hsgp_init

docs/source/contrib.rst

@@ -97,3 +97,202 @@ Stochastic Support
     :undoc-members:
     :show-inheritance:
     :member-order: bysource
+
+
+Hilbert Space Gaussian Processes Approximation
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+This module contains helper functions do use the  Hilbert Space Gaussian Process (HSGP) approximation method
+described in [1] and [2].
+
+.. warning::
+    This module is experimental. Currently, it only gaussian processes with one-dimensional inputs.
+
+
+**Why do we need an approximation?** 
+
+Gaussian processes do not scale well with the number of data points. Recall we had to invert the kernel matrix!
+The computational complexity of the Gaussian process model is :math:`\mathcal{O}(n^3)`, where :math:`n` is the number of data
+points. The HSGP approximation method is a way to reduce the computational complexity of the Gaussian process model
+to :math:`\mathcal{O}(mn + m)`, where :math:`m` is the number of basis functions used in the approximation.
+
+**Approximation Strategy Steps:**
+
+We strongly recommend reading [1] and [2] for a detailed explanation of the approximation method. In [3] you can find
+a practical approach using NumPyro and PyMC.
+
+Here we provide the main steps and ingredients of the approximation method:
+
+1. Each stationary kernel :math:`k` has an associated spectral density :math:`S(\omega)`. There are closed formulas for the most common kernels. These formulas depend on the hyperparameters of the kernel (e.g. amplitudes and length scales).
+2. We can approximate the spectral density :math:`S(\omega)` as a polynomial series in :math:`||\omega||`. We call :math:`\omega` the frequency.
+3. We can interpret these polynomial terms as "powers" of the Laplacian operator. The key observation is that the Fourier transform of the Laplacian operator is :math:`||\omega||^2`.
+4. Next, we impose Dirichlet boundary conditions on the Laplacian operator which makes it self-adjoint and with discrete spectrum.
+5. We identify the expansion in (2) with the sum of powers of the Laplacian operator in the eigenbasis of (4).
+
+For the one dimensional case the approximation formula, in the non-centered parameterization, is:
+
+.. math::
+
+    f(x) \approx \sum_{j = 1}^{m} 
+    \overbrace{\color{red}{\left(S(\sqrt{\lambda_j})\right)^{1/2}}}^{\text{all hyperparameters are here!}} 
+    \times
+    \underbrace{\color{blue}{\phi_{j}(x)}}_{\text{easy to compute!}}
+    \times
+    \overbrace{\color{green}{\beta_{j}}}^{\sim \: \text{Normal}(0,1)}
+
+where :math:`\lambda_j` are the eigenvalues of the Laplacian operator, :math:`\phi_{j}(x)` are the eigenfunctions of the
+Laplacian operator, and :math:`\beta_{j}` are the coefficients of the expansion (see Eq. (8) in [2]).
+
+.. note::
+    Even though the periodic kernel is not stationary, one can still adapt and find a similar approximation formula. 
+    See Appendix B in [2] for more details.
+
+**Example:**
+
+Here is an example of how to use the HSGP approximation method with NumPyro. We will use the squared exponential kernel.
+Other kernels can be used sim
+
+    .. code-block:: python
+
+        >>> from jax import random
+        >>> import jax.numpy as jnp
+
+        >>> import numpyro
+        >>> from numpyro.contrib.hsgp.approximation import hsgp_approximation_squared_exponential
+        >>> import numpyro.distributions as dist
+        >>> from numpyro.infer import MCMC, NUTS
+
+
+        >>> def generate_synthetic_data(rng_key, start, stop: float, num, scale):
+        ...     """Generate synthetic data."""
+        ...     x = jnp.linspace(start=start, stop=stop, num=num)
+        ...     y = jnp.sin(4 * jnp.pi * x) + jnp.sin(7 * jnp.pi * x)
+        ...     y_obs = y + scale * random.normal(rng_key, shape=(num,))
+        ...     return x, y_obs
+
+
+        >>> rng_key = random.PRNGKey(seed=42)
+        >>> rng_key, rng_subkey = random.split(rng_key)
+        >>> x, y_obs = generate_synthetic_data(
+        ...     rng_key=rng_subkey, start=0, stop=1, num=80, scale=0.3
+        >>> )
+
+
+        >>> def model(x, ell, m, non_centered, y=None):
+        ...     # --- Priors ---
+        ...     alpha = numpyro.sample("alpha", dist.InverseGamma(concentration=12, rate=10))
+        ...     length = numpyro.sample("length", dist.InverseGamma(concentration=6, rate=1))
+        ...     noise = numpyro.sample("noise", dist.InverseGamma(concentration=12, rate=10))
+        ...     # --- Parametrization ---
+        ...     f = hsgp_approximation_squared_exponential(
+        ...         x=x, alpha=alpha, length=length, ell=ell, m=m, non_centered=non_centered
+        ...     )
+        ...     # --- Likelihood ---
+        ...     with numpyro.plate("data", x.shape[0]):
+        ...         numpyro.sample("likelihood", dist.Normal(loc=f, scale=noise), obs=y)
+
+
+        >>> sampler = NUTS(model)
+        >>> mcmc = MCMC(sampler=sampler, num_warmup=500, num_samples=1_000, num_chains=2)
+
+        >>> rng_key, rng_subkey = random.split(rng_key)
+
+        >>> ell = 1.3
+        >>> m = 20
+        >>> non_centered = True
+
+        >>> mcmc.run(rng_subkey, x, ell, m, non_centered, y_obs)
+
+        >>> mcmc.print_summary()
+
+                      mean       std    median      5.0%     95.0%     n_eff     r_hat
+           alpha      1.24      0.34      1.18      0.72      1.74   1804.01      1.00
+         beta[0]     -0.10      0.66     -0.10     -1.24      0.92   1819.91      1.00
+         beta[1]      0.00      0.71     -0.01     -1.09      1.26   1872.82      1.00
+         beta[2]     -0.05      0.69     -0.03     -1.09      1.16   2105.88      1.00
+         beta[3]      0.25      0.74      0.26     -0.98      1.42   2281.30      1.00
+         beta[4]     -0.17      0.69     -0.17     -1.21      1.00   2551.39      1.00
+         beta[5]      0.09      0.75      0.10     -1.13      1.30   3232.13      1.00
+         beta[6]     -0.49      0.75     -0.49     -1.65      0.82   3042.31      1.00
+         beta[7]      0.42      0.75      0.44     -0.78      1.65   2885.42      1.00
+         beta[8]      0.69      0.71      0.71     -0.48      1.82   2811.68      1.00
+         beta[9]     -1.43      0.75     -1.40     -2.63     -0.21   2858.68      1.00
+        beta[10]      0.33      0.71      0.33     -0.77      1.51   2198.65      1.00
+        beta[11]      1.09      0.73      1.11     -0.23      2.18   2765.99      1.00
+        beta[12]     -0.91      0.72     -0.91     -2.06      0.31   2586.53      1.00
+        beta[13]      0.05      0.70      0.04     -1.16      1.12   2569.59      1.00
+        beta[14]     -0.44      0.71     -0.44     -1.58      0.73   2626.09      1.00
+        beta[15]      0.69      0.73      0.70     -0.45      1.88   2626.32      1.00
+        beta[16]      0.98      0.74      0.98     -0.15      2.28   2282.86      1.00
+        beta[17]     -2.54      0.77     -2.52     -3.82     -1.29   3347.56      1.00
+        beta[18]      1.35      0.66      1.35      0.30      2.46   2638.17      1.00
+        beta[19]      1.10      0.54      1.09      0.25      2.01   2428.37      1.00
+          length      0.07      0.01      0.07      0.06      0.09   2321.67      1.00
+           noise      0.33      0.03      0.33      0.29      0.38   2472.83      1.00
+
+        Number of divergences: 0
+
+
+.. note::
+    Additional examples with code can be found in [3], [4] and [5].
+
+**References:**
+
+1. Solin, A., Särkkä, S. Hilbert space methods for reduced-rank Gaussian process regression.
+Stat Comput 30, 419-446 (2020).
+
+2. Riutort-Mayol, G., Bürkner, PC., Andersen, M.R. et al. Practical Hilbert space
+approximate Bayesian Gaussian processes for probabilistic programming. Stat Comput 33, 17 (2023).
+
+3. `Orduz, J., A Conceptual and Practical Introduction to Hilbert Space GPs Approximation Methods <https://juanitorduz.github.io/hsgp_intro>`_.
+
+4. `Example: Hilbert space approximation for Gaussian processes <https://num.pyro.ai/en/stable/examples/hsgp.html>`_.
+
+5. `Gelman, Vehtari, Simpson, et al., Bayesian workflow book - Birthdays <https://avehtari.github.io/casestudies/Birthdays/birthdays.html>`_.
+
+.. note::
+    The code of this module is based on the code of the example `Example: Hilbert space approximation for Gaussian processes <https://num.pyro.ai/en/stable/examples/hsgp.html>`_.
+
+sqrt_eigenvalues
+----------------
+.. autofunction:: numpyro.contrib.hsgp.laplacian.sqrt_eigenvalues
+
+eigenfunctions
+--------------
+.. autofunction:: numpyro.contrib.hsgp.laplacian.eigenfunctions
+
+eigenfunctions_periodic
+-----------------------
+.. autofunction:: numpyro.contrib.hsgp.laplacian.eigenfunctions_periodic
+
+spectral_density_squared_exponential
+------------------------------------
+.. autofunction:: numpyro.contrib.hsgp.spectral_densities.spectral_density_squared_exponential
+
+spectral_density_matern
+-----------------------
+.. autofunction:: numpyro.contrib.hsgp.spectral_densities.spectral_density_matern
+
+diag_spectral_density_squared_exponential
+-----------------------------------------
+.. autofunction:: numpyro.contrib.hsgp.spectral_densities.diag_spectral_density_squared_exponential
+
+diag_spectral_density_matern
+----------------------------
+.. autofunction:: numpyro.contrib.hsgp.spectral_densities.diag_spectral_density_matern
+
+diag_spectral_density_periodic
+------------------------------
+.. autofunction:: numpyro.contrib.hsgp.spectral_densities.diag_spectral_density_periodic
+
+hsgp_approximation_squared_exponential
+--------------------------------------
+.. autofunction:: numpyro.contrib.hsgp.approximation.hsgp_approximation_squared_exponential
+
+hsgp_approximation_matern
+-------------------------
+.. autofunction:: numpyro.contrib.hsgp.approximation.hsgp_approximation_matern
+
+hsgp_approximation_periodic_non_centered
+-----------------------------------------
+.. autofunction:: numpyro.contrib.hsgp.approximation.hsgp_approximation_periodic_non_centered

docs/source/index.rst

@@ -37,6 +37,7 @@ NumPyro documentation
    tutorials/bad_posterior_geometry
    tutorials/truncated_distributions
    tutorials/censoring
+   tutorials/hsgp_example
 
 .. nbgallery::
    :maxdepth: 1

numpyro/contrib/hsgp/approximation.py

@@ -0,0 +1,176 @@
+"""
+This module contains the low-rank approximation functions of the Hilbert space Gaussian process.
+"""
+
+from jaxlib.xla_extension import ArrayImpl
+
+import jax.numpy as jnp
+
+import numpyro
+from numpyro.contrib.hsgp.laplacian import eigenfunctions, eigenfunctions_periodic
+from numpyro.contrib.hsgp.spectral_densities import (
+    diag_spectral_density_matern,
+    diag_spectral_density_periodic,
+    diag_spectral_density_squared_exponential,
+)
+import numpyro.distributions as dist
+
+
+def _non_centered_approximation(phi: ArrayImpl, spd: ArrayImpl, m: int) -> ArrayImpl:
+    with numpyro.plate("basis", m):
+        beta = numpyro.sample("beta", dist.Normal(loc=0.0, scale=1.0))
+
+    return phi @ (spd * beta)
+
+
+def _centered_approximation(phi: ArrayImpl, spd: ArrayImpl, m: int) -> ArrayImpl:
+    with numpyro.plate("basis", m):
+        beta = numpyro.sample("beta", dist.Normal(loc=0.0, scale=spd))
+
+    return phi @ beta
+
+
+def linear_approximation(
+    phi: ArrayImpl, spd: ArrayImpl, m: int, non_centered: bool = True
+) -> ArrayImpl:
+    """
+    Linear approximation formula of the Hilbert space Gaussian process.
+
+    See Eq. (8) in [1].
+
+    **References:**
+    1. Riutort-Mayol, G., Bürkner, PC., Andersen, M.R. et al. Practical Hilbert space
+       approximate Bayesian Gaussian processes for probabilistic programming. Stat Comput 33, 17 (2023).
+
+    :param ArrayImpl phi: laplacian eigenfunctions
+    :param ArrayImpl spd: square root of the diagonal of the spectral density evaluated at square
+        root of the first `m` eigenvalues.
+    :param int m: number of eigenfunctions in the approximation
+    :param bool non_centered: whether to use a non-centered parameterization
+    :return: The low-rank approximation linear model
+    :rtype: ArrayImpl
+    """
+    if non_centered:
+        return _non_centered_approximation(phi, spd, m)
+    return _centered_approximation(phi, spd, m)
+
+
+def hsgp_approximation_squared_exponential(
+    x: ArrayImpl,
+    alpha: float,
+    length: float,
+    ell: float,
+    m: int,
+    non_centered: bool = True,
+) -> ArrayImpl:
+    """
+    Hilbert space Gaussian process approximation using the squared exponential kernel.
+
+    The main idea of the approach is to combine the associated spectral density of the
+    squared exponential kernel and the spectrum of the Drichlet Laplacian operator to
+    obtain a low-rank approximation of the Gram matrix. For more details see [1, 2].
+
+    **References:**
+
+    1. Solin, A., Särkkä, S. Hilbert space methods for reduced-rank Gaussian process regression.
+    Stat Comput 30, 419-446 (2020).
+
+    2. Riutort-Mayol, G., Bürkner, PC., Andersen, M.R. et al. Practical Hilbert space
+    approximate Bayesian Gaussian processes for probabilistic programming. Stat Comput 33, 17 (2023).
+
+    :param ArrayImpl x: input data
+    :param float alpha: amplitude of the squared exponential kernel
+    :param float length: length scale of the squared exponential kernel
+    :param float ell: The length of the interval divided by 2
+    :param int m: The number of eigenvalues to compute
+    :param bool non_centered: Whether to use a non-centered parameterization. By default, it is set to True
+    :return: The low-rank approximation linear model
+    :rtype: ArrayImpl
+    """
+    phi = eigenfunctions(x=x, ell=ell, m=m)
+    spd = jnp.sqrt(
+        diag_spectral_density_squared_exponential(
+            alpha=alpha, length=length, ell=ell, m=m
+        )
+    )
+    return linear_approximation(phi=phi, spd=spd, m=m, non_centered=non_centered)
+
+
+def hsgp_approximation_matern(
+    x: ArrayImpl,
+    nu: float,
+    alpha: float,
+    length: float,
+    ell: float,
+    m: int,
+    non_centered: bool = True,
+):
+    """
+    Hilbert space Gaussian process approximation using the Matérn kernel.
+
+    The main idea of the approach is to combine the associated spectral density of the
+    Matérn kernel kernel and the spectrum of the Drichlet Laplacian operator to obtain
+    a low-rank approximation of the Gram matrix. For more details see [1, 2].
+
+    **References:**
+
+    1. Solin, A., Särkkä, S. Hilbert space methods for reduced-rank Gaussian process regression.
+    Stat Comput 30, 419-446 (2020).
+
+    2. Riutort-Mayol, G., Bürkner, PC., Andersen, M.R. et al. Practical Hilbert space
+    approximate Bayesian Gaussian processes for probabilistic programming. Stat Comput 33, 17 (2023).
+
+    :param ArrayImpl x: input data
+    :param float nu: smoothness parameter
+    :param float alpha: amplitude of the squared exponential kernel
+    :param float length: length scale of the squared exponential kernel
+    :param float ell: The length of the interval divided by 2
+    :param int m: The number of eigenvalues to compute
+    :param bool non_centered: Whether to use a non-centered parameterization. By default, it is set to True.
+    :return: The low-rank approximation linear model
+    :rtype: ArrayImpl
+    """
+    phi = eigenfunctions(x=x, ell=ell, m=m)
+    spd = jnp.sqrt(
+        diag_spectral_density_matern(nu=nu, alpha=alpha, length=length, ell=ell, m=m)
+    )
+    return linear_approximation(phi=phi, spd=spd, m=m, non_centered=non_centered)
+
+
+def hsgp_approximation_periodic_non_centered(
+    x: ArrayImpl, alpha: float, length: float, w0: float, m: int
+) -> ArrayImpl:
+    """
+    Low rank approximation for the periodic squared exponential kernel in the non-centered parametrization.
+
+    See Appendix B in [1].
+
+    **References:**
+
+    1. Riutort-Mayol, G., Bürkner, PC., Andersen, M.R. et al. Practical Hilbert space
+    approximate Bayesian Gaussian processes for probabilistic programming. Stat Comput 33, 17 (2023).
+
+    :param ArrayImpl x: input data
+    :param float alpha: amplitude
+    :param float length: length scale
+    :param float w0: frequency of the periodic kernel
+    :param int m: number of eigenfunctions in the approximation
+    :return: The low-rank approximation linear model
+    :rtype: ArrayImpl
+    """
+    q2 = diag_spectral_density_periodic(alpha=alpha, length=length, m=m)
+    cosines, sines = eigenfunctions_periodic(x=x, w0=w0, m=m)
+
+    with numpyro.plate("cos_basis", m):
+        beta_cos = numpyro.sample("beta_cos", dist.Normal(0, 1))
+
+    with numpyro.plate("sin_basis", m - 1):
+        beta_sin = numpyro.sample("beta_sin", dist.Normal(0, 1))
+
+    # The first eigenfunction for the sine component
+    # is zero, so the first parameter wouldn't contribute to the approximation.
+    # We set it to zero to identify the model and avoid divergences.
+    zero = jnp.array([0.0])
+    beta_sin = jnp.concatenate((zero, beta_sin))
+
+    return cosines @ (q2 * beta_cos) + sines @ (q2 * beta_sin)