A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipsc…
Given a piecewise linear (PL) function p defined on an open subset of Rn, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle Rn×Rn∗ representing the graph of the differential of p. Restricting to dimension 2, we show that any smooth functi…
Smoothing graphons improve link prediction in Bayesian SBM without increasing computational complexity.
problem Accurate modeling of exchangeable relational data with flexible and computationally efficient graphons.
method Introducing smoothing procedures to piecewise-constant graphons to create smoothing graphons, which allow continuous intensity values for relations.
result Smoothing graphons improve AUC and precision for link prediction in real-world data sets.
We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on t…
problem Measuring similarity between real-valued functions.
method Unified representation of real-valued functions via sets, derived general form of the kernel, explicit feature representation, and smooth approximation.
result General Tanimoto kernel for real-valued functions.
We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge posed by Thurston in dimension three and confirms a conjecture by Kwasik and L…
The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…
Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each …
Rham cohomology of Lie groupoids over triangulated manifolds is isomorphic to piecewise cohomology.
problem Establishing Rham cohomology for Lie groupoids over triangulated manifolds.
method Using isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, proving Rham cohomology is isomorphic to piecewise Rham cohomology.
result Piecewise de Rham cohomology of a Lie groupoid does not depend on the triangulation of the base.
The paper approximates smooth isotropic surfaces with piecewise linear ones.
problem Approximating smooth isotropic surfaces with piecewise linear ones.
method Using analogies with infinite dimensional moment map geometry, the authors prove the approximation of smooth isotropic immersions by piecewise linear ones.
result Smooth isotropic immersions can be approximated by piecewise linear isotropic maps.
The paper introduces a scalable unsupervised learning framework to improve deep neural networks.
problem Improving deep neural networks' performance and generalization in unsupervised settings.
method A scalable unsupervised regularization framework that constrains hypothesis space to non-trivial piecewise constant functions.
result The framework leads to a factually confident and smooth discriminative model, achieving state-of-the-art clustering results and generalization on both synthetic and real data.
Uniqueness of stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
problem Identifying stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
method Study of piecewise-smooth hypersurfaces with anisotropic energy, proving uniqueness of the Wulff shape under certain conditions.
result Closed stable equilibrium hypersurfaces are unique and the Wulff shape when the anisotropic energy density is twice continuously differentiable and convex.
We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of…
Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…