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REVIEW 3 major objections 4 minor 142 references

FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read FiGuRO estimates intrinsic dimension by letting bottleneck ranks grow or shrink under a fidelity budget; in multi-modal data, the converged ranks match the shared and private subspace IDs.

desk verdict Useful practical method for multi-modal rank estimation, but the theoretical ID claim is not supported as stated; still deserves peer review with major revision. read the letter →

arxiv 2608.10857 v1 pith:6FCE5PEH submitted 2026-08-11 cs.LG

Viktoria Schuster , Sana Tonekaboni , Caroline Uhler This is my paper · ORCID
classification cs.LG
keywords intrinsicdimensionestimationmulti-modalrepresentationlearningrate-distortiontheorytruncatedsingularvaluedecompositionlow-rankadaptationsharedandprivatesubspacesdisentanglementfidelity-guidedrankoptimization
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

This paper introduces FiGuRO, a training procedure that estimates the intrinsic dimension (ID, the minimum number of variables needed to describe data) of uni- and multi-modal data by letting the rank of low-rank projection layers grow or shrink under a fidelity budget. The central claim is that a rank driven by reconstruction fidelity converges to the true manifold dimension, and in multi-modal data the three returned ranks coincide with the IDs of the shared subspace and each modality's private subspace. If this works, ID estimation becomes a by-product of ordinary autoencoder training, and disentanglement of shared versus private information no longer needs auxiliary losses or fixed bottleneck sizes. The paper backs the claim with convergence theorems and with experiments on synthetic, biomedical, and real-world datasets, including latent spaces of pretrained models.

What carries the argument

The load-bearing mechanism is the bidirectional rank-update rule coupled to a rate-distortion objective: minimize total rank (the rate proxy) subject to per-modality reconstruction fidelity (distortion) below a budget λ. The implementation uses low-rank weight matrices $W \approx U^{(k^*)}S^{(k^*)}V^{(k^*)^\top}$ from truncated singular value decomposition, with ranks reduced by a cumulative energy threshold γ and increased by unmasking, checked every τ epochs with patience π. The coefficient of determination R² is the distortion metric, defining the minimum acceptable fidelity as $D_0 - \lambda$, and the explicit shared/private split of the latent space makes each subspace's rank interpretable as its ID. Theorem A.1.7 shows that this process converges to the manifold dimension, and Theorem A.2.2 shows that the unique stable optimum is the disentangled allocation of shared and private information.

What would settle it

Simulate two modalities with a known shared latent s and independent private latents p1 and p2, then add a strong linear correlation between s and p1 so that one low-rank direction carries much of both; if FiGuRO's converged total rank drops well below the true ks+k1 while the mutual-information-based split still requires separate dimensions, the rank-proxy assumption is falsified. A simpler check is to compare FiGuRO's ranks against direct mutual-information estimates on data with known generative variables: any disagreement in ordering between rank and mutual information would break the core equivalence.

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Extended reading notes

Core claim

FiGuRO is a fidelity-guided rank optimization algorithm. It places truncated-SVD low-rank layers in the bottleneck of an autoencoder, then periodically checks whether reconstruction fidelity, measured by the coefficient of determination R², has dropped more than a user-set distortion budget λ from its pretrained baseline. If fidelity is still acceptable, ranks are pruned by a cumulative singular-energy threshold γ; if it has degraded too much, ranks are increased. The paper's central assertion, formalized in Theorem A.1.7, is that for a sufficiently small budget λ the final rank satisfies $E[k^*] \approx r$, the true intrinsic dimension of the data manifold, provided the autoencoder is a sufficient approximator and R² is a valid fidelity proxy. In the multi-modal architecture with shared and private latent subspaces, Theorem A.2.2 states that minimizing total rank subject to reconstruction fidelity makes the unique stable optimum the disentangled allocation, so the converged ranks $(k_s,k_1,k_2)$ estimate the IDs of the shared and private subspaces. Empirically the paper reports that FiGuRO outperforms existing ID estimators on noisy and nonlinear data and can be applied post-hoc to frozen pretrained encoders.

Load-bearing premise

The load-bearing premise is that a subspace's rank measures its information content (rate, proportional to mutual information), so minimizing total rank under a fidelity budget separates shared from private information; if rank is not proportional to mutual information, the returned ranks are just compression dimensions at a user-chosen fidelity level.

Editorial extensions

If this is right

  • Under sufficient model capacity and sample size, the final rank equals the true intrinsic dimension for small distortion budgets, and the algorithm terminates in finite time.
  • In multi-modal data, the converged shared and private ranks estimate the IDs of the shared and modality-specific subspaces, with disentanglement emerging without auxiliary orthogonality or sparsity losses.
  • FiGuRO shows greater stability to hyperparameter changes than existing rank-reduction baselines and outperforms classical and neural ID estimators on noisy and nonlinear data.
  • The framework can be applied post-hoc to frozen pretrained uni-modal encoders, providing a scalable latent probe for large models.
  • Subspace ratios and shared-to-private scales are recovered on synthetic and real data, and downstream classification on the shared subspace matches or exceeds uni-modal embeddings.

Reading between the lines

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

  • The paper's own failure modes (underestimation on correlated generative variables, overestimation at low signal-to-noise ratios) suggest that FiGuRO estimates an effective rank at a chosen fidelity level rather than a universal manifold dimension; reporting an ID interval across a range of λ values would make this explicit.
  • If the rank-proxy assumption holds, the same machinery could serve as a principled data-dependent rank selector for low-rank adaptations, replacing fixed ranks in parameter-efficient fine-tuning.
  • Because the optimization runs online and bids ranks both up and down, FiGuRO is a natural fit for continual learning, where latent capacity must resize as new modalities or distributions arrive.
  • The sensitivity of the final ranks to the chosen distortion metric (the paper finds R² most robust) means any published ID estimate should include the metric and budget used to obtain it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces FiGuRO, a fidelity-guided rank optimization framework that estimates the intrinsic dimension (ID) of uni- and multi-modal data. The method embeds low-rank decomposable layers in an autoencoder, then iteratively reduces or increases subspace ranks based on reconstruction fidelity relative to a user-chosen distortion budget lambda and an SVD energy threshold gamma. The authors claim convergence to the true ID (Theorem A.1.7), emergent disentanglement of shared and private subspaces (Theorem A.2.2), and support these claims with an extensive empirical study: a 1080-run hyperparameter sweep, simulated datasets with known IDs, comparisons to classical and neural ID estimators and multi-view decomposition baselines, ablations, and applications to Audio MNIST, So2Sat, NYU Depth V2, and NInFEA. The central theoretical and conceptual claims are that the returned ranks equal the data's intrinsic dimensions and that rank minimization naturally separates shared from private information without auxiliary losses.

Significance. If the central claims held, FiGuRO would be a valuable contribution: it is one of the first neural methods to explicitly estimate shared and private subspace IDs in multi-modal data, and the demonstration on frozen pretrained encoders is practically appealing. The paper reports substantial empirical work, including diverse simulated datasets, a large hyperparameter sweep, multiple baselines, ablations, and publicly available code. These strengths are real and should be credited. However, the theoretical guarantees are not rigorous, and the interpretation of the returned ranks as information-theoretic intrinsic dimensions rests on an unproved equivalence between SVD-energy ranks and rate/information content. The empirical results, while showing improvements over baselines, are more consistent with the method finding a lambda-dependent operating point on a rate-distortion curve than with recovering a ground-truth generative dimension.

major comments (3)
  1. [Appendix A.1, Theorem A.1.7] The proof of Theorem A.1.7 is a heuristic rate-distortion argument, not a quantitative guarantee. It assumes a sharp 'cliff' at the true rank r, but under Assumption A.1.1, additive Gaussian noise makes the reconstruction distortion decrease smoothly with rank, so the returned rank is scale-dependent. More importantly, the theorem's 'there exists a sufficiently small lambda' is non-operational: lambda is a user-chosen hyperparameter, and in Section 4.2 and Appendix C.1 it is selected using dataset A with known ID=5. Figure 3A shows that estimates vary substantially with lambda (e.g., Bi2 shared, GT=2, returns about 7 at lambda=0.05), and Supplementary Table 1 shows monotone under-estimation as lambda increases. Without a data-driven rule for choosing lambda, the existence claim is circular, because the lambda that recovers r depends on the unknown r. Please either provide a quantitative bound linking lambda to r, noise variance, and manifold geometry, or explicitly reframe Theorem A.1.7 as a statement about the algorithm's operating point on the rate-distortion curve rather than about intrinsic dimension.
  2. [Appendix A.2, Assumption A.2.1 and Theorem A.2.2] The entire disentanglement theorem rests on Assumption A.2.1, which asserts that the learned rank k* is proportional to the information-theoretic rate R, approximately equal to I(Z;X). This equivalence is not proved and is doubtful in this setting: Algorithm 1 computes ranks via a cumulative SVD energy threshold on learned weight matrices (Algorithm 1, line 21), and a variance-explained rank is not an information-theoretic quantity. Supplementary Table 3 itself provides evidence of the mismatch: the Bi1 shared generative matrix has 20 generative variables but an effective rank of only 6.61. In addition, the proof of Theorem A.2.2 concedes that the 'all-shared' solution is rank-equivalent to the disentangled solution, so the claimed 'unique optimal' solution is not unique under the stated objective; the argument that the all-shared solution is 'optimizationally unstable' is informal. Please prove the rank-information equivalence under explicit assumptions, or soften the conclusions to describe FiGuRO as finding low-rank compressions that are useful for downstream tasks rather than as recovering intrinsic information dimensions.
  3. [Section 5.2, Table 2 and Table 3] The claim that FiGuRO 'outperforms' existing ID estimation techniques should be qualified by the actual numerical accuracy. In Table 3, on dataset Bs FiGuRO returns (3.6, 1.0, 6.2) for ground truth (2, 3, 5); on Bi1 the shared rank is 13.4 versus GT 20; and on Bl the private ranks are 12.8 and 15.2 versus GT 20. These results are better than the baselines' average deviations, but they are not close enough to support the interpretation that the returned ranks are the true generative dimensions. The paper's own text acknowledges underestimation and information leakage, but the abstract and introduction state the stronger claim without these caveats. Please report per-subspace deviations from ground truth consistently and adjust the central claim to match the observed accuracy.
minor comments (4)
  1. [Section 3.1, Preliminaries] The statement that the Minimum Description Length principle 'formalizes' a linear proportionality between description length and latent dimension is imprecise; MDL concerns code length and model complexity, not a simple linear relation. Please clarify or remove the appeal to MDL as a formal basis.
  2. [Section 5.1 / Appendix C.1] The choice of lambda=0.05 is justified by a sweep on dataset A with known ID=5. Since the paper correctly identifies lambda as the main driver of the estimate, the main text should explicitly state that lambda selection currently requires prior knowledge or a validation procedure, and should discuss how a practitioner without ground truth would choose it.
  3. [Table 2] The table formatting is difficult to parse: several rows have misaligned or overlapping values across the JIVE, AJIVE, SLIDE, ShIndICA, and FiGuRO columns. Please reformat the table so each reported rank is unambiguously associated with its method and subspace.
  4. [Appendix B.8.2, ARD-VAE] The ARD-VAE implementation is described as using 'a corrected loss function to ensure mathematical validity,' but the precise objective and hyperparameters are not fully specified. Please provide the exact ELBO formulation used, or a reference to the code, to make the baseline reproducible.

Circularity Check

3 steps flagged · score 7.0 of 10

FiGuRO's ranks are user-tuned rate-distortion operating points; Theorem A.1.7's 'exists a suitable λ' is non-operational because λ was selected on known-ID data, and the disentanglement theorem assumes the rank=information bridge it claims to prove.

  1. fitted input called prediction [Section 4.2 / Appendix A.1 Theorem A.1.7 / Supplementary Table 1]
    "Our hyperparameter sweep allowed us to select a single configuration balancing performance and speed {λ= 0.05, τ= 10, γ= 0.01, π= 10} for subsequent experiments unless stated otherwise. ... Theorem A.1.7. ... there exists a sufficiently small distortion budget λ such that the rank k∗ returned by FiGuRO satisfies E[k∗]≈r."

    The only operational content of the existence claim is that some λ reproduces r, yet λ was chosen on dataset A whose true ID is known (5); Supplementary Table 1 shows the estimate is strongly monotone in λ (6.56 at λ=0.005, 5.02 at λ=0.05, 3.57 at λ=0.2). The tuned λ=0.05 is exactly the value that makes the known-ID dataset come out at 5. The theorem therefore does not establish that FiGuRO measures r; it states that a user who already knows r can pick a tolerance that returns r. Estimating r on the same dataset used to select the tolerance is circular, and no data-driven rule for λ is provided.

  2. self definitional [Section 3.2, Algorithm 1 (lines 11-12 and 21)]
    "Our approach can be framed as a greedy algorithm for finding an efficient operating point on the rate-distortion curve. ... k t+1,i ←min(k: Pk j Ej ≥1−γ)."

    By the algorithm's own update rule, the final rank is the smallest k whose SVD energy reaches 1−γ, kept only if R2 stays above D0−λ. That is, by construction, the rate-distortion operating point R(D0−λ) for a user-chosen distortion budget, not an estimate of manifold dimension independent of the user's tolerance. Calling this quantity the 'intrinsic dimension' equates the estimator's output with its definition: the rank is literally defined as the minimal rank achieving a chosen reconstruction fidelity threshold. The thresholds λ and γ are external inputs, so the estimate inherits their arbitrariness.

1 more flagged steps
  1. self definitional [Appendix A.2, Assumption A.2.1 / Theorem A.2.2]
    "We assume that the rank k∗ of a latent subspace, as estimated by FiGuRO, serves as a proxy for its information content, or Rate R. This is an extension of the Minimum Description Length (MDL) principle, where the rank k∗ is the minimal number of parameters (degrees of freedom) required to describe the data, such that k∗ ∝ R ≈ I(Z;X)."

    The disentanglement theorem concludes I(Hs)→S, I(H1)→P1, I(H2)→P2, but its proof works by equating rank sums with information sums (R_total≈ks+k1+k2≈I(Hs)+I(H1)+I(H2)). That equation is precisely Assumption A.2.1, assumed rather than proved. A rank obtained from an SVD energy threshold (1−γ) and a fidelity budget λ has no proven relation to mutual information; the paper's own Table 3 shows divergence (Bi1 shared: 20 generative variables vs effective rank 6.61). Thus the central disentanglement claim reduces to the assumption that rank equals information, i.e., the conclusion is imported by definition.

full rationale

FiGuRO's central numerical claim is not self-contained: the returned rank is, by Algorithm 1's own update rules, the smallest rank satisfying a cumulative SVD energy threshold (1−γ) subject to a user-chosen fidelity tolerance R2_0−λ. The paper itself describes this as finding an efficient operating point on the rate-distortion curve. Consequently k* is a rate-distortion point R(D0−λ), not an estimator of a manifold dimension independent of the user's tolerance. Theorem A.1.7 asserts existence of a λ making E[k*]≈r; but the configuration λ=0.05 was selected precisely because the sweep on dataset A (known ID 5) returned about 5.02, and Supplementary Table 1 shows the estimate is strongly monotone in λ (6.56 at λ=0.005 down to 3.57 at λ=0.2). The 'exists a suitable λ' claim is therefore non-operational and does not establish that the algorithm recovers r without using r. The disentanglement theorem similarly rests on Assumption A.2.1, which postulates k*∝R≈I(Z;X); the proof's rank-sum comparisons simply re-label information quantities as ranks. Table 3's own effective-rank numbers (20 generative variables vs effective rank 6.61 for Bi1 shared) show that SVD-energy rank and generative/information content diverge, so the assumption is not a harmless definition but the load-bearing conclusion. The empirical comparison against baselines is legitimate evidence of competitiveness, but it does not rescue the definitional identification. No significant self-citation chain is involved; the circularity is in the definition of the estimator and the tuned λ.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method's load-bearing hyperparameters (λ, γ, τ, π) and the two ad hoc assumptions (fidelity proxy and rank-as-rate) do most of the work in producing the ID estimate; the theoretical guarantees are conditional on these choices.

free parameters (5)
  • λ (distortion budget) = 0.05 (selected via grid search on dataset A with GT ID=5)
    Controls the stopping criterion: ranks are pruned until reconstruction fidelity drops below D0 - λ. The value strongly affects the estimate (Table 1: λ=0.005→6.56, λ=0.05→5.02, λ=0.2→3.57), so it calibrates the absolute ID output.
  • γ (energy threshold) = 0.01 (default; swept over 1e-4 to 0.1)
    Determines how many singular values are pruned at once via cumulative energy. The paper reports negligible sensitivity on simulations, but γ=0.001 causes overestimation on Audio MNIST (Supplementary Table 15).
  • τ (reduction frequency) = 10 (default; swept over 5-20)
    How often ranks are re-evaluated. Higher τ slows adaptation; affects the estimate mildly.
  • π (patience) = 10 (default; swept over 5-100)
    Stopping patience: algorithm halts after π checks without rank changes. Affects convergence duration and final rank.
  • α (sample-to-capacity load) = ≥10 (from Schuster & Krogh 2021)
    Used in Corollary A.1.8 to state the sample-size requirement for the ID guarantee; cited from earlier work by an author of this paper.
assumptions (5)
  • domain assumption Assumption A.1.1: observed data X = Y + ε with Y on an r-dimensional compact manifold and zero-mean noise with covariance σ²I, σ² smaller than manifold geometric variance.
    Sets the generative model for the data; if the manifold/noise separation fails, the ID estimate targets the wrong quantity.
  • domain assumption Assumption A.1.3: the autoencoder family is a sufficient function approximator such that reconstruction distortion can be brought arbitrarily close to the noise floor σ²+ε.
    The rank estimate is only meaningful if the model can realize the manifold; the paper admits dependence on architecture capacity.
  • ad hoc to paper Assumption A.1.4: R2 (or chosen metric) is a stable proxy for reconstruction fidelity, so R2 ≥ R0² - λ bounds the true distortion.
    Justifies the stopping rule in Algorithm 1; not proven, and the paper observes it fails at high dropout/noise.
  • ad hoc to paper Assumption A.2.1: the learned rank k* is proportional to the information content/rate R≈I(Z;X).
    This postulate makes the disentanglement theorem an optimization over information content; asserted without derivation and generally not true for nonlinear decoders.
  • domain assumption Assumption A.1.5: manifold is compact and generative variables are i.i.d.
    Used for the guarantee in Theorem A.1.7; excludes sparse or dependent generative structures.

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Cite this review

Pith. "Pith review of FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data." pith.science (2026). https://pith.science/paper/6FCE5PEH

@misc{pith2026260810857,
  author       = {Pith},
  title        = {Pith review of: FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6FCE5PEH}},
  note         = {Machine review of arXiv:2608.10857}
}
read the original abstract

Determining the complexity, or Intrinsic Dimension (ID), of data is fundamental to efficient and interpretable representation learning. This is particularly challenging in multi-modal settings when trying to learn disentangled representations for shared and private information. Existing techniques leave a critical gap: they are often static, uni-modal, or in the case of contrastive methods, adapt only to the shared ID implicitly. We introduce Fidelity-Guided Rank Optimization (FiGuRO), a framework for approximating the ID of uni- and multi-modal data under constraints of model capacity and hyperparameters. FiGuRO learns the dimensions of low-rank projections using truncated singular value decomposition and an algorithm that determines when to reduce or increase dimension and in which latent space. Disentanglement of shared and private information arises as an emergent property of this optimization, eliminating the need for complex auxiliary loss functions. We demonstrate that FiGuRO outperforms existing ID estimation techniques and is more robust to hyperparameter changes. Across simulations and real-world data, FiGuRO captures distinct ID scales and varying subspace ratios, and decomposes shared and private information successfully. Furthermore, we show that FiGuRO can be applied to modern uni-modal pretrained models, enabling efficient, post-hoc disentanglement of multi-modal representations.

Figures

Figures reproduced from arXiv: 2608.10857 by the authors.

Figure 1
Figure 1. FiGuRO’s multi-modal adaptive fusion architecture. Uni-modal embeddings hm are decomposed into shared (zs) and private (zm) representations via pruned weight matrices W(k ∗ s ) s and W(k ∗m) m . When using frozen pretrained models, concatenated representations (zs, zm) are passed through a fusion layer to re￾construct hm. Adaptive rank reduction. ARR is a technique for dynam￾ically reducing the dimensionality of the… view at source ↗
Figure 2
Figure 2. FiGuRO’s ID estimation over time on the 3D Swiss Roll Manifold. From left to right: 3D data space, learned latent representation with 2 dimensions (seed 19), rank over epochs as the ID estimate (with a dashed line on true ID 2), training mean squared error (MSE) loss over epochs, and distortion metric (R 2 ) over epochs with the dashed line depicting the minimum acceptable R 2 = R 2 0 − λ. The orange vertical lines … view at source ↗
Figure 3
Figure 3. Multi-modal ID estimation and disentanglement of B with varying true IDs. The x axis presents the distortion threshold λ. (A) Log-scale estimated ranks (mean ±SEM, N = 5 seeds) for shared (blue) and modality-specific (orange, pink) subspaces of Bi1 and Bl. Ground truth (GT) IDs are depicted as dashed lines. Values closer to the GT lines of the same color indicate better performance. Results for the remaining subsets… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Disentanglement of semantic content from modality-specific style on Audio MNIST. (A) FiGuRO’s multi-modal adaptive fusion architecture. Uni-modal embeddings are decomposed into shared and private representations of learned dimensionality. Colors of matrices and decompo…
Figure 1
Figure 1. Figure 1: Robustness tests. We evaluated the method’s stability on simulated data (ground truth ID of 5, red dotted lines) by systematically varying key generative parameters and running for three random seeds. Top row: The top left plot shows that the estimated rank converges t…
Figure 2
Figure 2. Figure 2: Distortion metrics. We evaluated five different distortion metrics by measuring the mean deviation of their ID estimates from the ground truth (ID=5) across simulated datasets with varying levels of nonlinearity and sparsity. We report the average deviation from the gr…
Figure 3
Figure 3. Figure 3: Training dynamics on 3D Manifold datasets. Each row corresponds to a different dataset: Hypersphere, Swiss Roll, and S-Curve. For each dataset, the columns show: the original 3D data, the R² metric during rank optimization (red dashed line showing the distortion budget…
Figure 4
Figure 4. Figure 4: Multi-modal ID estimation and disentanglement of B with varying true IDs. The x axis presents the distortion threshold λ. (A) Log-scale estimated ranks (mean ±SEM, N = 5 seeds) for shared (blue) and modality-specific (orange, pink) subspaces. Ground truth (GT) IDs are …
Figure 5
Figure 5. Figure 5: Disentanglement evaluation of multi-modal baselines on dataset Bs. Each heatmap plots the predictability of the three ground truth labels (0: shared, 1: modality 1, 2: modality 2) from the decomposed joint and individual (private) subspaces per method. Predictability i…
Figure 6
Figure 6. Figure 6: Disentanglement evaluation of multi-modal baselines on dataset Bi1. For details see caption 5. 22 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png]
Figure 7
Figure 7. Figure 7: Disentanglement evaluation of multi-modal baselines on dataset Bi2. For details see caption 5 [PITH_FULL_IMAGE:figures/full_fig_p035_7.png]
Figure 8
Figure 8. Figure 8: Disentanglement evaluation of multi-modal baselines on dataset Bl. For details see caption 5. 23 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]
Figure 9
Figure 9. Figure 9: Training dynamics on NInFEA mECG-fPWD. The left plot shows the training and validation loss. The middle plot depicts the ranks of all three subspaces over epochs. The right plot shows the initial R 2 metrics per modality [PITH_FULL_IMAGE:figures/full_fig_p036_9.png]
Figure 10
Figure 10. Figure 10: Image reconstruction samples from Audio MNIST. The top row presents original test samples, the bottom its reconstructions from our pretrained Audio MNIST model (seed 0). 24 [PITH_FULL_IMAGE:figures/full_fig_p036_10.png]
Figure 11
Figure 11. Figure 11: Audio reconstruction samples from Audio MNIST. The top row presents original test samples, the bottom its reconstructions from our pretrained Audio MNIST model (seed 0) [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]
Figure 12
Figure 12. Figure 12: Reconstruction of the SAR modality (channel 5). Channel 5 contains the intensity of the refined Lee-filtered VH channel from Sentinel-1. Reconstructions are shown from the pretrained model (seed 0). 25 [PITH_FULL_IMAGE:figures/full_fig_p037_12.png]
Figure 13
Figure 13. Figure 13: Reconstruction of the RGB channels in the optical modality. Reconstructions are shown from the pretrained model (seed 0) [PITH_FULL_IMAGE:figures/full_fig_p038_13.png]
Figure 14
Figure 14. Figure 14: Image color is decomposed into the image-specific representation. Rows show PCA and UMAPs of each subspace (columns) from the train set. Color indicates the mean RGB values. 26 [PITH_FULL_IMAGE:figures/full_fig_p038_14.png]
Figure 15
Figure 15. Figure 15: Image depth is decomposed into the depth-specific representation. Rows show PCA and UMAPs of each subspace (columns) from the train set. Color indicates the mean depth values. 27 [PITH_FULL_IMAGE:figures/full_fig_p039_15.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.