DOptimal

sgptools.objectives.DOptimal

Bases: Objective

Computes the D-optimal design metric.

D-optimality seeks to minimize the determinant of the posterior covariance matrix \(|K(X, X)|\). The objective returns the negative log-determinant of \(K(X, X)\), which is maximized during optimization. tf.linalg.slogdet is used for numerical stability.

Source code in sgptools/objectives.py

class  DOptimal(Objective):            
    """
    Computes the D-optimal design metric.

    D-optimality seeks to minimize the determinant of the posterior
    covariance matrix $|K(X, X)|$. The objective returns
    the negative log-determinant of $K(X, X)$, which is maximized during
    optimization. `tf.linalg.slogdet` is used for numerical stability.
    """
    def __call__(self, X: tf.Tensor) -> tf.Tensor:
        """
        Computes the negative log-determinant of the covariance matrix $-log|K(X, X)|$.

        Args:
            X (tf.Tensor): The input points (e.g., sensing locations) for which
                           the objective is to be computed. Shape: (M, D).

        Returns:
            tf.Tensor: The computed D-optimal metric value.

        Usage:
            ```python
            import gpflow
            import numpy as np
            # Assume kernel is defined
            # X_objective = np.random.rand(100, 2) # Not used by D-Optimal but required by base class
            # kernel = gpflow.kernels.SquaredExponential()
            # noise_variance = 0.1

            d_optimal_objective = DOptimal(
                X_objective=X_objective,
                kernel=kernel,
                noise_variance=noise_variance
            )
            X_sensing = tf.constant(np.random.rand(10, 2))
            d_optimal_value = d_optimal_objective(X_sensing)
            ```
        """
        # K(X, X)
        K_X_X = self.kernel(X)
        _, logdet_K_X_X = tf.linalg.slogdet(self.jitter_fn(K_X_X))
        return -logdet_K_X_X

__call__(X)

Computes the negative log-determinant of the covariance matrix \(-log|K(X, X)|\).

Parameters:

Name	Type	Description	Default
X	Tensor	The input points (e.g., sensing locations) for which the objective is to be computed. Shape: (M, D).	required

Returns:

Type	Description
Tensor	tf.Tensor: The computed D-optimal metric value.

Usage

import gpflow
import numpy as np
# Assume kernel is defined
# X_objective = np.random.rand(100, 2) # Not used by D-Optimal but required by base class
# kernel = gpflow.kernels.SquaredExponential()
# noise_variance = 0.1

d_optimal_objective = DOptimal(
    X_objective=X_objective,
    kernel=kernel,
    noise_variance=noise_variance
)
X_sensing = tf.constant(np.random.rand(10, 2))
d_optimal_value = d_optimal_objective(X_sensing)

Source code in sgptools/objectives.py

def __call__(self, X: tf.Tensor) -> tf.Tensor:
    """
    Computes the negative log-determinant of the covariance matrix $-log|K(X, X)|$.

    Args:
        X (tf.Tensor): The input points (e.g., sensing locations) for which
                       the objective is to be computed. Shape: (M, D).

    Returns:
        tf.Tensor: The computed D-optimal metric value.

    Usage:
        ```python
        import gpflow
        import numpy as np
        # Assume kernel is defined
        # X_objective = np.random.rand(100, 2) # Not used by D-Optimal but required by base class
        # kernel = gpflow.kernels.SquaredExponential()
        # noise_variance = 0.1

        d_optimal_objective = DOptimal(
            X_objective=X_objective,
            kernel=kernel,
            noise_variance=noise_variance
        )
        X_sensing = tf.constant(np.random.rand(10, 2))
        d_optimal_value = d_optimal_objective(X_sensing)
        ```
    """
    # K(X, X)
    K_X_X = self.kernel(X)
    _, logdet_K_X_X = tf.linalg.slogdet(self.jitter_fn(K_X_X))
    return -logdet_K_X_X

__init__(X_objective, kernel, noise_variance, jitter=1e-06, **kwargs)

Initializes the base objective. This constructor primarily serves to define the expected parameters for all objective subclasses.

Parameters:

Name	Type	Description	Default
X_objective	ndarray	The fixed set of data points (e.g., candidate locations or training data points) against which MI is computed. Shape: (N, D).	required
kernel	Kernel	The GPflow kernel function to compute covariances.	required
noise_variance	float	The observed data noise variance, which is added to the jitter.	required
jitter	float	A small positive value to add for numerical stability to covariance matrix diagonals. Defaults to 1e-6.	1e-06
**kwargs	Any	Arbitrary keyword arguments.	{}

Source code in sgptools/objectives.py

def __init__(self,
             X_objective: np.ndarray,
             kernel: gpflow.kernels.Kernel,
             noise_variance: float,
             jitter: float = 1e-6,
             **kwargs: Any):
    """
    Initializes the base objective. This constructor primarily serves to define
    the expected parameters for all objective subclasses.

    Args:
        X_objective (np.ndarray): The fixed set of data points (e.g., candidate locations
                                  or training data points) against which MI is computed.
                                  Shape: (N, D).
        kernel (gpflow.kernels.Kernel): The GPflow kernel function to compute covariances.
        noise_variance (float): The observed data noise variance, which is added to the jitter.
        jitter (float): A small positive value to add for numerical stability to covariance
                        matrix diagonals. Defaults to 1e-6.
        **kwargs: Arbitrary keyword arguments.
    """
    self.X_objective = tf.constant(X_objective)
    self.kernel = kernel
    self.noise_variance = noise_variance
    # Total jitter includes the noise variance
    self._base_jitter = jitter
    self.jitter_fn = lambda cov: jitter_fn(
        cov, jitter=self._base_jitter + self.noise_variance)

update(kernel, noise_variance)

Updates the kernel and noise variance for the MI objective. This method is crucial for optimizing the GP hyperparameters externally and having the objective function reflect those changes.

Parameters:

Name	Type	Description	Default
kernel	Kernel	The updated GPflow kernel function.	required
noise_variance	float	The updated data noise variance.	required

Source code in sgptools/objectives.py

def update(self, kernel: gpflow.kernels.Kernel,
           noise_variance: float) -> None:
    """
    Updates the kernel and noise variance for the MI objective.
    This method is crucial for optimizing the GP hyperparameters externally
    and having the objective function reflect those changes.

    Args:
        kernel (gpflow.kernels.Kernel): The updated GPflow kernel function.
        noise_variance (float): The updated data noise variance.
    """
    # Update kernel's trainable variables (e.g., lengthscales, variance)
    for self_var, var in zip(self.kernel.trainable_variables,
                             kernel.trainable_variables):
        self_var.assign(var)

    self.noise_variance = noise_variance
    # Update the jitter function to reflect the new noise variance
    self.jitter_fn = lambda cov: jitter_fn(
        cov, jitter=self._base_jitter + self.noise_variance)