1 Introduction↩︎

Modern supervised learning is often constrained by label scarcity. This has motivated few-shot learning, meta-learning, self-supervision, and representation learning, all of which study how useful representations can be learned when labels are limited [1]–[6].

A standard answer is information. In the information bottleneck view, a good representation preserves target-relevant information while controlling complexity through compression [7], [8]. This view is influential, but it leaves open a structural issue. That is, two representations may retain similar task-relevant information while inducing very different latent geometries. In low-label settings, such geometric differences can change the effective complexity of the downstream learning problem and therefore affect generalization.

A separate line of work shows that geometry matters in learning. Manifold methods exploit low-dimensional structure in high-dimensional data [9], [10]. Learning theory under manifold assumptions shows that statistical complexity can depend on intrinsic rather than ambient dimension [11], [12]. Geometric deep learning further emphasizes structured latent spaces and geometry-aware inductive bias [13], [14]. What remains less developed is a representation-learning analysis in which geometry is treated not only as an assumption or diagnostic, but as an explicit component of representation quality.

This paper studies this geometric axis. We ask whether latent geometry can be formalized as a measurable part of representation quality under data scarcity, whether it enters low-label generalization bounds explicitly, whether it can be estimated in practice, and whether a practical training objective can improve the observed information–geometry trade-off.

We answer this question under explicit smoothness, transfer, and estimator assumptions. We define a geometry-aware population score combining task-relevant information with penalties on latent curvature and intrinsic dimension. Under these assumptions, geometry affects low-label generalization through the chain \[\text{encoder geometry} \;\Longrightarrow\; \text{latent covering numbers} \;\Longrightarrow\; \text{loss-class entropy} \;\Longrightarrow\; \text{generalization}.\] This yields non-asymptotic bounds in which latent intrinsic dimension and latent covering complexity enter explicitly. We also prove consistency of the empirical surrogate under concentration assumptions for the information, curvature, and intrinsic-dimension estimators.

These results motivate the Variational Geometric Information Bottleneck (V-GIB). V-GIB augments variational bottleneck training with tractable stochastic proxies for curvature and intrinsic dimension. The method is not proposed as uniformly dominant over all alternatives. Its role is to operationalize the theory and test whether the geometric quantities identified by the analysis track low-label behavior across datasets.

The paper makes four contributions.

1. We identify geometry as an explicit axis of representation quality under data scarcity and formalize it through a population score combining information, curvature, and intrinsic dimension.

2. We derive non-asymptotic low-label generalization bounds depending explicitly on latent intrinsic dimension and latent covering complexity, and characterize the associated information–geometry frontier.

3. We prove consistency of the empirical surrogate optimized by V-GIB under explicit concentration assumptions for the information, curvature, and intrinsic-dimension estimators.

4. We propose V-GIB and evaluate it on real low-label benchmarks to test whether geometric quantities track low-label behavior and whether geometry-aware training can improve the empirical frontier.

The claim is narrower than a general theory of representation learning and sharper than a generic regularization claim. We do not claim that geometry-aware training is uniformly best on every dataset or at every label fraction. We claim that, under the stated assumptions, geometry is a statistically meaningful variable of representation quality, and that accounting for it can improve the observed information–geometry trade-off in low-label regimes.

1.1 Related Work↩︎

The closest information-theoretic starting point is the Information Bottleneck, which studies representations that preserve target-relevant information while discarding irrelevant variation [7]. Variational relaxations made this principle practical for deep models [8]. In these formulations, complexity is controlled mainly through compression.

Geometry enters learning through a different route. Manifold learning and manifold regularization exploit low-dimensional structure in high-dimensional observations [9], [10]. Statistical learning under manifold assumptions shows that covering numbers and sample complexity can scale with intrinsic dimension rather than ambient dimension [11], [12]. Geometric deep learning extends these ideas to structured domains and geometry-aware architectures [13], [14].

Our setting combines these perspectives but is not identical to either. Unlike standard bottleneck formulations, we treat latent geometry as part of representation quality itself. Unlike manifold regularization, we do not only assume geometry in the input data and regularize along it. We study how encoder-induced geometry enters low-label generalization through latent covering complexity and loss-class entropy.

The paper is also related to interpretable and structured representations, although the objective is different. Many interpretability methods analyze trained models after the fact through attribution, concept activation, or neuron-level inspection [15]–[17]. Here, geometric structure is placed directly in the representation-learning criterion and studied before post hoc interpretation begins.

2 Geometry as a Missing Axis of Representation Quality↩︎

This section formalizes one claim, precisely, in low-label learning, predictive fit alone does not determine representation quality. Geometry provides an additional axis because it affects effective complexity and hence generalization. We define a geometry-aware population score, state the assumptions under which geometry enters low-label bounds, and identify the mechanism by which encoder geometry affects generalization. Full proofs for this section are given in Appendix 4.

All statements are conditional on the assumptions below. No claim is made for arbitrary data distributions or arbitrary network classes.

2.1 Setup↩︎

Let \(X \subset \mathbb{R}^D\) be the input space and \(Y\) the label space. A representation is an encoder \[\phi \colon X \to \mathbb{R}^m,\] combined with a predictor \[g \colon \mathbb{R}^m \to \mathcal{Y},\] so that predictions have the form \(g(\phi(x))\).

We use the standard representation-learning factorization, i.e., the encoder determines the latent geometry, and the predictor uses that geometry for the downstream task. This is compatible with information bottleneck learning, manifold-based learning, and geometric representation learning [7]–[10], [13], [14], [18]. The difference here is that geometry is not treated as a post hoc descriptor. In the low-label regime, latent geometry affects covering numbers, metric entropy, and uniform deviation, so it becomes a measurable variable of representation quality.

2.2 Geometry-aware representation quality↩︎

We use two geometric quantities. The first is the curvature proxy \[\mathcal{C}(\phi) = \mathbb{E}_{x \sim P_X}\bigl[\|\nabla^2 \phi(x)\|_F^2\bigr],\] which measures local variation of the encoder Jacobian. The second is the intrinsic dimension \(d_{\mathrm{int}}(\phi)\), the effective dimension of the representation \(\phi(X)\). In practice, it can be estimated by local PCA or nearest-neighbor methods [19]; in the theory it is treated as a population quantity.

The first term measures retained task-relevant information [7], [20]. The remaining terms penalize latent irregularity and latent effective dimension. Thus \(Q_{\beta,\gamma}\) treats geometry as part of representation quality itself, not as an after-training summary.

2.3 Assumptions↩︎

The analysis uses three assumptions.

This assumption specifies the geometric regime in which covering arguments are meaningful. Reach, curvature, and volume control the latent manifold, while Lipschitz regularity controls the encoder-induced distortion [11], [21], [22].

This is the structural bridge between latent geometry and statistical complexity. It is also where predictor-class complexity enters the analysis. The results are conditional on this transfer bound and do not claim it for arbitrary predictor classes.

This assumption isolates the estimation burden. It does not require a specific mutual-information estimator or intrinsic-dimension estimator. It only requires the uniform control used later in the surrogate consistency result.

2.4 Mechanism↩︎

The theory uses the following chain: \[\text{encoder geometry} \;\Longrightarrow\; \text{latent covering numbers} \;\Longrightarrow\; \text{loss-class entropy} \;\Longrightarrow\; \text{generalization}.\] Assumption 1 controls the covering numbers of the latent image \(\phi(\mathcal{M})\). Assumption 2 transfers this latent covering control to the entropy of the induced loss class. Empirical-process bounds then convert the entropy bound into a uniform deviation bound.

Lemma 1 gives the first part of the mechanism. Latent intrinsic dimension and the covering constant \(C_\phi\) control the size of the representation space seen by the downstream predictor. The proof is given in Appendix 4.1.

Theorem 1 is the main mechanism result. In this conditional regime, low-label generalization depends explicitly on latent intrinsic dimension and latent covering complexity, not only on predictive fit. Its proof is given in Appendix 4.2.

The theorem also gives a direct sample-complexity reading. At fixed task information, representations with smaller latent intrinsic dimension and smaller covering constant require fewer labels to reach the same uniform-deviation level. The formal corollary and proof are given in Appendix 4.3.

2.5 Information–geometry frontier↩︎

The previous results show how geometry enters generalization. The next result shows how it enters optimization.

The proof is given in Appendix 5.1. Theorem 2 is the optimization counterpart of Theorem 1. Geometry does not only enter the generalization bound; it parameterizes a trade-off curve between retained task information and geometric complexity. The slope identity says that \(\beta\) is the marginal information cost of increasing the allowed geometric complexity along the selected frontier.

2.6 Estimator concentration and surrogate consistency↩︎

The population score \(Q_{\beta,\gamma}\) is not optimized directly. The practical method optimizes the empirical surrogate \[\widehat Q_{N,K}(\phi) = \widehat I_N(\phi) - \beta \widehat{\mathcal{C}}_{N,K}(\phi) - \gamma \widehat d_N(\phi).\] Thus the optimizer targets a sample-based surrogate for \(Q_{\beta,\gamma}\), not the population score itself.

The fixed-encoder concentration statement for the stochastic curvature estimator is given in Appendix 5.2. The uniform surrogate result below assumes uniform control of the three empirical components over the feasible encoder class.

The proof is given in Appendix 5.3. Theorem 3 links the optimizer to the population theory. It does not claim that the empirical objective is exact. It states that, under explicit concentration assumptions, the empirical objective is a consistent surrogate for the geometry-aware population score.

2.7 What the theory establishes↩︎

Under the stated assumptions, geometry can be formalized as part of representation quality, enters low-label generalization through latent covering complexity, defines an information–geometry frontier, and can be targeted by a consistent empirical surrogate. The experiments test the corresponding empirical question, i.e., whether these geometric quantities track low-label behavior across datasets and whether geometry-aware training improves the observed trade-off.

3 Experimental Validation↩︎

The V-GIB family generally improves over ERM across the tested label fractions. The aim is not to show that V-GIB is uniformly dominant. The aim is to test the specific prediction of the theory, basically, when labels are scarce, latent geometry should act as a measurable axis of representation quality, and geometry-aware training can improve the information–geometry trade-off in some regimes.

The validation is organized around three questions. First, does V-GIB improve low-label predictive performance relative to ERM and VIB? Second, do the geometric diagnostics expose differences that are not visible from predictive fit alone? Third, do the ablations show that curvature and intrinsic dimension contribute differently across datasets?

3.1 Protocol↩︎

We evaluate on image and tabular benchmarks. The primary comparative results use Breast Cancer, FashionMNIST, and CIFAR-10, for which complete predictive summaries are available. We also include CovType as a supporting optimization-side check because the available CovType summaries report training loss rather than full validation and test metrics. We therefore use CovType only as supporting optimization evidence, not as primary generalization evidence.

All methods are trained on stratified subsets of the available training data with label fractions \[0.01,\qquad 0.05,\qquad 0.10,\qquad 0.20.\] Results are reported as mean \(\pm\) standard deviation over seeds \(13,29,47\).

We compare five methods. The first is deterministic empirical-risk minimization, denoted ERM. The second is the variational information bottleneck baseline, denoted VIB. The third is the full proposed method, denoted V-GIB. The remaining two are ablations, V-GIB-no-curv and V-GIB-no-dim, which remove the curvature and intrinsic-dimension penalties, respectively.

All methods use the same latent-classifier backbone. The common training objective is \[\mathcal{L} = \mathcal{L}_{\mathrm{CE}} + \beta_{\mathrm{KL}}\mathcal{L}_{\mathrm{KL}} + \beta_{\mathrm{curv}}\mathcal{L}_{\mathrm{curv}} + \gamma_{\mathrm{dim}}\mathcal{L}_{\mathrm{dim}}.\] For ERM, all regularization weights are zero and the encoder is deterministic. For VIB, only the KL term is active. For V-GIB, the KL, curvature, and dimension terms are active. The two ablations remove one geometric term at a time.

The curvature diagnostic is a stochastic Jacobian-norm proxy measuring the sensitivity of the latent mean \(\mu(x)\) to the input. The intrinsic-dimension diagnostic is the participation ratio of the latent covariance, reported as a fraction of the latent dimension. All geometry diagnostics are computed from \(\mu(x)\), not from sampled latent variables. Checkpoints are selected using validation performance and test metrics are evaluated only after checkpoint selection. Full architecture, preprocessing, hyperparameter, diagnostic, and checkpoint-selection details are given in Appendix 5.5.

3.2 Breast Cancer↩︎

The Breast Cancer benchmark gives a small-data tabular stress test. The \(1\%\) case is deliberately severe, because it contains only four labeled training examples. In that regime, ERM and VIB achieve the highest mean test accuracy, both \(0.8333\), while the full V-GIB model reaches \(0.8099\), as reported in Table 1 and Figure 1. We therefore do not treat the \(1\%\) case as evidence of dominance.

The clearer low-label signal appears from \(5\%\) onward. At \(5\%\), V-GIB improves test accuracy over ERM from \(0.8158\) to \(0.8977\), and improves macro-F1 from \(0.7960\) to \(0.8849\); see Table 1 and Figure 1 (b). At \(10\%\), V-GIB reaches \(0.9094\) test accuracy, matching VIB and improving over ERM by \(0.0556\), as shown in Table 1 and Figure 1 (a). At \(20\%\), V-GIB reaches \(0.9298\) test accuracy and \(0.9263\) macro-F1, while V-GIB-no-curv gives the best result in the geometric family; the corresponding predictive trends are shown in Figure 1.

The conclusion on this dataset is therefore precise. Geometry-aware variants improve low-label performance once the labeled set is large enough to stabilize training, but the full V-GIB objective is not uniformly best over all ablations, as seen in Table 1. This is consistent with the theory, which predicts a geometry-information trade-off rather than universal dominance of a single regularization setting.

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3.3 FashionMNIST↩︎

FashionMNIST gives a clearer image-domain test of the geometric hypothesis. Across all four label fractions, the full V-GIB model improves mean test accuracy over ERM, as shown in Table 2 and Figure 2. The gains are \(+0.0199\), \(+0.0480\), \(+0.0151\), and \(+0.0500\) at \(1\%\), \(5\%\), \(10\%\), and \(20\%\) labels, respectively.

The strongest mean predictive performance comes either from the full model or from V-GIB-no-dim. At \(1\%\), \(5\%\), and \(10\%\) labels, V-GIB-no-dim gives the best mean test accuracy, with \(0.6978\), \(0.7894\), and \(0.8415\) (Table 2; Figure 2 (a)). At \(20\%\), the full V-GIB model is best, reaching \(0.8692\) test accuracy, \(0.8679\) macro-F1, and \(0.9890\) AUROC (Table 2; Figure 2 (b)).

The geometry diagnostics show why this is not only a predictive-performance story. The full V-GIB model keeps test curvature substantially below ERM, as seen in Figure 2 (c). The curvature proxy for V-GIB is \(39.8209\), \(34.6334\), \(37.7687\), and \(30.2081\) across the four label fractions, compared with \(79.9008\), \(114.4583\), \(182.2521\), and \(142.1634\) for ERM. Thus, on FashionMNIST, the geometric regularizer improves or preserves predictive performance while producing a smoother latent representation; the corresponding utility behavior is shown in Figure 2 (d).

3.4 CIFAR-10↩︎

CIFAR-10 is the hardest primary benchmark in the validation suite. It is therefore the most important check that the geometric objective remains useful beyond small tabular data and simple image structure. The full V-GIB model improves over both ERM and VIB in test accuracy at every label fraction, as shown in Table 3 and Figure 3.

At \(1\%\) labels, V-GIB improves test accuracy from \(0.1323\) for ERM and \(0.1947\) for VIB to \(0.3405\) (Table 3; Figure 3 (a)). The corresponding macro-F1 improves to \(0.3277\), compared with \(0.0578\) for ERM and \(0.1380\) for VIB (Figure 3 (b)). At \(10\%\) and \(20\%\), the full V-GIB model is the strongest method among the compared variants. At \(20\%\), it reaches \(0.5929\) test accuracy and \(0.5926\) macro-F1, improving over ERM by \(0.0660\) accuracy and \(0.0770\) macro-F1, and over VIB by \(0.0398\) accuracy and \(0.0532\) macro-F1.

The ablations show a useful transition. Removing the curvature term gives the best result only at \(1\%\) and \(5\%\), while the full V-GIB model is best at \(10\%\) and \(20\%\) (Table 3). This suggests that the curvature term becomes more useful once the labeled set is large enough for geometric regularization to shape the representation without overwhelming the predictive signal. The AUROC results in Figure 3 (c) provide the corresponding ranking-quality check.

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3.5 Supporting CovType optimization check↩︎

The available CovType postprocessed summaries report training loss rather than full validation or test metrics. We therefore treat CovType as supporting optimization evidence rather than primary generalization evidence.

Across all methods, the mean training loss decreases as the label fraction increases from \(0.01\) to \(0.10\), indicating stable optimization under increased supervision (Table 4). ERM attains the lowest training loss at every reported fraction, which is expected because it has the least constrained objective. VIB remains below full V-GIB, while V-GIB-no-curv improves the optimization profile relative to full V-GIB. This pattern is useful rather than negative. It shows that the geometric terms impose a real optimization cost, so the predictive gains observed on FashionMNIST and CIFAR-10 are not a trivial consequence of easier training.

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3.6 Synthesis↩︎

The experiments support the theoretical claim in a specific and limited sense. They do not show that the full V-GIB objective is always best. They show that geometry is a consequential representation variable under label scarcity.

On Breast Cancer, geometry-aware variants improve over ERM once the labeled sample is not extremely small. On FashionMNIST, the V-GIB family improves predictive performance while substantially reducing curvature relative to ERM. On CIFAR-10, the full V-GIB model gives the strongest primary evidence, improving over both ERM and VIB at all tested label fractions. The ablations show that curvature and intrinsic dimension are not interchangeable. Removing the dimension term can help in very low-label FashionMNIST, while the full model becomes stronger at higher low-label budgets and on CIFAR-10. Removing curvature can ease optimization, as seen in CovType and some low-label regimes, but it also weakens the structural control that is central to the proposed objective.

These results support the paper’s main claim. Latent geometry is not only a post hoc diagnostic. Under label scarcity, it can be measured, regularized, and used to move the empirical information–geometry frontier. The evidence is strongest on FashionMNIST and CIFAR-10, where V-GIB improves predictive performance while controlling geometric complexity. The appropriate conclusion is therefore not that V-GIB universally dominates existing methods, but that it is a serious geometry-aware alternative to standard ERM and variational bottleneck training in data-scarce regimes.

Funding↩︎

The author received no financial support for the research, authorship, or publication of this article.

Competing interests↩︎

The author declares no competing interests.

Ethics approval and consent to participate↩︎

Not applicable. This study uses publicly available benchmark datasets and does not involve human participants, animal subjects, or newly collected personal data.

Consent for publication↩︎

Not applicable.

Data availability↩︎

The datasets used in this study are publicly available benchmark datasets. The primary experiments use the Breast Cancer Wisconsin Diagnostic dataset, FashionMNIST, CIFAR-10, and CovType. The processed experimental summaries used to generate the reported tables and figures are available through the project repository.

Materials availability↩︎

Not applicable.

Code availability↩︎

The code used to run the experiments and generate the reported tables and figures is publicly available at https://github.com/karjxenval/V-GIB.

Author contributions↩︎

R.K. conceived the study, developed the theoretical framework, proved the main results, designed and performed the experiments, analyzed and interpreted the results, and wrote the manuscript.

4 Proofs for the geometry-sensitive bounds↩︎

This appendix gives the proofs for the geometry-sensitive results in Sec. 2. The statements are conditional on Assumptions 1 and 2.

4.1 Proof of the latent covering bound↩︎

4.2 Proof of the geometry-sensitive uniform deviation bound↩︎

4.3 Sample-complexity consequence↩︎

5 Optimization and surrogate-consistency proofs↩︎

This appendix gives the proofs for Theorems 2 and 3, together with the curvature concentration statement used in the surrogate analysis.

5.1 Proof of the information–geometry frontier↩︎

5.2 Curvature estimator concentration↩︎

The following result is a fixed-encoder concentration statement for the stochastic curvature estimator. The uniform surrogate consistency theorem in the main text assumes the corresponding uniform control over \(\Phi\). Such uniform control can be obtained under additional complexity conditions on the encoder class, or imposed directly as in Theorem 3.

5.3 Proof of uniform surrogate consistency↩︎

5.4 Geometry penalties and diagnostics↩︎

The curvature proxy is computed from the sensitivity of the latent mean \(\mu(x)\) to the input. For a random probe vector \(r\), the default stochastic Jacobian proxy computes \[\left\|\nabla_x \langle \mu(x),r\rangle\right\|_2^2\] and averages over the batch and probes. A Hessian-based proxy is implemented, but the default experiments use the Jacobian proxy for stability and computational cost.

The intrinsic-dimension proxy is the participation ratio of the latent covariance. If \(\lambda_1,\ldots,\lambda_{d_z}\) are the eigenvalues of the empirical latent covariance, then \[d_{\mathrm{PR}} = \frac{\left(\sum_j \lambda_j\right)^2}{\sum_j \lambda_j^2}, \qquad \text{dimension ratio} = \frac{d_{\mathrm{PR}}}{d_z}.\] All geometry diagnostics are computed from \(\mu(x)\), not sampled latent variables. For computational stability, geometry penalties are evaluated on a capped sub-batch of \(32\) examples per training step.

5.5 Training protocol↩︎

All models are trained with AdamW using learning rate \(10^{-3}\), weight decay \(10^{-4}\), batch size \(128\), and gradient clipping at norm \(5.0\). Models are trained for at most \(25\) epochs with early stopping on validation performance using patience \(5\) and minimum improvement \(10^{-4}\).

The default regularization weights are \[\beta_{\mathrm{KL}}=10^{-3}, \qquad \beta_{\mathrm{curv}}=10^{-3}, \qquad \gamma_{\mathrm{dim}}=10^{-2}.\] The KL, curvature, and dimension penalties are linearly warmed up during the first \(5\) epochs.

5.6 Checkpoint selection↩︎

Checkpoints are selected by predictive validation performance rather than by the geometry-based utility score. In automatic mode, binary tasks use validation AUROC and multiclass tasks use validation macro-F1. Ties are broken by utility proxy, then lower curvature proxy, then lower dimension ratio. Final validation and test metrics are computed from the selected checkpoint.

5.7 Evaluation metrics↩︎

Predictive metrics include accuracy, balanced accuracy, macro-F1, AUROC, and cross-entropy. Binary AUROC uses the positive-class probability, while multiclass AUROC uses macro one-versus-rest scoring.

The information proxy is \[\widehat I_{\mathrm{proxy}} = \widehat H(Y_{\mathrm{train}}) - \widehat{\mathcal{L}}_{\mathrm{CE}},\] where \(\widehat H(Y_{\mathrm{train}})\) is the empirical label entropy of the labeled training subset and \(\widehat{\mathcal{L}}_{\mathrm{CE}}\) is the validation or test cross-entropy.

The utility proxy is \[\widehat U = \widehat I_{\mathrm{proxy}} - \beta_{\mathrm{KL}}\widehat{\mathcal{L}}_{\mathrm{KL}} - \beta_{\mathrm{curv}}\widehat{\mathcal{L}}_{\mathrm{curv}} - \gamma_{\mathrm{dim}}\widehat{\mathcal{L}}_{\mathrm{dim}}.\] Interpretive efficiency is \[\widehat E = \frac{\widehat U}{N_{\mathrm{lab}}},\] where \(N_{\mathrm{lab}}\) is the number of labeled training examples used in the run.

5.8 Saved outputs↩︎

For every dataset, method, label fraction, and seed, the implementation saves the training history, selected checkpoint, configuration, validation and test metrics, confusion matrix, and classification report. The runner also writes aggregate files containing individual runs, mean and standard-deviation summaries, and pairwise comparisons against ERM and VIB. These summaries are used to produce the final tables and figures.

6 Additional Experimental Results↩︎

This appendix provides the extended experimental summaries supporting Section 3. For each dataset, we report the additional predictive metrics, geometry diagnostics, utility proxies, scatter diagnostics, and pairwise comparisons that were omitted from the main text for space. Unless otherwise stated, all values are computed on the test split and reported as mean \(\pm\) standard deviation over seeds \(13,29,47\). The geometric diagnostics are used to interpret the information–geometry tradeoff and should not be read as direct substitutes for predictive performance.

6.1 Breast Cancer↩︎

The Breast Cancer benchmark is a small-data tabular setting. The \(1\%\) split contains only four labeled training examples, so the corresponding results are unstable and are included mainly for completeness. The more informative comparisons are from \(5\%\) labels onward, where the geometry-aware variants improve substantially over ERM in accuracy and macro-F1. The complete predictive and geometric summary is reported in Table 5, while the pairwise differences for the full V-GIB model are reported in Table 6.

Figure 4 reports the Breast Cancer geometry and utility diagnostics. The curvature proxy, dimension ratio, and utility proxy are shown separately in Figures 4 (a), 4 (b), and 4 (c). These diagnostics help interpret how the geometric objective changes the learned representation across label fractions.

Figure 5 gives the corresponding scatter diagnostics. The three panels compare accuracy with curvature, dimension ratio, and utility proxy, respectively; see Figures 5 (a), 5 (b), and 5 (c). These plots show that the predictive gains are not explained by a single monotone geometric statistic across all label fractions.

Table 5 reports the full Breast Cancer test summary across all label fractions and methods. It includes accuracy, macro-F1, AUROC, curvature, dimension ratio, and utility, so that the predictive and geometric behavior can be read together.

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Table 6 reports the pairwise differences between the full V-GIB model and the two main baselines, ERM and VIB. Positive entries indicate that V-GIB is higher than the comparator on the corresponding metric.

6.2 FashionMNIST↩︎

FashionMNIST provides a clear geometry-sensitive image benchmark. The additional diagnostics show that the main improvement is not only predictive. The full V-GIB model and the no-dimension ablation keep curvature substantially lower than ERM, VIB, and the no-curvature ablation, while remaining among the best predictive methods. The additional FashionMNIST diagnostics are shown in Figure 6, with macro-F1 in Figure 6 (a), dimension ratio in Figure 6 (b), and scatter diagnostics in Figures 6 (c), 6 (d), and 6 (e). The predictive metrics are reported in Table 7, and the geometric and utility diagnostics are reported in Table 8.

Table 7 reports the FashionMNIST predictive metrics.

Table 8 reports the FashionMNIST geometric and utility diagnostics.

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6.3 CIFAR-10↩︎

CIFAR-10 is the most challenging primary benchmark in this validation suite. in the validation suite. The additional results show that the full V-GIB model improves predictive performance while keeping curvature substantially lower than ERM and VIB from \(5\%\) labels onward. This supports the combined information–geometry interpretation rather than an explanation based on prediction alone. The geometry and utility diagnostics are reported in Figure 7, with the individual curvature, dimension-ratio, and utility panels shown in Figures 7 (a), 7 (b), and 7 (c). The scatter diagnostics are reported in Figure 8, with the individual panels shown in Figures 8 (a), 8 (b), and 8 (c). The full numerical summary is given in Table 9, and the pairwise differences for the full V-GIB model are given in Table 10.

Table 9 reports the full CIFAR-10 test summary across all label fractions and methods.

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Table 10 reports the pairwise differences for the full V-GIB model on CIFAR-10.

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6.4 CovType↩︎

CovType is included as supporting optimization evidence. The available CovType summaries report training loss rather than full validation or test performance, so these tables are used only to document optimization behavior and not as primary predictive comparisons. The results show that training loss decreases with increasing label fraction, ERM remains lowest because it solves the least constrained objective, and the curvature ablation improves the optimization profile relative to full V-GIB on this dataset. The per-seed results are reported in Table 11, while additional single-setting diagnostics are reported in Table 12.

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Table 12 reports additional CovType training-loss diagnostics.

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References↩︎