Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability

Latent component identification from unknown nonlinear mixtures is a foundational challenge in machine learning, with applications in tasks such as selfsupervised learning and causal representation learning. Prior work in nonlinear independent component analysis (nICA) has shown that auxiliary signals-such as weak supervision-can support identifiability of conditionally independent latent components. More recent approaches explore structural assumptions, e.g., sparsity in the Jacobian of the mixing function, to relax such requirements. In this work, we introduce Diverse Influence Component Analysis (DICA), a framework that exploits the convex geometry of the mixing function's Jacobian. We propose a Jacobian Volume Maximization (J-VolMax) criterion, which enables latent component identification by encouraging diversity in their influence on the observed variables. Under reasonable conditions, this approach achieves identifiability without relying on auxiliary information, latent component independence, or Jacobian sparsity assumptions. These results extend the scope of identifiability analysis and offer a complementary perspective to existing methods. the variations of u, one can fend against negative effects (e.g., the existence of measure-preserving automorphism (MPA) [6,43]) leading to non-identifiability under (1). The results are elegant and inspiring. Nonetheless, both u and conditional independence might not always be available.

nICA with Structured f . Another route to establish identifiability is to exploit prior structural information of f . For example, the works [9, 16, 18, 23, 44-48] used conformal, local isometry, close-to-linear, post-nonlinear, piecewise affine structures, and additive structures of f , respectively. Recent works, e.g., [14,15,18,23,48], also showed that some of these structures can be used for dependent component identification. Nonetheless, such explicit structures of f , e.g., post-nonlinear mixtures, only make sense when they are used for suitable applications (e.g., hyperspectral imaging [15] and audio separation [16]), yet most problems in generative model learning and representation learning may not have such structures.

Instead of imposing explicit structures directly on f , it is also plausible to exploit the structures of the Jacobian of f . Note that [J f (s)] i,j = ∂xi /∂sj characterizes how x i is influenced by the change of s j .

The aforementioned IMA approach [20], inspired by the principle of ICM [22], assumes orthogonal columns of J f (s) at each point s ∈ S. But these developments lack comprehensive identifiability characterizations (see [44]). On the other hand, the works [2, 3, 19, 49] assume that J f (s) exhibits a certain type of sparsity pattern. These works formulate the NMMI problem as Jacobian sparsityregularized data fitting problems. For example, the work [2] proposed the following:

where the first term finds a diffeomorphism f and its "inverse" g (in which g(x) is supposed to recover s), and the second term c sp (•) promotes sparsity of J f -see [2,3,19,49,50] for their respective ways of sparsity promotion. The most notable feature of this line of work lies in its relatively relaxed conditions on s for establishing identifiability of the model. For instance, the work [2] showed that identifiability can be established without using auxiliary variables or statistical independence of s. On the other hand, J f being sparse means that the observed features in x are only generated by subsets of s. This assumption is justifiable in some applications, e.g., object-centric image generation [2,23,24], but can be violated in other settings.