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arxiv: 2605.19073 · v1 · pith:3UML2GHYnew · submitted 2026-05-18 · 💻 cs.LG · cs.AI

Riemannian Networks over Full-Rank Correlation Matrices

Ziheng Chen , Xiaojun Wu , Bernhard Sch\"olkopf , Nicu Sebe This is my paper

Pith reviewed 2026-05-20 11:39 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords Riemannian networkscorrelation matricesmanifold learningdeep learning on manifoldsSPD manifoldGrassmannian manifoldbackpropagation on manifolds
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The pith

Neural networks can be defined directly on the manifold of full-rank correlation matrices using five geometries.

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

The paper establishes that Riemannian networks can operate on full-rank correlation matrices by adapting standard layers to five specific correlation geometries. It extends multinomial logistic regression, fully connected layers, and convolutional layers while supplying accurate backpropagation rules for two of the geometries. Experiments show these networks outperform prior approaches built on symmetric positive definite matrices or Grassmannian manifolds. A sympathetic reader would care because correlation matrices normalize away scale information that is often irrelevant, letting the network focus on the relational structure that many data sources actually encode.

Core claim

We introduce Riemannian networks over the manifold of full-rank correlation matrices by leveraging five recently developed correlation geometries. Basic layers including multinomial logistic regression, fully connected, and convolutional layers are extended to these geometries, and accurate backpropagation methods are presented for two of them. Experiments comparing the resulting networks against existing SPD and Grassmannian networks demonstrate their effectiveness.

What carries the argument

The five correlation geometries on the full-rank correlation manifold, which allow neural-network layers to be extended while remaining on the manifold and respecting its normalization.

If this is right

  • Convolutional layers can process correlation-matrix inputs directly without first embedding them into a larger SPD space.
  • Accurate gradients can be computed through the network for at least two of the correlation geometries.
  • Performance gains appear when the same task is solved on correlation matrices rather than on SPD or Grassmannian representations.
  • Correlation matrices function as a scale-normalized alternative to SPD matrices inside manifold-valued deep learning.

Where Pith is reading between the lines

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

  • Domains that already record pairwise correlations, such as financial time series or functional connectivity, could adopt these networks without first inflating the data to full SPD matrices.
  • The same layer-extension pattern might be applied to additional operations such as attention or pooling once the basic building blocks are shown to work.
  • If the correlation manifold admits a simpler or cheaper geometry than the SPD manifold for certain tasks, training cost per epoch could drop while accuracy rises.

Load-bearing premise

The five correlation geometries permit stable and useful extensions of neural-network operations including convolution and backpropagation that preserve manifold properties and produce measurable gains over prior manifold networks.

What would settle it

A dataset of correlation matrices on which a network built with these extended layers either fails to train stably or produces lower accuracy than an otherwise identical SPD-network baseline would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.19073 by Bernhard Sch\"olkopf, Nicu Sebe, Xiaojun Wu, Ziheng Chen.

Figure 1
Figure 1. Figure 1: Overview of our theoretical derivation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the Log-Euclidean 1D convolution with two kernels. The 3-channel input is first split into two receptive fields along the channel dimension. In each receptive field, two kernels are applied to the product space. 4. Poly-Hyperbolic-Cholesky Layers As discussed in Sec. 2, the space L n, consisting of the Cholesky factors of Cor+(n), can be identified with the product of n − 1 hyperbolic open … view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the PHCM convolution and MLR. The multi-channel input correlation matrices are denoted as {C i } c i=1. For the convolutional layer, the illustration focuses on the transformation within a receptive field and assumes a single-channel output. The multi-channel correlation matrices within a receptive field C = {C i ∈ Cor+(n)} c i=1 are first mapped to a β￾concatenated Poincare vector ´ x ∈ P … view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the decision hyperplanes in the correlation MLRs under five different geometries. The 3 × 3 correlation manifold can be embedded as an open elliptope in R 3 , by visualizing the strictly lower triangular part of each C ∈ Cor+(3). The black dots denote the boundary. The PHCM hyperplane is defined by the one in the β-concatenated Poincare space. ´ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of per-sample coefficients of variation of diagonal variances on FPHA. Higher values indicate stronger diagonal variability, which could cause nuisance noise. and the mean magnitude of off-diagonal entries O = 1 N(N − 1) X i̸=j |Σij |. (92) We then form the sample-wise ratio R = D O , (93) which measures how much larger the diagonal amplitudes are compared to the off-diagonal correlations. Eac… view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of per-sample coefficients of variation of diagonal variances on HDM05. Higher values indicate stronger diagonal variability, which could cause nuisance noise. denoted by “-EigN”. We report results on the Radar, HDM05, and FPHA datasets for representative SPD models: SPDNet, SPDNetBN, SPDResNet, SPDNetLieBN, SPDNetMLR, GyroAI, and GyroSPD++. Here, SPDResNet is implemented under the LEM, while … view at source ↗
Figure 8
Figure 8. Figure 8: Distribution of ratios of diagonal to off-diagonal entries on FPHA. to [20, 20] by the FC layer and then classified into 10 classes by the MLR layer. For each 30 ≤ n ≤ 1000, we randomly generate 30 correlation matrices and record the average runtime of a single forward pass. As implied by Tab. 7, the runtime is governed by two factors: the co-domain computation (Euclidean or hyperbolic) and the complexity … view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of ratios of diagonal to off-diagonal entries on HDM05 [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the Euclidean spaces LT0 (m), Hol(m) and Row0(m), where ⋆ can be obtained by symmetry. Now, we present the proof of Thm. 3.5. Proof of Thm. 3.5. As ECM, LECM, OLM, and LSM are pullback metrics from Euclidean spaces, we resort to Thm. I.2 and its extension Thm. I.3. Denoting the zero matrix as 0, we have the following: ϕ EC(In) = log ◦Θ(In) = 0 ∈ LT0 (n), (131) Log◦ (In) = 0 ∈ Hol(n), (132)… view at source ↗
read the original abstract

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.

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.

Referee Report

1 major / 1 minor

Summary. The paper introduces Riemannian networks over the manifold of full-rank correlation matrices by leveraging five recently developed correlation geometries. It systematically extends basic layers including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers to these geometries, presents methods for accurate backpropagation for two of the geometries, and reports experiments comparing the approach to existing SPD and Grassmannian networks to demonstrate effectiveness.

Significance. If the central constructions hold and the experiments are reproducible, the work would be a useful contribution by providing a normalized alternative to SPD-based manifold networks. The systematic layer extensions across multiple correlation geometries and the explicit backpropagation derivations for two geometries represent concrete engineering progress that could enable new applications in domains where correlation structure (rather than scale) is the primary signal.

major comments (1)
  1. Abstract: the central claim requires stable, manifold-preserving extensions of layers and backpropagation across all five correlation geometries, yet the manuscript provides explicit accurate backpropagation derivations for only two geometries. For the remaining three, no equivalent procedure is given for computing Riemannian gradients or performing updates while enforcing the correlation-matrix constraints (unit diagonal, positive semidefinite). This gap directly threatens the weakest assumption that the five geometries permit stable and useful extensions yielding measurable gains over prior manifold networks.
minor comments (1)
  1. The abstract provides no quantitative results, dataset descriptions, or implementation details, which limits the ability to assess the strength of the experimental claims from the summary alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claim requires stable, manifold-preserving extensions of layers and backpropagation across all five correlation geometries, yet the manuscript provides explicit accurate backpropagation derivations for only two geometries. For the remaining three, no equivalent procedure is given for computing Riemannian gradients or performing updates while enforcing the correlation-matrix constraints (unit diagonal, positive semidefinite). This gap directly threatens the weakest assumption that the five geometries permit stable and useful extensions yielding measurable gains over prior manifold networks.

    Authors: We appreciate the referee pointing out this important aspect. The manuscript does extend the basic layers (MLR, FC, and convolutional) to all five correlation geometries in a manifold-preserving manner, as described in Sections 3 and 4 of the paper. The backpropagation methods with explicit derivations are provided for two geometries to ensure accurate and efficient computation of Riemannian gradients. For the other three geometries, the Riemannian gradients can be obtained by projecting the Euclidean gradients onto the tangent space using the metric tensor defined for each geometry, followed by retraction operations that inherently enforce the unit diagonal and positive semidefiniteness constraints, as these are built into the manifold definitions. We will revise the manuscript to include a more detailed general framework for backpropagation applicable to all five geometries, along with pseudocode or additional explanations for the remaining three to improve clarity and address this concern directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations build on external prior geometries

full rationale

The paper takes five correlation geometries as given inputs from recent external developments and extends standard layers (MLR, FC, convolution) plus backpropagation procedures to operate on the correlation manifold. No equations or claims reduce the network constructions or experimental results back to the input geometries by definition or self-fit. Backpropagation derivations are presented as new contributions for two geometries rather than tautological renamings. The chain remains self-contained against the cited external manifold structures.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the work relies on five externally developed correlation geometries whose details and any associated assumptions are not specified here.

pith-pipeline@v0.9.0 · 5628 in / 1190 out tokens · 63202 ms · 2026-05-20T11:39:53.293900+00:00 · methodology

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Reference graph

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