Weighting vectors for machine learning: numerical harmonic analysis
applied to boundary detection
- URL: http://arxiv.org/abs/2106.00827v1
- Date: Tue, 1 Jun 2021 22:14:22 GMT
- Title: Weighting vectors for machine learning: numerical harmonic analysis
applied to boundary detection
- Authors: Eric Bunch, Jeffery Kline, Daniel Dickinson, Suhaas Bhat, Glenn Fung
- Abstract summary: We show that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection.
We demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets.
- Score: 3.8848561367220276
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Metric space magnitude, an active field of research in algebraic topology, is
a scalar quantity that summarizes the effective number of distinct points that
live in a general metric space. The {\em weighting vector} is a closely-related
concept that captures, in a nontrivial way, much of the underlying geometry of
the original metric space. Recent work has demonstrated that when the metric
space is Euclidean, the weighting vector serves as an effective tool for
boundary detection. We recast this result and show the weighting vector may be
viewed as a solution to a kernelized SVM. As one consequence, we apply this new
insight to the task of outlier detection, and we demonstrate performance that
is competitive or exceeds performance of state-of-the-art techniques on
benchmark data sets. Under mild assumptions, we show the weighting vector,
which has computational cost of matrix inversion, can be efficiently
approximated in linear time. We show how nearest neighbor methods can
approximate solutions to the minimization problems defined by SVMs.
Related papers
- Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities [0.5592394503914488]
We introduce a method for identifying and approximating a target measure $pi$ as a perturbation of a given reference measure $mu$.
Our main contribution unveils a connection between the emphdimensional logarithmic Sobolev inequality (LSI) and approximations with this ansatz.
arXiv Detail & Related papers (2024-06-18T20:02:44Z) - An Efficient Algorithm for Clustered Multi-Task Compressive Sensing [60.70532293880842]
Clustered multi-task compressive sensing is a hierarchical model that solves multiple compressive sensing tasks.
The existing inference algorithm for this model is computationally expensive and does not scale well in high dimensions.
We propose a new algorithm that substantially accelerates model inference by avoiding the need to explicitly compute these covariance matrices.
arXiv Detail & Related papers (2023-09-30T15:57:14Z) - An evaluation framework for dimensionality reduction through sectional
curvature [59.40521061783166]
In this work, we aim to introduce the first highly non-supervised dimensionality reduction performance metric.
To test its feasibility, this metric has been used to evaluate the performance of the most commonly used dimension reduction algorithms.
A new parameterized problem instance generator has been constructed in the form of a function generator.
arXiv Detail & Related papers (2023-03-17T11:59:33Z) - Linearized Wasserstein dimensionality reduction with approximation
guarantees [65.16758672591365]
LOT Wassmap is a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space.
We show that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size.
We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.
arXiv Detail & Related papers (2023-02-14T22:12:16Z) - Weight Vector Tuning and Asymptotic Analysis of Binary Linear
Classifiers [82.5915112474988]
This paper proposes weight vector tuning of a generic binary linear classifier through the parameterization of a decomposition of the discriminant by a scalar.
It is also found that weight vector tuning significantly improves the performance of Linear Discriminant Analysis (LDA) under high estimation noise.
arXiv Detail & Related papers (2021-10-01T17:50:46Z) - Analysis of Truncated Orthogonal Iteration for Sparse Eigenvector
Problems [78.95866278697777]
We propose two variants of the Truncated Orthogonal Iteration to compute multiple leading eigenvectors with sparsity constraints simultaneously.
We then apply our algorithms to solve the sparse principle component analysis problem for a wide range of test datasets.
arXiv Detail & Related papers (2021-03-24T23:11:32Z) - Random Matrix Based Extended Target Tracking with Orientation: A New
Model and Inference [0.0]
We propose a novel extended target tracking algorithm which is capable of representing the extent of dynamic objects as an ellipsoid with a time-varying orientation angle.
A diagonal positive semi-definite matrix is defined to model objects' extent within the random matrix framework.
It is not possible to find a closed-form analytical expression for the true posterior because of the absence of conjugacy.
arXiv Detail & Related papers (2020-10-17T16:33:06Z) - Practical applications of metric space magnitude and weighting vectors [8.212024590297894]
The magnitude of a metric space is a real number that aims to quantify the effective number of distinct points in the space.
The contribution of each point to a metric space's global magnitude, which is encoded by the em weighting vector, captures much of the underlying geometry of the original metric space.
Surprisingly, when the metric space is Euclidean, the weighting vector also serves as an effective tool for boundary detection.
arXiv Detail & Related papers (2020-06-24T21:30:55Z) - Semiparametric Nonlinear Bipartite Graph Representation Learning with
Provable Guarantees [106.91654068632882]
We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution.
We show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate.
Our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.
arXiv Detail & Related papers (2020-03-02T16:40:36Z) - Statistical Outlier Identification in Multi-robot Visual SLAM using
Expectation Maximization [18.259478519717426]
This paper introduces a novel and distributed method for detecting inter-map loop closure outliers in simultaneous localization and mapping (SLAM)
The proposed algorithm does not rely on a good initialization and can handle more than two maps at a time.
arXiv Detail & Related papers (2020-02-07T06:34:44Z)
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.