Learning Along the Arrow of Time: Hyperbolic Geometry for Backward-Compatible Representation Learning

Leonardo Neves; Menglin Yang; Mingxuan Ju; Neil Shah; Ngoc Bui; Rex Ying; Runjin Chen; Tong Zhao

REVIEW 4 major objections 4 minor 70 references

Learning Along the Arrow of Time: Hyperbolic Geometry for Backward-Compatible Representation Learning

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By lifting embeddings into hyperbolic space and constraining new ones inside old entailment cones, this paper claims to keep upgraded models compatible with existing galleries while letting uncertain old embeddings be downweighted.

desk verdict A promising but under-validated hyperbolic approach to backward-compatible retrieval; the uncertainty mechanism is the main risk. read the letter →

arxiv 2506.05826 v1 pith:MVMQKNS2 submitted 2025-06-06 cs.LG

Ngoc Bui , Menglin Yang , Runjin Chen , Leonardo Neves , Mingxuan Ju , Rex Ying , Neil Shah , Tong Zhao This is my paper

classification cs.LG

keywords backward-compatiblerepresentationlearninghyperbolicgeometryentailmentconeuncertaintyestimationcontrastivealignmentmodelevolutionimageretrievalLorentz

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

Backward-compatible representation learning aims to let an upgraded embedding model query a gallery built by its predecessor, so stored data never needs to be re-embedded. The paper claims this compatibility is more naturally achieved in hyperbolic space: embeddings are lifted onto a hyperboloid, each new embedding is constrained to lie inside the entailment cone of its old counterpart, and the alignment loss weights each old–new pair by a norm-derived uncertainty of the old embedding. Uncertain old embeddings get wider cones and lower alignment weight, so the new model is free to improve rather than being forced to reproduce outdated representations. The reported gains are a 21.4% improvement in CMC@1 compatibility and a 44.8% improvement in mAP compatibility over the strongest Euclidean baseline, with little loss in new-model performance. The paper frames this as treating model evolution as movement along a time-like dimension of hyperbolic space.

What carries the argument

The load-bearing object is the hyperbolic entailment cone in the Lorentz model of hyperbolic space. For an old embedding $h_o$, the cone has half-aperture $\mathrm{aper}(h_o)=\sin^{-1}(2\epsilon\sqrt K/\|h_{o,\mathrm{space}}\|)$, which narrows as the embedding moves away from the origin; the loss $\mathcal L_{\mathrm{entail}}=\max(0,\mathrm{ext}(h_o,h_n)-\mathrm{aper}(h_o))$ penalizes new embeddings that fall outside it. Uncertainty is measured by $\mathrm{Uncertainty}(h)=1-\frac{1}{\sqrt K}\tanh(\sqrt K\|z\|)$, which equals the norm shrinkage caused by the exponential map and is bounded in $[0,1]$ for $K=1$. A robust contrastive alignment loss (RINCE) with $q(h_o)=\mathrm{Uncertainty}(h_o)$ downweights positive pairs whose old embedding is uncertain. The full objective is $\mathcal L=\mathcal L_{\mathrm{base}}+\lambda(\mathcal L_{\mathrm{entail}}+\mathcal L_{\mathrm{contrast}})$, applied after exponential-map lifting of any Euclidean encoder's output.

What would settle it

Measure, on a held-out set, the correlation between the old model's norm-based hyperbolic uncertainty and its actual retrieval errors for each gallery item; if low-uncertainty (high-norm) embeddings are equally or more error-prone, or if downweighting low-norm embeddings does not improve new-to-old retrieval, the central mechanism is falsified. A direct experiment would inject label noise into high-norm training samples so that old-model errors concentrate there, and check whether HBCT's gains invert.

Watch

Extended reading notes

Core claim

The central claim is that the radial structure of hyperbolic embeddings provides a usable signal for model evolution: the same coordinate that encodes an old embedding's uncertainty also controls how much the new embedding may deviate from it. Concretely, HBCT defines uncertainty as $1-\frac{1}{\sqrt K}\tanh(\sqrt K\|z\|)$ for the pre-exponential Euclidean vector $z$, and sets the entailment-cone aperture inversely to the old embedding's norm, so unreliable old embeddings produce wider permissible cones. The contrastive term uses RINCE with $q(h_o)=\mathrm{Uncertainty}(h_o)$, dynamically lowering the pull toward uncertain old embeddings. The paper demonstrates across extended-data, extended-class, new-architecture, and combined scenarios that this yields stronger new-to-old retrieval compatibility than Euclidean baselines such as BCT, Hot-Refresh, AdvBCT, and HOC, and that the advantage persists through five consecutive model updates.

Load-bearing premise

The load-bearing premise is that an old embedding's norm in hyperbolic space reliably measures how uncertain or unreliable that embedding is; if low-norm embeddings are not the ones causing retrieval failures, the adaptive weighting and cone-widening would be miscalibrated and the compatibility gains could vanish.

Editorial extensions

If this is right

An upgraded model can query the existing gallery immediately, so backfilling the vector database becomes optional rather than mandatory.
Old embeddings are no longer treated as ground truth: high-uncertainty old embeddings are downweighted and allowed to drift more, so the new model can learn better representations without sacrificing compatibility.
The compatibility advantage is reported to survive multiple consecutive updates, where Euclidean alignment methods' compatibility declines rapidly.
The method is described as applicable to any base loss, meaning the same hyperbolic alignment could wrap supervised, self-supervised, or foundation-model training.
Even the hyperbolic method without the entailment loss outperforms Euclidean baselines in several settings, suggesting the geometry itself, not just the cone constraint, drives part of the gain.

Reading between the lines

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

One natural test is to replace the norm-based uncertainty with a calibrated measure, such as the old model's own softmax entropy on a probe set, and check whether compatibility improves further; the geometry would stay intact while the weight $q(h_o)$ becomes task-grounded.
Because the objective is agnostic to the base loss, HBCT could be applied to self-supervised and multimodal foundation-model updates; a concrete extension would be fine-tuning a CLIP-style model with the hyperbolic alignment and measuring zero-shot retrieval compatibility.
The paper's own limitation about growing clipping thresholds implies a stress test: run fifty or more sequential updates and observe whether norm growth destabilizes training, or whether periodic rescaling of the time coordinate restores stability.
The exponential volume growth of hyperbolic space suggests the approach may scale naturally to continual learning with many emerging classes, a regime where Euclidean compatibility methods degrade, though the paper only demonstrates five update steps.

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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

HBCT is a sensible combination of existing ingredients—Lorentz embeddings, entailment cones, and RINCE—with a new twist: the norm of the old hyperbolic embedding is used as a per-sample uncertainty to widen the cone and downweight the contrastive alignment. The method is simple, has few hyperparameters, and reports large compatibility gains over strong Euclidean baselines across most settings. To my knowledge the specific combination is new, and the idea of treating the old embedding's norm as a quality signal in backward-compatible training is worth exploring.

What the paper does well: the geometry is mostly careful, the entailment cone loss is borrowed from Desai et al. and applied sensibly, the RINCE-based adaptive weighting is clearly motivated, and the experiments cover four realistic scenarios including architecture change and new classes. The ablation in Table 2 shows that both the cone and the robust loss add something, though the robust loss's contribution is modest (RINCE w/ entail vs InfoNCE w/ entail: 0.495 vs 0.475 CMC@1 Pcom). The code is promised on GitHub, which is good practice.

The soft spots are real. First, the uncertainty assumption is not validated at the per-example level. Eq (10) defines a deterministic function of the norm, and Figure 2 only shows that unseen classes have lower norms on average. That doesn't establish that low norm correlates with the old-model errors that actually hurt new-to-old retrieval. If the ranking is wrong, the adaptive weighting and cone widening downweight the wrong pairs. The small RINCE-vs-InfoNCE gap in the ablation makes this more, not less, urgent. Second, the experimental protocol is weaker than the claims: baselines are tuned over 10 hyperparameter combinations with the best run on the test metrics, no validation split is mentioned, and no error bars or multi-seed results are given. That can easily inflate gaps, especially the 21.4% and 44.8% headline numbers. Third, there's a minor algebraic slip in Eq (10): the factor 1/sqrt(K) appears misplaced for K != 1 (the time coordinate already carries a 1/sqrt(K)), and the uncertainty formula should be 1 - tanh(sqrt(K)||z||) rather than 1 - (1/sqrt(K)) tanh(...). The experiments use K=1, so it's cosmetic, but it should be fixed. Finally, HBCT is not uniformly best: in the Both/TinyImageNet row, Hot-Refresh has higher CMC@1 Pcom (0.365 vs 0.348).

I think the core idea is sound and the gains may be real, but the paper doesn't yet prove that the uncertainty mechanism is the source of the gains. It's a good candidate for serious peer review, not a desk reject. A referee should ask for per-example calibration analysis, multi-seed runs with a validation split for hyperparameter selection, and a corrected Eq (10).

Referee Report

4 major / 4 minor

Summary. The paper proposes HBCT, a method for backward-compatible representation learning that operates in hyperbolic space (Lorentz model). The old and new embedding models are mapped to the hyperboloid, and training enforces two geometric constraints: (i) an entailment-cone loss that keeps each new embedding inside the cone defined by its old counterpart, and (ii) a robust contrastive (RINCE) loss in which the per-pair weighting exponent q is set to a hyperbolic uncertainty estimate of the old embedding (Eq. 10). The uncertainty estimate is a monotone function of the old embedding’s norm, and the paper argues that lower-norm old embeddings are less reliable, so they are downweighted and their cones are widened. Experiments on CIFAR-100 and Tiny-ImageNet across four update scenarios report that HBCT improves backward-compatibility metrics (Pcom) over Euclidean baselines, with claimed average gains of 21.4% in CMC@1 and 44.8% in mAP, at small cost to new-model performance. Ablations and appendices probe the influence of curvature, clipping thresholds, alignment weights, and distance functions.

Significance. If the reported gains are robust, the paper would introduce a useful new perspective: hyperbolic geometry provides a principled way to express both representation uncertainty and the ‘evolution’ of an embedding space across model updates, and the entailment cone offers a natural partial-order constraint for compatibility. The paper ships code, which aids reproducibility, and it is among the first to apply hyperbolic geometry to backward-compatible training. However, the significance is tempered by two factors: the experimental evidence comes from two small image datasets with no large-scale validation, and the central uncertainty mechanism, which is the paper’s main conceptual novelty, has only weak empirical support. The method’s gains appear to come substantially from the hyperbolic geometry/entailment constraint rather than from the uncertainty-aware weighting, and the comparison against Euclidean baselines is confounded by the use of different (and often better) hyperbolic old models. The core idea is promising and worth developing, but the current evidence falls short of establishing that the uncertainty-driven mechanism is the source of the benefits.

major comments (4)

[§4.1, Eq. (10), Fig. 2] The uncertainty measure q(ho)=1-(1/sqrt(K))tanh(sqrt(K)||z||) is used per-sample to set the RINCE exponent in Eq. (12) and to widen entailment cones via Eq. (11), but the only evidence that this quantity tracks old-model reliability is the aggregate, class-conditional histogram in Fig. 2. Old-model error is per-example: an embedding can have low norm because it is out-of-distribution, because it is a hard example, or because it lies near a classification boundary, and these cases do not necessarily coincide with the errors that actually hurt new-to-old retrieval. The authors should validate the calibration directly, for example by correlating Uncertainty(ho) with per-sample retrieval errors of the old model, or with the distance between same-class old and new embeddings. Without such evidence, the central mechanism of the paper—adaptive downweighting of unreliable old embeddings—remains an unsupported assumption.
[§5.1, Eq. (14), Table 1] The headline claim that HBCT improves CMC@1 compatibility by 21.4% over the strongest Euclidean baseline is not clearly reproducible from Table 1. In the Both/TinyImageNet scenario the HBCT Pcom (0.348) is actually lower than the best Euclidean baseline Hot-Refresh (0.365), and across all eight cells the relative improvements vary from -4.7% to +53%. Moreover, Pcom as defined in Eq. (14) is not invariant to the choice of old model: its denominator M(phi*, phi*_G) - M(phi_o, phi_o_G) shrinks when the old model improves, so the stronger hyperbolic old models (e.g., 0.425 vs 0.376 on Ext-class CIFAR CMC@1) inflate Pcom even for identical absolute compatibility gains. The paper acknowledges that old models differ but claims Pcom is ‘calibrated’; it is not fully calibrated. The authors should either use the same old model across geometries (e.g., lift a Euclidean old model’s embeddings into hyperbolic space) or report raw cross-retrieval metrics with a careful sensitivity analysis.
[§5.1, Implementation Details] The experimental protocol has no validation split: for each Euclidean baseline, 10 hyperparameter combinations are evaluated and the run that best performs on the test metrics (subject to not degrading the new model) is selected, while HBCT’s hyperparameters are fixed based on an ablation. No error bars, standard deviations, or multiple seeds are reported. This makes it impossible to assess whether the reported margins are statistically meaningful or whether the baseline tuning procedure overfits the test set. The authors should select hyperparameters on a hold-out validation set and report mean ± std over at least three independent runs for all methods.
[§5.4, Table 2] The ablation in Table 2 shows that replacing the uncertainty-weighted RINCE loss with a standard InfoNCE loss changes CMC@1 Pcom only from 0.495 to 0.475, while removing the entailment loss reduces it to 0.435. The difference attributable to the uncertainty-aware weighting (0.02) is small relative to the overall improvement over Euclidean baselines (e.g., 0.495 vs 0.360 for the best Euclidean baseline in the same setting). This suggests that the paper’s central narrative—that dynamically adjusting alignment weights based on uncertainty is the key to the gains—is not supported by the ablation. The authors should either provide additional experiments that isolate the effect of q(ho) (e.g., random or reverse uncertainty weighting) or temper the claims about the uncertainty mechanism.

minor comments (4)

[Abstract and §5.2] The 21.4% and 44.8% improvements are stated without a precise definition of how the average is computed across settings and datasets. Specify the aggregation rule and report the per-setting numbers.
[§4.1, Eq. (10)] The displayed formula for Uncertainty(h) is garbled in the manuscript, with several alignment artifacts. Please rewrite it clearly, and also clarify the relationship between the Lorentz-model expression and the Poincaré-ball uncertainty used in prior work (the factor 1/sqrt(K) appears inconsistent with the stated equivalence for K ≠ 1).
[§4.2, Eq. (11)] The aperture constant ε is fixed to 0.1 without discussion. A one-sentence sensitivity analysis or a reference to the original derivation would help.
[§5.3] The sequential-update experiment reports that HBCT ‘demonstrates superior compatibility maintenance’ but gives no quantitative summary of the compatibility matrices. Include the average or diagonal-to-off-diagonal decay metrics.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the headline compatibility gains are held-out empirical results with fixed hyperparameters, leaving only the unvalidated norm-as-uncertainty premise of Eq. (10) as a quasi-circular conceptual loop.

full rationale

I found no circular step that meets the evidence bar. The paper's central claims, a 21.4% average CMC@1 Pcom improvement and a 44.8% mAP Pcom improvement over the strongest Euclidean baseline, are empirical measurements computed with the held-out Pcom/Pup metrics of Eqs. (14)-(15) on test galleries; they do not reduce to any fitted parameter or to any equation equal to the method's input by construction. Hyperparameters are fixed constants (K=1.0, lambda=0.3, tau=0.5, beta=0.01, zeta_o=1.0, zeta_n=1.2), and the Euclidean baselines were actually given more tuning freedom ('For each baseline, we run with 10 combinations of these hyperparameters and choose the best run'), so the fitted-input-called-prediction pattern is inverted if anything. The uncertainty definition in Eq. (10) is transparently imported from external prior work (Franco et al. 2023; Atigh et al. 2022), not from the authors' own papers; the only co-authored references (Yang et al. 2023, 2024) appear in related work and are not load-bearing. The statement that the entailment cone widens as embeddings become more uncertain is a direct, explicitly derived consequence of Eqs. (10)-(11), both of which are monotone functions of the same norm, and the paper presents it as a design note rather than as an empirical discovery; the active contributions of the components are tested by ablation (Table 2: RINCE with entailment Pcom 0.495 vs InfoNCE 0.475 vs 0.431 without entailment). The genuine weakness is calibration, not circularity: the premise that low hyperbolic norm tracks per-example old-model error is supported only by the aggregate class-conditional histogram of Figure 2, so the uncertainty weighting could be miscalibrated and the attributed gains could partly come from the geometry rather than the uncertainty mechanism. That is a correctness risk, which the rules direct to correctness review rather than to a circularity finding. The score of 2 reflects the mild conceptual loop, where 'uncertainty' is defined through the norm and then invoked as the cause of norm-based reweighting, rather than any demonstrated derivation equivalent to its inputs.

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

The central claim rests on three groups of assumptions: norm-based hyperbolic uncertainty is valid and calibrated; the entailment-cone partial order is the correct model of embedding evolution; and the exponential-map lift makes old and new embeddings comparable. These are adaptations of prior work, not derived here. The method's free hyperparameters, curvature, clipping thresholds, lambda, tau, beta, epsilon, are fixed by hand or small sweeps, and they affect the reported performance.

free parameters (4)

Curvature K = 1.0
Global curvature of the Lorentz model; fixed at 1.0 following prior work. Ablation in Table 3 shows performance declines for K=0.1 and K=1.5, and learnable curvature hurts compatibility. The central claim depends on this choice.
Clipping thresholds zeta_o and zeta_n = 1.0 and 1.2
Norm clipping thresholds for numerical stability. Chosen by hand; ablation in Table 4 shows 1.2 balances self and cross performance, while 1.0 and 1.3 are worse. Affects the geometry and hence the entailment cones.
Entailment cone epsilon = 0.1
Small constant preventing degenerate aperture near the origin in Eq. (11); taken from Desai et al. 2023.
Contrastive and alignment hyperparameters lambda, tau, beta = lambda=0.3, tau=0.5, beta=0.01
Chosen by hand or small sweep over values; fixed across settings for HBCT. Ablation in Figure 7 shows lambda and tau trade-offs. These are not fit to a held-out validation set.

assumptions (4)

domain assumption The L2 norm of a hyperbolic embedding, equivalently Eq. (10), is a valid monotone measure of model uncertainty, so q(ho)=Uncertainty(ho) correctly downweights unreliable old embeddings.
Adopted from Atigh et al. 2022 and Franco et al. 2023. Figure 2 illustrates the heuristic on CIFAR100, but no calibrated uncertainty benchmark or theoretical justification is provided. The adaptive weighting relies on this.
ad hoc to paper The entailment cone partial order is the right model for embedding-space evolution over time: new embeddings should lie inside the cone of old embeddings.
This is a design choice, not derived. The entailment cone is borrowed from Desai et al. 2023; applying it to model updates is new and unproven. Ablations show it is critical, since removing it drops cross CMC@1 from 0.575 to 0.188 in the Both scenario.
domain assumption Both old and new Euclidean features can be lifted to the same Lorentz hyperboloid via the exponential map at the origin, making cross-model geodesic distances meaningful before alignment.
The hybrid encoder in Section 4.1 applies expm to [0,z]; comparability of old and new embeddings is assumed and only enforced by losses. No guarantees are given.
standard math Hyperbolic MLR classification with decision hyperplanes, Eq. (9), is a valid base loss for training old and new encoders.
From Bdeir et al. 2023; used without proof in this paper.

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Cite this review

Pith. "Pith review of Learning Along the Arrow of Time: Hyperbolic Geometry for Backward-Compatible Representation Learning." pith.science (2026). https://pith.science/paper/MVMQKNS2

@misc{pith2026250605826,
  author       = {Pith},
  title        = {Pith review of: Learning Along the Arrow of Time: Hyperbolic Geometry for Backward-Compatible Representation Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MVMQKNS2}},
  note         = {Machine review of arXiv:2506.05826}
}

read the original abstract

Backward compatible representation learning enables updated models to integrate seamlessly with existing ones, avoiding to reprocess stored data. Despite recent advances, existing compatibility approaches in Euclidean space neglect the uncertainty in the old embedding model and force the new model to reconstruct outdated representations regardless of their quality, thereby hindering the learning process of the new model. In this paper, we propose to switch perspectives to hyperbolic geometry, where we treat time as a natural axis for capturing a model's confidence and evolution. By lifting embeddings into hyperbolic space and constraining updated embeddings to lie within the entailment cone of the old ones, we maintain generational consistency across models while accounting for uncertainties in the representations. To further enhance compatibility, we introduce a robust contrastive alignment loss that dynamically adjusts alignment weights based on the uncertainty of the old embeddings. Experiments validate the superiority of the proposed method in achieving compatibility, paving the way for more resilient and adaptable machine learning systems.

Figures

Figures reproduced from arXiv: 2506.05826 by the authors.

**Figure 1.** a) A typical setting of the backward-compatible training problem. A large gallery set is embedded and indexed into a vector database using the old model. Updating to the model may require re-indexing the entire vector database (backfilling). Backwardcompatible training allows the new model to query and retrieve from the vector database of old embeddings directly. b) Simulating the model evolution in hyperbolic spac… view at source ↗

**Figure 2.** The distribution of uncertainty measures on hyperbolic embeddings of CIFAR100 produced by the old model ϕo. The old model is ResNet18 trained with the first 50 classes (old identities) and then used to embed the upcoming 50 classes (new identities) into the vector database. The gallery is unseen samples from the test set. The curvature parameter K is set to 1.0. section presents our base hyperbolic encoder with an u… view at source ↗

**Figure 4.** Comparison between InfoNCE and RINCE losses for varying distances between old and new embedding ho and hn. As q increases, the gradient norm decreases for noisy positive pairs (−dL ∈ [−5, −3]) and increases for clean positive pairs (−dL ∈ [−2, −0]). For this illustration, we choose one negative sample with the distance dL(hn, h ′ o) = 4.0 and β = 0.01. the inherent trade-off between maintaining compatibility and ach… view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: The compatibility matrix of different alignment methods after five consecutive model updates. The value at column j and row i quantifies the CMC@1 compatibility Pcom where the queries are embedded by the model ϕi and the gallery is embedded by ϕj . with only minimal co…

**Figure 6.** Figure 6: Visualization of old and new gallery embeddings in CIFAR100. We compress 128-dimensional embeddings into a 2- dimensional hyperboloid using UMAP (McInnes et al., 2018) and visualize them in the tangent space T0¯L 2 . The top histogram is the distribution of the uncert…

**Figure 7.** Figure 7: Comparison of self vs. cross retrieval mAP across three hyperparameters. Distance function self cross RINCE - Geodesic 0.6552 0.3982 RINCE - Lorentz inner 0.6550 0.3985 RINCE - Squared Lorentz 0.6529 0.3979 Mean distortion - Geodesic 0.6487 0.3970 Mean distortion - Squ…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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