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### Kernel PCA: Find most important variables for each PC

To find the most important variable for each Principal Component is easy with PCA: With data->X and variables->variable_names ...
18 views

### Denoising and pre-images in Kernel PCA

In "Pattern Recognition and Machine Learning" by Bishop, the following problem about Kernel PCA is laid out : In linear PCA, we can approximate data points by projecting them onto the $L < D$-...
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### Transforming the Kernel principal components to original space

According to my understanding, we obtain the kernel/gram matrix eigenvectors/values in kernel PCA. We can use the kernel matrix for transforming the data however is there a way to transform those ...
67 views

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### Reference request about feature maps in ML

Can someone kindly link to some recent papers on understanding feature maps in ML? It would help to get an idea of what are the recent issues there that people have been working on with regards to ...
93 views

### Does it make sense to do PCA before kernel regression?

I have a set of features extracted from the same samples and I'm learning a kernel ridge regression. Now, especially for feature fusion, reducing the number of features before combining them seems ...
2k views

### What's the physical meaning of the eigenvectors of the Gram/Kernel matrix?

If we have some centered dataset $X$ then the eigenvectors of $X^TX$ represent the principal components of the dataset, and their physical meaning is the directions that data follow in the original ...
651 views

### Kernel PCA increases dimensionality compared with PCA?

I was trying to use sklearn to perform kernel PCA with 28*28 = 784 dims data. At first I used PCA to reduce dimensionality and I chose to reduce to k dimensions where k could explain 95% of the ...
3k views

### Projecting to lower/higher-dimensional space for classification: dimensionality reduction vs kernel trick

Whilst learning about classification, I have seen two different arguments. One is that projecting the data to a lower-dimensional space, such as with PCA, makes the data more easily separable. The ...
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### Why can kernel PCA with Gaussian kernel separate half-moon shapes and concentric circles but not Swiss Roll?

According to this website, kernel PCA with RBF (Gaussian) kernel can separate half-moon shapes and concentric circles effectively but not Swiss Roll shapes (in 3-D). I don't understand why it doesn't ...
261 views

### How to compare PCA with KPCA for dimension reduction?

Both linear principal component analysis (PCA) and kernel principal component analysis (KPCA) are unsupervised dimension reduction methods. I have a dataset with $4000$ training samples and $40000$ ...
598 views

### PCA vs. Spectral Clustering with Linear Kernel

Consider a feature vector matrix $X := [x_1 x_2 \dots x_d] \in \mathbb {R}^{n\times d}$ that I hope to use as part of some supervised learning procedure, say, regression. Suppose that also, $d \gg n$...
220 views

### Kernel PCA vs principal curve analysis

Both principal curve analysis and kernel PCA provide the ability to find nonlinear PCA. Kernel PCA does this by finding principal components in a higher dimensional space. Principal curve analysis is ...
504 views

### Kernel PCA for feature selection for various machine learning algorithms [duplicate]

I would like to forecast stock index returns with SVM, k-NN, and Neural Networks. In advance I want to select my inputs via kernel PCA (kPCA). Everything is performed in R. For the KPCA I use ...
1k views

### A problem with kernel-PCA implementation

Linear PCA and kPCA with linear kernel should produce exactly the same results ( good explanation is in this post ). As I am learning to use PCA family methods I try to write my own functions ...
313 views

### Generalization error of PCA and kernel PCA

I've been recently reading Shawe-Taylor et al. 2005, On the Eigenspectrum of the Gram Matrix and the Generalization Error of Kernel PCA, where the authors analyze the squared residual of kernel ...
2k views

### Kernel PCA and classification

I need to perform kernel PCA on the colon-­‐cancer dataset and then I need to plot number of principal components vs classification accuracy with PCA data. For the first part I am ...
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### How to choose a kernel for kernel PCA?

What are the ways to choose what kernel would result in good data separation in the final data output by kernel PCA (principal component analysis), and what are the ways to optimize parameters of the ...
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### What exactly is the procedure to compute principal components in kernel PCA?

In kernel PCA (principal component analysis) you first choose a desired kernel, use it to find your $K$ matrix, center the feature space via the $K$ matrix, find its eigenvalues and eigenvectors, then ...
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### Integrating length for input-space feature PC projections in kernel PCA

I read a paper detailing the algebraic process of kernel PCA. I have question though: the paper details the projection of new points onto the new eigenvectors in the feature space, but what if I want ...
36k views

### What makes the Gaussian kernel so magical for PCA, and also in general?

I was reading about kernel PCA (1, 2, 3) with Gaussian and polynomial kernels. How does the Gaussian kernel separate seemingly any sort of nonlinear data exceptionally well? Please give an intuitive ...
157 views

### How to fit a single quadratic term to a regression

I have a high dimensional multivariate model and am fitting linear weights to each of the $N$ free variables using a classic stable SVD matrix solver. This works. I want to improve the fit by using a ...
6k views

### How to project a new vector onto the PC space using kernel PCA?

Let $X_{N \times d}$ be the data matrix, where $N$ is the number of samples and $d$ the size of the features space. Using kernel PCA (kPCA), one first computes a kernel matrix $K_{N \times N}$, and ...
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### Non-decaying eigenvalues in Kernel PCA with small kernel width

I noticed that when I use a small width kernel (RBF) with PCA, I get my desired result (clustering in this case), but I do not get a decay in the eigenvalues (they stay about the same value). Is that ...
759 views

### Which PCA (or kernel PCA) basis better describes a single test sample?

I have two PCA bases obtained by decomposition of two groups of training data. I also have some samples of test data. How can I decide which PCA basis fits better each test sample? I tried to ...
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### How to apply a Gaussian radial basis function kernel PCA to nonlinear data?

I have an assignment to implement a Gaussian radial basis function-kernel principal component analysis (RBF-kernel PCA) and have some challenges here. It would be great if someone could point me to ...
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### Are eigenvectors obtained in Kernel PCA orthogonal?

As Kernel PCA is the same as PCA in higher dimension space, shouldn't the eigenvectors obtained be orthogonal? Suppose, I have $n$ data points and let $a$ and $b$ be two eigenvectors of covariance ...
8k views

### Is Kernel PCA with linear kernel equivalent to standard PCA?

If in kernel PCA I choose a linear kernel $K(\mathbf{x},\mathbf{y}) = \mathbf x^\top \mathbf y$, is the result going to be different from the ordinary linear PCA? Are the solutions fundamentally ...
15k views

### What are the advantages of kernel PCA over standard PCA?

I want to implement an algorithm in a paper which uses kernel SVD to decompose a data matrix. So I have been reading materials about kernel methods and kernel PCA etc. But it still is very obscure to ...
2k views

### How to select a number of components to retain in kernel PCA?

I'm using kpca function from kernlab and try to get the proportion of variance explained by each component as in standard PCA. I ...
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### Can one use eigenvalues to choose a number of components to retain in kernel PCA?

When using Kernel PCA for dimensionality reduction, is there any simple criterion which can be used to determine the number of components to use? I am using Kernel PCA with linear kernel, which would ...
550 views

### Kernel PCA with an SVD algo

Suppose that I have a great algo for calculating the SVD and I want to do Kernel PCA. It is possible to first apply the Kernel function to my data and then run the SVD algo on the transformed data?