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Subspace Pursuit

Solver ID: SP

Usage

from invert import Solver

# fwd = ...    (mne.Forward object)
# evoked = ... (mne.Evoked object)

solver = Solver("SP")
solver.make_inverse_operator(fwd)
stc = solver.apply_inverse_operator(evoked)
stc.plot()

Overview

Greedy sparse recovery that alternates between support expansion and pruning, solving a least-squares fit on the candidate support each iteration.

References

  1. Dai, W., & Milenkovic, O. (2009). Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 55(5), 2230–2249.
  2. Duarte, M. F., & Eldar, Y. C. (2011). Structured compressed sensing: From theory to applications. IEEE Transactions on Signal Processing, 59(9), 4053–4085.

API Reference

Bases: BaseSolver

Class for the Subspace Pursuit (SP) inverse solution [1]. The algorithm as described by [2] was implemented.

References

[1] Dai, W., & Milenkovic, O. (2009). Subspace pursuit for compressive sensing signal reconstruction. IEEE transactions on Information Theory, 55(5), 2230-2249.

[2] Duarte, M. F., & Eldar, Y. C. (2011). Structured compressed sensing: From theory to applications. IEEE Transactions on signal processing, 59(9), 4053-4085.

Source code in invert/solvers/matching_pursuit/sp.py 
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class SolverSP(BaseSolver):
    """Class for the Subspace Pursuit (SP) inverse solution [1]. The algorithm
        as described by [2] was implemented.

    References
    ----------
    [1] Dai, W., & Milenkovic, O. (2009). Subspace pursuit for compressive
    sensing signal reconstruction. IEEE transactions on Information Theory,
    55(5), 2230-2249.

    [2] Duarte, M. F., & Eldar, Y. C. (2011). Structured compressed sensing:
    From theory to applications. IEEE Transactions on signal processing, 59(9),
    4053-4085.
    """

    meta = SolverMeta(
        acronym="SP",
        full_name="Subspace Pursuit",
        category="Matching Pursuit",
        description=(
            "Greedy sparse recovery that alternates between support expansion and pruning, "
            "solving a least-squares fit on the candidate support each iteration."
        ),
        references=[
            "Dai, W., & Milenkovic, O. (2009). Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 55(5), 2230–2249.",
            "Duarte, M. F., & Eldar, Y. C. (2011). Structured compressed sensing: From theory to applications. IEEE Transactions on Signal Processing, 59(9), 4053–4085.",
        ],
    )

    def __init__(self, name="Subspace Pursuit", **kwargs):
        self.name = name
        return super().__init__(**kwargs)

    def make_inverse_operator(self, forward, *args, alpha="auto", verbose=0, **kwargs):
        """Calculate inverse operator.

        Parameters
        ----------
        forward : mne.Forward
            The mne-python Forward model instance.
        alpha : float
            The regularization parameter.

        Return
        ------
        self : object returns itself for convenience
        """
        super().make_inverse_operator(forward, *args, alpha=alpha, **kwargs)
        # Store original leadfield for coefficient estimation
        self.leadfield_original = self.leadfield.copy()
        # Use robust normalization from base class for atom selection
        self.leadfield_normed = self.robust_normalize_leadfield(self.leadfield)

        self.inverse_operators = []
        return self

    def apply_inverse_operator(self, mne_obj, K="auto") -> mne.SourceEstimate:  # type: ignore
        """Apply the SP inverse solution.

        Parameters
        ----------
        mne_obj : [mne.Evoked, mne.Epochs, mne.io.Raw]
            The MNE data object.
        K : int
            The number of atoms to select per iteration.

        Return
        ------
        stc : mne.SourceEstimate
            The source estimate containing the inverse solution.
        """
        data = self.unpack_data_obj(mne_obj)
        source_mat = np.stack([self.calc_sp_solution(y, K=K) for y in data.T], axis=1)
        stc = self.source_to_object(source_mat)
        return stc

    def calc_sp_solution(self, y, K="auto"):
        """Calculates the Subspace Pursuit (SP) inverse solution.

        Parameters
        ----------
        y : numpy.ndarray
            The data matrix (channels,).
        K : ["auto", int]
            The number of atoms to select per iteration.

        Return
        ------
        x_hat : numpy.ndarray
            The inverse solution (dipoles,)
        """
        n_chans = len(y)
        _, n_dipoles = self.leadfield.shape

        if K == "auto":
            K = int(n_chans / 2)

        # Use robust residual from base class
        def resid(y, phi):
            return self.robust_residual(y, phi)

        # Use normalized leadfield for initial atom selection
        b = self.leadfield_normed.T @ y
        T0 = np.where(thresholding(b, K) != 0)[0]

        # Use original leadfield for residual calculation
        R = resid(y, self.leadfield_original[:, T0])

        T_list = [
            T0,
        ]
        R_list = [
            R,
        ]

        for i in range(1, n_chans + 1):
            # Use normalized leadfield for atom selection
            b = self.leadfield_normed.T @ R_list[-1]

            new_T = np.where(thresholding(b, K) != 0)[0]
            T_tilde = np.unique(np.concatenate([T_list[i - 1], new_T]))

            # Use original leadfield for coefficient estimation with numerical stability
            if len(T_tilde) > 0:
                L_tilde = self.leadfield_original[:, T_tilde]
                cond_num = np.linalg.cond(L_tilde)
                if cond_num > 1e12:
                    regularization = 1e-12 * np.eye(L_tilde.shape[1])
                    xp = np.linalg.solve(
                        L_tilde.T @ L_tilde + regularization, L_tilde.T @ y
                    )
                else:
                    xp = np.linalg.pinv(L_tilde) @ y

                T_l = T_tilde[np.where(thresholding(xp, K) != 0)[0]]
            else:
                T_l = T_tilde

            T_list.append(T_l)
            R = resid(y, self.leadfield_original[:, T_l])

            R_list.append(R)

            # Early stopping with improved criterion
            if len(R_list) > 1 and (
                np.linalg.norm(R_list[-1]) >= np.linalg.norm(R_list[-2]) or i == n_chans
            ):
                T_l = T_list[-2]
                break
            else:
                T_l = T_list[-1]

        x_hat = np.zeros(n_dipoles)
        if len(T_l) > 0:
            L_final = self.leadfield_original[:, T_l]
            x_hat[T_l] = self.robust_inverse_solution(L_final, y)
        return x_hat

__init__ 

__init__(name='Subspace Pursuit', **kwargs)
Source code in invert/solvers/matching_pursuit/sp.py 
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def __init__(self, name="Subspace Pursuit", **kwargs):
    self.name = name
    return super().__init__(**kwargs)

make_inverse_operator 

make_inverse_operator(
    forward, *args, alpha="auto", verbose=0, **kwargs
)

Calculate inverse operator.

Parameters:

Name Type Description Default
forward Forward

The mne-python Forward model instance.

 required
alpha float

The regularization parameter.

 'auto'
Return

self : object returns itself for convenience

Source code in invert/solvers/matching_pursuit/sp.py 
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def make_inverse_operator(self, forward, *args, alpha="auto", verbose=0, **kwargs):
    """Calculate inverse operator.

    Parameters
    ----------
    forward : mne.Forward
        The mne-python Forward model instance.
    alpha : float
        The regularization parameter.

    Return
    ------
    self : object returns itself for convenience
    """
    super().make_inverse_operator(forward, *args, alpha=alpha, **kwargs)
    # Store original leadfield for coefficient estimation
    self.leadfield_original = self.leadfield.copy()
    # Use robust normalization from base class for atom selection
    self.leadfield_normed = self.robust_normalize_leadfield(self.leadfield)

    self.inverse_operators = []
    return self

apply_inverse_operator 

apply_inverse_operator(
    mne_obj, K="auto"
) -> mne.SourceEstimate

Apply the SP inverse solution.

Parameters:

Name Type Description Default
mne_obj [Evoked, Epochs, Raw]

The MNE data object.

 required
K int

The number of atoms to select per iteration.

 'auto'
Return

stc : mne.SourceEstimate The source estimate containing the inverse solution.

Source code in invert/solvers/matching_pursuit/sp.py 
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def apply_inverse_operator(self, mne_obj, K="auto") -> mne.SourceEstimate:  # type: ignore
    """Apply the SP inverse solution.

    Parameters
    ----------
    mne_obj : [mne.Evoked, mne.Epochs, mne.io.Raw]
        The MNE data object.
    K : int
        The number of atoms to select per iteration.

    Return
    ------
    stc : mne.SourceEstimate
        The source estimate containing the inverse solution.
    """
    data = self.unpack_data_obj(mne_obj)
    source_mat = np.stack([self.calc_sp_solution(y, K=K) for y in data.T], axis=1)
    stc = self.source_to_object(source_mat)
    return stc