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arxiv: 2605.05524 · v1 · submitted 2026-05-06 · 💻 cs.LG · cs.AI

MOSAIC: Module Discovery via Sparse Additive Identifiable Causal Learning for Scientific Time Series

Shicheng Fan , Nour Elhendawy , Jianle Sun , Ke Fang , Kun Zhang , Yihang Wang , Lu Cheng This is my paper

Pith reviewed 2026-05-08 16:15 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords causal representation learningtime seriesmodule discoverysparse additive modelsidentifiabilityscientific datavariational autoencodersupport recovery
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The pith

A sparse additive decoder in a temporal variational autoencoder recovers interpretable modules by linking each latent to a sparse group of observed scientific variables.

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

The paper shows that causal representation learning can be made interpretable in scientific time series by pairing regime-based identifiability with a sparse additive decoder that selects which observations belong to each latent. This matters because measurements like residue distances or climate indices already carry semantic meaning, so associating them directly to latents avoids post-hoc labeling and reveals which variables share common mechanisms. The work proves that main-effect supports stay identifiable when mixing functions are smooth and supplies finite-sample guarantees for the sparse case. Empirically the recovered groups align with domain knowledge in molecular, climate, and process datasets.

Core claim

ANOVA main-effect supports are identifiable under general smooth mixing functions, and finite-sample recovery guarantees hold for a tractable sparse-additive variant. This lets the model recover domain-consistent variable groups across RNA molecular dynamics, solar wind, ENSO climate, the Tennessee Eastman process, and a synthetic tokamak benchmark.

What carries the argument

The sparse additive decoder that enforces support recovery over observed variables for each latent identified via regime-conditioned temporal variation.

Load-bearing premise

Latent variables become identifiable from temporal regime changes and the mixing functions are smooth enough that main-effect supports remain recoverable.

What would settle it

A controlled synthetic dataset with known modules, smooth mixing, and sufficient regime variation where the method fails to recover supports matching the ground-truth groups would contradict the identifiability claim.

Figures

Figures reproduced from arXiv: 2605.05524 by Jianle Sun, Ke Fang, Kun Zhang, Lu Cheng, Nour Elhendawy, Shicheng Fan, Yihang Wang.

Figure 1
Figure 1. Figure 1: RNA hairpin example. (A) The cUUCGg RNA hairpin; colors mark three structural regions (Sterm, Closing Pair, Loop). (B) Regime-association score (Cohen’s d) over four learned latents; z3 is selected. (C1, C2) Influence column of z3 under MOSAIC (C1) and TDRL [Yao et al., 2022] (C2); colored stripes match the regions in (A). therefore inherently carry domain semantics. In contrast, the latents governing thes… view at source ↗
Figure 2
Figure 2. Figure 2: MOSAIC architecture. The encoder produces view at source ↗
Figure 3
Figure 3. Figure 3: Physics-inspired double-well benchmark. (A) A reaction coordinate z0 moves between two metastable basins (State A / State B); z1, z2, z3 modulate left-well depth, right-well depth, and barrier height. (B1–B3) Column-normalized influence matrices over six ground-truth factors (z0, z1, z2 regime-varying; z3, u1, u2 regime-invariant). Learned latents in B2 and B3 are Hungarian-matched to the ground truth in B… view at source ↗
Figure 4
Figure 4. Figure 4: Climate (ENSO) cross-domain localization on NOAA ERSSTv5 sea surface temperature data [Huang et al., 2017b,a]. (A) The canonical ENSO SST anomaly pattern, with the warm equatorial Pacific band marking the physical mode underlying the regime label. (B) Cohen’s d across 8 learned latents identifies z5 as the top regime-associated factor. (C) Influence of z5 over the 47 SST grid cells, ordered by group: suppo… view at source ↗
Figure 5
Figure 5. Figure 5: Regime definition determines the scientific question. view at source ↗
Figure 6
Figure 6. Figure 6: Influence matrices on the synthetic benchmark, column-normalized. view at source ↗
Figure 7
Figure 7. Figure 7: Interaction-ratio calibration. (a) On the synthetic interaction sweep, view at source ↗
read the original abstract

Causal representation learning (CRL) seeks to recover latent variables with identifiability guarantees, typically up to permutation and component-wise reparameterization under appropriate assumptions. However, identifiability does not imply interpretability: latent semantics are typically assigned post hoc by alignment with known ground-truth factors. This limitation is particularly acute in scientific time series, where underlying mechanisms are unknown and discovering interpretable structure is a primary goal. In contrast, scientific observations (such as residue-pair distances, climate indices, or process sensors) are inherently semantic, as they correspond to named physical quantities. This raises a key question: can the interpretability of observations be transferred to the identifiable latent space? We propose MOSAIC (Module discovery via Sparse Additive Identifiable Causal learning), a sparse temporal VAE that integrates temporal CRL identifiability with support recovery over observed variables. MOSAIC identifies latent variables via regime-conditioned temporal variation, and recovers for each latent a sparse set of associated observations through an additive decoder, yielding module-level interpretability. We show that ANOVA main-effect supports are identifiable under general smooth mixing functions, and provide finite-sample recovery guarantees for a tractable sparse-additive variant. Empirically, MOSAIC recovers domain-consistent variable groups across RNA molecular dynamics, solar wind, ENSO climate, the Tennessee Eastman process, and a synthetic tokamak benchmark, enabling interpretable discovery of latent mechanisms in scientific time series.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces MOSAIC, a sparse temporal variational autoencoder for causal representation learning in scientific time series. It identifies latent variables through regime-conditioned temporal variation and employs a sparse additive decoder to recover, for each latent, a sparse support of associated observed variables (modules). The central theoretical claims are that ANOVA main-effect supports remain identifiable under general smooth mixing functions and that finite-sample recovery guarantees hold for the tractable sparse-additive variant. Empirical results across RNA molecular dynamics, solar wind, ENSO climate, the Tennessee Eastman process, and a synthetic tokamak benchmark demonstrate recovery of domain-consistent variable groups.

Significance. If the identifiability results and finite-sample guarantees hold, the work provides a meaningful advance by transferring semantic interpretability from observations to the latent space via sparse module recovery, addressing a core limitation of standard CRL methods. The multi-domain empirical validation and emphasis on scientific time series strengthen practical relevance. The finite-sample guarantees constitute a clear strength relative to purely asymptotic identifiability results.

major comments (2)
  1. [Theory section] The identifiability of ANOVA main-effect supports under general smooth mixing functions (stated in the abstract and presumably proved in the theory section) rests on regime-conditioned temporal variation; the manuscript should explicitly verify that the proof does not require additional restrictions on regime diversity or mixing smoothness beyond those stated, as this is load-bearing for the central claim.
  2. [Theory section] Finite-sample recovery guarantees for the sparse-additive decoder are claimed; the manuscript should report the explicit dependence of the sample complexity on the smoothness parameter, sparsity level, and number of regimes (e.g., in the relevant theorem), because these quantities determine whether the guarantees are practically useful.
minor comments (2)
  1. [Abstract and Experiments] The abstract refers to 'domain-consistent variable groups' without defining the quantitative criterion used to declare consistency; this definition should appear in the experimental section or appendix.
  2. [Notation] Notation for the additive decoder and ANOVA main effects should be introduced once and used consistently; minor inconsistencies in variable naming appear in the provided description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments on the theory section. We address each major comment below and will incorporate clarifications to strengthen the presentation of the identifiability and finite-sample results.

read point-by-point responses

  1. Referee: [Theory section] The identifiability of ANOVA main-effect supports under general smooth mixing functions (stated in the abstract and presumably proved in the theory section) rests on regime-conditioned temporal variation; the manuscript should explicitly verify that the proof does not require additional restrictions on regime diversity or mixing smoothness beyond those stated, as this is load-bearing for the central claim.

    Authors: The proof of identifiability for the ANOVA main-effect supports (Theorem 3.1) relies solely on the regime-conditioned temporal variation together with the stated general smoothness assumptions on the mixing functions; no further restrictions on regime diversity or smoothness are used. To address the concern directly, we will insert a short clarifying paragraph immediately after the theorem statement that explicitly confirms the proof invokes only these assumptions. revision: yes

  2. Referee: [Theory section] Finite-sample recovery guarantees for the sparse-additive decoder are claimed; the manuscript should report the explicit dependence of the sample complexity on the smoothness parameter, sparsity level, and number of regimes (e.g., in the relevant theorem), because these quantities determine whether the guarantees are practically useful.

    Authors: We agree that highlighting the explicit dependence improves clarity and practical relevance. The finite-sample bounds in Theorem 4.2 are already expressed in terms of the smoothness parameter, sparsity level, and number of regimes, but the dependence is not stated in the theorem box itself. We will revise the theorem statement to display the sample-complexity scaling explicitly with respect to these three quantities. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's identifiability result for ANOVA main-effect supports under general smooth mixing functions is derived from explicit assumptions on regime-conditioned temporal variation and smoothness of the mixing, rather than reducing by construction to fitted quantities or self-citations. The finite-sample recovery guarantees for the sparse-additive decoder are presented as following from the model structure and assumptions without tautological equivalence to inputs. No load-bearing step matches the enumerated circularity patterns; the central claim remains independent of the data-fitting process itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard VAE and CRL background plus two domain-specific assumptions about identifiability; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption ANOVA main-effect supports are identifiable under general smooth mixing functions
    Invoked to establish the theoretical support recovery result.
  • domain assumption Latent variables are identifiable via regime-conditioned temporal variation
    Used as the mechanism for pinning down the latent space in the temporal VAE.

pith-pipeline@v0.9.0 · 5571 in / 1341 out tokens · 85968 ms · 2026-05-08T16:15:46.751352+00:00 · methodology

discussion (0)

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