.. ****************************************************************************** .. * Copyright 2019 Intel Corporation .. * .. * Licensed under the Apache License, Version 2.0 (the "License"); .. * you may not use this file except in compliance with the License. .. * You may obtain a copy of the License at .. * .. * http://www.apache.org/licenses/LICENSE-2.0 .. * .. * Unless required by applicable law or agreed to in writing, software .. * distributed under the License is distributed on an "AS IS" BASIS, .. * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. .. * See the License for the specific language governing permissions and .. * limitations under the License. .. *******************************************************************************/ .. _kernel: Kernel Functions ---------------- .. note:: Kernel functions are also available with oneAPI interfaces: - :ref:`alg_linear_kernel` - :ref:`alg_rbf_kernel` .. toctree:: :glob: :maxdepth: 4 Kernel functions form a class of algorithms for pattern analysis. The main characteristic of kernel functions is a distinct approach to this problem. Instead of reducing the dimension of the original data, kernel functions map the data into higher-dimensional spaces in order to make the data more easily separable there. Linear Kernel ============= A linear kernel is the simplest kernel function. Problem Statement ***************** Given a set :math:`X` of :math:`n` feature vectors :math:`x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})` of dimension :math:`p` and a set :math:`Y` of :math:`m` feature vectors :math:`y_1 = (y_{11}, \ldots, y_{1p}), \ldots, y_m = (y_{m1}, \ldots, x_{mp})`, the problem is to compute the linear kernel function :math:`K(x_i,, y_i)` for any pair of input vectors: :math:`K(x_i, y_i) = k {X_i}^T y_i + b`. Batch Processing **************** Algorithm Input +++++++++++++++ The linear kernel function accepts the input described below. Pass the ``Input ID`` as a parameter to the methods that provide input for your algorithm. .. tabularcolumns:: |\Y{0.2}|\Y{0.8}| .. list-table:: Algorithm Input for Linear Kernel (Batch Processing) :header-rows: 1 :align: left :widths: 10 60 :class: longtable * - Input ID - Input * - X - Pointer to the :math:`n \times p` numeric table that represents the matrix X. This table can be an object of any class derived from NumericTable. * - Y - Pointer to the :math:`m \times p` numeric table that represents the matrix Y. This table can be an object of any class derived from NumericTable. Algorithm Parameters ++++++++++++++++++++ The linear kernel function has the following parameters: .. tabularcolumns:: |\Y{0.15}|\Y{0.15}|\Y{0.7}| .. list-table:: Algorithm Parameters for Linear Kernel (Batch Processing) :header-rows: 1 :align: left :widths: 10 10 60 :class: longtable * - Parameter - Default Value - Description * - ``algorithmFPType`` - ``float`` - The floating-point type that the algorithm uses for intermediate computations. Can be ``float`` or ``double``. * - ``method`` - ``defaultDense`` - Available computation methods: + ``defaultDense`` - default performance-oriented method + ``fastCSR`` - performance-oriented method for CSR numeric tables * - ``computationMode`` - ``matrixMatrix`` - Computation mode for the kernel function. Can be: For CPU: + ``vectorVector`` - compute the kernel function for given feature vectors :math:`x_i` and :math:`y_j` + ``matrixVector`` - compute the kernel function for all vectors in the set :math:`X` and a given feature vector :math:`y_j` + ``matrixMatrix`` - compute the kernel function for all vectors in the sets :math:`X` and :math:`Y`. In |product|, this mode requires equal numbers of observations in both input tables: :math:`n = m`. For GPU: + ``matrixMatrix`` - compute the kernel function for all vectors in the sets :math:`X` and :math:`Y`. In |product|, this mode requires equal numbers of observations in both input tables: :math:`n = m`. * - ``rowIndexX`` - :math:`0` - Index i of the vector in the set :math:`X` for the ``vectorVector`` computation mode. * - ``rowIndexY`` - :math:`0` - Index :math:`j` of the vector in the set :math:`Y` for the ``vectorVector`` or ``matrixVector`` computation mode. * - ``rowIndexResult`` - :math:`0` - Row index in the values numeric table to locate the result of the computation for the ``vectorVector`` computation mode. * - :math:`k` - :math:`1.0` - The coefficient :math:`k` of the linear kernel. * - :math:`b` - :math:`0.0` - The coefficient :math:`b` of the linear kernel. Algorithm Output ++++++++++++++++ The linear kernel function calculates the results described below. Pass the ``Result ID`` as a parameter to the methods that access the results of your algorithm. .. tabularcolumns:: |\Y{0.2}|\Y{0.8}| .. list-table:: Algorithm Output for Linear Kernel (Batch Processing) :header-rows: 1 :align: left :widths: 10 60 * - Result ID - Result * - ``values`` - Pointer to the :math:`n \times m` numeric table with the values of the kernel function. .. note:: By default, this result is an object of the ``HomogenNumericTable`` class, but you can define the result as an object of any class derived from ``NumericTable`` except ``PackedSymmetricMatrix``, ``PackedTriangularMatrix``, and ``CSRNumericTable``. Examples ++++++++ .. tabs:: .. tab:: oneAPI DPC++ Batch Processing: - :ref:`dpc_linear_kernel_dense_batch.cpp` .. tab:: oneAPI C++ Batch Processing: - :ref:`cpp_linear_kernel_dense_batch.cpp` .. tab:: C++ (CPU) Batch Processing: - :cpp_example:`kernel_func_lin_dense_batch.cpp ` - :cpp_example:`kernel_func_lin_csr_batch.cpp ` .. Python*: .. - kernel_func_lin_dense_batch.py .. - kernel_func_lin_csr_batch.py Radial Basis Function Kernel ============================ The Radial Basis Function (RBF) kernel is a popular kernel function used in kernelized learning algorithms. Problem Statement ***************** Given a set :math:`X` of :math:`n` feature vectors :math:`x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})` of dimension :math:`p` and a set :math:`Y` of :math:`m` feature vectors :math:`y_1 = (y_{11}, \ldots, y_{1p}), \ldots, y_m = (y_{m1}, \ldots, x_{mp})`, the problem is to compute the RBF kernel function :math:`K(x_i,, y_i)` for any pair of input vectors: .. math:: K\left({x}_{i},{y}_{j}\right)=exp\left(-\frac{{\left(\|{x}_{i}-{y}_{j}\|\right)}^{2}}{2{\sigma }^{2}}\right) Batch Processing **************** Algorithm Input +++++++++++++++ The RBF kernel accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. .. tabularcolumns:: |\Y{0.2}|\Y{0.8}| .. list-table:: Algorithm Input for Radial Basis Function Kernel (Batch Processing) :header-rows: 1 :align: left :widths: 10 60 :class: longtable * - Input ID - Input * - :math:`X` - Pointer to the :math:`n \times p` numeric table that represents the matrix :math:`X`. This table can be an object of any class derived from ``NumericTable``. * - :math:`Y` - Pointer to the :math:`m \times p` numeric table that represents the matrix :math:`Y`. This table can be an object of any class derived from ``NumericTable``. Algorithm Parameters ++++++++++++++++++++ The RBF kernel has the following parameters: .. tabularcolumns:: |\Y{0.15}|\Y{0.15}|\Y{0.7}| .. list-table:: Algorithm Parameters for Radial Basis Function Kernel (Batch Processing) :header-rows: 1 :align: left :widths: 10 10 60 :class: longtable * - Parameter - Default Value - Description * - ``algorithmFPType`` - ``float`` - The floating-point type that the algorithm uses for intermediate computations. Can be ``float`` or ``double``. * - ``method`` - ``defaultDense`` - Available computation methods: + ``defaultDense`` - default performance-oriented method + ``fastCSR`` - performance-oriented method for CSR numeric tables * - ``computationMode`` - ``matrixMatrix`` - Computation mode for the kernel function. Can be: For CPU: + ``vectorVector`` - compute the kernel function for given feature vectors :math:`x_i` and :math:`y_j` + ``matrixVector`` - compute the kernel function for all vectors in the set :math:`X` and a given feature vector :math:`y_j` + ``matrixMatrix`` - compute the kernel function for all vectors in the sets :math:`X` and :math:`Y`. In |product|, this mode requires equal numbers of observations in both input tables: :math:`n = m`. For GPU: + ``matrixMatrix`` - compute the kernel function for all vectors in the sets :math:`X` and :math:`Y`. In |product|, this mode requires equal numbers of observations in both input tables: :math:`n = m`. * - ``rowIndexX`` - :math:`0` - Index :math:`i` of the vector in the set :math:`X` for the ``vectorVector`` computation mode. * - ``rowIndexY`` - :math:`0` - Index :math:`j` of the vector in the set :math:`Y` for the ``vectorVector`` or ``matrixVector`` computation mode. * - ``rowIndexResult`` - :math:`0` - Row index in the values numeric table to locate the result of the computation for the ``vectorVector`` computation mode. * - ``sigma`` - :math:`1.0` - The coefficient :math:`\sigma` of the RBF kernel. Algorithm Output ++++++++++++++++ The RBF kernel calculates the results described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm. .. tabularcolumns:: |\Y{0.2}|\Y{0.8}| .. list-table:: Algorithm Output for Radial Basis Function Kernel (Batch Processing) :header-rows: 1 :align: left :widths: 10 60 * - Result ID - Result * - ``values`` - Pointer to the :math:`n \times m` numeric table with the values of the kernel function. .. note:: By default, this result is an object of the ``HomogenNumericTable`` class, but you can define the result as an object of any class derived from ``NumericTable`` except ``PackedSymmetricMatrix``, ``PackedTriangularMatrix``, and ``CSRNumericTable``. Examples ******** .. tabs:: .. tab:: oneAPI DPC++ Batch Processing: - :ref:`dpc_rbf_kernel_dense_batch.cpp` .. tab:: oneAPI C++ Batch Processing: - :ref:`cpp_rbf_kernel_dense_batch.cpp` .. tab:: C++ (CPU) Batch Processing: - :cpp_example:`kernel_func_rbf_dense_batch.cpp ` - :cpp_example:`kernel_func_rbf_csr_batch.cpp ` .. Python*: .. - kernel_func_rbf_dense_batch.py .. - kernel_func_rbf_csr_batch.py