# Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis

@article{Mohammed2017ManifoldLU, title={Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis}, author={Kitty Mohammed and Hariharan Narayanan}, journal={arXiv: Statistics Theory}, year={2017} }

We consider the problem of recovering a $d-$dimensional manifold $\mathcal{M} \subset \mathbb{R}^n$ when provided with noiseless samples from $\mathcal{M}$. There are many algorithms (e.g., Isomap) that are used in practice to fit manifolds and thus reduce the dimensionality of a given data set. Ideally, the estimate $\mathcal{M}_\mathrm{put}$ of $\mathcal{M}$ should be an actual manifold of a certain smoothness; furthermore, $\mathcal{M}_\mathrm{put}$ should be arbitrarily close to $\mathcal{M… Expand

#### 12 Citations

Non-Parametric Estimation of Manifolds from Noisy Data

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This work considers the problem of estimating a d dimensional sub-manifold of RD from a finite set of noisy samples and presents an algorithm that takes a point r from the tubular neighborhood and outputs p̂n ∈ RD, and T̂p⩽nM an element in the Grassmanian Gr(d,D). Expand

Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem

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Algorithmic procedures are developed that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and the computational complexity of these procedures are estimated. Expand

Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimension Estimation

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This work presents new approach to manifold hypothesis checking and underlying manifold dimension estimation and uses two very different methods simultaneously — one geometric, another probabilistic — and checks whether they give the same result. Expand

Manifold Fitting under Unbounded Noise

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- ArXiv
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This paper introduces a new manifold-fitting method, by which the output manifold is constructed by directly estimating the tangent spaces at the projected points on the underlying manifold, rather than at the sample points, to decrease the error caused by the noise. Expand

Manifold Fitting in Ambient Space

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This work extends the idea of subsampling to noisy datasets in high dimensional space and utilize the Moving Least Squares approach to approximate the underlying manifold using the Laplace-Beltrami operator and the MLS approach. Expand

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- Computer Science, Mathematics
- ArXiv
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A method to explain embedding coordinates on a manifold as compositions of functions from a user-defined dictionary is proposed, which can be set up as a sparse Group Lasso recovery problem, find sufficient recovery conditions, and demonstrate its effectiveness on data. Expand

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- 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)
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This work presents a new generic algorithm for unfolding manifolds that have been estimated by local linear approximations and is a combination of ideas from principal curves and density ridge estimation and tools from classical differential geometry. Expand

Fitting a Putative Manifold to Noisy Data

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In the present work, we give a solution to the following question from manifold learning. Suppose data belonging to a high dimensional Euclidean space is drawn independently, identically distributed… Expand

Manifold Coordinates with Physical Meaning

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This paper proposes a method to explain embedding coordinates of a manifold as non-linear compositions of functions from a user-defined dictionary, and shows that it can be set up as a sparse linear Group Lasso recovery problem, find sufficient recovery conditions, and demonstrate its effectiveness on data. Expand

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A theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori is presented, and a convolution operator for a manifold of arbitrary topology and dimension is defined. Expand

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