SQPn

SQPn, short for Squared Polynomial Networks, is a class of machine learning models that represent nonlinear relationships using a basis of squared polynomial features. The approach maps input data into a feature space spanned by squared terms and their interactions, such as x_i^2 and cross-squared terms, enabling linear or convex optimization in the transformed space while capturing nonlinearity through the squared polynomials. SQPn aims to provide a balance between interpretability and expressiveness, since each feature corresponds to an explicit polynomial term.

In model construction, the prediction is formed as a linear combination of squared polynomial basis functions: y ≈ w0 + sum_j w_j φ_j(x), where φ_j are predefined squared polynomial features derived from the input coordinates. The feature set grows with the number of input variables and the chosen degree, making feature selection an important consideration for practical use. Regularization is commonly employed to control model size and prevent overfitting.

Training SQPn can be framed as solving a sequence of quadratic programming subproblems. At each iteration, a quadratic objective is minimized with respect to the coefficient vector under constraints such as nonnegativity, sparsity, or bounded coefficients. This sequential quadratic programming–like approach often yields stable convergence and leverages convex subproblems, even if the overall problem may be nonconvex when complex regularization is included.

Variants of SQPn include sparse SQPn, which emphasizes a small subset of polynomial features, and kernelized versions that approximate the squared polynomial space in high dimensions. Applications span regression and binary classification, particularly when interpretability and tractable optimization are priorities. Limitations include combinatorial growth of features with degree and dimensionality, which can necessitate feature screening or dimensionality reduction.

See also: polynomial networks, kernel methods, quadratic programming, feature engineering.