# Locally Defined Principal Curves and Surfaces

@article{Ozertem2011LocallyDP, title={Locally Defined Principal Curves and Surfaces}, author={Umut Ozertem and Deniz Erdoğmuş}, journal={J. Mach. Learn. Res.}, year={2011}, volume={12}, pages={1249-1286} }

Principal curves are defined as self-consistent smooth curves passing through the middle of the data, and they have been used in many applications of machine learning as a generalization, dimensionality reduction and a feature extraction tool. We redefine principal curves and surfaces in terms of the gradient and the Hessian of the probability density estimate. This provides a geometric understanding of the principal curves and surfaces, as well as a unifying view for clustering, principal…

## 199 Citations

### Local linear approximation of principal curve projections

- Computer Science, Mathematics2012 IEEE International Workshop on Machine Learning for Signal Processing
- 2012

A local linear approximation is proposed to achieve data dimension reduction using numerical integration based differential equation solvers without significant loss of accuracy while reducing computational complexity.

### Principal Curves in High Dimensions

- Computer Science
- 2021

L-SCMS is presented, an approximation algorithm for SCMS which reduces the computational complexity from n 2 to n , where n is the dimensionality of the data and the number of steps required.

### Manifold unwrapping using critical surfaces

- Computer Science, Mathematics2015 IEEE 25th International Workshop on Machine Learning for Signal Processing (MLSP)
- 2015

It is proposed that local nonlinear coordinate transformations to unwrap the so-called manifold are based on curvilinear coordinate systems defined in a manner that is consistent with the critical surfaces, and provided a convenient characterization of all critical surfaces in the form of the zero level set of the determinant of a matrix.

### Modified Subspace Constrained Mean Shift Algorithm

- Computer ScienceJ. Classif.
- 2021

It is shown that the modification of the MS algorithm guarantees its convergence and also implies the convergence of different variations of the SCMS algorithm, which together estimate the underlying low-dimensional principal curve, embedded in a high-dimensional space.

### Manifold unwrapping using density ridges

- Mathematics
- 2016

Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and…

### Curve Estimation Based on Localised Principal Components - Theory and Applications

- Mathematics
- 2011

In this work, basic theory and some proposed developments to localised principal components and curves are introduced. In addition, some areas of application for local principal curves are explored.…

### On principal curves with a length constraint

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2020

Principal curves are defined as parametric curves passing through the ``middle'' of a probability distribution in R^d. In addition to the original definition based on self-consistency, several points…

### Principal Polynomial Analysis

- Computer ScienceInt. J. Neural Syst.
- 2014

The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines, which makes it computationally feasible and robust.

### Incremental algorithm for finding principal curves

- Computer ScienceIET Signal Process.
- 2015

An incremental version of the SCMS algorithm that trains using a sequence of observations is proposed that shows the effectiveness of the proposed algorithm to find a principal curve using a stream of observations.

### Machine Learning using Principal Manifolds and Mode Seeking

- Computer Science
- 2016

This thesis presents a fast and exact kernel density derivative estimator, a novel algorithm for manifold unwrapping based on tracing the gradient flow along a manifold estimated using density derivatives, and a novel framework for robust mode seeking based on ensemble clustering and resampling techniques.

## References

SHOWING 1-10 OF 149 REFERENCES

### Principal Curves and Surfaces

- Mathematics
- 1984

Principal curves are smooth one dimensional curves that pass through the middle of a p dimensional data set. They minimise the distance from the points, and provide a non-linear summary of the data.…

### Geometric Properties of Principal Curves in the Plane

- Mathematics
- 1996

Principal curves were introduced to formalize the notion of “a curve passing through the middle of a dataset”. Vaguely speaking, a curve is said to pass through the middle of a dataset if every point…

### A Unified Model for Probabilistic Principal Surfaces

- MathematicsIEEE Trans. Pattern Anal. Mach. Intell.
- 2001

A unified covariance model is introduced that implements the probabilistic principal surface (PPS), and it is shown in two different comparisons that the PPS outperforms the GTM under identical parameter settings.

### Principal curves and principal oriented points

- Mathematics
- 1998

Principal curves have been defined Hastie and Stuetzle (JASA, 1989) as smooth curves passing through the middle of a multidimensional data set. They are nonlinear generalizations of the first…

### Principal Curves

- Mathematics
- 2007

Principal curves are smooth one-dimensional curves that pass through the middle of a p-dimensional data set, providing a nonlinear summary of the data. They are nonparametric, and their shape is…

### Learning and Design of Principal Curves

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 2000

This work defines principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution, making it possible to theoretically analyze principal curve learning from training data and it also leads to a new practical construction.

### Extremal properties of principal curves in the plane

- Mathematics
- 1996

Principal curves were introduced to formalize the notion of "a curve passing through the middle of a dataset." Vaguely speaking, a curve is said to pass through the middle of a dataset if every point…

### Principal curves: learning, design, and applications

- Computer Science, Mathematics
- 2000

The main result here is the first known consistency proof of a principal curve estimation scheme, and an application of the polygonal line algorithm to hand-written character skeletonization.

### Principal curves with bounded turn

- Mathematics2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060)
- 2000

The existence of principal curves with bounded length for any distribution that satisfies some minimal regularity conditions has been shown and a learning algorithm for them is presented.

### Manifold Parzen Windows

- Computer ScienceNIPS
- 2002

A new non-parametric kernel density estimation method which captures the local structure of an underlying manifold through the leading eigenvectors of regularized local covariance matrices, yielding classification rates similar to SVMs and much superior to the Parzen classifier.