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Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability
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Proposes Diverse Influence Component Analysis (DICA) with a Jacobian volume maximization (J-VolMax) criterion that identifies nonlinear mixture latent components without auxiliary variables, independence, or Jacobian sparsity, via a sufficiently diverse influence (SDI) geometric condition.
Problem
Identifiability of latent components from unknown nonlinear mixtures (nonlinear ICA) without auxiliary information, statistical independence, or Jacobian sparsity assumptions.
Approach
Exploits convex geometry of the mixing function's Jacobian: if the rows of the Jacobian are sufficiently scattered (SDI, an extension of SSC from polytopic matrix factorization), maximizing the Jacobian log-determinant volume while fitting an autoencoder recovers the latent components up to permutation and element-wise invertible maps.
Key results
- On synthetic Mixture A/B with (d,m)=(2,30), DICA achieves R^2 0.92 and MCC 0.93 versus 0.79/0.83 for the sparse baseline.
- On SERGIO single-cell data (d=5, m=178), DICA reaches mean R^2 approximately 1 and MCC approximately 1, versus 0.4544 R^2 for Sparse and 0.3381 for Base.
- In the m/d ablation with d=3, DICA's R^2 drops from 0.90 at m=40 to 0.55 at m=3, and matches the vanilla autoencoder for m<2d.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add Jacobian volume maximization | autoencoder on Mixtures A and B · synthetic nonlinear mixture identification | R²: DICA scores are mostly above 0.92 | Helped |
| add Jacobian volume maximization | autoencoder on Mixture C · synthetic nonlinear mixture identification | R² | Helped |
| add Jacobian volume maximization | autoencoder on single-cell TF inference · single-cell transcriptomics | R²: mean R² ≈ 1; Sparse mean R² ≈ 0.4544; Base mean R² ≈ 0.3381 | Helped |
| increase m/d from 1 to approximately 2 (ablation) | DICA on Mixture C (d=3) · Mixture C | R²: 0.55 ± 0.07 -> 0.90 ± 0.10 | Helped |
| reduce m below 2d (ablation) | DICA on Mixture C · Mixture C | R²: 0.55 ± 0.07 | Hurt |
Surprising findings
- The authors report that the sparse-Jacobian baseline is outperformed by DICA even on Mixture C, where the sparsity-favoring structure is partially present.
Limitations
- Identifiability analysis assumes noise-free data, unlimited samples, and that the learner class contains the target function.
- The log-det volume term is costly to optimize and must be evaluated at every latent sample.
- SDI is harder to satisfy when m is close to 2d, and performance degrades in that regime.
Experimental setup
Datasets: synthetic mixtures A, B, C SERGIO simulated single-cell gene expression TRRUST mouse TF-gene network
Benchmarks: synthetic mixtures A, B, C SERGIO TF activity inference
Metrics: nonlinear R^2 MCC
Baselines: IMA Sparse Jacobian autoencoder unregularized autoencoder
Builds on: nonlinear ICA sufficiently scattered condition polytopic matrix factorization independent mechanism analysis Jacobian sparsity regularization
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