GeoERM: Geometry-Aware Multi-Task Representation Learning on Riemannian Manifolds

Aoran Chen; Yang Feng

REVIEW 4 major objections 5 minor 54 references

GeoERM: Geometry-Aware Multi-Task Representation Learning on Riemannian Manifolds

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Treating task representations as points on the Stiefel manifold and updating them with Riemannian gradients and polar retraction improves multi-task estimation accuracy and robustness.

desk verdict A reasonable geometry-aware twist on pERM whose robustness claim overreaches; worth reviewing and revising, but not a shift. read the letter →

arxiv 2505.02972 v1 pith:4WS7UKKM submitted 2025-05-05 stat.ML cs.LGstat.ME

Aoran Chen , Yang Feng This is my paper · ORCID

classification stat.MLcs.LGstat.ME MSC 68T0565K10

keywords multi-tasklearningrepresentationStiefelmanifoldRiemannianoptimizationpolarretractionnegativetransferoutlierrobustnessactivityrecognition

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

Multi-task learning usually treats each task's shared representation as an unconstrained matrix in Euclidean space, even when the representation is required to be orthonormal. GeoERM instead treats each representation matrix as a point on the Stiefel manifold, the curved space of $p \times r$ matrices with orthonormal columns, and updates it with Riemannian gradients followed by a polar retraction that returns it to the manifold. The paper claims this geometry-aware update improves estimation accuracy on the hardest normal tasks, reduces negative transfer, and remains stable under adversarial label noise, outperforming leading multi-task and single-task baselines. If true, the result would make manifold geometry a standard part of MTL optimization rather than a post-hoc constraint, at no extra per-iteration cost.

What carries the argument

The load-bearing object is the Stiefel manifold $\mathrm{St}(p,r)$, the set of $p \times r$ matrices with orthonormal columns, which gives a curved search space for task representations. The argument runs on two explicit manifold operations: the Riemannian gradient $\tilde{\nabla}_{A^{(t)}} f = \nabla_{A^{(t)}} \bar f - A^{(t)} \mathrm{sym}((A^{(t)})^\top \nabla_{A^{(t)}} \bar f)$, which projects the Euclidean gradient onto the tangent space, and the polar retraction $R_{A^{(t)}}(H) = (A^{(t)}+H)(I_r+H^\top H)^{-1/2}$, which maps the update back onto the manifold while preserving orthonormality. Together they replace the Euclidean update plus naive projection step that can fail Riemannian convergence conditions, and they keep the per-iteration cost comparable to Euclidean baselines.

What would settle it

Run a synthetic experiment at fixed $n=100$, $p=50$, $r=5$, $h=0.5$ with an increasing fraction of outlier tasks whose coefficient vectors are drawn with large magnitude but whose feature distributions match the normal tasks; if GeoERM's maximum error on the normal tasks approaches or exceeds that of pERM once the outlier fraction passes some threshold, the claim that manifold geometry alone shields the shared representation from outlier distortion would be contradicted.

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Extended reading notes

Core claim

The central claim is that explicitly enforcing the Stiefel-manifold geometry of task representation matrices, rather than imposing orthogonality after the fact, yields better and more robust multi-task estimates. GeoERM factorizes each task parameter as $\beta^{(t)} = A^{(t)}\theta^{(t)}$ with $A^{(t)} \in \mathrm{St}(p,r)$, adds a penalty pulling each $A^{(t)}$ toward a shared center $A$ on the same manifold, and optimizes the objective by projecting Euclidean gradients onto the tangent space and applying the polar retraction $R_A(H) = (A+H)(I_r+H^\top H)^{-1/2}$. The paper reports that on synthetic heterogeneous-task experiments and on the Human Activity Recognition benchmark, this procedure achieves the lowest maximum estimation error among non-outlier tasks and the lowest classification error across all tested representation dimensions, while remaining robust when a fraction of tasks are outliers.

Load-bearing premise

The paper's robustness claim rests on the assumption that outlier tasks, optimized jointly with normal tasks in a single objective, do not distort the learned shared manifold structure; no outlier-identification or down-weighting step enforces this, so if outliers pull the shared center the guarantee fails.

Editorial extensions

If this is right

Every iterate of the representation matrices stays exactly on the Stiefel manifold, so orthonormality is preserved during training rather than repaired afterwards.
The two-step structure inherited from pERM-style MTL is preserved, so GeoERM can be applied to any matrix-factorized MTL model with the same per-iteration cost as Euclidean updates.
Empirically, the method lowers maximum estimation error on the hardest normal tasks, which is the metric the authors use to show reduced negative transfer.
On the Human Activity Recognition benchmark, GeoERM reports the lowest classification error at every tested representation dimension, with its best result about $1.02\%$ at $r=10$.
Robustness to outliers and adversarial label noise is attributed to the geometry-aware update itself, since the algorithm optimizes all tasks in a single objective without a separate outlier-removal stage.

Reading between the lines

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

The authors leave implicit that polar retraction is a drop-in replacement for the Euclidean update in existing pERM-style pipelines, so the robustness gain could be tested by ablating only the projection-and-retraction step while keeping the penalty structure fixed.
A natural extension is to ask whether outlier robustness would improve further if Step 1 first identified or down-weighted outlier tasks before fitting the shared center A; the paper's current algorithm does not do this, so such an extension is a testable modification rather than a claim.
If the empirical gains generalize, geometry-aware updates could be applied to other matrix-valued representation spaces, such as Grassmann manifolds for subspaces or SPD manifolds for covariance representations, where the same projection-and-retraction pattern applies.
The paper's own discussion points to the need for Riemannian concentration inequalities; a concrete next step is to prove a finite-sample error bound for the Stiefel-constrained estimator, which would turn the empirical robustness claim into a testable statistical statement.

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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

The new thing here is straightforward: run pERM-style multi-task learning with the representation matrices constrained to the Stiefel manifold via Riemannian gradient descent and polar retraction. The math for the tangent-space projection and the retraction is correct, and the paper is honest that the geometry tools come from Absil et al. and the objective from Tian et al. (2023). What is new is applying them to the representation matrices in this MTL setup, and that integration seems sensible. The experiments are fairly extensive: synthetic runs across p, n, T, plus the HAR benchmark, and the code link is provided. On the core question, does geometry-aware optimization help, the answer is plausibly yes; the gains over pERM on HAR are small percentage points but consistent across r.

The soft spots are real but not fatal. The biggest one is the robustness claim. The abstract and Section 2.1 say the method stays stable under outliers and adversarial conditions, but there is no test for adversarial label noise. The synthetic outliers are tasks with arbitrary coefficients and different feature distributions, not label noise. More importantly, neither the objective in Eq. (1) nor Algorithm 1 has any outlier flag, robust loss, trimming, or downweighting. Every task enters the average equally, so an outlier task with a large loss can pull its A(t) and thereby shift the shared center A. The sentence in Section 2.1 claiming the distinction 'prevents outlier tasks from distorting the learned manifold-based structure' is simply not backed by a mechanism or a bound. The experiments may still come out fine because the outlier fraction is small, but the paper needs either a robustification step or a much more careful claim. Second, most figures have no error bars; Table 1 has standard deviations, but the simulation figures don't, which makes it hard to judge whether the gaps are noise. Third, the theoretical section is mostly standard manifold facts, and the paper's own limitations section admits there are no error bounds; the discussion's phrase 'our theoretical analysis established' oversells it. Finally, there is a small inconsistency in the algorithm description: the text says Adam is used, but the update lines show a fixed step size and retraction, no adaptive moments.

None of this kills the central idea. The optimization is sound and the empirical trend is consistent enough to be worth engineering. This paper is for readers working on multi-task or transfer learning who want an alternative to Euclidean-only methods. I would send it to referees, but ask for a revision that either adds a robust objective or removes the strong robustness claims, adds error bars, and dials down the theory framing.

For peer review: yes, engage with it, but expect meaningful revisions.

Referee Report

4 major / 5 minor

Summary. The paper proposes GeoERM, a multi-task representation learning framework that constrains task-specific representation matrices A^(t) to the Stiefel manifold and optimizes them with Riemannian gradient projection followed by polar retraction. The objective (Eq. (1)) combines per-task losses with a penalty that aligns each A^(t) to a shared center A; a second step refines task coefficients by ridge-type shrinkage toward the learned low-rank structure. The authors claim that this geometry-aware optimization improves estimation accuracy, reduces negative transfer, and remains stable under adversarial label noise, and they support this with synthetic experiments under task heterogeneity and outlier contamination, plus a real-data study on the UCI HAR dataset.

Significance. If the empirical claims held, the paper would provide a simple, general recipe for injecting manifold geometry into existing matrix-factorized MTL methods at the same per-iteration cost. Positive features include the clear algorithm, the release of code, the use of a public benchmark, and the explicit use of retraction to maintain orthonormality. However, the advertised robustness claim is not supported by the algorithm as written or by the experiments: outlier tasks are included in the joint optimization without any robust mechanism, and no adversarial label-noise experiment is actually conducted. The theoretical content is limited to standard retraction facts and a citation of Riemannian gradient descent, which does not cover the Adam-based implementation. The central idea is reasonable and independently testable, but the current evidence is conditional.

major comments (4)

[Section 2.1, Eq. (1), Algorithm 1] The claim that distinguishing normal from outlier tasks "prevents outlier tasks from distorting the learned manifold-based structure" is not realized by the algorithm. Equation (1) minimizes an unweighted average over all T tasks, with no outlier indicator, robust loss, trimming, or down-weighting, and Algorithm 1 lines 5-12 updates every A^(t) and the shared center A using gradients of that same objective. An outlier task with arbitrary beta^(t) can therefore pull its own A^(t), and through the penalty term, shift the shared center A away from the normal tasks' subspace. No mechanism or theoretical bound is provided to show this shift is negligible. The paper needs either a robustification mechanism (e.g., robust loss, trimming, or outlier detection) or a substantially weakened claim.
[Abstract and Section 3.2] The abstract advertises stability "under adversarial label noise," but no experiment in the manuscript manipulates labels. The simulation's "outlier tasks" draw coefficients from Uniform(-3,3) and features from N(0,2), which is coefficient and feature contamination, not label noise. The HAR experiment contains no label corruption either. To support the advertised claim, the authors should add experiments with actual flipped or adversarially corrupted labels, or amend the abstract and contribution statements.
[Section 2.2 and Algorithm 1] The convergence discussion in Section 2.2 invokes the Riemannian gradient descent result of Absil et al. (2008), but the implemented GeoERM uses the Adam optimizer (Section 3.1) combined with polar retraction. The descent-type inequality and stationarity conclusion for Riemannian gradient descent do not automatically transfer to Adam with retraction, and no separate convergence analysis is given for Algorithm 1. The paper should either provide a convergence guarantee for the actual algorithm or explicitly limit the theoretical claim to the geometric invariance of the iterates.
[Section 3.2 and Figures 2-7] The figures reporting maximum estimation error contain no error bars, confidence bands, or other measures of variability, even though the text says results were averaged over 100 iterations. Without variability information, the claimed consistent improvements over baselines cannot be assessed statistically. Adding error bars or standard-error tables is necessary to support the comparative claims.

minor comments (5)

[Eq. (1)] The displayed objective is missing a summation over t in the penalty term; as written, the penalty depends on an unbound index t. It should be sum over t of the norm term.
[Algorithm 1, line 5] The algorithm refers to "Euclidean gradients of \bar f (Eq. (1))", but Eq. (1) defines the full objective and the notation f versus \bar f is only introduced in Section 2.3. Please make the notation consistent.
[Section 3.1] The citation "Kingma and Adam (2015)" should be "Kingma and Ba (2015)".
[Section 4] The sentence "Our theoretical analysis established the geometric foundations" overstates the paper's theoretical content; the paper contains retraction properties and a citation of a standard descent result, but no convergence or error-bound theorems for GeoERM itself.
[Section 3.2] The text says results are averaged over 100 iterations but does not clarify whether these are independent data-generating repetitions or optimization runs; please specify.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: GeoERM's geometry claim is independently testable; self-citations to Tian et al. are transparent and non-load-bearing.

full rationale

GeoERM's central claim—that Riemannian updates on Stiefel-manifold representation matrices improve MTL—is not derived from the quantities it predicts. The objective (Eq. (1)) and Algorithm 1 are stated explicitly; no fitted parameter is renamed as a prediction, and the evaluation metric (max error over regular tasks) is not an input to the optimization. The paper repeatedly cites Tian et al. (2023), a preprint co-authored by Yang Feng, for the two-step framework, the penalty form, the hyperparameters (lambda = sqrt(r(p+log T)), gamma = sqrt(p+log T)), and the n ≳ p + log T condition. These citations are transparent and not load-bearing for the geometric contribution, which is tested against external baselines on synthetic and HAR data. One internal weakness is flagged rather than scored as circular: Section 2.1 says the normal/outlier distinction 'prevents outlier tasks from distorting the learned manifold-based structure,' but Eq. (1) and Algorithm 1 average over all T tasks with no outlier down-weighting; this is a correctness/robustness gap, not a circular reduction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the task-parameter factorization with orthonormal A, on the unproven behavior of outliers under joint optimization, on the implicit transfer of Riemannian gradient descent guarantees to an Adam-style update, and on the specific penalty form borrowed from prior MTL work. The geometric tools themselves are standard matrix-manifold results.

free parameters (4)

lambda = sqrt(r(p + logT))
Penalty weight on the distance between each A(t) and the shared center A. Set by hand following Tian et al. (2023), not fitted to the reported data.
gamma = sqrt(p + logT)
Regularization weight in the Step 2 refinement toward the low-rank structure. Chosen by hand following Tian et al. (2023).
learning_rate_alpha = 0.01
Adam step size used in all experiments; selected by the authors, not tuned per dataset.
rank_r = 5 in simulations; 5, 10, 15 in HAR
Rank of the representation. The method's performance depends on this model dimension, explored rather than jointly chosen.

assumptions (6)

domain assumption Each task's true coefficient vector factorizes as beta = A theta with A orthonormal, for normal tasks.
This structural model is stated in Section 2.1 and underlies the Stiefel manifold formulation; it is not derived from data.
ad hoc to paper Outlier tasks may take arbitrary coefficients and still do not distort the learned manifold structure when included in the joint Step 1 optimization.
Section 2.1 asserts this distinction, but Algorithm 1 has no mechanism to detect or exclude outliers; this is an unproven premise about the optimization dynamics.
ad hoc to paper Adam combined with polar retraction retains the convergence and geometric fidelity formalized for Riemannian gradient descent.
The paper cites Absil et al. (2008) for Riemannian gradient descent convergence but uses Adam in experiments without a proof that the Adam-update-plus-retraction inherits these properties.
domain assumption The Frobenius penalty ||A(t)(A(t))^T - A A^T||^2 adequately enforces subspace alignment for the shared representation.
This penalty form is borrowed from Duan and Wang (2023) and Tian et al. (2023); it is a modeling choice rather than an exact measure of subspace distance.
standard math The Stiefel manifold St(p,r) with the Euclidean metric has tangent and normal spaces as described in Section C.1, and the orthogonal projection onto the tangent space is given by Eq. (3).
This is a classical result from matrix manifold theory (Absil et al., 2008), proven in the appendix; used throughout Section 2 for the Riemannian gradient computation.
standard math The polar retraction R_A(H) = (A+H)(I + H^T H)^{-1/2} is a valid smooth retraction on St(p,r).
Propositions C.5 and C.6 in the appendix establish this using standard arguments; the paper relies on it for the update rule.

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Cite this review

Pith. "Pith review of GeoERM: Geometry-Aware Multi-Task Representation Learning on Riemannian Manifolds." pith.science (2026). https://pith.science/paper/4WS7UKKM

@misc{pith2026250502972,
  author       = {Pith},
  title        = {Pith review of: GeoERM: Geometry-Aware Multi-Task Representation Learning on Riemannian Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4WS7UKKM}},
  note         = {Machine review of arXiv:2505.02972}
}

read the original abstract

Multi-Task Learning (MTL) seeks to boost statistical power and learning efficiency by discovering structure shared across related tasks. State-of-the-art MTL representation methods, however, usually treat the latent representation matrix as a point in ordinary Euclidean space, ignoring its often non-Euclidean geometry, thus sacrificing robustness when tasks are heterogeneous or even adversarial. We propose GeoERM, a geometry-aware MTL framework that embeds the shared representation on its natural Riemannian manifold and optimizes it via explicit manifold operations. Each training cycle performs (i) a Riemannian gradient step that respects the intrinsic curvature of the search space, followed by (ii) an efficient polar retraction to remain on the manifold, guaranteeing geometric fidelity at every iteration. The procedure applies to a broad class of matrix-factorized MTL models and retains the same per-iteration cost as Euclidean baselines. Across a set of synthetic experiments with task heterogeneity and on a wearable-sensor activity-recognition benchmark, GeoERM consistently improves estimation accuracy, reduces negative transfer, and remains stable under adversarial label noise, outperforming leading MTL and single-task alternatives.

Figures

Figures reproduced from arXiv: 2505.02972 by the authors.

**Figure 2.** Maximum error across varying heterogeneity parameter [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** repeats the experiment with p = 50, increasing feature dimensionality. As expected, the task becomes more challenging, and performance differences between methods become more pronounced. GeoERM again achieves the lowest maximum error, demonstrating strong adaptability under higher dimensionality. Notably, pERM’s performance is closer to GeoERM’s with outliers, indicating stable behavior under these scenarios. Spec… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Maximum error across varying h, under ϵ = 0 (left) and ϵ = 0.1 (right). Simulations: n = 100, T = 50, p = 80, r = 5. To restore the balance n ≳ p + log T, we increase the sample size to n = 150. As shown in [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]

**Figure 5.** Figure 5: Maximum error across varying h, under ϵ = 0 (left) and ϵ = 0.1 (right). Simulations: n = 150, T = 50, p = 80, r = 5. A.1 Simulation with Different Sample Size n We next vary n ∈ [60, 200] with fixed h = 0.5. As shown in [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

**Figure 6.** Figure 6: Simulation with varying sample size n. Simulation with varying sample size n ∈ [60, 200] at fixed h = 0.5, T = 50, p = 50, and r = 5. Left: ϵ = 0; Right: ϵ = 0.1. See Section 3.2 for metric definitions and additional details. A.2 Simulation with Different Task Number T…

**Figure 7.** Figure 7: Maximum error across varying task number [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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