Nonlinear principal component analysis (NLPCA) is commonly seen as a nonlinear generalization of standard principal component analysis (PCA). It generalizes the principal components from straight lines to curves (nonlinear). Thus, the subspace in the original data space which is described by all nonlinear components is also curved.
Nonlinear PCA can be achieved by using a neural network with an autoassociative architecture also known as autoencoder, replicator network, bottleneck or sandglass type network. Such autoassociative neural network is a multi-layer perceptron that performs an identity mapping, meaning that the output of the network is required to be identical to the input. However, in the middle of the network is a layer that works as a bottleneck in which a reduction of the dimension of the data is enforced. This bottleneck-layer provides the desired component values (scores).
Nonlinear Principal Component Analysis is a simple algorithm that uses this nonlinear dimensionality reduction for face recognition. This approach does not require the detection of any reference point and it can be used for real-time applications.
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In order to reduce dimensionality and to preserve the local features, a principal component analysis (PCA) method is proposed. For a dataset of m observations of n features, PCA is an unsupervised learning technique that describes the data set as a linear combination of a set of orthonormal bases. The principal components of the data are extracted by performing an eigen decomposition of the covariance matrix of the dataset. The first principal component has the largest eigenvalue, and the principal components are computed in descending order. The covariance matrix of the dataset is given by C = [ Σ ′ Σ ε
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This library is based on the keymacro algorithm, which is a c++ implementation of the MACRO (Method of Activating Conference) cipher (MACE) (Brown, D. G., McArdle, J., and van der Meulen, J. V. “A. M.”), which is a block cipher (in the sense that it operates on blocks of data) devised to be used in spread spectrum radio applications (e.g., cell phone communications). Keymacro was originally developed by J. V. van der Meulen of Stanford University as part of the development of the S-PACE (Spread Spectrum Positioning System) system.
The MACRO was defined by Brown, et al. in 1983 and received more attention when it was rediscovered by Harada and Tsudik in 1998. Like many other block ciphers (e.g., RC4 and DES), the MACRO was first described in the literature in the form of a ciphertext. It has two ciphertext formats: rectangular and multi-column. For a given plaintext block, a MACRO ciphertext block is obtained by performing three basic operations: add, subtract, and modulo-2. The operation of the MACRO is a block cipher. The output of the MACRO, an encrypted block, depends on the plaintext block, the key, and a specific bit position (“offset”) within the key. Note that the key is a long bit string of fixed length; the key length is equal to the block size and the offset is a constant integer which controls the position within the key.
The operation of the rectangular MACRO is as follows: 1. For each input plaintext block, compute a MACRO ciphertext block by performing a 3-term Fourier series expansion of the plaintext block and adding the Fourier series terms. (For a 3-term Fourier series expansion, use the algorithm described by R. van Stee of Carnegie Mellon University. The offset can be set to zero.) 2. XOR each plaintext block, with its corresponding ciphertext block. 3. XOR each XORed plaintext block with its corresponding XORed ciphertext block. 4. If the offset is non-zero, modulo-2 each of the
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Nonlinear Principal Component Analysis Download 2022 [New]
For a given image I, we want to find a subspace C such that I ~ ( C ) = ∑ j = 1 N I j ~ ( C ) ·
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Nonlinear Principal Component Analysis (NLPCA) is a simple nonlinear dimensionality reduction method. It works by performing the linear projection (PCA) in nonlinear space. The process of PCA is to find a basis in the high-dimensional space that describes the most of the variance in the data. This basis is obtained by projecting the data into a lower dimensional space. In this approach, the manifold is converted into a convex compact domain. Thus, an inverse mapping, which takes the points in the lower dimensional space to the manifold can be calculated. The inverse of this mapping is called Principal Component.
Algorithm
The following is a description of the algorithm:
Procedure: The following algorithm is the nonlinear PCA algorithm that consists of two main steps:
Step 1: A set of training samples are linearly transformed into a feature space of a lower dimension.
Step 2: From the reduced dimension, a basis for the original data is calculated.
Step 1: Extract the input data X.
Step 2: Perform the linear PCA on X.
Step 3: Reduce the dimension of the data by computing the eigenvectors of the covariance matrix and use them as components.
Step 4: If the user needs a smaller number of components, he should repeat the PCA several times (k is the number of times) and retain the largest eigenvalues (singular values) to form a reduced set of components.
Step 5: Reconstruct X to get the input vector.
Generalization of the method to higher dimensions
NLPCA does not reduce data in the real nonlinear space, instead, it works in the tangent space and the space of the gradient. The conversion from the tangent space to the original space is done by mapping the points in the tangent space to the real space through a nonlinear mapping. The conversion of the data from the tangent space to the original space is done by the principal component mapping.
The main principle of the algorithm is to map the data in a lower-dimensional space and then reconstruct it back to the original data space. In contrast to linear PCA, in the NLPCA the nonlinear mapping is used instead of the linear projection.
The following algorithm performs the nonlinear dimensionality reduction:
Procedure: The following algorithm is the nonlinear PCA algorithm that consists of two main steps:
Step 1: A set of training samples are linearly transformed into a feature space of a lower dimension.
Step 2: From the reduced dimension, a basis for the original data is calculated.
Step 1: Extract the input data X.
Step 2: Perform the linear PCA on X.
Step 3: Reduce the dimension of the data by computing the eigenvectors of the covariance matrix and use them
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System Requirements For Nonlinear Principal Component Analysis:
– OS: Win98, Win2000, Win7, Vista, Win8, Win10
– Processor: Intel Pentium III 800 MHz or greater
– Memory: 256 MB or greater
– Hard Disk: 80 MB or greater
Download link (v1.3):
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Demo Version –
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iKnife 2.0 runs on Windows 98, Windows 2000, Windows 7, Windows 8 and Windows 10. It is also supported in Internet Explorer 6, Internet Explorer 8 and Google Chrome
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