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Tracking Dynamic Gaussian Density with a Theoretically Optimal Sliding Window Approach

Yinsong Wang, Yu Ding, Shahin Shahrampour

TL;DR

This work theoretically characterizes the exact MISE, which can be formulated as constrained quadratic programming, and presents empirical evidence with synthetic datasets to show that the weighting scheme indeed improves the tracking performance compared to heuristic approaches.

Abstract

Dynamic density estimation is ubiquitous in many applications, including computer vision and signal processing. One popular method to tackle this problem is the "sliding window" kernel density estimator. There exist various implementations of this method that use heuristically defined weight sequences for the observed data. The weight sequence, however, is a key aspect of the estimator affecting the tracking performance significantly. In this work, we study the exact mean integrated squared error (MISE) of "sliding window" Gaussian Kernel Density Estimators for evolving Gaussian densities. We provide a principled guide for choosing the optimal weight sequence by theoretically characterizing the exact MISE, which can be formulated as constrained quadratic programming. We present empirical evidence with synthetic datasets to show that our weighting scheme indeed improves the tracking performance compared to heuristic approaches.

Tracking Dynamic Gaussian Density with a Theoretically Optimal Sliding Window Approach

TL;DR

This work theoretically characterizes the exact MISE, which can be formulated as constrained quadratic programming, and presents empirical evidence with synthetic datasets to show that the weighting scheme indeed improves the tracking performance compared to heuristic approaches.

Abstract

Dynamic density estimation is ubiquitous in many applications, including computer vision and signal processing. One popular method to tackle this problem is the "sliding window" kernel density estimator. There exist various implementations of this method that use heuristically defined weight sequences for the observed data. The weight sequence, however, is a key aspect of the estimator affecting the tracking performance significantly. In this work, we study the exact mean integrated squared error (MISE) of "sliding window" Gaussian Kernel Density Estimators for evolving Gaussian densities. We provide a principled guide for choosing the optimal weight sequence by theoretically characterizing the exact MISE, which can be formulated as constrained quadratic programming. We present empirical evidence with synthetic datasets to show that our weighting scheme indeed improves the tracking performance compared to heuristic approaches.
Paper Structure (10 sections, 3 theorems, 20 equations, 1 figure)

This paper contains 10 sections, 3 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation and Algorithm
  3. Theoretical Result: Optimal Weight Sequence
  4. Empirical Results
  5. Synthetic Dataset Design
  6. Experimental Settings
  7. Performance
  8. Conclusion
  9. Appendix: Proof of Theorem \ref{['the: mise']}
  10. Acknowledgements

Key Result

theorem 1.1

Estimating the evolving Gaussian density $p_t(x)$ with the "sliding window" Gaussian Kernel Density Estimator DKDE results in the following exact mean integrated squared error where $\boldsymbol{\Lambda} = \boldsymbol{\Phi} + \mathbf{D}$, and $\boldsymbol{\Phi} \in \mathbb{R}^{T \times T}$ is such that $[\boldsymbol{\Phi}]_{ij}=\phi_{(\sigma^2+\gamma^2_i+\sigma^2+\gamma^2_j)^{1/2}}(\mu_i-\mu_j)$.

Figures (1)

  • Figure 1: Left: The estimated MISE versus different window sizes. Right: The estimated MISE versus different kernel bandwidths.

Theorems & Definitions (3)

  • theorem 1.1
  • corollary thmcountercorollary
  • lemma thmcounterlemma