FPVTn

FPVTn stands for Finite Polynomial Vector Transform in n dimensions. It is a theoretical class of nonlinear vector transforms used in data representation and signal processing. FPVTn describes a family of mappings that take an input vector x in R^d and produce an output vector y in R^m by applying a linear projection followed by a polynomial nonlinearity of degree up to n, or by applying a polynomial expansion directly to the projected coordinates.

A common formalization begins with a projection matrix A in R^{m×d}. The transform then applies a vector-valued polynomial p of degree up to n to the projected vector z = Ax, resulting in FPVTn(x) = p(z). Each component p_i of p can be expressed as a polynomial in z with coefficients that define the transform. When n = 1, FPVT1 reduces to a linear transform; higher degrees introduce nonlinearity and greater expressive power through polynomial terms and cross-terms.

Variants of FPVTn include applying the polynomial expansion before or after projection, using tensorized or cross-term bases to capture interactions among input features, and kernelized versions that approximate the polynomial map in high-dimensional feature spaces. The transform can be designed to be invertible under certain conditions on A and the polynomial coefficients, and its stability depends on the norms of these components.

Applications are found in feature extraction, nonlinear dimensionality reduction, and preprocessing for machine learning or signal-processing pipelines. FPVTn is discussed in theoretical contexts as a conceptual bridge between linear transforms and more complex nonlinear feature mappings, with related concepts including polynomial feature maps and kernel methods.