Visible to Intel only — GUID: GUID-F356EC66-0299-4E03-8629-81B5E57620C1
basic_statistics_dense_batch.cpp
column_accessor_homogen.cpp
cor_dense_batch.cpp
cov_dense_batch.cpp
dbscan_brute_force_batch.cpp
df_cls_hist_batch.cpp
df_cls_traverse_model.cpp
df_reg_hist_batch.cpp
df_reg_traverse_model.cpp
kmeans_init_dense.cpp
kmeans_lloyd_dense_batch.cpp
knn_cls_brute_force_dense_batch.cpp
knn_reg_brute_force_dense_batch.cpp
knn_search_brute_force_dense_batch.cpp
linear_kernel_dense_batch.cpp
linear_regression_dense_batch.cpp
pca_cor_dense_batch.cpp
pca_precomputed_cor_dense_batch.cpp
pca_precomputed_cov_dense_batch.cpp
rbf_kernel_dense_batch.cpp
svm_two_class_thunder_dense_batch.cpp
basic_statistics_dense_batch.cpp
column_accessor_homogen.cpp
connected_components_batch.cpp
cor_dense_batch.cpp
cov_dense_batch.cpp
dbscan_brute_force_batch.cpp
df_cls_dense_batch.cpp
df_reg_dense_batch.cpp
directed_graph.cpp
graph_service_functions.cpp
jaccard_batch.cpp
jaccard_batch_app.cpp
kmeans_init_dense.cpp
kmeans_lloyd_dense_batch.cpp
knn_cls_brute_force_dense_batch.cpp
knn_cls_kd_tree_dense_batch.cpp
knn_search_brute_force_dense_batch.cpp
linear_kernel_dense_batch.cpp
linear_regression_dense_batch.cpp
louvain_batch.cpp
pca_dense_batch.cpp
pca_precomputed_dense_batch.cpp
polynomial_kernel_dense_batch.cpp
rbf_kernel_dense_batch.cpp
shortest_paths_batch.cpp
sigmoid_kernel_dense_batch.cpp
subgraph_isomorphism_batch.cpp
svm_multi_class_thunder_dense_batch.cpp
svm_nu_cls_thunder_dense_batch.cpp
svm_nu_reg_thunder_dense_batch.cpp
svm_reg_thunder_dense_batch.cpp
svm_two_class_smo_dense_batch.cpp
svm_two_class_thunder_dense_batch.cpp
triangle_counting_batch.cpp
K-Means Clustering
Density-Based Spatial Clustering of Applications with Noise
Correlation and Variance-Covariance Matrices
Principal Component Analysis
Principal Components Analysis Transform
Singular Value Decomposition
Association Rules
Kernel Functions
Expectation-Maximization
Cholesky Decomposition
QR Decomposition
Outlier Detection
Distance Matrix
Distributions
Engines
Moments of Low Order
Quantile
Quality Metrics
Sorting
Normalization
Optimization Solvers
Decision Forest
Decision Trees
Gradient Boosted Trees
Stump
Linear and Ridge Regressions
LASSO and Elastic Net Regressions
k-Nearest Neighbors (kNN) Classifier
Implicit Alternating Least Squares
Logistic Regression
Naïve Bayes Classifier
Support Vector Machine Classifier
Multi-class Classifier
Boosting
Visible to Intel only — GUID: GUID-F356EC66-0299-4E03-8629-81B5E57620C1
Principal Components Analysis (PCA)
Principal Component Analysis (PCA) is an algorithm for exploratory data analysis and dimensionality reduction. PCA transforms a set of feature vectors of possibly correlated features to a new set of uncorrelated features, called principal components. Principal components are the directions of the largest variance, that is, the directions where the data is mostly spread out.
Operation |
Computational methods |
Programming Interface |
|||
Mathematical formulation
Training
Given the training set 
of p-dimensional feature vectors and the number of principal components r, the problem is to compute r principal directions (p-dimensional eigenvectors [Lang87]) for the training set. The eigenvectors can be grouped into the 
matrix T that contains one eigenvector in each row.
Training method: Covariance
This method uses eigenvalue decomposition of the covariance matrix to compute the principal components of the datasets. The method relies on the following steps:
Computation of the covariance matrix
Computation of the eigenvectors and eigenvalues
Formation of the matrices storing the results
Covariance matrix computation is performed in the following way:
Compute the vector-column of sums
.Compute the cross-product
.Compute the covariance matrix
.
To compute eigenvalues 
and eigenvectors 
, the implementer can choose an arbitrary method such as [Ping14].
The final step is to sort the set of pairs 
in the descending order by and form the resulting matrix 
. Additionally, the means and variances of the initial dataset are returned.
Training method: SVD
This method uses singular value decomposition of the dataset to compute its principal components. The method relies on the following steps:
Computation of the singular values and singular vectors
Formation of the matrices storing the results
To compute singular values and singular vectors 
and 
, the implementer can choose an arbitrary method such as [Demmel90].
The final step is to sort the set of pairs 
in the descending order by and form the resulting matrix 
. Additionally, the means and variances of the initial dataset are returned.
Sign-flip technique
Eigenvectors computed by some eigenvalue solvers are not uniquely defined due to sign ambiguity. To get the deterministic result, a sign-flip technique should be applied. One of the sign-flip techniques proposed in [Bro07] requires the following modification of matrix T:
where 
is i-th row, 
is the element in the i-th row and j-th column, 
is the signum function,
Inference
Given the inference set 
of p-dimensional feature vectors and the matrix T produced at the training stage, the problem is to transform 
to the set 
, where 
is an r-dimensional feature vector, 
.
The feature vector is computed through applying linear transformation [Lang87] defined by the matrix T to the feature vector 
,
Inference methods: Covariance and SVD
Covariance and SVD inference methods compute according to (1).
Programming Interface
Distributed mode
The algorithm supports distributed execution in SMPD mode (only on GPU).
Usage example
Training
pca::model<> run_training(const table& data) {
const auto pca_desc = pca::descriptor<float>{}
.set_component_count(5)
.set_deterministic(true);
const auto result = train(pca_desc, data);
print_table("means", result.get_means());
print_table("variances", result.get_variances());
print_table("eigenvalues", result.get_eigenvalues());
print_table("eigenvectors", result.get_eigenvectors());
return result.get_model();
}
Inference
table run_inference(const pca::model<>& model,
const table& new_data) {
const auto pca_desc = pca::descriptor<float>{}
.set_component_count(model.get_component_count());
const auto result = infer(pca_desc, model, new_data);
print_table("labels", result.get_transformed_data());
}
Examples
oneAPI DPC++
Batch Processing:
dpc_pca_cor_dense_batch.cpp
oneAPI C++
Batch Processing:
cpp_pca_dense_batch.cpp
Python* with DPC++ support
Batch Processing: