arxiv: 2604.07925 · v1 · submitted 2026-04-09 · 💻 cs.LG · cs.AI· math.OC

Recognition: unknown

Sinkhorn doubly stochastic attention rank decay analysis

Michela Lapenna , Rita Fioresi , Bahman Gharesifard

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:46 UTC · model grok-4.3

classification 💻 cs.LG cs.AImath.OC

keywords attention mechanismstransformersrank collapsedoubly stochastic matricesSinkhorn algorithmself-attentionneural network depthentropy regularization

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The pith

Doubly stochastic attention matrices normalized using the Sinkhorn algorithm maintain higher rank across multiple layers of a Transformer compared to standard softmax attention.

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

The paper examines how attention matrices in Transformers lose rank as the network gets deeper, leading to less informative token representations. It argues that making the attention matrix doubly stochastic using the Sinkhorn algorithm helps preserve this rank better than the usual row-stochastic softmax normalization. Although both methods show rank dropping to one in a doubly exponential way with depth, the Sinkhorn version does better in practice, especially when skip connections are used. This matters because rank collapse can degrade the model's ability to distinguish between different inputs over many layers, affecting performance on tasks like classifying text sentiment or images. The work provides both theoretical bounds and experiments to support the claim.

Core claim

Sinkhorn normalization produces doubly stochastic attention that preserves rank more effectively than softmax row-stochastic attention. The paper derives that the rank of the product of such matrices decays doubly exponentially to one with network depth, matching the known behavior for softmax, yet empirical results indicate slower effective decay and better task performance when using Sinkhorn, with skip connections playing a key role in mitigation.

What carries the argument

The Sinkhorn algorithm, which iteratively normalizes rows and columns of the attention matrix to achieve double stochasticity, ensuring equal marginals that counteract the concentration leading to rank collapse.

If this is right

Using Sinkhorn attention can allow for deeper Transformer models without as rapid loss of representational capacity.
Skip connections become even more important in pure attention stacks to avoid collapse.
Empirical improvements on sentiment analysis and image classification tasks suggest practical benefits from this normalization.
Theoretical analysis shows the decay rate is the same order as softmax, pointing to finite-depth advantages.

Where Pith is reading between the lines

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

The advantage might become more pronounced in very deep models where finite effects accumulate.
This approach could be combined with other techniques like layer normalization to further stabilize representations.
Investigating the interaction with other attention variants such as multi-head might show how rank preservation translates to overall model capacity.

Load-bearing premise

That the difference in rank preservation between Sinkhorn and softmax arises primarily from the double stochasticity rather than implementation details or finite precision effects in the normalization process.

What would settle it

A direct comparison measuring the singular values or rank of attention matrices at each layer in identical network setups with and without Sinkhorn normalization, checking if the decay curves match exactly or diverge.

Figures

Figures reproduced from arXiv: 2604.07925 by Bahman Gharesifard, Michela Lapenna, Rita Fioresi.

**Figure 1.** Figure 1: Attention matrices from the first layer and a single attention head of a Vision Transformer [2] trained on MNIST [21], for one sampled input image. See Appendix F for details on the experimental setup. Row-stochastic attention (Softmax) concentrates on a few key tokens, while doubly stochastic attention (Sinkhorn) distributes attention more uniformly across tokens. Both matrices are visualized on a shared … view at source ↗

**Figure 2.** Figure 2: Rank collapse for the product of attention matrices in a path [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Normalized spectral norm of res(Pt) in 11 as a function of path depth t, with the y-axis on a logarithmic scale. For each depth, results are estimated from 100 sampled attention paths in trained Transformer architectures. The central line indicates the median, while the box spans the interquartile range, and the whiskers extend to non-outlier values. Lower values indicate rank closer to one. 8 [PITH_FULL_… view at source ↗

**Figure 4.** Figure 4: Normalized spectral norm of res(SAN(X)ℓ) in 12 as a function of layer depth ℓ. Results show the mean over the batch, with shaded regions indicating one standard deviation. We use SAN(X) in the y-axis label to denote all four settings: a pure self-attention network, a SAN(X) with skip connections, a SAN(X) with feed-forward layers, and a Transformer with both skip connections and feed-forward layers. 10 [P… view at source ↗

**Figure 5.** Figure 5: Normalized spectral norm of res(Pt) in 11 as a function of path depth t. The attention matrices in Pt are randomly generated rather than extracted from a trained Transformer. For each depth, results are estimated from 100 sampled attention paths. Lower values indicate rank closer to one. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

read the original abstract

The self-attention mechanism is central to the success of Transformer architectures. However, standard row-stochastic attention has been shown to suffer from significant signal degradation across layers. In particular, it can induce rank collapse, resulting in increasingly uniform token representations, as well as entropy collapse, characterized by highly concentrated attention distributions. Recent work has highlighted the benefits of doubly stochastic attention as a form of entropy regularization, promoting a more balanced attention distribution and leading to improved empirical performance. In this paper, we study rank collapse across network depth and show that doubly stochastic attention matrices normalized with Sinkhorn algorithm preserve rank more effectively than standard Softmax row-stochastic ones. As previously shown for Softmax, skip connections are crucial to mitigate rank collapse. We empirically validate this phenomenon on both sentiment analysis and image classification tasks. Moreover, we derive a theoretical bound for the pure self-attention rank decay when using Sinkhorn normalization and find that rank decays to one doubly exponentially with depth, a phenomenon that has already been shown for Softmax.

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.

Desk Editor's Note private letter to a colleague

The paper derives the same doubly exponential rank decay bound for Sinkhorn attention as was already known for Softmax, so the claim of better rank preservation rests only on the experiments.

read the letter

The core takeaway is that Sinkhorn-normalized attention matrices reach rank one at the same doubly exponential rate with depth as row-stochastic Softmax ones. The paper restates this bound for the Sinkhorn case and then points to empirical results on sentiment analysis and image classification to argue that the normalization still preserves rank more effectively in practice. Skip connections are shown to help, which aligns with earlier findings for Softmax.

Referee Report

2 major / 2 minor

Summary. The manuscript studies rank collapse in Transformer self-attention, claiming that Sinkhorn-normalized doubly stochastic attention matrices preserve rank more effectively than standard Softmax row-stochastic ones. It derives a theoretical bound showing that rank decays to one doubly exponentially with depth under pure Sinkhorn self-attention (a form previously shown for Softmax), stresses the mitigating role of skip connections, and reports empirical validation on sentiment analysis and image classification tasks.

Significance. If the empirical results demonstrate clear rank-preservation benefits with appropriate controls, the work would usefully extend the analysis of attention-induced rank collapse to an alternative normalization and could motivate Sinkhorn use in deep models for better signal propagation. The provision of a matching theoretical bound for Sinkhorn is a positive step toward unifying the analysis of row- versus doubly-stochastic attention, even though the asymptotic form is identical.

major comments (2)

[theoretical bound derivation] Abstract and theoretical bound section: the derived rank-decay bound is stated to take the same doubly exponential form previously obtained for Softmax, with no explicit comparison of contraction bases, leading constants, or finite-depth behavior. This leaves the central claim that Sinkhorn 'preserve[s] rank more effectively' without theoretical support and shifts the entire burden onto the empirical sections.
[empirical validation] Empirical validation sections: the abstract reports results on sentiment analysis and image classification but supplies no information on controls for entropy-regularization strength, skip-connection scaling factors, or optimization trajectory differences between Sinkhorn and Softmax runs. Without these, observed rank differences cannot be confidently attributed to the normalization choice.

minor comments (2)

The manuscript should explicitly cite the prior Softmax rank-collapse results it reuses and clarify whether the Sinkhorn analysis re-derives the bound from scratch or directly imports the earlier proof structure.
Clarify the precise definition of 'rank' employed (e.g., numerical rank, effective rank via singular-value threshold) and whether error bars or multiple random seeds are reported for the empirical rank-decay curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the contributions and limitations of our analysis. We address each major point below and describe the revisions we will make.

read point-by-point responses

Referee: Abstract and theoretical bound section: the derived rank-decay bound is stated to take the same doubly exponential form previously obtained for Softmax, with no explicit comparison of contraction bases, leading constants, or finite-depth behavior. This leaves the central claim that Sinkhorn 'preserve[s] rank more effectively' without theoretical support and shifts the entire burden onto the empirical sections.

Authors: We agree that the derived bound for pure Sinkhorn self-attention takes the same doubly exponential form as the known Softmax bound, and the manuscript does not include an explicit comparison of contraction bases, leading constants, or finite-depth rates. The central claim that Sinkhorn attention preserves rank more effectively is therefore supported by the empirical results rather than by a stricter theoretical contraction rate. In the revised manuscript we will (i) clarify in the abstract and introduction that the asymptotic decay form is identical while the practical advantage is empirical, (ii) add a short discussion comparing the explicit constants appearing in our Sinkhorn derivation with those reported for Softmax, and (iii) note any limitations in directly comparing finite-depth behavior from the two proofs. These changes will make the division of labor between theory and experiments transparent. revision: partial
Referee: Empirical validation sections: the abstract reports results on sentiment analysis and image classification but supplies no information on controls for entropy-regularization strength, skip-connection scaling factors, or optimization trajectory differences between Sinkhorn and Softmax runs. Without these, observed rank differences cannot be confidently attributed to the normalization choice.

Authors: We accept that the current empirical sections lack sufficient detail on these controls. In the revised manuscript we will expand the experimental sections to report: the entropy-regularization parameter used for Sinkhorn normalization, the scaling coefficients applied to skip connections, learning-rate schedules, and any observed differences in optimization trajectories (e.g., loss curves or convergence epochs). We will also add a brief ablation or controlled comparison that holds all other hyperparameters fixed while varying only the attention normalization, thereby strengthening the attribution of rank-preservation differences to the choice of Sinkhorn versus Softmax. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives a new theoretical bound on rank decay specifically for Sinkhorn-normalized doubly stochastic attention and states that the resulting doubly exponential form matches the one previously shown for Softmax. This is presented as an independent derivation for the Sinkhorn case rather than a reduction to prior inputs by construction. The claim of more effective rank preservation is positioned as an empirical observation (validated on sentiment and image tasks) rather than a direct consequence of the asymptotic bound. Skip-connection mitigation is noted as previously shown for Softmax but is not used to derive the Sinkhorn result. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations that force the central result appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the mathematical equivalence of Sinkhorn normalization to doubly stochastic projection and on the transfer of rank-decay analysis techniques from prior Softmax literature.

axioms (2)

domain assumption Attention matrices admit Sinkhorn normalization to doubly stochastic form without altering the underlying attention scores in a way that invalidates rank analysis.
Invoked when replacing Softmax with Sinkhorn in the self-attention computation.
domain assumption The rank-decay proof framework previously developed for row-stochastic Softmax matrices applies directly once the matrix is made doubly stochastic.
Used to obtain the doubly exponential bound for the Sinkhorn case.

pith-pipeline@v0.9.0 · 5478 in / 1345 out tokens · 32844 ms · 2026-05-10T16:46:32.643986+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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