Nonsmooth Analysis & Operator Methods

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Abstractions about generalized function spaces, inequalities, potentials, proximal and subdifferential operators, ridge functions, and matrix-free computation.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Barron space — A function space characterized by integral representations or spectral moment bounds that control approximation by two-layer neural networks with dimension-favorable error rates.
• Colombeau algebra — A differential algebra of generalized functions that embeds distributions while permitting nonlinear multiplication and retaining compatibility with smooth-function products.
• Korn's inequality — A rigidity inequality bounding the full gradient of a vector field, modulo rigid motions, by its symmetric gradient under specified domain and boundary conditions.
• Matrix-Free Methods — Solve large linear, eigenvalue, or nonlinear subproblems through an operator-application interface that computes matrix–vector products on demand without assembling or storing the full coefficient or Jacobian matrix.
• Proximal operator — The operator mapping a point to the unique minimizer of a function plus one-half the squared distance to that point, under standard proper lower-semicontinuous convex assumptions.
• Ridge function — Factor a multivariate function through one linear or affine projection, so its value varies only along a selected direction and remains constant across every orthogonal affine slice.
• Riesz potential — Apply the convolution kernel proportional to |x|^{α−n} to realize a fractional inverse power of the Laplacian on Euclidean space.
• Subderivative — A supporting slope or covector that generalizes the derivative of a convex function at a point where an ordinary derivative may not exist.