Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation

Jarek Duda

REVIEW 3 major objections 5 minor 10 references

Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Rotation-invariant shape features go beyond PCA's ellipsoid.

desk verdict Honest proposal, but the headline claim outruns what is actually shown: the invariants are necessary, not sufficient, and no experiment demonstrates they separate shapes. read the letter →

arxiv 2601.03326 v3 pith:RBK7T5QI submitted 2026-01-06 cs.CV cs.LG

Jarek Duda This is my paper · ORCID

classification cs.CVcs.LG

keywords rotationinvariantsshapedescriptorscentralmomentstensorspolynomialexpansionHermitepolynomialssimilarityPCA

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 proposes extending PCA's covariance-ellipsoid shape descriptor to higher-order central moment tensors (order 3 and up) and to polynomial-times-Gaussian expansions. The aim is a rotation-invariant shape description that can be made as detailed as needed and that can be decoded back into a shape. If the approach works, two shapes that differ only by rotation can be recognized and compared by computing a fixed vector of invariants, without costly optimization over rotations. This matters for molecular shape description in chemoinformatics, 2D image and 3D scan recognition, and cheap shape-similarity metrics where rotation should be ignored.

What carries the argument

The load-bearing object is the symmetric tensor of central moments of order r, defined by averaging products of r centered coordinates, together with the graph-based invariants obtained by contracting such tensors over paired indices; each contracted edge remains invariant under a common orthogonal rotation because O^T O = I. For decodable and complete representations, the paper uses an orthonormal basis of Hermite polynomials times a Gaussian, whose coefficients are expectations of basis functions over the shape.

What would settle it

Construct two non-congruent 3D point sets with the same covariance ellipsoid but different third-order structure, then compute all graph-based invariants up to a chosen order; if the two sets produce identical invariant vectors, the descriptor fails to separate shape modulo rotation.

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Extended reading notes

Core claim

The central claim is that extending PCA to order-r central moments gives rotation-invariant features that describe a shape modulo rotation with as much detail as needed, and that polynomial-times-Gaussian expansions make the description decodable and complete in the limit as the polynomial degree grows. The paper shows how to build these tensors from densities, volumes, or weighted point sets, and how to generate graph-based invariants—such as sums of squared entries of the order-3 tensor—that are unchanged by rotation because each contracted edge pairs matching indices. It notes that these invariants are necessary conditions, and that a complete set for order 3 and higher is an open problem

Load-bearing premise

The argument depends on a finite set of graph-based invariants for order-3 and higher tensors being discriminative enough—only necessary conditions are known, and the paper itself says a complete set is a difficult open problem; if two non-congruent shapes share all tested invariants, the descriptor no longer describes shape modulo rotation.

Editorial extensions

If this is right

Shape similarity modulo rotation can be computed as a distance between invariant feature vectors, with no iterative rotation search.
The polynomial-times-Gaussian representation gives a continuous, reconstruction-capable shape descriptor whose accuracy increases with the maximum polynomial degree.
The same invariants apply to densities, volumes, weighted point sets (e.g. atoms), and grayscale images.
Variability of a shape, such as molecular dynamics, can be encoded as a distribution of invariant vectors rather than a single vector.

Reading between the lines

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

Because completeness is open for order 3 and higher, a practical risk is that two non-congruent shapes share the same invariant set; this could be probed empirically by testing discrimination on datasets of rotated versus non-congruent 2D and 3D shapes.
A natural extension is to train machine-learning models directly on these invariant vectors, using them as fixed-dimensional features alongside existing descriptors.
In d=2 and d=3, where the rotation group has only 1 or 3 dimensions, even redundant invariant sets may be discriminative, so a direct comparison against alignment-based methods could test their practical sufficiency.

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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

Jarek's paper proposes extending PCA's covariance ellipsoid to higher-order moment tensors and to polynomial-times-Gaussian expansions, with rotation invariants from graph contractions. The genuinely useful part is the framing: a single recipe that covers central moments, decodable polynomial approximations, and a similarity metric based on invariant vectors, plus the honest discussion of what is known. The MNIST reconstruction figures give a nice intuition for how expressive the polynomial representation is.

The mathematics that is stated is mostly correct. Traces of powers of the covariance matrix do determine the matrix up to rotation, and the diagrammatic contractions are rotation invariant. The paper also clearly flags that a complete set of invariants for order ≥3 is open and that the independence problem is hard. That is more candid than most.

The soft spots are the ones the paper itself names. The central claim of describing shape modulo rotation is not established for order ≥3. A finite vector of necessary conditions can easily collide for non-congruent shapes; no experiment or argument shows the invariants are discriminative in practice. The stress-test point about degree-3 invariants in d=2 is a concrete instance: the graph invariants shown are quadratic, and it is not shown that they separate SO(2) orbits. Also, the diagrammatic machinery is from the author's earlier [7], and classical moment invariants (Hu, Zernike) are not cited. So the novelty is mostly the application and the polynomial-Gaussian decoding idea, not the invariant construction.

That said, the paper is explicitly an initial proposal and says so. The open-problem section is honest, and the proposed framework is coherent enough that someone could test it. It does not overclaim in the conclusion; it lists exactly what remains.

Who is this for? Someone working on rotation-invariant shape descriptors who wants a compact summary of this particular approach and its open problems. It is not a validation paper. If it came to a journal, I would send it to a careful referee, mainly because the proposal is concrete and the limitations are stated well enough that a referee could ask for targeted experiments. But it should not be published as-is; it needs either a proof of separation for a useful invariant set or empirical demonstration on a recognition/similarity task.

I would cite it if I were writing about higher-order moment invariants, because it cleanly lays out the recipe and the open question, and I would bring it up in a reading group as an example of a paper that is honest about its own limits.

Referee Report

3 major / 5 minor

Summary. The paper proposes to extend PCA's covariance-matrix shape descriptor to higher-order central moments and to polynomial-times-Gaussian expansions, and to use graph-based tensor contractions as rotation-invariant features. These invariants are gathered into a feature vector intended to describe shapes modulo rotation and to define an inexpensive shape-similarity metric. The paper illustrates the polynomial representation on MNIST digits but provides no empirical evaluation, and it explicitly acknowledges that for order ≥3 a complete set of rotation invariants is an open problem.

Significance. The underlying mathematics is largely sound: traces of powers determine eigenvalues for symmetric matrices, and graph contractions indeed yield rotation invariants. The idea of extending PCA beyond order-2 moments is natural and could be of practical use in chemoinformatics and 2D/3D recognition. However, the central claim—that the proposed features describe shape modulo rotation—is not established for order ≥3, because the graph invariants are only necessary conditions, not sufficient ones. The paper is honest about this limitation, but the abstract and conclusion overstate the result. The MNIST figure demonstrates representation fidelity, not invariance discriminative power.

major comments (3)

[Section III.B] The paper admits that for order ≥3 a complete set of invariants is a difficult open problem and that agreement of graph invariants is only a necessary condition. Thus the proposed finite invariant vectors cannot be guaranteed to separate rotation orbits, and two non-congruent shapes may share the same descriptor. In d=2, the space of order-3 tensors has dimension 4 while SO(2) has dimension 1, so at least 3 independent invariants are needed; the paper provides no such set nor any argument that its chosen invariants are sufficiently discriminative. This gap directly undermines the 'shape modulo rotation' claim in the title and abstract.
[Section III.C] The proposed shape similarity metric is not actually specified. The text says only 'some distance between two such vectors' without defining a concrete distance, the set of features to use, or how to handle the redundancy and weighting of invariants. Without a specific metric, claims that the approach enables 'inexpensive comparison' and avoids costly rotation optimization cannot be tested or reproduced.
[General] No experiments are reported. Despite the stated applications (molecular shape, 2D/3D object recognition), there is no empirical evidence that the higher-order invariants separate non-congruent shapes or that the feature vector is robust to noise and deformations. Given that the theoretical completeness is open, a synthetic benchmark measuring orbit separation or false match rates for d=2 and d=3 tensors is essential to support the practical significance of the method.

minor comments (5)

[Section II.E] The Hermite polynomial list contains a typo: the degree-5 polynomial should be 32x^5 - 160x^3 + 120x, not '32x^2 - 160x^3 + 120x'. Also, the notation for Hermite polynomials is inconsistent (h_i vs f_j).
[Section II.B] The notation [p] is used for the covariance matrix but also for tensors in later sections; this should be clarified to avoid confusion.
[Abstract] The formula for p_ab is incomplete, ending with an ellipsis. The abstract would benefit from a cleaner presentation of the central equations.
[Section III.C] Typo: 'mane of them' should be 'many of them'.
[Section III.B] Figures 3 and 4 are referenced but not described in sufficient detail in the text; the graph construction should be defined with explicit equations so that the invariants are unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the invariance argument is derived in-text and completeness limits are explicitly acknowledged.

full rationale

The paper is a proposal rather than a derivation-from-fit. Its central invariance claim is self-contained: Eq. (10) defines p~q via existence of an orthogonal O, and Section III.A proves graph contractions are invariant using the identity sum_i O_ai O_bi = delta_ab, with the sentence 'Applying any orthogonal matrix O... for each such edge it multiplies from one side by O, from the other by O^T, not changing the summation outcome - therefore, each such graph indeed defines rotation invariant.' No parameter is fitted to data and then renamed as a prediction; the MNIST examples are illustrative reconstructions from Hermite expansions (Eqs. 4-6), not fitted predictions. The paper explicitly declines to claim completeness for order >=3: 'while agreement of such invariants is necessary conditions ... for higher orders it seems a difficult open problem' (Sec. III.B). That limitation weakens the practical claim of describing shape modulo rotation, but it is an acknowledged open problem rather than a circular reduction. Self-citations [3], [5], [6], and [7] supply context, diagrams, and a basis-expansion citation, but the load-bearing invariance proof is reproduced in the text, so the citations are not load-bearing circularity. The main caveat is novelty overlap with the author's earlier [7], which is a novelty/priority concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central mathematical claims rely on standard facts: orthogonality relations, power sums/eigenvalues, and completeness of Hermite basis. The main ad hoc assumption is that an incomplete set of high-order invariants is discriminative enough in practice. No free parameters are fitted; the choices (degree m, normalization scale) are tuning choices for a proposed method.

assumptions (5)

standard math Rotation invariance of graph-based contractions follows from Σ_i O_ai O_bi = δ_ab for orthogonal matrices.
Section III.B states that applying O to both indices of an edge does not change the summation, which is true because O^T O = I.
standard math For order-2 tensors, equality of Tr([p]^i) for i=1..d determines the eigenvalue multiset via Newton's identities.
Section II.B claims agreement of these traces ensures the covariance matrices differ only by rotation; this relies on Newton's identities for symmetric matrices.
standard math The Hermite polynomial times Gaussian basis is orthonormal and complete for continuous density estimation.
Section II.E uses the orthonormal basis (5) to compute coefficients u_j = E[f_j(x)], which requires completeness and orthonormality of the basis.
domain assumption The product basis remains approximately orthonormal when integration is replaced by summation over a finite lattice (as for MNIST).
Figure 2 caption acknowledges 'the basis slightly loses orthonormality' and suggests Gram-Schmidt; the reconstruction assumes the error is acceptable.
ad hoc to paper A finite set of redundant graph-based rotation invariants is sufficient for practical shape discrimination despite completeness being open.
Section III.B states completeness for order ≥3 is a difficult open problem but argues that redundant invariants suffice for applications in d=2,3. This is a heuristic, not proven.

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Cite this review

Pith. "Pith review of Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation." pith.science (2026). https://pith.science/paper/RBK7T5QI

@misc{pith2026260103326,
  author       = {Pith},
  title        = {Pith review of: Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RBK7T5QI}},
  note         = {Machine review of arXiv:2601.03326}
}

abstract

PCA can be used for rotation invariant features, describing a shape with its $p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$ covariance matrix approximating shape by ellipsoid, allowing for rotation invariants like its traces of powers. However, real shapes are usually much more complicated, hence there is proposed its extension to e.g. $p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$ order-3 or higher tensors describing central moments, or polynomial times Gaussian allowing decodable shape descriptors of arbitrarily high accuracy, and their analogous rotation invariants. Its practical applications could be rotation-invariant features to include shape modulo rotation e.g. for molecular shape descriptors, or for up to rotation object recognition in 2D images/3D scans maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.

Figures

Figures reproduced from arXiv: 2601.03326 by the authors.

**Figure 2.** Visualization of accuracy of Gaussian times polynomial represen [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 4.** Some possibilities for systematic generation of large numbers of [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: Diagrammatic representations ([6], [7]) of some first rotation invari [PITH_FULL_IMAGE:figures/full_fig_p003_3.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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10 extracted references · 4 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [7]

[Online]

------, `` P=? NP as minimization of degree 4 polynomial, or Grassmann number problem ,'' arXiv preprint arXiv:1703.04456, 2017. [Online]. Available: https://arxiv.org/pdf/1703.04456

arXiv 2017

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11em plus .33em minus .07em 4000 4000 100 4000 4000 500 `\.=1000 = #1 \@IEEEnotcompsoconly \@IEEEcompsoconly #1 * [1] 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEcompsocnotconfonly \@IEEEcompsocconfonly \@IEEEauthorblockAstyle \@IEEEcompsocnotconfonly \@IEEEcompsocconfonly \@IEEEcompsocconfonly \@IEEEauthordefaulttextstyle \@IEEEcompsocnotc...

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Mavridis, B

L. Mavridis, B. D. Hudson, and D. W. Ritchie, ``Toward high throughput 3d virtual screening using spherical harmonic surface representations,'' Journal of chemical information and modeling, vol. 47, no. 5, pp. 1787--1796, 2007

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