
A generalization of a theorem of Hurewicz for quasiPolish spaces
We identify four countable topological spaces S_2, S_1, S_D, and S_0 whi...
read it

Residuated implications derived from quasioverlap functions on lattices
In this paper, we introduce the concept of residuated implications deriv...
read it

kNN Regression Adapts to Local Intrinsic Dimension
Many nonparametric regressors were recently shown to converge at rates t...
read it

DimensionFree Bounds on Chasing Convex Functions
We consider the problem of chasing convex functions, where functions arr...
read it

Topological Drawings meet Classical Theorems from Convex Geometry
In this article we discuss classical theorems from Convex Geometry in th...
read it

SVM Learning Rates for Data with Low Intrinsic Dimension
We derive improved regression and classification rates for support vecto...
read it

Measuring the Intrinsic Dimension of Objective Landscapes
Many recently trained neural networks employ large numbers of parameters...
read it
A Topological Approach to Inferring the Intrinsic Dimension of Convex Sensing Data
We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasiconvex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasiconvex functions, under natural generic assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We introduce a construction of a filtration of Dowker complexes, associated to measurements by quasiconvex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural generic assumptions. We also illustrate the usability of this method in simulations.
READ FULL TEXT
Comments
There are no comments yet.