Inverting Self-Organizing Maps: A Unified Activation-Based Framework

Self-Organizing Maps (SOMs) provide topology-preserving projections of high-dimensional data, yet their use as generative models remains largely unexplored. We show that the activation pattern of a SOM -- the squared distances to its prototypes -- can be inverted to recover the exact input, following from a classical result in Euclidean distance geometry: a point in D dimensions is uniquely determined by its distances to D+1 affinely independent references. We derive the corresponding linear system and characterize the conditions under which inversion is well-posed. Building on this mechanism, we introduce the Manifold-Aware Unified SOM Inversion and Control (MUSIC) update rule, which modifies squared distances to selected prototypes while preserving others, producing controlled, semantically meaningful trajectories aligned with the SOM's piecewise-linear structure. Tikhonov regularization stabilizes the update and ensures smooth motion in high dimensions. Unlike variational or diffusion-based generative models, MUSIC requires no sampling, latent priors, or learned decoders: it operates entirely on prototype geometry. If no perturbation is applied, inversion recovers the exact input; when a target prototype or cluster is specified, MUSIC produces coherent semantic transitions. We validate the framework on synthetic Gaussian mixtures, MNIST digits, and the Labeled Faces in the Wild dataset. Across all settings, MUSIC trajectories maintain high classifier confidence, produce significantly sharper intermediate images than linear interpolation, and reveal an interpretable geometric structure of the learned map.