Softmaxiga

Softmaxiga is a theoretical framework in machine learning that aims to generalize the classical softmax activation by incorporating ideas from isogeometric analysis (IGA). In this approach, probability distributions over discrete categories are allowed to vary smoothly across a domain, such as a image grid or spatial field, by using a basis of smooth functions to shape the latent scores before normalization.

Mathematically, softmaxiga assumes latent category scores z_i(x) that depend on a location x in a domain X. Instead of treating z_i(x) as independent scalars, the framework represents them as linear combinations of isogeometric basis functions phi_k(x) with learned weights w_i,k. The category probabilities at x are then defined by a softmax normalization: p_i(x) = exp(sum_k w_i,k phi_k(x)) / sum_j exp(sum_k w_j,k phi_k(x)). The isogeometric basis functions, such as B-splines or NURBS, enforce smooth variation of probabilities across x, producing spatially coherent outputs suitable for structured prediction tasks.

Learning in softmaxiga proceeds through maximizing a likelihood or cross-entropy objective over labeled data, using gradient-based optimization to update the weight tensors w_i,k. The approach integrates naturally with convolutional or grid-based architectures, with the basis evaluation added as a differentiable layer. Regularization can be applied to the weights or to the basis coefficients to control smoothness and prevent overfitting.

Applications of softmaxiga are envisioned in tasks where category probabilities should change gradually over space, such as semantic segmentation, geospatial classification, and certain medical imaging problems. Benefits include smoother probability maps and improved consistency across neighboring locations, while limitations involve additional computational cost and the need to design appropriate isogeometric bases. See also softmax, isogeometric analysis, and structured prediction.