19-EFT.WP.Methods.SynthData v1.0 | Chapter 9 — Multimodal Synthesis & Balancing (Tabular / Image / Text / Audio / Graph)

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Chapter 9 — Multimodal Synthesis & Balancing (Tabular / Image / Text / Audio / Graph)

I. Scope & Targets

1. Goals
   • Establish a unified synthesis convention across five modalities—tabular, image, text, audio, graph—covering joint generation p({x_m}|c), cross-modal consistency, and balancing (coverage/ratio/quality).
   • Organize both decoupled and coupled generation paths with shared latent z and condition c, supporting one-to-one, one-to-many, and many-to-one pairing rules.
   • Use tau_mono as the unified internal time base and publish on ts; share offset/skew/J across modalities. Any modality involving arrival must record both T_arr formulations and delta_form.
   • Output a multimodal bundle and manifest.synth.bundle.* that downstream evaluation and audit systems can consume directly.
2. Inputs
   • Per-modality schemas SRef_m, reference data or target statistics ref_m, pairing relationships and cardinality constraints (e.g., 1:1, 1:N).
   • A family of per-modality engines {engine_m} or a joint engine engine_joint, with shared prior p(z) and condition c.
   • Consistency rules Rules = { g_j(x_m,x_n, t) ≤ 0 } and quality thresholds.
3. Outputs
   Synthesized samples {x_m}, cross-modal links link_id and matching matrix pi_{mn}, consistency & balancing report, and manifest.synth.bundle.

II. Terms & Symbols

• Modalities & objects: M = {tab, img, txt, aud, graph}, x_m ∈ X_m, condition c, latent z ~ p(z).
• Encoders & decoders: E_m: X_m → U_m, D_m: Z × C → X_m, embeddings u_m = E_m(x_m).
• Joint factorization: p({x_m}|c) = ( ∫ p(z|c) ∏_{m ∈ M} p(x_m|z,c) dz ).
• Pairing & alignment: pi_{mn} ∈ {0,1}^{N_m × N_n} (or soft matches in [0,1]), mapping A_{m→n}.
• Time & arrival: tau_mono, ts, T_arr, gamma(ell), delta_form, offset/skew/J.
• Rules & constraints: g_j(x_m,x_n,t) ≤ 0 (geometric/semantic/physical/referential), unit(x), dim(x).
• Distances & metrics: KL, W1, MMD, FID/KID (image), BLEU/BERTScore (text), PESQ/STOI (audio), spec_MMD/triad_dist (graph), covg (coverage).

*III. Axioms P409- **

• P409-1 (Schema Fidelity): Each modality must satisfy its SRef_m and pass check_dim(expr).
• P409-2 (Shared Latent): Use z as the shared representation for joint generation; allow modality-specific p(x_m|z,c) while coupling with consistency terms.
• P409-3 (Cross-Modal Alignability): Provide a computable mapping A_{m→n} or matching pi_{mn} with stated error bounds.
• P409-4 (Unified Time Base): Align all modal timelines on tau_mono, publish on ts, and record offset/skew/J.
• P409-5 (Dual Arrival Forms): Any propagation/path modality must record both T_arr formulations and delta_form.
• P409-6 (Measurable Balancing): Publish coverage/ratio and quality weights w_m, ensuring sum(w_m)/|M| ≈ 1.
• P409-7 (Reproducibility & Signature): Persist seed/rng/model_spec and link_id in the manifest and sign them.
• P409-8 (Privacy Budget): Accumulate multimodal eps_total per composition rules and persist the accounting trail.
• P409-9 (Referential Integrity): Cross-modal foreign keys and references must resolve; no orphan links are allowed.
• P409-10 (Name Collisions): Do not mix T_fil with T_trans; strictly distinguish n from n_eff (volume-wide rule).

*IV. Minimal Equations S409- **

1. S409-1 (Joint Objective)
   • L_joint = ( ∑_{m} w_m * D_m( p_model^m || p_ref^m ) ) + ( ∑_{m<n} w_{mn} * R_{mn}( u_m, u_n ) )。
   • R_{mn} may be contrastive (InfoNCE), semantic cosine 1 - cos(u_m,u_n), or cycle-consistency || z - z' ||_2.
2. S409-2 (Pairing via Optimal Transport)
   pi_{mn} = arg min_{Pi ∈ U(a,b)} < C_{mn}, Pi > + λ * Ω(Pi)，U(a,b) 为边际约束集合，C_{mn} 为跨模态代价。
3. S409-3 (Cycle Consistency)
   x_m → z = Enc_m(x_m) → x_n' = D_n(z,c) → z' = Enc_n(x_n')，|| z - z' ||_2 ≤ tol_cycle。
4. S409-4 (Time Mapping)
   ts^m = a_m * tau_mono + b_m，发布 offset_m = a_m - 1，skew_m = b_m / T_h 与抖动 J_m。
5. S409-5 (Dual Arrival Forms)
   T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )；T_arr = ( ∫ ( n_eff / c_ref ) d ell )；
   delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |。
6. S409-6 (Balancing Metrics)
   covg_m = ( N_syn^m / N_target^m )；imbalance = || p_model^m - p_ref^m ||_{W1}；
   balance_score = 1 - sigmoid( α * imbalance )。
7. S409-7 (Graph Topology Fidelity)
   MMD_spec = MMD( spec(G_syn), spec(G_ref) )；|| deg_dist_syn - deg_dist_ref ||_1 ≤ tol_deg。

V. Metrology Flow M40-9 (Closed Loop for Multimodal Synthesis & Balancing)

1. Ready
   Consolidate SRef_m, ref_m, and condition set C; define pairing cardinalities and Rules; set thresholds: tol_clip, tol_cycle, tol_w1^m, tol_Tarr, J_max.
2. Pretrain & Align
   Train or select {E_m} and align cross-modal embeddings (e.g., contrastive learning); calibrate scales and temperatures for u_m.
3. Joint Sampling
   Sample z ~ p(z|c) and decode per modality x_m = D_m(z,c); for scenes not jointly generated, synthesize independently then solve pi_{mn} via optimal matching.
4. Rules & Constraints
   Enforce g_j(x_m,x_n,t) ≤ 0: geometry (e.g., bbox/pose), semantics (cos(u_img,u_txt) ≥ tol_clip), physics, and unit consistency.
5. Time & Arrival
   Run align_cross_time to produce ts^m and record offset/skew/J; for path-involved modalities, write both T_arr forms and delta_form.
6. Balancing
   Compute covg_m and imbalance; apply reweight | mapping | domain_randomization to balance; update w_m and n_eff.
7. Fidelity Evaluation
   • Image: FID/KID; Text: BLEU/BERTScore; Audio: PESQ/STOI; Graph: MMD_spec/triad_dist; Tabular: W1/MMD.
   • Cross-modal: semantic cos(u_m,u_n) and cycle consistency ||z - z'||.
8. Compliance & Privacy
   Evaluate eps_total and attack surfaces (membership/linkability); if unmet, roll back or downgrade.
9. Persist & Freeze
   Emit bundle and manifest.synth.bundle, including TraceID, link_id, seed/rng, metrics.*, contracts.*, and signature.

VI. Contracts & Assertions C40-9xx

• C40-901 (Schema & References): validate_schema(x_m)=true; foreign_key(link_id) resolves; drop_orphan=0.
• C40-902 (Semantic Consistency): cos(u_img,u_txt) ≥ tol_clip; InfoNCE_gap ≤ tol_ince; cycle ||z - z'|| ≤ tol_cycle.
• C40-903 (Temporal Consistency): For all time-stamped modalities, |offset_m| ≤ off_max, |skew_m| ≤ skew_max, J_m ≤ J_max.
• C40-904 (Arrival Consistency): For propagation modalities, delta_form ≤ tol_Tarr.
• C40-905 (Balancing Targets): covg_m ≥ covg_min, imbalance_m ≤ tol_w1^m, balance_score ≥ b_min.
• C40-906 (Graph Topology): MMD_spec ≤ tol_spec, ||deg_dist_gap||_1 ≤ tol_deg.
• C40-907 (Units & Dimensions): check_dim(expr)=true, with shared physical quantities consistent across modalities.
• C40-908 (Privacy Budget): eps_total ≤ eps_max, manifest contains the accounting trace.
• C40-909 (Reproducibility & Signature): hash_sha256(bundle) == signature.payload.
• C40-910 (SLO): latency_ms_p99 ≤ SLO_bundle, oom_rate ≤ oom_max.

VII. Implementation Bindings I40-9*

• compose_multimodal(syn_specs, coherence_rules) -> bundle
• learn_cross_modal_embeddings(datasets, model) -> {E_m}
• sample_joint(engine_bundle, n, condition, pairing) -> bundle'
• match_modalities(objs, cost, cardinality) -> pi_{mn}
• enforce_cross_rules(bundle, rules) -> bundle''
• align_cross_time(bundle, sync_ref) -> bundle_timed (write offset/skew/J and T_arr/delta_form)
• balance_multimodal(ref, bundle_timed, method) -> map|weights
• measure_multimodal_fidelity(bundle_timed, ref, metrics) -> report
• emit_bundle_manifest(bundle_timed, policy) -> manifest.synth.bundle
• Invariants: unique(link_id); sum(w_m)/|M| ≈ 1; non_decreasing(ts); delta_form ≤ tol_Tarr; units/dimensions pass checks; privacy budget within limits.

VIII. Cross-References

• This volume: Chapter 5 (deep generation), Chapter 6 (scene graphs & constraints), Chapter 8 (time series & events), Chapter 12 (fidelity/utility evaluation), Chapter 13 (release & manifests).
• Methods.Cleaning v1.0: Chapters 9/10 (de-duplication/referential integrity, release freeze) and 5/6 (time & arrival).
• Methods.Imaging v1.0: Chapters 13 (time/path gating) and 14 (imaging quality metrics).
• Methods.CrossStats v1.0: Chapters 7/14 (drift detection & statistical SLOs).

IX. Quality SLIs & Risk Control

1. Key SLIs
   FID/KID (image), BLEU/BERTScore (text), PESQ/STOI (audio), W1/MMD (tabular), MMD_spec/triad_dist (graph), cos(u_m,u_n), ||z - z'||, covg_m, imbalance_m, offset/skew/J, delta_form, latency_ms_p99, eps_total.
2. Common risks & mitigations
   • Semantic drift: increase w_{mn} or introduce hard g_j constraints; apply temperature calibration and re-ranking.
   • Pairing imbalance: re-match with optimal transport or adjust sampling ratios and modality weights w_m.
   • Time misalignment: re-run align_cross_time and publish corrective offset/skew/J.
   • Graph topology bias: project or regularize degree distribution and spectral properties.
   • Privacy leakage: downgrade to DP(eps,delta), increase noise, or apply sampling capping.

Summary