Estimation and Quantization of Expected Persistence Diagrams

• URL: http://arxiv.org/abs/2105.04852v1
• Date: Tue, 11 May 2021 08:12:18 GMT
• Title: Estimation and Quantization of Expected Persistence Diagrams
• Authors: Vincent Divol (DATASHAPE, LMO), Th\'eo Lacombe (DATASHAPE)
• Abstract summary: We study two such summaries, the Persistence Diagram (EPD) and its quantization. EPD is a measure supported on R 2. We prove that this estimator is optimal from a minimax standpoint on a large class of models with a parametric rate of convergence.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Persistence diagrams (PDs) are the most common descriptors used to encode the topology of structured data appearing in challenging learning tasks; think e.g. of graphs, time series or point clouds sampled close to a manifold. Given random objects and the corresponding distribution of PDs, one may want to build a statistical summary-such as a mean-of these random PDs, which is however not a trivial task as the natural geometry of the space of PDs is not linear. In this article, we study two such summaries, the Expected Persistence Diagram (EPD), and its quantization. The EPD is a measure supported on R 2 , which may be approximated by its empirical counterpart. We prove that this estimator is optimal from a minimax standpoint on a large class of models with a parametric rate of convergence. The empirical EPD is simple and efficient to compute, but possibly has a very large support, hindering its use in practice. To overcome this issue, we propose an algorithm to compute a quantization of the empirical EPD, a measure with small support which is shown to approximate with near-optimal rates a quantization of the theoretical EPD.

Related papers

• A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams [0.3968603035422276]
  We introduce a class of pseudodistances called Extended Topological Pseudodistances (ETD) ETDs have tunable complexity, and can approximate Sliced and classical Wasserstein distances at the high-complexity extreme. We experimentally verify that ETDs outperform PSs in terms of accuracy and outperform Wasserstein and Sliced Wasserstein distances in terms of computational complexity.
  arXiv  Detail & Related papers  (2024-02-22T12:27:35Z)
• Online Heavy-tailed Change-point detection [6.7643284029102295]
  We present an algorithm based on clipped Gradient Descent (SGD), that works even if we only assume that the second moment of the data generating process is bounded. We derive guarantees on worst-case, finite-sample false-positive rate (FPR) over the family of all distributions with bounded second moment. Our method is the first OCPD algorithm that guarantees finite-sample FPR, even if the data is high dimensional and the underlying distributions are heavy-tailed.
  arXiv  Detail & Related papers  (2023-06-15T23:39:05Z)
• Dimensionality Reduction as Probabilistic Inference [10.714603218784175]
  Dimensionality reduction (DR) algorithms compress high-dimensional data into a lower dimensional representation while preserving important features of the data. We introduce the ProbDR variational framework, which interprets a wide range of classical DR algorithms as probabilistic inference algorithms in this framework.
  arXiv  Detail & Related papers  (2023-04-15T23:48:59Z)
• FaDIn: Fast Discretized Inference for Hawkes Processes with General Parametric Kernels [82.53569355337586]
  This work offers an efficient solution to temporal point processes inference using general parametric kernels with finite support. The method's effectiveness is evaluated by modeling the occurrence of stimuli-induced patterns from brain signals recorded with magnetoencephalography (MEG) Results show that the proposed approach leads to an improved estimation of pattern latency than the state-of-the-art.
  arXiv  Detail & Related papers  (2022-10-10T12:35:02Z)
• Scaling Structured Inference with Randomization [64.18063627155128]
  We propose a family of dynamic programming (RDP) randomized for scaling structured models to tens of thousands of latent states. Our method is widely applicable to classical DP-based inference. It is also compatible with automatic differentiation so can be integrated with neural networks seamlessly.
  arXiv  Detail & Related papers  (2021-12-07T11:26:41Z)
• Sampling from Arbitrary Functions via PSD Models [55.41644538483948]
  We take a two-step approach by first modeling the probability distribution and then sampling from that model. We show that these models can approximate a large class of densities concisely using few evaluations, and present a simple algorithm to effectively sample from these models.
  arXiv  Detail & Related papers  (2021-10-20T12:25:22Z)
• Nonparametric estimation of continuous DPPs with kernel methods [0.0]
  Parametric and nonparametric inference methods have been proposed in the finite case, i.e. when the point patterns live in a finite ground set. We show that a restricted version of this maximum likelihood (MLE) problem falls within the scope of a recent representer theorem for nonnegative functions in an RKHS. We propose, analyze, and demonstrate a fixed point algorithm to solve this finite-dimensional problem.
  arXiv  Detail & Related papers  (2021-06-27T11:57:14Z)
• Random Persistence Diagram Generation [4.435094091999926]
  Topological data analysis (TDA) studies the shape patterns of data. Persistent homology (PH) is a widely used method in TDA that summarizes homological features of data at multiple scales and stores this in persistence diagrams (PDs) We propose random persistence diagram generation (RPDG), a method that generates a sequence of random PDs from the ones produced by the data.
  arXiv  Detail & Related papers  (2021-04-15T19:33:01Z)
• On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification [101.0377583883137]
  Projection robust (PR) OT seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces.
  arXiv  Detail & Related papers  (2020-06-22T14:35:33Z)
• Projection Robust Wasserstein Distance and Riemannian Optimization [107.93250306339694]
  We show that projection robustly solidstein (PRW) is a robust variant of Wasserstein projection (WPP) This paper provides a first step into the computation of the PRW distance and provides the links between their theory and experiments on and real data.
  arXiv  Detail & Related papers  (2020-06-12T20:40:22Z)
• Neural Methods for Point-wise Dependency Estimation [129.93860669802046]
  We focus on estimating point-wise dependency (PD), which quantitatively measures how likely two outcomes co-occur. We demonstrate the effectiveness of our approaches in 1) MI estimation, 2) self-supervised representation learning, and 3) cross-modal retrieval task.
  arXiv  Detail & Related papers  (2020-06-09T23:26:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.