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:class:`PCA` with Kernel PCA ---------- -:class:`KernelPCA` is an extension of PCA which achieves non-linear -dimensionality reduction through the use of kernels (see :ref:`metrics`). It -has many applications including denoising, compression and structured +:class:`KernelPCA` is an extension of PCA which achieves non-linear dimensionality +reduction by first projecting the data :math:`X` ``(n_samples,n_features)`` into a higher +dimensional space using :math:`\phi(X)` ``(n_samples,m_features)`` before finding the projection axes. + +Although this might sound computationally expensive (especially when ``m_features >> n_features``), +it is possible to make this technique feasible by using kernel functions (see :ref:`metrics`) and the so-called kernel +trick which allow us to model those higher dimensional space without having to compute the coordinates +of each points. + +It is interesting to use :class:`KernelPCA` when dealing with non-linear datasets. +In fact linear subspaces might not adequately represent the underlying manifold if the dataset has some +non linear structure. + +Here in this example, we have in the upper-left a 2-d surface embedded in a 3-d space. + + +.. image:: images/pca_vs_manifold.png + :align: center + :scale: 75% + +Applying linear dimensionality reduction (i.e projecting all data points onto a hyperplane) induces a high +loss of information (top right figure). +In contrast, non-linear approaches are capable of modeling and unfolding the manifold to represent the data adequately. + +Kernel approaches rely on the intuition that many datasets which are not linearly separable in their original +attributes can be made so by projecting them into a higher dimensional space. +It is famously used with Support Vector Machines. + +Recall that in PCA, we define the projection matrix as the eigenvectors of the centered covariance +matrix associated with the highest eigenvalues. Equivalently, SVD (see :class:`TruncatedSVD`) can be used +where the projection matrix becomes the left singular vectors of :math:`U` associated with the highest +singular values of :math:`\Sigma`. The latter approach proves to be more efficient when dealing with large datasets. + +Here, instead of using :math:`X` ``(n_features,n_samples)``, we use :math:`\phi(X)` which is an embedding +of the data into a higher dimensional space (``(m_features,n_samples)`` where ``m_features >> n_features)``). We want to avoid computing +the latter as it becomes infeasible for large values of ``m`` : This is where the kernel trick is used. +We define and use a kernel function :math:`k(.,.)` (see :ref:`metrics`) which enables us to operate in +higher-dimensional space without ever computing the coordinates :math:`\phi(X)` of the data. +Instead, we simply compute the inner products :math:`\phi(X)^T \phi(X)` between all pairs of data points projected in the feature space. +This means that we can implicitly model any non linear transformation :math:`\phi(X)` as long as we can +calculate the dot product between the data points. + +Therefore, instead of computing the covariance matrix :math:`\phi(X)\phi(X)^T` ``(m_features,m_features``, we use the kernel matrix +:math:`K` ``(n_samples,n_samples)`` associated with the kernel function :math:`k(.,.)` which contains all pairwise evaluations +of the dot product. Just like for PCA, the matrix :math:`K` has to be centered before solving the eigenproblem : + +.. math:: + + K \alpha_{i} = \lambda_{i} \alpha_{i} + +whose eigenvectors :math:`\alpha_{i}` ``(n_samples,)`` give us +our new projection axes. All that's left is to pick a number of components ``n_components`` just like in PCA. +Each data point :math:`x` can then be projected by computing its coordinates for each new dimension :math:`j` : + +.. math:: + + y_{j} = \sum\limits_{i=1}^{n_{samples}} \alpha_{ij} k(x_{i},x) \qquad \text{for }j = 1 , \dots, n_{components} + +.. note:: + + Because we’re using the kernel matrix :math:`K` ``(n_samples,n_samples)`` instead of the covariance matrix :math:`C` ``(n_features,n_features)``, + KPCA scales with the number of data points ``n_samples`` whereas + PCA does so with the number of input features ``n_features``. + +It has many applications including denoising, compression and structured prediction (kernel dependency estimation). :class:`KernelPCA` supports both ``transform`` and ``inverse_transform``. @@ -188,6 +250,8 @@ prediction (kernel dependency estimation). :class:`KernelPCA` supports both :align: center :scale: 75% + + .. topic:: Examples: * :ref:`sphx_glr_auto_examples_decomposition_plot_kernel_pca.py`