Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks
We derive bounds for the volume of tubular neighbourhoods of smooth Pfaffian hypersurfaces, generalising known results for algebraic varieties. The bounds are given in terms of the Pfaffian format of the defining functions. As an application, we obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions, in both the uniform and Gaussian settings. In the special case of single-hidden-layer sigmoid networks with rational weights, we derive polynomial-in-width bounds for tubular neighbourhoods of the decision boundary.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Tubular Curvature Filter: Implicit Pointwise Curvature Calculation Method for Tubular Objects
Curvature estimation methods are important as they capture salient features for various applications in image processing, especially within medical domains where tortuosity of vascular structures is of significant intere…
On sampling determinantal and Pfaffian point processes on a quantum computer
DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsampling tools in statistics and computer science. Most applications require sampling from a …
Point ProcessesSolution of matching equations of IDA-PBC by Pfaffian differential equations
Finding the general solution of partial differential equations (PDEs) is essential for controller design in newly developed methods. Interconnection and damping assignment passivity based control (IDA-PBC) is one of such…
Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations
Neural wave functions accomplished unprecedented accuracies in approximating the ground state of many-electron systems, though at a high computational cost. Recent works proposed amortizing the cost by learning generaliz…
Approximate inference on planar graphs using Loop Calculus and Belief Propagation
We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z of a gr…