The estimation of the gradient of a density function, with applications in pattern recognition

@article{Fukunaga1975TheEO,
  title={The estimation of the gradient of a density function, with applications in pattern recognition},
  author={Keinosuke Fukunaga and Larry D. Hostetler},
  journal={IEEE Trans. Inf. Theory},
  year={1975},
  volume={21},
  pages={32-40}
}
Nonparametric density gradient estimation using a generalized kernel approach is investigated. Conditions on the kernel functions are derived to guarantee asymptotic unbiasedness, consistency, and uniform consistency of the estimates. The results are generalized to obtain a simple mcan-shift estimate that can be extended in a k -nearest-neighbor approach. Applications of gradient estimation to pattern recognition are presented using clustering and intrinsic dimensionality problems, with the… 

Figures from this paper

On the Estimation of the Gradient Lines of a Density and the Consistency of the Mean-Shift Algorithm
We consider the problem of estimating the gradient lines of a density, which can be used to cluster points sampled from that density, for example via the mean-shift algorithm of Fukunaga and
Generalized Mode and Ridge Estimation
TLDR
A method is proposed for studying the geometric structure of generalized densities using a modification of the mean shift algorithm and its variant, subspace constrained mean shift, which can be used to perform clustering and to calculate a measure of connectivity between clusters.
Applying neighborhood consistency for fast clustering and kernel density estimation
  • Kai Zhang, Ming Tang, J. Kwok
  • Computer Science
    2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05)
  • 2005
TLDR
This paper derives the fast mean shift algorithm (FMS), a conceptually novel approach of density estimation, the fast kernel density estimation (FKDE) for clustering that greatly reduces the complexity of feature space analysis, resulting satisfactory precision of classification.
ERRATA: On the Estimation of the Gradient Lines of a Density and the Consistency of the Mean-Shift Algorithm
We consider the problem of estimating the gradient lines of a density, which can be used to cluster points sampled from that density, for example via the mean-shift algorithm of Fukunaga and
Direct Density Derivative Estimation
TLDR
A novel method that directly estimates density derivatives without going through density estimation is proposed, which provides computationally efficient estimation for the derivatives of any order on multidimensional data with a hyperparameter tuning method and achieves the optimal parametric convergence rate.
A Histogram Transform for ProbabilityDensity Function Estimation
  • Ezequiel López-Rubio
  • Computer Science, Mathematics
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 2014
TLDR
A new nonparametric approach is presented which is based on the integration of several multivariate histograms, computed over affine transformations of the training data, and belongs to the class of averaged histogram density estimators.
Density Estimation Using Mixtures of Mixtures of Gaussians
TLDR
A new density estimation algorithm using mixtures of mixture of Gaussians, based on the Jensen-Shannon divergence, which overcomes the limitations of the popular Expectation Maximization algorithm and introduces a new model selection criterion called the Penalty-less Information Criterion.
Nonparametric Ridge Estimation
TLDR
Ridge estimation is an extension of mode finding and is useful for understanding the structure of a density and can be used to find hidden structure in point cloud data.
Uniform Convergence Rates for Kernel Density Estimation
TLDR
Finite-sample high-probability density estimation bounds for multivariate KDE are derived under mild density assumptions which hold uniformly in x ∈ R and bandwidth matrices and uniform convergence results for local intrinsic dimension estimation are given.
...
...

References

SHOWING 1-10 OF 14 REFERENCES
Estimation of a probability density function and its derivatives
We investigate statistical estimates of a probability density distribution function and its derivatives. As the starting point of the investigation we take a priori assumptions about the degree of
Remarks on Some Nonparametric Estimates of a Density Function
1. Summary. This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild
On Estimation of a Probability Density Function and Mode
Abstract : Given a sequence of independent identically distributed random variables with a common probability density function, the problem of the estimation of a probability density function and of
A finite-memory deterministic algorithm for the symmetric hypothesis testing problem
TLDR
It is shown how members of this class of irreducible deterministic finite-memory algorithms can be constructed to give a steady-state probability of error that decreases asymptotically faster in the number of states than the best previously known deterministic algorithm.
...
...