Where Latents Belong and How They Flow: Joint Identification of Structural Roles

1 INTRODUCTION

• A structural role combines group expression with incoming and outgoing temporal influence for the same latent coordinate.The example role ((1, 1, 0), 1, 1) records expression in two groups, receipt of past influence, and influence on the next state.Source: §1 · p. 1
• Earlier work identifies latent components, observation-side structure, or temporal causal structure, but leaves the complete latent role unresolved.The paper contrasts this gap with work on group incidence and temporal support [1?].Source: §1 · p. 2
• Separate recovery can assign group expression and temporal influence to different learned coordinates, so joint identification requires one common permutation σt.Under the common alignment, recovered attributes define the role of the same latent coordinate.Source: §1 · p. 2
• The paper establishes joint identifiability under structural and variability conditions and develops ROLETRACE to estimate roles in a shared coordinate system.The framework uses decoder Jacobians for group expression and transition-density derivatives for temporal roles.Source: §1 · p. 2

2 PRELIMINARIES

• The data-generating process maps a latent sequence to observations partitioned into K groups and uses first-order latent dynamics.The stated model assumes a one-step time lag; instantaneous dependencies and higher-order lags are discussed in Appendix B.Source: §2 · pp. 2–3
• Observational equivalence means the true and learned models induce the same distribution over observed sequences.The preliminaries also define coordinate subspaces and support spaces for the identifiability analysis.Source: §2 · p. 3
• The framework defines observation, incoming, and outgoing roles for each latent coordinate, then combines them into a joint role signature.The signature retains the group-incidence pattern together with the two temporal roles.Source: §2 · p. 3

3 IDENTIFIABILITY THEORY

• The section proceeds from identifying observation supports to temporal supports, then aligns the results to identify complete signatures.The proofs are provided in Appendix A.Source: §3 · p. 3

3.1 IDENTIFYING GROUPED OBSERVATION SUPPORT

• Theorem 3.1 recovers the full group-expression pattern under one permutation, given invertibility, sufficient within-group variation, and structural minimality.These conditions exclude indistinguishable directions and extra links between groups and latents.Source: §3.1 · p. 4

3.2 IDENTIFYING INCOMING AND OUTGOING TEMPORAL SUPPORT

• Theorem 3.2 identifies outgoing temporal support, while Corollary 3.3 identifies incoming support from the transition cross-Hessian’s column and row supports.The outgoing guarantee requires detectable cross derivatives, enough transition variation, and minimality in the number of outgoing-active latents; the incoming result applies corresponding conditions to row support.Source: §3.2 · pp. 4–5

3.3 JOINT IDENTIFICATION OF STRUCTURAL ROLES

• Lemma 3.4 aligns the separately identified structures because they constrain the same time-t reparameterization Jacobian at a common point.This yields a single permutation associating group, incoming, and outgoing roles with the same true latent.Source: §3.3 · p. 5
• Theorem 3.5 establishes joint identifiability of the complete latent roles under the conditions of Lemma 3.4.The result obtains a common correspondence without a path-connectedness assumption and allows ambiguity among coordinates with the same structural role.Source: §3.3 · pp. 5–6

4 ROLETRACE: ESTIMATION OF TEMPORAL LATENT ROLES

• ROLETRACE first learns a latent representation and transition density, keeping the representation fixed during temporal density estimation to preserve its geometry.An encoder–decoder estimates the latent representation; a differentiable conditional density models transitions between latent states.Source: §4 · p. 6
• Decoder Jacobians and transition cross-Hessians supply derivative slices for estimating group expression and incoming and outgoing temporal influence.A common root-mean-square score is used, and the resulting scores are thresholded to estimate the three role types.Source: §4 · p. 6
• A single orthogonal transformation at each time point jointly controls the observation-Jacobian columns and incoming and outgoing transition cross-Hessian supports.The refinement searches for coordinates that make all three structural views jointly sparse.Source: §4 · p. 6
• Deterministic Jacobi coordinate rotations minimize the finite-sample sparsity objective, followed by a transition-density refit and final Jacobi sweep.The additional refit recomputes cross-Hessians after the latent coordinates change.Source: §4 · p. 7

5.1 SYNTHETIC EXPERIMENTS

• The synthetic study uses 16 randomly generated mechanisms for RQ1 and controlled mechanisms with N = 4 for RQ2 and RQ3.RQ1 uses N = 6, K = 3, T = 5, and six observed dimensions per group.Source: §5.1 · p. 7
• At n = 3200, ROLETRACE reaches 0.854 Joint accuracy versus 0.708 for both THOUGHTCOMM and NO-REFIT, and 0.973 F1 for each temporal role.THOUGHTCOMM has slightly higher A F1; ROLETRACE’s improvement over NO-REFIT supports refitting the transition after coordinate refinement.Source: §5.1 · p. 7, Table 1 · p. 8

5.2 REAL-WORLD CASE: MULTI-PARTY MEETINGS

• The AMI evaluation includes 118 scenario-based meetings from 30 teams, partitioned into 15-second windows for next-speaker prediction.The corpus contains 100 hours of recorded meetings; the scenario-based meetings feature four participants collaborating on a shared design project.Source: §5.2 · pp. 7–8

5.1 SYNTHETIC EXPERIMENTS

• Joint alignment yields 0.950 Joint accuracy, versus 0.700, 0.475, and 0.458 for alignment using A, B, and C individually.As identifiability conditions weaken, mean Joint accuracy falls from 1 to 0.633, 0.750, and 0.500 at the respective endpoints; recovery stays perfect in the cancellation experiment until the selected edge has zero net influence.Source: §5.1 · p. 7, Figure 3 · p. 8, Figure 4 · p. 9

5.2 REAL-WORLD CASE: MULTI-PARTY MEETINGS

• At matched latent budgets, Role C performs best at k = 4, 8, and 12; at k = 24, temporal selectors share 51.475 ± 0.119% accuracy.All methods select coordinates from the same representation and use the same downstream classifier; at k = 16, Role C remains comparable to the strongest density readout.Source: Table 2 · p. 9, §5.2 · p. 9

6 CONCLUSION

• Under suitable structural and variability conditions, A, B, and C are jointly identifiable under a common permutation that recovers their correspondence at each latent coordinate.ROLETRACE estimates these roles from finite data in a shared latent coordinate system.Source: §6 · p. 9
• Synthetic experiments support structural recovery and highlight the importance of common alignment, while AMI results suggest benefits for downstream tasks.The conclusion frames the work as a foundation for understanding where hidden factors are expressed and how their influence propagates.Source: §6 · p. 9
• The paper has not empirically explored a broader range of real-world applications, although its formulation applies to a broader class of grouped sequential systems.Source: §6 · p. 9

A.1 PROOF OF THEOREM 3.1

• The proof assumes the true and learned observation maps are C1 embeddings onto the same observation manifold, yielding a C1 diffeomorphism between latent spaces.Source: §A.1 · p. 15
• The learned grouped supports are assumed no denser overall than the true grouped supports.Source: §A.1 · p. 15
• At least one permutation is supported by the nonzero pattern, and every permutation satisfying that condition aligns all group-specific observation supports.The proof applies the same permutation simultaneously across groups.Source: §A.1 · p. 16

A.2 PROOFS OF THEOREM 3.2 AND COROLLARY 3.3

• For outgoing support, a learned cross-Hessian column that vanishes everywhere implies the corresponding true column vanishes, using the stated variation condition.The conditional-density change of variables removes the log-Jacobian term when differentiated with respect to the previous state.Source: §A.2 · pp. 17–18
• A supported permutation of the outgoing-support matrix aligns the learned and true outgoing supports under the theorem’s assumptions.The proof uses nonsingularity, cross-Hessian faithfulness, and a support-cardinality condition.Source: §A.2 · pp. 17–18
• The incoming-support corollary applies the row-side counterpart of the outgoing argument to obtain a supported permutation of incoming supports.It assumes the smoothness and faithfulness conditions for the transition into time t and a learned-support cardinality condition.Source: §A.2 · p. 19

A.3 PROOF OF LEMMA 3.4

• At least one supported permutation exists, and the alignment argument applies to every permutation satisfying the stated support condition.The proof uses the common reference point and the support and sparsity conditions of the preceding results.Source: §A.3 · pp. 19–20
• A single permutation supported by the nonzero pattern of D⋆_t simultaneously aligns grouped observation, incoming, and outgoing supports.The proof obtains the permutation from the same matrix and applies the support arguments for all three structures.Source: §A.3 · pp. 19–20

A.4 PROOF OF THEOREM 3.5

• Under Lemma 3.4’s conditions, Theorem A.5 identifies the joint structural signature under one permutation at each time point.The proof combines the aligned group-expression, incoming, and outgoing indicators.Source: §A.4 · pp. 20–21

A.5 COMMON STRUCTURAL ALIGNMENT AND GLOBAL PERMUTATION INVARIANCE

• Common alignment does not require a unique permutation: coordinates with the same structural signature may be exchanged without changing their roles.In the example, both permutations preserve group, incoming, and outgoing roles.Source: §A.5 · p. 22
• The guarantee permits time-specific permutations, so tracking a persistent latent identity across the sequence requires additional conditions linking reparameterizations across time.Examples include a shared latent reparameterization or suitable continuity conditions over a common temporal context.Source: §A.5 · p. 22

B EXTENSIONS OF THE TEMPORAL MODEL

• The extensions section describes adapting the same structural objects to richer temporal models.Figure 5 contrasts local permutation switching on disconnected regions with continuity-based prevention and common role alignment.Source: B EXTENSIONS OF THE TEMPORAL MODEL · p. 22, Figure 5 · p. 23

B.1 HIGHER-ORDER TEMPORAL DEPENDENCIES

• For higher-order dynamics, incoming and outgoing roles summarize whether a coordinate has any delayed parent or child within the lag range.The method conditions derivatives on the full history to distinguish direct delayed edges from dependencies induced by marginalizing intermediate states.Source: §B.1 · p. 23
• Under the stated variation and support conditions, higher-order temporal views constrain one matrix and preserve joint signatures under a common relabeling at time t.Estimating the extension can condition the transition model on L previous states and score roles from blocks involving Zt.Source: §B.1 · p. 24

B.2 INSTANTANEOUS DEPENDENCIES

• With instantaneous dependencies, an added restriction is needed because instantaneous parents can create incoming cross-Hessian rows for coordinates without direct delayed parents.The restriction requires every instantaneous parent of a delayed receiver to have a delayed parent as well.Source: §B.2 · p. 25

C RELATED WORK

• The paper separates group expression, incoming influence, and outgoing influence, distinguishing coordinates with the same observation profile but different temporal roles.This factorized target does not require recovery of every delayed causal edge.Source: C RELATED WORK · p. 26

• RQ1 uses 16 mechanisms with state-dependent transition means and variances, and nested training prefixes ranging from 50 to 3200 sequences.Observation maps and transition laws remain fixed across time within each mechanism.Source: §D.1 · p. 27
• Evaluation uses one shared permutation for all three roles and distinguishes individual-support recovery from complete-signature Joint and Hamming accuracy.The coordinate transformation is learned without ground truth; evaluation matching only relabels the fitted model.Source: §D.1 · p. 28
• OURS Joint accuracy rises from 0.396 at n = 50 to 0.708 at 100 and 0.844 at 150, then ranges from 0.802 to 0.875 for n ≥150.OURS exceeds TEMPORAL-ONLY at every displayed sample size.Source: Figure 6 · p. 28, §D.1 · p. 28
• The RQ2 controlled alignment experiment uses ten mechanisms and evaluates the middle time point with different coordinate ambiguities across the two transitions.It uses N = 4, K = 3, T = 3, four observed dimensions per group, and 3200 training sequences.Source: §D.1 · pp. 28–29
• Figure 8’s colors show relative score strength within each role or observation group, not probabilities or binary support.The example mechanism’s generating structure is shown in Figure 7.Source: §D.1 · p. 28, Figure 8 · p. 29, Figure 7 · p. 29
• Common alignment gives 0.950±0.033 Joint accuracy, versus 0.700±0.033, 0.475±0.017, and 0.458±0.028 under A-, B-, and C-guided alignments.NO STRUCTURAL gives 0.100 Joint accuracy.Source: §D.1 · p. 29
• Joint refinement reaches 0.900 Joint accuracy, compared with 0.875, 0.450, and 0.575 using A, B, and C alone.The comparison uses fixed representations and pre-refit derivatives across ten mechanisms.Source: Table 3 · p. 30, §D.1 · p. 30
• Mean Joint accuracy drops from one to 0.633, 0.750, and 0.500 at group-span, temporal-span, and cancellation endpoints, but these paths do not establish universal necessity.Group degradation primarily affects A; temporal paths affect B and C, while cancellation remains perfect at positive ratios and drops only at zero.Source: §D.1 · p. 31

• The experiment uses 118 scenario meetings from 30 teams, with four participant roles defining observation groups; 20 held-out meetings are not used.Team-level folds keep meetings from the same team within a single partition.Source: §D.2 · p. 31
• Meetings are divided into non-overlapping 15-second windows, with each participant represented by text and activity-related features; silent windows remain in the sequence.The 118 meetings yield 14,513 observation windows.Source: §D.2 · p. 31
• The full 32-dimensional Role C representation achieves 51.500 ± 0.135% accuracy, exceeding the strongest fixed-input reference.These reference methods use different input dimensions and do not isolate coordinate selection at a matched budget.Source: Table 4 · p. 32, §D.2 · p. 32
• The grouped observations combine PCA-projected text and three activity features into 140 dimensions, which the representation model maps to a 32-dimensional latent state.A GRU summarizes the preceding three latent states, and training predicts future latent states and linguistic content without next-speaker labels.Source: §D.2 · p. 32
• The downstream task predicts the participant linked to the first valid dialogue act starting at or after each 15-second boundary, using logistic regression on selected latent coordinates.The task yields 14,028 prediction samples from 14,513 observation windows, with final evaluation on held-out teams.Source: §D.2 · p. 32
• The experiment treats participant roles as observation groups and identifies group-specific expression and temporal propagation roles of latent information factors.This targets latent structure rather than speakers, utterances, discourse relations, or content spans as the structural units.Source: §D.2 · p. 33