Topology, Convergence, and Reconstruction of Predictive States
Predictive equivalence in discrete stochastic processes have been applied with great success to identify randomness and structure in statistical physics and chaotic dynamical systems and to inferring hidden Markov models. We examine the conditions under which they can be reliably reconstructed from time-series data, showing that convergence of predictive states can be achieved from empirical samples in the weak topology of measures. Moreover, predictive states may be represented in Hilbert spaces that replicate the weak topology. We mathematically explain how these representations are particularly beneficial when reconstructing high-memory processes and connect them to reproducing kernel Hilbert spaces.
Code (0)
등록된 구현이 없습니다.
Tasks
Time SeriesTime Series AnalysisSimilar Papers 제목 키워드 기반
Convergence of maximum likelihood supertree reconstruction
Supertree methods are tree reconstruction techniques that combine several smaller gene trees (possibly on different sets of species) to build a larger species tree. The question of interest is whether the reconstructed s…
Boundary Sampling to Learn Predictive Safety Filters via Pontryagin's Maximum Principle
Safety filters provide a practical approach for enforcing safety constraints in autonomous systems. While learning-based tools scale to high-dimensional systems, their performance depends on informative data that include…
STITCH: Surface reconstrucTion using Implicit neural representations with Topology Constraints and persistent Homology
We present STITCH, a novel approach for neural implicit surface reconstruction of a sparse and irregularly spaced point cloud while enforcing topological constraints (such as having a single connected component). We deve…
Surface ReconstructionTopological Data AnalysisExploring Predictive States via Cantor Embeddings and Wasserstein Distance
Predictive states for stochastic processes are a nonparametric and interpretable construct with relevance across a multitude of modeling paradigms. Recent progress on the self-supervised reconstruction of predictive stat…
ClusteringDimensionality ReductionTime SeriesTime Series AnalysisSequential Topological Representations for Predictive Models of Deformable Objects
Deformable objects present a formidable challenge for robotic manipulation due to the lack of canonical low-dimensional representations and the difficulty of capturing, predicting, and controlling such objects. We constr…