geometric_kernels.kernels ========================= .. py:module:: geometric_kernels.kernels .. autoapi-nested-parse:: This module provides the abstract base class for geometric kernels and specialized classes for various types of spaces. Unless you know exactly what you are doing, always use the :class:`MaternGeometricKernel` that "just works". Submodules ---------- .. toctree:: :maxdepth: 1 /autoapi/geometric_kernels/kernels/base/index /autoapi/geometric_kernels/kernels/feature_map/index /autoapi/geometric_kernels/kernels/karhunen_loeve/index /autoapi/geometric_kernels/kernels/matern_kernel/index /autoapi/geometric_kernels/kernels/product/index Package Contents ---------------- .. py:class:: BaseGeometricKernel(space) Bases: :py:obj:`abc.ABC` Abstract base class for geometric kernels. :param space: The space on which the kernel is defined. .. py:method:: K(params, X, X2 = None, **kwargs) :abstractmethod: Compute the cross-covariance matrix between two batches of vectors of inputs, or batches of matrices of inputs, depending on the space. :param params: A dict of kernel parameters, typically containing two keys: `"lengthscale"` for length scale and `"nu"` for smoothness. The types of values in the params dict determine the output type and the backend used for the internal computations, see the warning below for more details. .. note:: The values `params["lengthscale"]` and `params["nu"]` are typically (1,)-shaped arrays of the suitable backend. This serves to point at the backend to be used for internal computations. In some cases, for example, when the kernel is :class:`~.kernels.ProductGeometricKernel`, the values of `params` may be (s,)-shaped arrays instead, where `s` is the number of factors. .. note:: Finite values of `params["nu"]` typically correspond to the generalized (geometric) Matérn kernels. Infinite `params["nu"]` typically corresponds to the heat kernel (a.k.a. diffusion kernel, generalized squared exponential kernel, generalized Gaussian kernel, generalized RBF kernel). Although it is often considered to be a separate entity, we treat the heat kernel as a member of the Matérn family, with smoothness parameter equal to infinity. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :param X2: A batch of M inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. `X2=None` sets `X2=X1`. Defaults to None. :return: The N x M cross-covariance matrix. .. warning:: The types of values in the `params` dict determine the backend used for internal computations and the output type. Even if, say, `geometric_kernels.jax` is imported but the values in the `params` dict are NumPy arrays, the output type will be a NumPy array, and NumPy will be used for internal computations. To get a JAX array as an output and use JAX for internal computations, all the values in the `params` dict must be JAX arrays. .. py:method:: K_diag(params, X, **kwargs) :abstractmethod: Returns the diagonal of the covariance matrix `self.K(params, X, X)`, typically in a more efficient way than actually computing the full covariance matrix with `self.K(params, X, X)` and then extracting its diagonal. :param params: Same as for :meth:`~.K`. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :return: The N-dimensional vector representing the diagonal of the covariance matrix `self.K(params, X, X)`. .. py:method:: init_params() :abstractmethod: Initializes the dict of the trainable parameters of the kernel. It typically contains only two keys: `"nu"` and `"lengthscale"`. This dict can be modified and is passed around into such methods as :meth:`~.K` or :meth:`~.K_diag`, as the `params` argument. .. note:: The values in the returned dict are always of the NumPy array type. Thus, if you want to use some other backend for internal computations when calling :meth:`~.K` or :meth:`~.K_diag`, you need to replace the values with the analogs typed as arrays of the desired backend. .. py:property:: space :type: beartype.typing.Union[geometric_kernels.spaces.Space, beartype.typing.List[geometric_kernels.spaces.Space]] The space on which the kernel is defined. .. py:class:: MaternFeatureMapKernel(space, feature_map, key, normalize = True) Bases: :py:obj:`geometric_kernels.kernels.base.BaseGeometricKernel` This class computes a (Matérn) kernel based on a feature map. .. math :: k_{\nu, \kappa}(x, y) = \langle \phi_{\nu, \kappa}(x), \phi_{\nu, \kappa}(y) \rangle_{\mathbb{R}^n} where $\langle \cdot , \cdot \rangle_{\mathbb{R}^n}$ is the standard inner product in $\mathbb{R}^n$ and $\phi_{\nu, \kappa}: X \to \mathbb{R}^n$ is an arbitrary function called *feature map*. We assume that it depends on the smoothness and length scale parameters $\nu$ and $\kappa$, respectively, which makes this kernel specifically Matérn. .. note:: A brief introduction into feature maps and related kernels can be found on :doc:`this page `. Note that the finite-dimensional feature maps this kernel is meant to be used with are, in most cases, some approximations of the intractable infinite-dimensional feature maps. :param space: The space on which the kernel is defined. :param feature_map: A :class:`~.feature_maps.FeatureMap` object that represents an arbitrary function $\phi_{\nu, \kappa}: X \to \mathbb{R}^n$, where $X$ is the `space`, $n$ can be an arbitrary finite integer, and $\nu, \kappa$ are the smoothness and length scale parameters. :param key: Random state, either `np.random.RandomState`, `tf.random.Generator`, `torch.Generator` or `jax.tensor` (which represents a random state). Many feature maps used in the library are randomized, thus requiring a `key` to work. The :class:`MaternFeatureMapKernel` uses this `key` to make them (and thus the kernel) deterministic, applying the utility function :func:`~.make_deterministic` to the pair `feature_map, key`. .. note:: Even if the `feature_map` is deterministic, you need to provide a valid key, although it will essentially be ignored. In the future, we should probably make the `key` parameter optional. :param normalize: This parameter is directly passed on to the `feature_map` as a keyword argument "normalize". If normalize=True, then either $k(x, x) = 1$ for all $x \in X$, or $\int_X k(x, x) d x = 1$, depending on the type of the feature map and on the space $X$. .. note:: For many kernel methods, $k(\cdot, \cdot)$ and $a k(\cdot, \cdot)$ are indistinguishable, whatever the positive constant $a$ is. For these, it makes sense to use normalize=False to save up some computational overhead. For others, like for the Gaussian process regression, the normalization of the kernel might be important. In these cases, you will typically want to set normalize=True. .. py:method:: K(params, X, X2 = None, **kwargs) Compute the cross-covariance matrix between two batches of vectors of inputs, or batches of matrices of inputs, depending on the space. :param params: A dict of kernel parameters, typically containing two keys: `"lengthscale"` for length scale and `"nu"` for smoothness. The types of values in the params dict determine the output type and the backend used for the internal computations, see the warning below for more details. .. note:: The values `params["lengthscale"]` and `params["nu"]` are typically (1,)-shaped arrays of the suitable backend. This serves to point at the backend to be used for internal computations. In some cases, for example, when the kernel is :class:`~.kernels.ProductGeometricKernel`, the values of `params` may be (s,)-shaped arrays instead, where `s` is the number of factors. .. note:: Finite values of `params["nu"]` typically correspond to the generalized (geometric) Matérn kernels. Infinite `params["nu"]` typically corresponds to the heat kernel (a.k.a. diffusion kernel, generalized squared exponential kernel, generalized Gaussian kernel, generalized RBF kernel). Although it is often considered to be a separate entity, we treat the heat kernel as a member of the Matérn family, with smoothness parameter equal to infinity. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :param X2: A batch of M inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. `X2=None` sets `X2=X1`. Defaults to None. :return: The N x M cross-covariance matrix. .. warning:: The types of values in the `params` dict determine the backend used for internal computations and the output type. Even if, say, `geometric_kernels.jax` is imported but the values in the `params` dict are NumPy arrays, the output type will be a NumPy array, and NumPy will be used for internal computations. To get a JAX array as an output and use JAX for internal computations, all the values in the `params` dict must be JAX arrays. .. py:method:: K_diag(params, X, **kwargs) Returns the diagonal of the covariance matrix `self.K(params, X, X)`, typically in a more efficient way than actually computing the full covariance matrix with `self.K(params, X, X)` and then extracting its diagonal. :param params: Same as for :meth:`~.K`. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :return: The N-dimensional vector representing the diagonal of the covariance matrix `self.K(params, X, X)`. .. py:method:: init_params() Initializes the dict of the trainable parameters of the kernel. Returns `dict(nu=np.array([np.inf]), lengthscale=np.array([1.0]))`. This dict can be modified and is passed around into such methods as :meth:`~.K` or :meth:`~.K_diag`, as the `params` argument. .. note:: The values in the returned dict are always of the NumPy array type. Thus, if you want to use some other backend for internal computations when calling :meth:`~.K` or :meth:`~.K_diag`, you need to replace the values with the analogs typed as arrays of the desired backend. .. py:class:: MaternGeometricKernel This class represents a Matérn geometric kernel that "just works". Unless you really know what you are doing, you should always use this kernel class. Upon creation, unpacks into a specific geometric kernel based on the provided space, and, optionally, the associated (approximate) feature map. .. py:method:: __new__(space, num = None, normalize = True, return_feature_map = False, **kwargs) Construct a kernel and (if `return_feature_map` is `True`) a feature map on `space`. .. note:: See :doc:`this page ` for a brief introduction into feature maps. :param space: Space to construct the kernel on. :param num: If provided, controls the "order of approximation" of the kernel. For the discrete spectrum spaces, this means the number of "levels" that go into the truncated series that defines the kernel (for example, these are unique eigenvalues for the :class:`~.spaces.Hypersphere` or eigenvalues with repetitions for the :class:`~.spaces.Graph` or for the :class:`~.spaces.Mesh`). For the non-compact symmetric spaces (:class:`~.spaces.NoncompactSymmetricSpace`), this is the number of random phases used to construct the kernel. If num=None, we use a (hopefully) reasonable default, which is space-dependent. :param normalize: Normalize the kernel (and the feature map). If normalize=True, then either $k(x, x) = 1$ for all $x \in X$, where $X$ is the `space`, or $\int_X k(x, x) d x = 1$, depending on the space. Defaults to True. .. note:: For many kernel methods, $k(\cdot, \cdot)$ and $a k(\cdot, \cdot)$ are indistinguishable, whatever the positive constant $a$ is. For these, it makes sense to use normalize=False to save up some computational overhead. For others, like for the Gaussian process regression, the normalization of the kernel might be important. In these cases, you will typically want to set normalize=True. :param return_feature_map: If `True`, return a feature map (needed e.g. for efficient sampling from Gaussian processes) along with the kernel. Default is False. :param ``**kwargs``: Any additional keyword arguments to be passed to the kernel (like `key`). .. note:: For non-compact symmetric spaces, like :class:`~.spaces.Hyperbolic` or :class:`~.spaces.SymmetricPositiveDefiniteMatrices`, the `key` **must** be provided in ``**kwargs``. .. py:class:: MaternKarhunenLoeveKernel(space, num_levels, normalize = True) Bases: :py:obj:`geometric_kernels.kernels.base.BaseGeometricKernel` This class approximates Matérn kernel by its truncated Mercer decomposition, in terms of the eigenfunctions & eigenvalues of the Laplacian on the space. .. math:: k(x, x') = \sum_{l=0}^{L-1} S(\sqrt\lambda_l) \sum_{s=1}^{d_l} f_{ls}(x) f_{ls}(x'), where $\lambda_l$ and $f_{ls}(\cdot)$ are the eigenvalues and eigenfunctions of the Laplacian such that $\Delta f_{ls} = \lambda_l f_{ls}$, and $S(\cdot)$ is the spectrum of the Matérn kernel. The eigenvalues and eigenfunctions belong to the :class:`~.spaces.DiscreteSpectrumSpace` instance. We denote .. math:: G_l(\cdot, \cdot') = \sum_{s=1}^{d_l} f_{ls}(\cdot) f_{ls}(\cdot') and term the sets $[f_{ls}]_{s=1}^{d_l}$ *"levels"*. For many spaces, like the sphere, we can employ addition theorems to efficiently compute $G_l(\cdot, \cdot')$ without calculating the individual $f_{ls}(\cdot)$. Note that $\lambda_l$ are not required to be unique: it is possible that for some $l,l'$, $\lambda_l = \lambda_{l'}$. In other words, the "levels" do not necessarily correspond to full eigenspaces. A level may even correspond to a single eigenfunction. .. note:: A brief introduction into the theory behind :class:`MaternKarhunenLoeveKernel` can be found in :doc:`this ` & :doc:`this ` documentation pages. :param space: The space to define the kernel upon. :param num_levels: Number of levels to include in the summation. :param normalize: Whether to normalize kernel to have unit average variance. .. py:method:: K(params, X, X2 = None, **kwargs) Compute the cross-covariance matrix between two batches of vectors of inputs, or batches of matrices of inputs, depending on the space. :param params: A dict of kernel parameters, typically containing two keys: `"lengthscale"` for length scale and `"nu"` for smoothness. The types of values in the params dict determine the output type and the backend used for the internal computations, see the warning below for more details. .. note:: The values `params["lengthscale"]` and `params["nu"]` are typically (1,)-shaped arrays of the suitable backend. This serves to point at the backend to be used for internal computations. In some cases, for example, when the kernel is :class:`~.kernels.ProductGeometricKernel`, the values of `params` may be (s,)-shaped arrays instead, where `s` is the number of factors. .. note:: Finite values of `params["nu"]` typically correspond to the generalized (geometric) Matérn kernels. Infinite `params["nu"]` typically corresponds to the heat kernel (a.k.a. diffusion kernel, generalized squared exponential kernel, generalized Gaussian kernel, generalized RBF kernel). Although it is often considered to be a separate entity, we treat the heat kernel as a member of the Matérn family, with smoothness parameter equal to infinity. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :param X2: A batch of M inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. `X2=None` sets `X2=X1`. Defaults to None. :return: The N x M cross-covariance matrix. .. warning:: The types of values in the `params` dict determine the backend used for internal computations and the output type. Even if, say, `geometric_kernels.jax` is imported but the values in the `params` dict are NumPy arrays, the output type will be a NumPy array, and NumPy will be used for internal computations. To get a JAX array as an output and use JAX for internal computations, all the values in the `params` dict must be JAX arrays. .. py:method:: K_diag(params, X, **kwargs) Returns the diagonal of the covariance matrix `self.K(params, X, X)`, typically in a more efficient way than actually computing the full covariance matrix with `self.K(params, X, X)` and then extracting its diagonal. :param params: Same as for :meth:`~.K`. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :return: The N-dimensional vector representing the diagonal of the covariance matrix `self.K(params, X, X)`. .. py:method:: eigenvalues(params, normalize = None) Eigenvalues of the kernel. :param params: Parameters of the kernel. Must contain keys `"lengthscale"` and `"nu"`. The shapes of `params["lengthscale"]` and `params["nu"]` are `(1,)`. :param normalize: Whether to normalize kernel to have unit average variance. If None, uses `self.normalize` to decide. Defaults to None. :return: An [L, 1]-shaped array. .. py:method:: init_params() Initializes the dict of the trainable parameters of the kernel. Returns `dict(nu=np.array([np.inf]), lengthscale=np.array([1.0]))`. This dict can be modified and is passed around into such methods as :meth:`~.K` or :meth:`~.K_diag`, as the `params` argument. .. note:: The values in the returned dict are always of the NumPy array type. Thus, if you want to use some other backend for internal computations when calling :meth:`~.K` or :meth:`~.K_diag`, you need to replace the values with the analogs typed as arrays of the desired backend. .. py:property:: eigenfunctions :type: geometric_kernels.spaces.eigenfunctions.Eigenfunctions Eigenfunctions of the kernel. .. py:property:: eigenvalues_laplacian :type: lab.Numeric Eigenvalues of the Laplacian. .. py:property:: space :type: geometric_kernels.spaces.DiscreteSpectrumSpace The space on which the kernel is defined. .. py:class:: ProductGeometricKernel(*kernels, dimension_indices = None) Bases: :py:obj:`geometric_kernels.kernels.base.BaseGeometricKernel` Product kernel, defined as the product of a sequence of kernels. See :doc:`this page ` for a brief account on theory behind product kernels and the :doc:`Torus.ipynb ` notebook for a tutorial on how to use them. :param ``*kernels``: A sequence of kernels to compute the product of. Cannot contain another instance of :class:`ProductGeometricKernel`. We denote the number of factors, i.e. the length of the "sequence", by s. :param dimension_indices: Determines how a product kernel input vector `x` is to be mapped into the inputs `xi` for the factor kernels. `xi` are assumed to be equal to `x[dimension_indices[i]]`, possibly up to a reshape. Such a reshape might be necessary to accommodate the spaces whose elements are matrices rather than vectors, as determined by `element_shapes`. The transformation of `x` into the list of `xi`\ s is performed by :func:`~.project_product`. If None, assumes the each input is layed-out flattened and concatenated, in the same order as the factor spaces. In this case, the inverse to :func:`~.project_product` is :func:`~.make_product`. Defaults to None. .. note:: `params` of a :class:`ProductGeometricKernel` are such that `params["lengthscale"]` and `params["nu"]` are (s,)-shaped arrays, where `s` is the number of factors. Basically, `params["lengthscale"][i]` stores the length scale parameter for the `i`-th factor kernel. Same goes for `params["nu"]`. Importantly, this enables *automatic relevance determination*-like behavior. .. py:method:: K(params, X, X2=None, **kwargs) Compute the cross-covariance matrix between two batches of vectors of inputs, or batches of matrices of inputs, depending on the space. :param params: A dict of kernel parameters, typically containing two keys: `"lengthscale"` for length scale and `"nu"` for smoothness. The types of values in the params dict determine the output type and the backend used for the internal computations, see the warning below for more details. .. note:: The values `params["lengthscale"]` and `params["nu"]` are typically (1,)-shaped arrays of the suitable backend. This serves to point at the backend to be used for internal computations. In some cases, for example, when the kernel is :class:`~.kernels.ProductGeometricKernel`, the values of `params` may be (s,)-shaped arrays instead, where `s` is the number of factors. .. note:: Finite values of `params["nu"]` typically correspond to the generalized (geometric) Matérn kernels. Infinite `params["nu"]` typically corresponds to the heat kernel (a.k.a. diffusion kernel, generalized squared exponential kernel, generalized Gaussian kernel, generalized RBF kernel). Although it is often considered to be a separate entity, we treat the heat kernel as a member of the Matérn family, with smoothness parameter equal to infinity. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :param X2: A batch of M inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. `X2=None` sets `X2=X1`. Defaults to None. :return: The N x M cross-covariance matrix. .. warning:: The types of values in the `params` dict determine the backend used for internal computations and the output type. Even if, say, `geometric_kernels.jax` is imported but the values in the `params` dict are NumPy arrays, the output type will be a NumPy array, and NumPy will be used for internal computations. To get a JAX array as an output and use JAX for internal computations, all the values in the `params` dict must be JAX arrays. .. py:method:: K_diag(params, X) Returns the diagonal of the covariance matrix `self.K(params, X, X)`, typically in a more efficient way than actually computing the full covariance matrix with `self.K(params, X, X)` and then extracting its diagonal. :param params: Same as for :meth:`~.K`. :param X: A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the `self.space` are represented. :return: The N-dimensional vector representing the diagonal of the covariance matrix `self.K(params, X, X)`. .. py:method:: init_params() Returns a dict `params` where `params["lengthscale"]` is the concatenation of all `self.kernels[i].init_params()["lengthscale"]` and same for `params["nu"]`. .. py:property:: space :type: beartype.typing.List[geometric_kernels.spaces.Space] The list of spaces upon which the factor kernels are defined. .. py:function:: default_feature_map(*, space = None, num = None, kernel = None) Constructs the default feature map for the specified space or kernel. :param space: A space to construct the feature map on. If provided, kernel must either be omitted or set to None. :param kernel: A kernel to construct the feature map from. If provided, `space` and `num` must either be omitted or set to None. :param num: Controls the number of features (dimensionality of the feature map). If omitted or set to None, the default value for each respective space is used. Must only be provided when constructing a feature map on a space (not from a kernel). :return: Callable which is the respective feature map.