Remarkably, the fast rate almost coincides with the minimax rate of nonparametric regression. The validity of our deep nonlinear sufficient dimension reduction methods is demonstrated through simulations and real data analysis.Chen, YinFengEast China Normal UnivJiao, YuLingWuhan UnivQiu, RuiHu, ZhouThe Annals of Statistics: An Official Journal of...
On the other hand, nonlinear dimension reduction techniques like t-SNE and UMAP, have been shown to be effective in a variety of applications and are commonly used in single-cell data processing [146]. These methods also have some drawbacks, such as lacking robustness to random sampling, ...
Nonlinear dimensionality reduction (NLDR) plays an important role in feature extraction and visualization of high dimensional data. It transforms patterns of interest, or manifolds, in data to regions in a lower-dimensional latent space. When the manifolds are highly complicated, the NLDR transformati...
Autoencoder based data-driven modeling approach is proposed for nonlinear materials. • Autoencoders enable noise filtering and dimensionality reduction of material data. • Convexity-preserving interpolation is employed for enhanced stability in data search. • Improved generalization capability is demo...
If a section is not properly executed, make sure that all required fields are populated to execute a particular step. If you have not given sufficient information, the module will not proceed. It is also important to ensure that only the required fields for a particular section are populated....
2. The LLTSA algorithm takes the global and local structures of the dataset fully into account, which enables better clustering of irregular and inhomogeneous nonlinear data after performing dimensionality reduction. Consequently, LLTSA is chosen for feature dimensionality reduction in this paper. To ...
(2018). More precisely: we use the dimensionality reduction described in their article and solve the remaining auxiliary problem using a ten- sorised Gauss-Hermite quadrature. Due to the smoothness of the auxiliary problem as small number of 9 quadrature points per dimension is sufficient for a ...
Each neuron is a function for performing two mathematical operations for one or more inputs x, then performing a nonlinear mapping \(y=f\left( \sum _{i} w_{i} x_{i}+b\right)\), and finally outputs y, where w and b are trainable parameters, referred to as weights and biases. f...
When input data is highly nonlinear, more hidden layers are required to deal with this complexity. They are mostly used in dimensionality reduction, like a non linear Principal Component Analysis. From a more mathematical approach, the encoder takes a given input x and transforms it into a ...
Deep Neural Networks (DNNs), also called convolutional networks, are composed of multiple levels of nonlinear operations, such as neural nets with many hidden layers (Bengio et al., 2007; Krizhevsky et al., 2012). Deep learning methods aim at learning feature hierarchies, where features at hi...