be injective. On the positive side, we show that when restricted to multisets without multiplicities, even simple deep-sets models suffice for injectivity and bi-Lipschitzness.

Figures

Figures reproduced from arXiv: 2505.20150 by the authors.

**Figure 1.** This figure illustrates that the assumption that multisets do not have (near)-repeated points is realistic for small molecule datasets. In (a) we show an example multiset from the QM9 [Ruddigkeit et al., 2012, Ramakrishnan et al., 2014] small molecule dataset. This example shows visually that different set elements are not very close together. In (b) we see the statistics of the minimal distance within each multiset… view at source ↗

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