AOptimal

sgptools.objectives.AOptimal

Bases: Objective

Computes the A-optimal design metric.

A-optimality aims to minimize the trace of the covariance matrix \(Tr(K(X, X))\). Since optimization algorithms typically minimize a function, this objective returns the negative trace, which is then maximized.

Source code in sgptools/objectives.py

class  AOptimal(Objective):          
    """
    Computes the A-optimal design metric.

    A-optimality aims to minimize the trace of the
    covariance matrix $Tr(K(X, X))$. Since optimization algorithms typically
    minimize a function, this objective returns the negative trace, which
    is then maximized.
    """  
    def __call__(self, X: tf.Tensor) -> tf.Tensor:
        """
        Computes the negative trace of the covariance matrix $-Tr(K(X, X))$.

        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 A-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 A-Optimal but required by base class
            # kernel = gpflow.kernels.SquaredExponential()
            # noise_variance = 0.1

            a_optimal_objective = AOptimal(
                X_objective=X_objective,
                kernel=kernel,
                noise_variance=noise_variance
            )
            X_sensing = tf.constant(np.random.rand(10, 2), dtype=tf.float64)
            a_optimal_value = a_optimal_objective(X_sensing)
            ```
        """
        # K(X, X)
        K_X_X = self.kernel(X)
        trace_K_X_X = tf.linalg.trace(self.jitter_fn(K_X_X))
        return -trace_K_X_X

__call__(X)

Computes the negative trace of the covariance matrix \(-Tr(K(X, X))\).

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 A-optimal metric value.

Usage

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

a_optimal_objective = AOptimal(
    X_objective=X_objective,
    kernel=kernel,
    noise_variance=noise_variance
)
X_sensing = tf.constant(np.random.rand(10, 2), dtype=tf.float64)
a_optimal_value = a_optimal_objective(X_sensing)

Source code in sgptools/objectives.py

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

    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 A-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 A-Optimal but required by base class
        # kernel = gpflow.kernels.SquaredExponential()
        # noise_variance = 0.1

        a_optimal_objective = AOptimal(
            X_objective=X_objective,
            kernel=kernel,
            noise_variance=noise_variance
        )
        X_sensing = tf.constant(np.random.rand(10, 2), dtype=tf.float64)
        a_optimal_value = a_optimal_objective(X_sensing)
        ```
    """
    # K(X, X)
    K_X_X = self.kernel(X)
    trace_K_X_X = tf.linalg.trace(self.jitter_fn(K_X_X))
    return -trace_K_X_X