Proposing Equivalence Between K-Means and RBF

By defining the responsibility of the RBF Network via entropy regularization, it rewrites it in the same variational structure as K-Means.

• Optimizing responsibility on the simplex yields exactly the Softmax solution.
• In the σ→0 limit, responsibility concentrates on the nearest centroid, and the objective function converges to the K-Means distortion via Γ-convergence.
• It claims that gradient-based center updates recover K-Means' mean centroid update under certain conditions.
• It proposes Entmax-1.5 to reduce numerical instability in the low-temperature regime.
• In synthetic geometry experiments, it confirmed a tendency for the soft RBF centroid to monotonically converge to the K-Means fixed point.

In summary, it interprets K-Means not as a separate, discrete clustering stage but as the low-temperature limit of end-to-end differentiable clustering.