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Identifiability in Blind Source Separation through Stabilizer Shrinkage: Unifying Non-Gaussianity and Observation Diversity

Tomomi Ogawa, Hiroki Matsumoto

Abstract

Identifiability is a central issue in blind source separation (BSS), determining whether latent sources can be uniquely recovered from observed mixtures. Classical approaches address identifiability either by exploiting source non-Gaussianity via higher-order statistics (HOS) or by enriching the observation structure through temporal, spatial, or multi-channel diversity using second-order statistics (SOS), and these routes are often regarded as fundamentally different. In this paper, we revisit identifiability in BSS from a structural perspective, interpreting it as constraint-induced reduction of residual ambiguity in the mixing model. Within this framework, the observation mechanism is viewed broadly to include both input-side statistical constraints and output-side observation structures. HOS-based and SOS-based approaches are then unified as mechanisms of stabilizer shrinkage, in which observation-induced constraints reduce an initially continuous ambiguity to a finite residual one. To connect this structural viewpoint with finite-sample regimes, we introduce a Jacobian-based sensitivity probe as a numerical diagnostic of local identifiability. Numerical experiments show that increasing non-Gaussianity or observation diversity suppresses the same residual symmetry, revealing a structural trade-off between source statistics and observation design. These results provide a unified interpretation of classical BSS methods and clarify how observation constraints govern identifiability.

Identifiability in Blind Source Separation through Stabilizer Shrinkage: Unifying Non-Gaussianity and Observation Diversity

Abstract

Identifiability is a central issue in blind source separation (BSS), determining whether latent sources can be uniquely recovered from observed mixtures. Classical approaches address identifiability either by exploiting source non-Gaussianity via higher-order statistics (HOS) or by enriching the observation structure through temporal, spatial, or multi-channel diversity using second-order statistics (SOS), and these routes are often regarded as fundamentally different. In this paper, we revisit identifiability in BSS from a structural perspective, interpreting it as constraint-induced reduction of residual ambiguity in the mixing model. Within this framework, the observation mechanism is viewed broadly to include both input-side statistical constraints and output-side observation structures. HOS-based and SOS-based approaches are then unified as mechanisms of stabilizer shrinkage, in which observation-induced constraints reduce an initially continuous ambiguity to a finite residual one. To connect this structural viewpoint with finite-sample regimes, we introduce a Jacobian-based sensitivity probe as a numerical diagnostic of local identifiability. Numerical experiments show that increasing non-Gaussianity or observation diversity suppresses the same residual symmetry, revealing a structural trade-off between source statistics and observation design. These results provide a unified interpretation of classical BSS methods and clarify how observation constraints govern identifiability.
Paper Structure (42 sections, 1 theorem, 36 equations, 3 figures, 2 tables)

This paper contains 42 sections, 1 theorem, 36 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation and Notation
  3. Blind Source Separation Model
  4. Ambiguities and Equivalence up to Inherent Transformations
  5. Observation Statistics as Constraint-Inducing Information
  6. Scope and Assumptions
  7. Background: HOS-Based and SOS-Based Blind Source Separation
  8. HOS-Based Approaches
  9. SOS-Based Approaches
  10. Structural Comparison
  11. Structural Viewpoint: Observation Constraints and Symmetry
  12. Observation-Induced Constraints on the Mixing Model
  13. Residual Ambiguities under Multiple Constraints
  14. Accumulation of Constraints and Structural Ambiguity Reduction
  15. Local Identifiability via Jacobian Sensitivity
  16. ...and 27 more sections

Key Result

Lemma 1

Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be index sets corresponding to two different families of observation operators. Then the stabilizer associated with their joint constraint satisfies

Figures (3)

  • Figure 1: Effect of source non-Gaussianity on structural identifiability in the HOS route. Primary axis: Jacobian-based probe $\sigma_{\min}(J)$ computed from fourth-order cumulant constraints. The probe collapses near the Gaussian boundary ($p=2.0$) and increases for non-Gaussian sources. Secondary axis: Amari performance index (API). Error bars indicate Monte Carlo dispersion. Note that the absolute scales of both the primary probe and the secondary separation index are experiment-dependent and should not be compared across different figures or constraint families.
  • Figure 2: Effect of observation diversity on structural identifiability in the SOS route. Primary axis: Jacobian-based probe $\sigma_{\min}(J)$ for lag counts $L=1,2,\dots,7$. Secondary axis: Amari performance index (API). Increasing lag diversity strengthens the probe, while gains saturate for larger $L$. Note that the absolute scales of both the primary probe and the secondary separation index are experiment-dependent and should not be compared across different figures or constraint families.
  • Figure 3: Trade-off between source statistics and observation diversity under second-order observation constraints. Heatmap: $\mathrm{probe}_{\mathrm{SOS}}(p,L)$. Markers: $L_{\min}(p;\varepsilon)$ defined in \ref{['eq:Lmin']}. This visualization highlights structurally equivalent configurations rather than performance optima.

Theorems & Definitions (1)

  • Lemma 1: Intersection of stabilizers under multiple constraints