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 \emph{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 \emph{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.

翻译：自组织映射（SOMs）能够提供高维数据的拓扑保持投影，但其作为生成模型的用途在很大程度上仍未得到探索。我们证明，SOM的激活模式——即到其原型（prototype）的平方距离——可以被*反演*以精确恢复输入数据，这源于欧几里得距离几何学中的一个经典结果：一个$D$维空间中的点，由它到$D{+}1$个仿射无关参考点的距离唯一确定。我们推导了相应的线性系统，并刻画了反演适定（well-posed）的条件。基于此机制，我们引入了*流形感知统一SOM反演与控制*（MUSIC）更新规则，该规则在修改到选定原型的平方距离的同时，保持到其他原型的距离不变，从而产生受控的、语义上有意义的轨迹，这些轨迹与SOM的分段线性结构对齐。Tikhonov正则化稳定了更新过程，并确保了高维空间中的平滑运动。与基于变分或扩散的生成模型不同，MUSIC不需要采样、潜在先验或学习解码器：它完全基于原型几何进行操作。如果不施加扰动，反演可精确恢复输入；当指定目标原型或簇时，MUSIC会产生连贯的语义转换。我们在合成高斯混合、MNIST手写数字以及Labeled Faces in the Wild数据集上验证了该框架。在所有设置中，MUSIC轨迹保持了较高的分类器置信度，产生的中间图像比线性插值显著更清晰，并揭示了所学映射的可解释几何结构。