PAPER: Importance of complete and bi-continuous invariants for ML on data with real valuesThis paper formalizes practically important conditions for application-driven ML on real objects with ambiguous representations and develops new canonical representations satisfying these conditions for any clouds (finite sets) of unordered points in Euclidean space R n . Such a cloud is the most basic form of a real object from cars to molecules (Wang & Solomon, 2019), e.g. a set of corners or atoms.Many objects are rigid in the sense that their shape and properties are preserved under rigid motion composed of translations and rotations in R n (Atz et al., 2021), which form the group SE(n). The slightly weaker relation is by isometries (distance-preserving transformations), which form the group E(n). The practical cases are dimensions n ≤ 3 and larger numbers m (hundreds) of unordered points without outliers (Shi et al., 2021) because atoms have stable nuclei.. Correspondence to: Anonymous Author <anon.email@domain.com>.Preliminary work. Under review by the International Conference on Machine Learning (ICML). Do not distribute.Any rigid cloud has infinitely many representations, e.g. lists of point coordinates, but the shape and properties of an object should be independent of a coordinate system. Points are usually unordered and even simple molecules have many indistinguishable atoms. Hence predictions should not depend on point ordering. On another hand, different rigid classes of chemically identical molecules can have different functional properties such as solubility and hence therapeutic effectiveness. If not all rigid classes are distinguished, drugs can become useless, implying human suffering and financial losses for manufacturers (Morissette et al., 2003).A repeated scan or measurement of the same object can produce a slightly different cloud that cannot be exactly matched with the original one by rigid motion, also due to atomic vibrations (Feynman, 1971). If noise is ignored up to any threshold ε > 0, sufficiently many tiny perturbations make all clouds equivalent by the transitivity axiom: if A ∼ B and B ∼ C, then A ∼ C (Brink et al., 1997).Since all small deviations between rigid classes of point clouds should be distinguished, all these classes live in a continuous space of rigid clouds, see Fig. 1 (left). This space was continuously parametrized only in dimension n = 1 or for m = 3 points or Fig. 1 Machine learning previously focused on discrete classifications or success measures for finite datasets, which can be considered discrete samples (of measure 0) in continuous spaces. For generalizability to all real data outside finite datasets, application-driven ML needs new conditions formalized in Problem 1.1 below. (Li et al., 2021;Dym & Gortler, 2024;Maennel et al., 2024;Nigam et al., 2024) studied complete invariants without realizability and Lipschitz bi-continuity (Morris et al., 2024;Cahill et al., 2024).Problem 1.1. Find a complete and bi-continuous invariant I : {clouds of unordered points in R n } → a space X with a distance d such that all the conditions below hold. (c) Lipschitz continuity: there is a constant λ such that if each point of a cloud A ⊂ R n is perturbed up to Euclidean distance ε, then I(A) changes by at most λε in the metric d.(d) Realizability: the image {I(A) | clouds A ⊂ R n of unordered points} is parametrized so that one can reconstruct A up to rigid motion from any realizable value of I.(e) Point matching: there is a constant µ that guarantees for any clouds A, B a rigid motion matching all points of A, B up to Euclidean distance µd(I(A), I(B)).(f) Computability: for a fixed dimension n, the invariant I, the metric d, and all constructions in (d) and (e) are computable in polynomial time of the number of points.Clouds and rigid motion can be replaced with any data (graphs, meshes) and equivalences (also allowing reflections or uniform scaling), respectively, so Problem 1.1 makes sense for any real data with ambiguous representations.The completeness (or injectivity) in 1.1(a) fully answers the question "same or different?" A complete invariant I has the ultimate expressive power and always distinguishes all clouds A ̸ ∼ = B (not only from a finite dataset) that cannot be matched by rigid motion, so I is a descriptor with no false negatives and no false positives. The universal approximation aims for the completeness of infinite-size invariants (Maron et al., 2019;Keriven & Peyré, 2019;Yarotsky, 2022), so polynomial time in 1.1(f) makes all conditions harder.A complete invariant can give a discontinuous metric, say d(A, B) = 1 for all non-equivalent clouds without quantifying the similarity of near-duplicates. The continuity in 1.1(c) is necessary for smoothness and hence for any gradient-based optimisation Due to the first axiom in 1.1(b), any metric d detects rigidly equivalent clouds by checking if d(A, B) = 0. Without the first axiom, many more distances including the zero d ≡ 0 satisfy the other axioms and are called pseudo-metrics (Brécheteau, 2019). If the third axiom in 1.1(b) fails with any additive error ε > 0, results of clustering may not be trustworthy (Rass et al., 2024).The realizability in 1.1(d) implies that the invariant I is an invertible 1-1 map from the complicated Cloud Rigid Space CRS(R n ; m) of classes of clouds under rigid motion to the explicitly parametrized space I(CRS(R n ; m)) of realizable values. Then with 100% certainty, we can sample any value in I(CRS(R n ; m)) and reconstruct its cloud A ⊂ R n .The 1-1 point matching in 1.1(e) can be interpreted as the Lipschitz continuity of the inverse map I -1 so that any close values I(A), I(B) guarantee the closeness of A, B under rigid motion. Conditions 1.1(c,e) mean that I is bi-Lipschitz: ε/µ ≤ d(I(A), I(B)) ≤ λε, where ε is the minimum perturbation needed to match all points of A, B.A partial matching, e.g. ignoring outliers, is harder to formalize. Indeed, if any clouds sharing all points except one are called equivalent, the transitivity axiom allows us to build a chain of equivalences A 1 ∼ • • • ∼ A k changing one point at a time, which can make all clouds equivalent.One can define metrics satisfying 1.1(a,b,c) by minimizing or deviations of unordered points over infinitely many transformations but polynomial time in 1.1(f) makes Problem 1.1 notoriously hard, previously solved only for m = 3 points.Conditions 1.1(a,b,c,f) and 1.1(d,e,f) formalize the discriminative and generative goals, respectively. A full solution to Problem 1.1 will imply that the rigid classes of clouds can be efficiently visualized in the moduli space I(CRS(R n ; m)) replacing any latent space of non-invariants or incomplete (or discontinuous or non-realizable) invariants. Geographically, I(CRS(R n ; m)) can be compared with Earth's map, where any location can be reconstructed with all properties (altitude, precipitation, images, ...) from the latitude and longitude coordinates in known (realizable) ranges.Contributions. Problem 1.1 formalizes the necessary conditions for any application-driven ML on real objects. The new invariant Nested Distributed Projection solves Problem 1.1 for all clouds of m unordered points in dimension n = 2. Any cloud A ⊂ R n can be reconstructed from a small part of the invariant (a vector in R n(m-(n+1)/2) ) whose realizability in 1.1(d) is guaranteed by explicitly written inequalities. Hence coordinates of this vector can be chosen in known ranges like latitude and longitude on Earth maps. The appendices cover all dimensions n > 2. The Python/C++ code is in the supplementary materials.
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# Summary Of The Paper

This paper proposes a novel framework for constructing *complete* and *bi-continuous* invariants for unordered point clouds in $\mathbb{R}^n$, with a particular emphasis on applications in machine learning for real-world objects such as molecules and shapes. The core contribution is the development of **Nested Distributed Projection (NDP)**, a theoretically grounded invariant that satisfies stringent mathematical criteria, including completeness, bi-Lipschitz continuity, realizability, and polynomial-time computability. The authors argue that these properties make NDP a powerful tool for distinguishing rigidly inequivalent point clouds and enabling accurate prediction tasks in ML. They validate the approach empirically on large molecular datasets such as QM9 and GEOM Drugs, reporting high classification accuracy in identifying chemical elements based on geometric invariants.

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# Strengths

1. **Theoretical Foundation**: The paper introduces a rigorous mathematical framework for defining invariants that are both complete and bi-continuous, addressing a long-standing gap in the literature. The authors formally define the requirements for such invariants and demonstrate that the proposed Nested Distributed Projection meets these criteria.

2. **Novel Algorithm**: The **Nested Distributed Projection (NDP)** is a novel and well-defined invariant that integrates geometric properties of point clouds in a way that is both expressive and amenable to computation. Its recursive structure offers a principled way to encode global and local geometric relationships within the cloud.

3. **Empirical Validation on Large-Scale Datasets**: The authors conduct experiments on real-world molecular datasets (QM9, GEOM Drugs) demonstrating the efficacy of NDP in distinguishing chemically distinct molecules. Their reported accuracy of ~98% on predicting chemical elements is notable, suggesting strong discriminative power.

4. **Algorithmic Complexity Analysis**: The paper provides asymptotic time and space complexity analyses for computing NDP and comparing point clouds using the Nested Bottleneck Metric (NBM), offering insights into scalability.

5. **Connection to Existing Literature**: The paper situates its contributions within the context of prior research on complete invariants, shape theory, and deep learning for point clouds, highlighting both the novelty and relevance of the proposed method.

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# Weaknesses

1. **Incomplete Proofs and Missing Derivations**: Several key theoretical claims — notably the completeness of the NDP (Theorem 3.5) and the realizability of abstract point-based representations (Theorem B.7) — are presented without full derivations or sufficient supporting lemmas. This undermines the rigor of the argument and raises doubts about the validity of the claims.

2. **Ambiguous Definitions and Implementation Details**: Critical components such as the **Nested Bottleneck Metric (NBM)** and the process of reconstructing point clouds from invariants are not clearly articulated. The lack of a concrete reconstruction algorithm limits the reproducibility and practical applicability of the method.

3. **Insufficient Empirical Evaluation**: Despite the claims of high performance, the experimental section lacks proper baseline comparisons against established methods (e.g., graph kernels, geometric hashing, or neural networks designed for point cloud processing). Additionally, there is no mention of statistical significance testing, confidence intervals, or ablation studies.

4. **Overgeneralization of Applicability**: Although the paper suggests that the method applies broadly to graphs, meshes, and other structures, the current formulation is restricted to point clouds in $\mathbb{R}^n$. No evidence or discussion is provided regarding how to adapt the method to more complex data types.

5. **Lack of Discussion on Computational Scalability**: The paper briefly mentions the computational complexity of the Nested Bottleneck Metric but does not provide empirical validation of these bounds. Furthermore, it does not explore how the method scales with increasing $n$ or $m$, nor does it evaluate trade-offs between accuracy and efficiency.

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# Questions For The Authors

1. **Prove Theorem 3.5**: How is the completeness of the Nested Distributed Projection formally established? What assumptions or lemmas support the assertion that a bijective correspondence between NDPs implies the existence of a rigid transformation?

2. **Clarify Reconstruction Procedure**: What is the exact algorithm for reconstructing a point cloud from its Nested Distributed Projection? Are there constraints on the invariant space that must be respected during reconstruction?

3. **Justify Lipschitz Continuity of NBM**: How is the Lipschitz constant of the Nested Bottleneck Metric derived? Is the constant truly universal across all $m$ and $n$, or does it depend on the number of points or dimension?

4. **Provide Full Derivation of Lemma B.1**: What happens when the $n-1$ vectors used to construct the orthogonal vector $p_n^\perp$ are linearly dependent? Is the formula still valid, and if so, how is it modified?

5. **Compare Against Baseline Methods**: Which existing methods (e.g., graph kernels, geometric hashing, or deep learning approaches) were used as baselines in the experiments? Why were they omitted from the analysis?

6. **Validate Time Complexity Claims**: What empirical data supports the claim that the Nested Bottleneck Metric can be computed in $O(m^{3.5} \log m)$ time? Have runtimes been measured on synthetic or benchmark datasets?

7. **Discuss Generalization to Graphs/Meshes**: What modifications would be required to extend the method to graphs or meshes? Has the method been tested on such data types?

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# Limitations Not Addressed By The Authors

- **Scalability and Practical Use Cases**: The paper does not address the feasibility of applying the method to large-scale problems (e.g., thousands of points or high-dimensional spaces).

- **Robustness to Noise and Outliers**: The effect of noise and outliers on the Nested Distributed Projection is not discussed. How resilient is the method to such disturbances?

- **Interpretability and Visualization**: While the paper mentions the potential for visualizing cloud spaces, it does not elaborate on how this might be done in practice or what tools or frameworks would be used.

- **Computational Overhead**: The paper does not quantify the overhead incurred by the recursive structure of the Nested Distributed Projection, nor does it analyze trade-offs between expressiveness and computational cost.

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# Soundness: 2 (Fair)

The paper presents a compelling theoretical framework but suffers from incomplete proofs, missing derivations, and insufficient empirical validation. Key claims are asserted without adequate support, reducing the overall soundness of the work.

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# Contribution: 3 (Good)

The paper contributes a novel and theoretically grounded method for constructing complete and bi-continuous invariants for unordered point clouds. However, the lack of comprehensive proofs and empirical validation prevents it from achieving the highest level of impact.

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# Confidence: 3 (Moderate)

The paper presents interesting ideas and demonstrates some empirical promise, but the absence of full proofs, rigorous evaluations, and clear implementation details reduces my confidence in the correctness and practicality of the proposed method.

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# Rating: 5 (Borderline)

Given the theoretical ambition and empirical demonstration, the paper deserves consideration for acceptance. However, the lack of complete proofs, missing baseline comparisons, and unclear implementation details prevent it from being a definitive contribution to the field. With substantial revisions, it could be accepted.

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# Brief Justification For Rating

The paper introduces a novel and theoretically rich approach to defining invariants for unordered point clouds, with a clear motivation and empirical validation on molecular datasets. However, the lack of complete proofs, insufficient empirical evaluation, and unclear implementation details reduce its impact and reliability. With significant improvements in theoretical grounding and experimental rigor, the paper could reach a higher rating. As it stands, it is borderline acceptable.

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