{"id":"7345a6a8-bc85-43b5-b513-2be0c931b30e","arxiv_id":"2608.10857","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"FiGuRO estimates the intrinsic dimensionality of shared and private subspaces in multi-modal data by adaptively growing or shrinking low-rank bottleneck layers guided by a reconstruction-fidelity budget.","lead":"This paper introduces FiGuRO, an algorithm that automatically finds how many dimensions are really needed to represent data from several modalities at once, splitting shared and modality-specific information. It is a new tool for making multi-modal machine learning models smaller, faster, and more interpretable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"FiGuRO's returned ranks are λ/γ-dependent rate-distortion operating points, not intrinsic dimensions; Theorem A.1.7's 'exists a suitable λ' is non-operational because λ is tuned on known-ID data and estimates shift with λ.","rationale":"Read in good faith, FiGuRO is a serious empirical contribution: it provides broad benchmarks, ablations, real-world applications, and a public codebase, and its rank-optimization dynamics are clearly described. My concern does not challenge the reproducibility or practical usefulness of FiGuRO as an adaptive rank-reduction and multi-modal fusion method. It challenges the interpretation of k* as the intrinsic dimension. The reader's weakest assumption (A.2.1, rank as a proxy for information content) is real and important, but I see a more upstream issue: before rank can proxy mutual information, the algorithm must actually return the generative dimension. Algorithm 1's update rule is driven by an SVD energy threshold γ and a user-fixed distortion budget λ, both extrinsic to the data's generative process. The paper's own Table 3 and Figure 3A show rank estimates that track effective variance rank and vary with λ, so the 'exists λ' statement in Theorem A.1.7 is not an estimator unless λ is chosen with knowledge of the answer. A controlled imbalanced-variance experiment would settle whether k* is an intrinsic dimension or a variance-explained count. If the latter, the paper should be reframed as estimating effective dimensionality at a chosen fidelity budget, with a principled selection rule for λ, and the disentanglement claims should be presented as empirical inductive-bias results rather than consequences of rank-information proportionality. This is a substantive revision but not a rejection, so the conditional verdict remains appropriate.","tokens_in":36947,"tokens_out":6924,"duration_ms":74087,"concrete_test":"Run FiGuRO on a uni-modal dataset with known generative ID r=5 but strongly imbalanced latent variances, e.g., generative variables whose mixing-matrix singular values decay geometrically so the fifth explains <1% of total variance. Use the paper's default λ=0.05, γ=0.01 from §4.2. If the returned k* is ≈1–2 (the variance-explained rank at the γ threshold) rather than ≈5, then FiGuRO is estimating an energy/variance-explained dimension, not the generative ID, and Theorem A.1.7 fails on this data. Repeat the sweep over λ∈{0.005,0.01,0.05,0.1}: if the λ required to recover r changes with the dataset's variance profile, the 'there exists λ' guarantee is not an operational estimator.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical claim, Theorem A.1.7, asserts that there exists a sufficiently small distortion budget λ such that the returned rank satisfies E[k*]≈r. But Algorithm 1 does not directly measure a generative dimension: rank decreases are made to min(k: Σ_{j≤k} E_j ≥ 1−γ), a cumulative-energy threshold on the SVD of learned weight matrices (Algorithm 1, line 21), and the increase/decrease decision is triggered by comparing R² to D0−λ. Thus k* is the smallest rank whose singular values explain (1−γ) of the learned weight energy subject to a λ-tolerance on reconstruction fidelity. The proof of Theorem A.1.7 assumes a sharp rate-distortion cliff at r, yet under Assumption A.1.1 with additive noise the distortion decreases smoothly with rank, so the 'intrinsic dimension' is scale-dependent. Moreover, λ=0.05 was selected on dataset A with known ID=5 (§4.2, C.1), and Figure 3A shows estimates varying strongly with λ (e.g., Bi2 shared, GT=2, returns ≈7 at λ=0.05). Without a data-driven rule for choosing λ, the existence claim is circular: the λ that recovers r depends on the unknown r. The same root issue underlies Assumption A.2.1: a rank obtained by energy/variance thresholds is not an information-theoretic rate, and Table 3's own effective-rank computations (e.g., Bi1 shared: 20 generative variables but effective rank 6.61) demonstrate that generative count and variance-explained rank diverge.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":37301,"tokens_out":3573,"duration_ms":38231,"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":[{"comment":"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.","section":"Appendix A.1, Theorem A.1.7"},{"comment":"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.","section":"Appendix A.2, Assumption A.2.1 and Theorem A.2.2"},{"comment":"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.","section":"Section 5.2, Table 2 and Table 3"}],"minor_comments":[{"comment":"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.","section":"Section 3.1, Preliminaries"},{"comment":"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.","section":"Section 5.1 / Appendix C.1"},{"comment":"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.","section":"Table 2"},{"comment":"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.","section":"Appendix B.8.2, ARD-VAE"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: FiGuRO is a real, usable method for estimating the ranks of shared and private subspaces in a multi-modal autoencoder, and it beats the linear baselines on the simulated benchmarks. But the paper's headline claim—that the returned ranks are intrinsic dimensions, supported by Theorem A.1.7—does not hold as stated. The estimates are operating points on a rate-distortion curve set by the user's λ choice, and λ is tuned on known-ID data.\n\nWhat's new: the bidirectional rank update (both reduce and increase) on top of SVD-based low-rank layers, combined with the shared/private architecture, is a sensible combination that prior work didn't have. The 1080-run hyperparameter sweep and the ablation showing rank-reduction-only collapses to rank 1 are genuinely informative. The NInFEA shared-over-total ratios line up with clinical expectation; that's a nice sanity check. Credit where due: the empirical work is extensive and the method is reproducible (code on GitHub).\n\nWhere it's soft: Theorem A.1.7 is basically an assertion that there exists a λ that makes the rank come out right. But Algorithm 1 computes ranks via cumulative energy threshold γ on SVDs of weight matrices, and the stopping rule only checks whether fidelity is above D0−λ. Nothing in the algorithm measures manifold dimension directly. So the proof is a heuristic plausibility argument, not a guarantee; the paper even says 'exists a sufficiently small λ' without specifying how to pick it. In practice, λ=0.05 was chosen by testing dataset A with known ID=5, and Figure 3A shows estimates swing strongly with λ (Bi2 shared returns ≈7 at λ=0.05 when GT=2). That's circular if the goal is to estimate unknown IDs.\n\nAssumption A.2.1, that rank ∝ mutual information, is the load-bearing piece for the disentanglement theorem, and the paper gives no evidence for it. The effective-rank vs generative-variable gap in Table 3 (Bi1 shared: 20 generative variables, effective rank 6.61) is the paper's own admission of that gap. So the disentanglement result is better described as 'the architecture and objective tend to separate shared and private information' rather than a proven optimality theorem.\n\nAlso, the deviations from ground truth in Table 2 are not tiny (e.g., Bi1 shared estimated 13.4 vs 20). The paper says 'preserve scales and relationships' which is fair, but 'close-to-accurate' overstates it.\n\nThe citation pattern is honest; they cite the relevant ARR/LoRA/JIVE literature and don't seem to be hiding prior work.\n\nWho this is for: anyone building multi-modal autoencoders or doing representation learning on paired data will find this a practical tool for choosing subspace sizes. A reader who wants a provable ID estimator should look elsewhere.\n\nMy advice: send it to peer review. The empirical method is solid enough to be worth referee time, and the theoretical overclaim is addressable by reframing A.1.7 and A.2.2 as conjectures, and by adding a principled selection rule for λ. As it stands, I'd want major revision before acceptance, but it deserves the round.","headline":"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.","tokens_in":37836,"tokens_out":2362,"would_cite":true,"duration_ms":23013,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["intrinsic dimension estimation","multi-modal representation learning","rate-distortion theory","truncated singular value decomposition","low-rank adaptation","shared and private subspaces","disentanglement","fidelity-guided rank optimization"],"falsifier":"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.","tokens_in":36682,"feed_emoji":"📉","tokens_out":9138,"duration_ms":75680,"temperature":0.7,"pith_summary":"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.","feed_headline":"Ranks converge to true dimension; shared/private subspaces disentangle","feed_subtitle":"A fidelity budget guides low-rank projections, so multi-modal IDs emerge without extra losses.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the rate-distortion function R(D) that motivates minimizing rank subject to a fidelity budget.","marker":"[Berger, 1975]"},{"why":"Eckart-Young-Mirsky theorem justifies truncated SVD as the optimal low-rank approximation used for pruning.","marker":"[Eckart & Young, 1936; Mirsky, 1960]"},{"why":"Provides adaptive rank reduction, the dynamic SVD-based pruning mechanism that FiGuRO extends with bidirectional updates.","marker":"[Mounayer et al., 2025]"},{"why":"LoRA's low-rank weight decomposition inspires the trainable low-rank layers in FiGuRO's bottleneck.","marker":"[Hu et al., 2021]"},{"why":"Minimum Description Length principle supplies the link between latent dimension and rate that underpins the rank-proxy assumption.","marker":"[Grünwald, 2007]"},{"why":"Supports the claim that globally optimized autoencoders can provably recover manifold dimension, used in the proof of Theorem A.1.7.","marker":"[Zheng et al., 2023]"},{"why":"Defines the sample-to-capacity ratio α used in Corollary A.1.8 for the sample size requirement.","marker":"[Schuster & Krogh, 2021]"}],"fun_headline_variants":["Fidelity budget guides rank optimization to true dimension","Emergent disentanglement via rank pruning on multi-modal data","FiGuRO: pruned ranks estimate intrinsic dimension and subspaces","Truncated SVD plus R² budget estimates multi-modal ID","Rank pruning reveals true dimension without auxiliary losses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fidelity budget guides rank optimization to true dimension","Emergent disentanglement via rank pruning on multi-modal data","FiGuRO: pruned ranks estimate intrinsic dimension and subspaces","Truncated SVD plus R² budget estimates multi-modal ID","Rank pruning reveals true dimension without auxiliary losses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000434,"raw_usage":{"total_tokens":2243,"prompt_tokens":1008,"completion_tokens":1235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":1155}},"tokens_in":624,"tokens_out":1235,"duration_ms":9236,"temperature":1.0,"reasoning_tokens":1155,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:36:26.439842+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}