Run the (extended) Infomax ICA decomposition on raw data

    based on the publications of Bell & Sejnowski 1995 (Infomax)
    and Lee, Girolami & Sejnowski, 1999 (extended Infomax)

    Parameters
    ----------
    data : np.ndarray, shape (n_samples, n_features)
        The data to unmix.
    w_init : np.ndarray, shape (n_features, n_features)
        The initialized unmixing matrix. Defaults to None. If None, the
        identity matrix is used.
    l_rate : float
        This quantity indicates the relative size of the change in weights.
        Note. Smaller learining rates will slow down the procedure.
        Defaults to 0.010d / alog(n_features ^ 2.0)
    block : int
        The block size of randomly chosen data segment.
        Defaults to floor(sqrt(n_times / 3d))
    w_change : float
        The change at which to stop iteration. Defaults to 1e-12.
    anneal_deg : float
        The angle at which (in degree) the learning rate will be reduced.
        Defaults to 60.0
    anneal_step : float
        The factor by which the learning rate will be reduced once
        ``anneal_deg`` is exceeded:
            l_rate *= anneal_step
        Defaults to 0.9
    extended : bool
        Wheather to use the extended infomax algorithm or not. Defaults to
        True.
    n_subgauss : int
        The number of subgaussian components. Only considered for extended
        Infomax.
    kurt_size : int
        The window size for kurtosis estimation. Only considered for extended
        Infomax.
    ext_blocks : int
        The number of blocks after which to recompute Kurtosis.
        Only considered for extended Infomax.
    max_iter : int
        The maximum number of iterations. Defaults to 200.
    verbose : bool, str, int, or None
        If not None, override default verbose level (see mne.verbose).

    Returns
    -------
    unmixing_matrix : np.ndarray of float, shape (n_features, n_features)
        The linear unmixing operator.

def infomax(data, weights=None, l_rate=None, block=None, w_change=1e-12,
            anneal_deg=60., anneal_step=0.9, extended=False, n_subgauss=1,
            kurt_size=6000, ext_blocks=1, max_iter=200,
            random_state=None, verbose=None):
    """Run the (extended) Infomax ICA decomposition on raw data

    based on the publications of Bell & Sejnowski 1995 (Infomax)
    and Lee, Girolami & Sejnowski, 1999 (extended Infomax)

    Parameters
    ----------
    data : np.ndarray, shape (n_samples, n_features)
        The data to unmix.
    w_init : np.ndarray, shape (n_features, n_features)
        The initialized unmixing matrix. Defaults to None. If None, the
        identity matrix is used.
    l_rate : float
        This quantity indicates the relative size of the change in weights.
        Note. Smaller learining rates will slow down the procedure.
        Defaults to 0.010d / alog(n_features ^ 2.0)
    block : int
        The block size of randomly chosen data segment.
        Defaults to floor(sqrt(n_times / 3d))
    w_change : float
        The change at which to stop iteration. Defaults to 1e-12.
    anneal_deg : float
        The angle at which (in degree) the learning rate will be reduced.
        Defaults to 60.0
    anneal_step : float
        The factor by which the learning rate will be reduced once
        ``anneal_deg`` is exceeded:
            l_rate *= anneal_step
        Defaults to 0.9
    extended : bool
        Wheather to use the extended infomax algorithm or not. Defaults to
        True.
    n_subgauss : int
        The number of subgaussian components. Only considered for extended
        Infomax.
    kurt_size : int
        The window size for kurtosis estimation. Only considered for extended
        Infomax.
    ext_blocks : int
        The number of blocks after which to recompute Kurtosis.
        Only considered for extended Infomax.
    max_iter : int
        The maximum number of iterations. Defaults to 200.
    verbose : bool, str, int, or None
        If not None, override default verbose level (see mne.verbose).

    Returns
    -------
    unmixing_matrix : np.ndarray of float, shape (n_features, n_features)
        The linear unmixing operator.
    """
    rng = check_random_state(random_state)

    # define some default parameter
    max_weight = 1e8
    restart_fac = 0.9
    min_l_rate = 1e-10
    blowup = 1e4
    blowup_fac = 0.5
    n_small_angle = 20
    degconst = 180.0 / np.pi

    # for extended Infomax
    extmomentum = 0.5
    signsbias = 0.02
    signcount_threshold = 25
    signcount_step = 2
    if ext_blocks > 0:  # allow not to recompute kurtosis
        n_subgauss = 1  # but initialize n_subgauss to 1 if you recompute

    # check data shape
    n_samples, n_features = data.shape
    n_features_square = n_features ** 2

    # check input parameter
    # heuristic default - may need adjustment for
    # large or tiny data sets
    if l_rate is None:
        l_rate = 0.01 / math.log(n_features ** 2.0)

    if block is None:
        block = int(math.floor(math.sqrt(n_samples / 3.0)))

    logger.info('computing%sInfomax ICA' % ' Extended ' if extended is True
                else ' ')

    # collect parameter
    nblock = n_samples // block
    lastt = (nblock - 1) * block + 1

    # initialize training
    if weights is None:
        # initialize weights as identity matrix
        weights = np.identity(n_features, dtype=np.float64)

    BI = block * np.identity(n_features, dtype=np.float64)
    bias = np.zeros((n_features, 1), dtype=np.float64)
    onesrow = np.ones((1, block), dtype=np.float64)
    startweights = weights.copy()
    oldweights = startweights.copy()
    step = 0
    count_small_angle = 0
    wts_blowup = False
    blockno = 0
    signcount = 0

    # for extended Infomax
    if extended is True:
        signs = np.identity(n_features)
        signs.flat[slice(0, n_features * n_subgauss, n_features)]
        kurt_size = min(kurt_size, n_samples)
        old_kurt = np.zeros(n_features, dtype=np.float64)
        oldsigns = np.zeros((n_features, n_features))

    # trainings loop
    olddelta, oldchange = 1., 0.
    while step < max_iter:

        # shuffle data at each step
        permute = list(range(n_samples))
        rng.shuffle(permute)

        # ICA training block
        # loop across block samples
        for t in range(0, lastt, block):
            u = np.dot(data[permute[t:t + block], :], weights)
            u += np.dot(bias, onesrow).T

            if extended is True:
                # extended ICA update
                y = np.tanh(u)
                weights += l_rate * np.dot(weights,
                                           BI - np.dot(np.dot(u.T, y), signs) -
                                           np.dot(u.T, u))
                bias += l_rate * np.reshape(np.sum(y, axis=0,
                                            dtype=np.float64) * -2.0,
                                            (n_features, 1))

            else:
                # logistic ICA weights update
                y = 1.0 / (1.0 + np.exp(-u))
                weights += l_rate * np.dot(weights,
                                           BI + np.dot(u.T, (1.0 - 2.0 * y)))
                bias += l_rate * np.reshape(np.sum((1.0 - 2.0 * y), axis=0,
                                            dtype=np.float64), (n_features, 1))

            # check change limit
            max_weight_val = np.max(np.abs(weights))
            if max_weight_val > max_weight:
                wts_blowup = True

            blockno += 1
            if wts_blowup:
                break

            # ICA kurtosis estimation
            if extended is True:

                n = np.fix(blockno / ext_blocks)

                if np.abs(n) * ext_blocks == blockno:
                    if kurt_size < n_samples:
                        rp = np.floor(rng.uniform(0, 1, kurt_size) *
                                      (n_samples - 1))
                        tpartact = np.dot(data[rp.astype(int), :], weights).T
                    else:
                        tpartact = np.dot(data, weights).T

                    # estimate kurtosis
                    kurt = kurtosis(tpartact, axis=1, fisher=True)

                    if extmomentum != 0:
                        kurt = (extmomentum * old_kurt +
                                (1.0 - extmomentum) * kurt)
                        old_kurt = kurt

                    # estimate weighted signs
                    signs.flat[::n_features + 1] = ((kurt + signsbias) /
                                                    np.abs(kurt + signsbias))

                    ndiff = ((signs.flat[::n_features + 1] -
                              oldsigns.flat[::n_features + 1]) != 0).sum()
                    if ndiff == 0:
                        signcount += 1
                    else:
                        signcount = 0
                    oldsigns = signs

                    if signcount >= signcount_threshold:
                        ext_blocks = np.fix(ext_blocks * signcount_step)
                        signcount = 0

        # here we continue after the for
        # loop over the ICA training blocks
        # if weights in bounds:
        if not wts_blowup:
            oldwtchange = weights - oldweights
            step += 1
            angledelta = 0.0
            delta = oldwtchange.reshape(1, n_features_square)
            change = np.sum(delta * delta, dtype=np.float64)
            if step > 1:
                angledelta = math.acos(np.sum(delta * olddelta) /
                                       math.sqrt(change * oldchange))
                angledelta *= degconst

            # anneal learning rate
            oldweights = weights.copy()
            if angledelta > anneal_deg:
                l_rate *= anneal_step    # anneal learning rate
                # accumulate angledelta until anneal_deg reached l_rates
                olddelta = delta
                oldchange = change
                count_small_angle = 0  # reset count when angle delta is large
            else:
                if step == 1:  # on first step only
                    olddelta = delta  # initialize
                    oldchange = change
                count_small_angle += 1
                if count_small_angle > n_small_angle:
                    max_iter = step

            # apply stopping rule
            if step > 2 and change < w_change:
                step = max_iter
            elif change > blowup:
                l_rate *= blowup_fac

        # restart if weights blow up
        # (for lowering l_rate)
        else:
            step = 0  # start again
            wts_blowup = 0  # re-initialize variables
            blockno = 1
            l_rate *= restart_fac  # with lower learning rate
            weights = startweights.copy()
            oldweights = startweights.copy()
            olddelta = np.zeros((1, n_features_square), dtype=np.float64)
            bias = np.zeros((n_features, 1), dtype=np.float64)

            # for extended Infomax
            if extended:
                signs = np.identity(n_features)
                signs.flat[slice(0, n_features * n_subgauss, n_features)]
                oldsigns = np.zeros((n_features, n_features))

            if l_rate > min_l_rate:
                if verbose:
                    logger.info('... lowering learning rate to %g'
                                '\n... re-starting...' % l_rate)
            else:
                raise ValueError('Error in Infomax ICA: unmixing_matrix matrix'
                                 'might not be invertible!')

    # prepare return values
    return weights.T