Python keras.backend.cast() Examples

def yolo_correct_boxes(box_xy, box_wh, input_shape, image_shape):
    '''Get corrected boxes'''
    box_yx = box_xy[..., ::-1]
    box_hw = box_wh[..., ::-1]
    input_shape = K.cast(input_shape, K.dtype(box_yx))
    image_shape = K.cast(image_shape, K.dtype(box_yx))
    new_shape = K.round(image_shape * K.min(input_shape/image_shape))
    offset = (input_shape-new_shape)/2./input_shape
    scale = input_shape/new_shape
    box_yx = (box_yx - offset) * scale
    box_hw *= scale

    box_mins = box_yx - (box_hw / 2.)
    box_maxes = box_yx + (box_hw / 2.)
    boxes =  K.concatenate([
        box_mins[..., 0:1],  # y_min
        box_mins[..., 1:2],  # x_min
        box_maxes[..., 0:1],  # y_max
        box_maxes[..., 1:2]  # x_max
    ])

    # Scale boxes back to original image shape.
    boxes *= K.concatenate([image_shape, image_shape])
    return boxes 

def call(self, x, mask=None):
        eij = dot_product(x, self.W)

        if self.bias:
            eij += self.b

        eij = K.tanh(eij)

        a = K.exp(eij)

        if mask is not None:
            a *= K.cast(mask, K.floatx())

        a /= K.cast(K.sum(a, axis=1, keepdims=True) + K.epsilon(), K.floatx())

        weighted_input = x * K.expand_dims(a)

        result = K.sum(weighted_input, axis=1)

        if self.return_attention:
            return [result, a]
        return result 

def call(self, x, mask=None):
        uit = dot_product(x, self.W)

        if self.bias:
            uit += self.b

        uit = K.tanh(uit)
        ait = dot_product(uit, self.u)

        a = K.exp(ait)

        # apply mask after the exp. will be re-normalized next
        if mask is not None:
            # Cast the mask to floatX to avoid float64 upcasting in theano
            a *= K.cast(mask, K.floatx())

        # in some cases especially in the early stages of training the sum may be almost zero
        # and this results in NaN's. A workaround is to add a very small positive number ε to the sum.
        # a /= K.cast(K.sum(a, axis=1, keepdims=True), K.floatx())
        a /= K.cast(K.sum(a, axis=1, keepdims=True) + K.epsilon(), K.floatx())

        a = K.expand_dims(a)
        weighted_input = x * a
        return K.sum(weighted_input, axis=1) 

def call(self,x,mask=None):
        conv_input,theta = x
        s = theta.shape
        theta = T.reshape(theta,[-1,s[2]])
        m = K.not_equal(conv_input,0.)

        #### For translation
        trans = _trans(theta)
        output = _transform_trans(trans, conv_input)
        output = output * K.cast(m,K.floatx())

        ### For rotation
        M = _fusion(theta)
        output = _transform_rot(M,output)

        return output 

def _correct_boxes(
            self, box_xy, box_wh, input_shape, image_shape):
        """Get corrected boxes, which are scaled to original shape."""
        box_yx = box_xy[..., ::-1]
        box_hw = box_wh[..., ::-1]
        input_shape = K.cast(input_shape, K.dtype(box_yx))
        image_shape = K.cast(image_shape, K.dtype(box_yx))
        new_shape = K.round(image_shape * K.min(input_shape / image_shape))
        offset = (input_shape - new_shape) / 2. / input_shape
        scale = input_shape / new_shape
        box_yx = (box_yx - offset) * scale
        box_hw *= scale

        box_mins = box_yx - (box_hw / 2.)
        box_maxes = box_yx + (box_hw / 2.)
        boxes = K.concatenate([
            box_mins[..., 0:1],  # y_min
            box_mins[..., 1:2],  # x_min
            box_maxes[..., 0:1],  # y_max
            box_maxes[..., 1:2]  # x_max
        ])

        # Scale boxes back to original image shape.
        boxes *= K.concatenate([image_shape, image_shape])
        return boxes 

def rpn_class_loss_graph(rpn_match, rpn_class_logits):
    """RPN anchor classifier loss.
    rpn_match: [batch, anchors, 1]. Anchor match type. 1=positive,
               -1=negative, 0=neutral anchor.
    rpn_class_logits: [batch, anchors, 2]. RPN classifier logits for FG/BG.
    """
    # Squeeze last dim to simplify
    rpn_match = tf.squeeze(rpn_match, -1)
    # Get anchor classes. Convert the -1/+1 match to 0/1 values.
    anchor_class = K.cast(K.equal(rpn_match, 1), tf.int32)
    # Positive and Negative anchors contribute to the loss,
    # but neutral anchors (match value = 0) don't.
    indices = tf.where(K.not_equal(rpn_match, 0))
    # Pick rows that contribute to the loss and filter out the rest.
    rpn_class_logits = tf.gather_nd(rpn_class_logits, indices)
    anchor_class = tf.gather_nd(anchor_class, indices)
    # Cross entropy loss
    loss = K.sparse_categorical_crossentropy(target=anchor_class,
                                             output=rpn_class_logits,
                                             from_logits=True)
    loss = K.switch(tf.size(loss) > 0, K.mean(loss), tf.constant(0.0))
    return loss 

def rpn_loss_regr(num_anchors):
	def rpn_loss_regr_fixed_num(y_true, y_pred):
		if K.image_dim_ordering() == 'th':
			x = y_true[:, 4 * num_anchors:, :, :] - y_pred
			x_abs = K.abs(x)
			x_bool = K.less_equal(x_abs, 1.0)
			return lambda_rpn_regr * K.sum(
				y_true[:, :4 * num_anchors, :, :] * (x_bool * (0.5 * x * x) + (1 - x_bool) * (x_abs - 0.5))) / K.sum(epsilon + y_true[:, :4 * num_anchors, :, :])
		else:
			x = y_true[:, :, :, 4 * num_anchors:] - y_pred
			x_abs = K.abs(x)
			x_bool = K.cast(K.less_equal(x_abs, 1.0), tf.float32)

			return lambda_rpn_regr * K.sum(
				y_true[:, :, :, :4 * num_anchors] * (x_bool * (0.5 * x * x) + (1 - x_bool) * (x_abs - 0.5))) / K.sum(epsilon + y_true[:, :, :, :4 * num_anchors])

	return rpn_loss_regr_fixed_num 

def __init__(self, lr=0.001, beta_1=0.9, beta_2=0.999,
                 epsilon=None, decay=0., amsgrad=False, accum_iters=1, **kwargs):
        if accum_iters < 1:
            raise ValueError('accum_iters must be >= 1')
        super(AdamAccumulate, self).__init__(**kwargs)
        with K.name_scope(self.__class__.__name__):
            self.iterations = K.variable(0, dtype='int64', name='iterations')
            self.lr = K.variable(lr, name='lr')
            self.beta_1 = K.variable(beta_1, name='beta_1')
            self.beta_2 = K.variable(beta_2, name='beta_2')
            self.decay = K.variable(decay, name='decay')
        if epsilon is None:
            epsilon = K.epsilon()
        self.epsilon = epsilon
        self.initial_decay = decay
        self.amsgrad = amsgrad
        self.accum_iters = K.variable(accum_iters, K.dtype(self.iterations))
        self.accum_iters_float = K.cast(self.accum_iters, K.floatx()) 

def rpn_class_loss_graph(rpn_match, rpn_class_logits):
    """RPN anchor classifier loss.

    rpn_match: [batch, anchors, 1]. Anchor match type. 1=positive,
               -1=negative, 0=neutral anchor.
    rpn_class_logits: [batch, anchors, 2]. RPN classifier logits for FG/BG.
    """
    # Squeeze last dim to simplify
    rpn_match = tf.squeeze(rpn_match, -1)
    # Get anchor classes. Convert the -1/+1 match to 0/1 values.
    anchor_class = K.cast(K.equal(rpn_match, 1), tf.int32)
    # Positive and Negative anchors contribute to the loss,
    # but neutral anchors (match value = 0) don't.
    indices = tf.where(K.not_equal(rpn_match, 0))
    # Pick rows that contribute to the loss and filter out the rest.
    rpn_class_logits = tf.gather_nd(rpn_class_logits, indices)
    anchor_class = tf.gather_nd(anchor_class, indices)
    # Crossentropy loss
    loss = K.sparse_categorical_crossentropy(target=anchor_class,
                                             output=rpn_class_logits,
                                             from_logits=True)
    loss = K.switch(tf.size(loss) > 0, K.mean(loss), tf.constant(0.0))
    return loss 

def call(self, x, mask=None):
        # computes a probability distribution over the timesteps
        # uses 'max trick' for numerical stability
        # reshape is done to avoid issue with Tensorflow
        # and 1-dimensional weights
        logits = K.dot(x, self.W)
        x_shape = K.shape(x)
        logits = K.reshape(logits, (x_shape[0], x_shape[1]))
        ai = K.exp(logits - K.max(logits, axis=-1, keepdims=True))

        # masked timesteps have zero weight
        if mask is not None:
            mask = K.cast(mask, K.floatx())
            ai = ai * mask
        att_weights = ai / (K.sum(ai, axis=1, keepdims=True) + K.epsilon())
        weighted_input = x * K.expand_dims(att_weights)
        result = K.sum(weighted_input, axis=1)
        if self.return_attention:
            return [result, att_weights]
        return result 

def yolo_correct_boxes(box_xy, box_wh, input_shape, image_shape):
    '''Get corrected boxes'''
    box_yx = box_xy[..., ::-1]
    box_hw = box_wh[..., ::-1]
    input_shape = K.cast(input_shape, K.dtype(box_yx))
    image_shape = K.cast(image_shape, K.dtype(box_yx))
    new_shape = K.round(image_shape * K.min(input_shape/image_shape))
    offset = (input_shape-new_shape)/2./input_shape
    scale = input_shape/new_shape
    box_yx = (box_yx - offset) * scale
    box_hw *= scale

    box_mins = box_yx - (box_hw / 2.)
    box_maxes = box_yx + (box_hw / 2.)
    boxes =  K.concatenate([
        box_mins[..., 0:1],  # y_min
        box_mins[..., 1:2],  # x_min
        box_maxes[..., 0:1],  # y_max
        box_maxes[..., 1:2]  # x_max
    ])

    # Scale boxes back to original image shape.
    boxes *= K.concatenate([image_shape, image_shape])
    return boxes 

def yolo_correct_boxes(box_xy, box_wh, input_shape, image_shape):
    '''Get corrected boxes'''
    box_yx = box_xy[..., ::-1]
    box_hw = box_wh[..., ::-1]
    input_shape = K.cast(input_shape, K.dtype(box_yx))
    image_shape = K.cast(image_shape, K.dtype(box_yx))
    new_shape = K.round(image_shape * K.min(input_shape/image_shape))
    offset = (input_shape-new_shape)/2./input_shape
    scale = input_shape/new_shape
    box_yx = (box_yx - offset) * scale
    box_hw *= scale

    box_mins = box_yx - (box_hw / 2.)
    box_maxes = box_yx + (box_hw / 2.)
    boxes =  K.concatenate([
        box_mins[..., 0:1],  # y_min
        box_mins[..., 1:2],  # x_min
        box_maxes[..., 0:1],  # y_max
        box_maxes[..., 1:2]  # x_max
    ])

    # Scale boxes back to original image shape.
    boxes *= K.concatenate([image_shape, image_shape])
    return boxes 

def gen_adv_loss(logits, y, loss='logloss', mean=False):
    """
    Generate the loss function.
    """

    if loss == 'training':
        # use the model's output instead of the true labels to avoid
        # label leaking at training time
        y = K.cast(K.equal(logits, K.max(logits, 1, keepdims=True)), "float32")
        y = y / K.sum(y, 1, keepdims=True)
        out = K.categorical_crossentropy(y, logits, from_logits=True)
    elif loss == 'logloss':
        out = K.categorical_crossentropy(y, logits, from_logits=True)
    else:
        raise ValueError("Unknown loss: {}".format(loss))

    if mean:
        out = K.mean(out)
    # else:
    #     out = K.sum(out)
    return out 

def gen_adv_loss(logits, y, loss='logloss', mean=False):
    """
    Generate the loss function.
    """

    if loss == 'training':
        # use the model's output instead of the true labels to avoid
        # label leaking at training time
        y = K.cast(K.equal(logits, K.max(logits, 1, keepdims=True)), "float32")
        y = y / K.sum(y, 1, keepdims=True)
        out = K.categorical_crossentropy(logits, y, from_logits=True)
    elif loss == 'logloss':
        # out = K.categorical_crossentropy(logits, y, from_logits=True)
        out = tf.nn.softmax_cross_entropy_with_logits(logits=logits, labels=y)
        out = tf.reduce_mean(out)
    else:
        raise ValueError("Unknown loss: {}".format(loss))

    if mean:
        out = tf.mean(out)
    # else:
    #     out = K.sum(out)
    return out 

def yolo_correct_boxes(box_xy, box_wh, input_shape, image_shape):
    '''Get corrected boxes'''
    box_yx = box_xy[..., ::-1]
    box_hw = box_wh[..., ::-1]
    input_shape = K.cast(input_shape, K.dtype(box_yx))
    image_shape = K.cast(image_shape, K.dtype(box_yx))
    new_shape = K.round(image_shape * K.min(input_shape/image_shape))
    offset = (input_shape-new_shape)/2./input_shape
    scale = input_shape/new_shape
    box_yx = (box_yx - offset) * scale
    box_hw *= scale

    box_mins = box_yx - (box_hw / 2.)
    box_maxes = box_yx + (box_hw / 2.)
    boxes =  K.concatenate([
        box_mins[..., 0:1],  # y_min
        box_mins[..., 1:2],  # x_min
        box_maxes[..., 0:1],  # y_max
        box_maxes[..., 1:2]  # x_max
    ])

    # Scale boxes back to original image shape.
    boxes *= K.concatenate([image_shape, image_shape])
    return boxes 

def yolo_correct_boxes(box_xy, box_wh, input_shape, image_shape):
    '''Get corrected boxes'''
    box_yx = box_xy[..., ::-1]
    box_hw = box_wh[..., ::-1]
    input_shape = K.cast(input_shape, K.dtype(box_yx))
    image_shape = K.cast(image_shape, K.dtype(box_yx))
    new_shape = K.round(image_shape * K.min(input_shape/image_shape))
    offset = (input_shape-new_shape)/2./input_shape
    scale = input_shape/new_shape
    box_yx = (box_yx - offset) * scale
    box_hw *= scale

    box_mins = box_yx - (box_hw / 2.)
    box_maxes = box_yx + (box_hw / 2.)
    boxes =  K.concatenate([
        box_mins[..., 0:1],  # y_min
        box_mins[..., 1:2],  # x_min
        box_maxes[..., 0:1],  # y_max
        box_maxes[..., 1:2]  # x_max
    ])

    # Scale boxes back to original image shape.
    boxes *= K.concatenate([image_shape, image_shape])
    return boxes 

def yolo_correct_boxes(box_xy, box_wh, input_shape, image_shape):
    '''Get corrected boxes'''
    box_yx = box_xy[..., ::-1]
    box_hw = box_wh[..., ::-1]
    input_shape = K.cast(input_shape, K.dtype(box_yx))
    image_shape = K.cast(image_shape, K.dtype(box_yx))
    new_shape = K.round(image_shape * K.min(input_shape/image_shape))
    offset = (input_shape-new_shape)/2./input_shape
    scale = input_shape/new_shape
    box_yx = (box_yx - offset) * scale
    box_hw *= scale

    box_mins = box_yx - (box_hw / 2.)
    box_maxes = box_yx + (box_hw / 2.)
    boxes =  K.concatenate([
        box_mins[..., 0:1],  # y_min
        box_mins[..., 1:2],  # x_min
        box_maxes[..., 0:1],  # y_max
        box_maxes[..., 1:2]  # x_max
    ])

    # Scale boxes back to original image shape.
    boxes *= K.concatenate([image_shape, image_shape])
    return boxes 

def add_boundary_energy(x, b_start=None, b_end=None, mask=None):
    '''Given the observations x, it adds the start boundary energy b_start (resp.
    end boundary energy b_end on the start (resp. end) elements and multiplies
    the mask.'''
    if mask is None:
        if b_start is not None:
            x = K.concatenate([x[:, :1, :] + b_start, x[:, 1:, :]], axis=1)
        if b_end is not None:
            x = K.concatenate([x[:, :-1, :], x[:, -1:, :] + b_end], axis=1)
    else:
        mask = K.cast(mask, K.floatx())
        mask = K.expand_dims(mask, 2)
        x *= mask
        if b_start is not None:
            mask_r = K.concatenate([K.zeros_like(mask[:, :1]), mask[:, :-1]], axis=1)
            start_mask = K.cast(K.greater(mask, mask_r), K.floatx())
            x = x + start_mask * b_start
        if b_end is not None:
            mask_l = K.concatenate([mask[:, 1:], K.zeros_like(mask[:, -1:])], axis=1)
            end_mask = K.cast(K.greater(mask, mask_l), K.floatx())
            x = x + end_mask * b_end
    return x 

def rpn_class_loss_graph(rpn_match, rpn_class_logits):
    """RPN anchor classifier loss.

    rpn_match: [batch, anchors, 1]. Anchor match type. 1=positive,
               -1=negative, 0=neutral anchor.
    rpn_class_logits: [batch, anchors, 2]. RPN classifier logits for FG/BG.
    """
    # Squeeze last dim to simplify
    rpn_match = tf.squeeze(rpn_match, -1)
    # Get anchor classes. Convert the -1/+1 match to 0/1 values.
    anchor_class = K.cast(K.equal(rpn_match, 1), tf.int32)
    # Positive and Negative anchors contribute to the loss,
    # but neutral anchors (match value = 0) don't.
    indices = tf.where(K.not_equal(rpn_match, 0))
    # Pick rows that contribute to the loss and filter out the rest.
    rpn_class_logits = tf.gather_nd(rpn_class_logits, indices)
    anchor_class = tf.gather_nd(anchor_class, indices)
    # Cross entropy loss
    loss = K.sparse_categorical_crossentropy(target=anchor_class,
                                             output=rpn_class_logits,
                                             from_logits=True)
    loss = K.switch(tf.size(loss) > 0, K.mean(loss), tf.constant(0.0))
    return loss 

def rpn_loss_regr(num_anchors):
	def rpn_loss_regr_fixed_num(y_true, y_pred):
		if K.image_dim_ordering() == 'th':
			x = y_true[:, 4 * num_anchors:, :, :] - y_pred
			x_abs = K.abs(x)
			x_bool = K.less_equal(x_abs, 1.0)
			return lambda_rpn_regr * K.sum(
				y_true[:, :4 * num_anchors, :, :] * (x_bool * (0.5 * x * x) + (1 - x_bool) * (x_abs - 0.5))) / K.sum(epsilon + y_true[:, :4 * num_anchors, :, :])
		else:
			x = y_true[:, :, :, 4 * num_anchors:] - y_pred
			x_abs = K.abs(x)
			x_bool = K.cast(K.less_equal(x_abs, 1.0), tf.float32)

			return lambda_rpn_regr * K.sum(
				y_true[:, :, :, :4 * num_anchors] * (x_bool * (0.5 * x * x) + (1 - x_bool) * (x_abs - 0.5))) / K.sum(epsilon + y_true[:, :, :, :4 * num_anchors])

	return rpn_loss_regr_fixed_num 

def _forward(x, reduce_step, initial_states, U, mask=None):
    '''Forward recurrence of the linear chain crf.'''

    def _forward_step(energy_matrix_t, states):
        alpha_tm1 = states[-1]
        new_states = reduce_step(K.expand_dims(alpha_tm1, 2) + energy_matrix_t)
        return new_states[0], new_states

    U_shared = K.expand_dims(K.expand_dims(U, 0), 0)

    if mask is not None:
        mask = K.cast(mask, K.floatx())
        mask_U = K.expand_dims(K.expand_dims(mask[:, :-1] * mask[:, 1:], 2), 3)
        U_shared = U_shared * mask_U

    inputs = K.expand_dims(x[:, 1:, :], 2) + U_shared
    inputs = K.concatenate([inputs, K.zeros_like(inputs[:, -1:, :, :])], axis=1)

    last, values, _ = K.rnn(_forward_step, inputs, initial_states)
    return last, values 

def trim_zeros_graph(boxes):
    """Often boxes are represented with matricies of shape [N, 4] and
    are padded with zeros. This removes zero boxes.

    boxes: [N, 4] matrix of boxes.

    TODO: use this function to reduce code duplication
    """
    area = tf.boolean_mask(boxes, tf.cast(tf.reduce_sum(tf.abs(boxes), axis=1),
                                          tf.bool)) 

def chain_crf_loss(y, x, U, b_start=None, b_end=None, mask=None):
    '''Variant of sparse_chain_crf_loss but with one-hot encoded tags y.'''
    y_sparse = K.argmax(y, -1)
    y_sparse = K.cast(y_sparse, 'int32')
    return sparse_chain_crf_loss(y_sparse, x, U, b_start, b_end, mask) 

def mrcnn_bbox_loss_graph(target_bbox, target_class_ids, pred_bbox):
    """Loss for Mask R-CNN bounding box refinement.

    target_bbox: [batch, num_rois, (dy, dx, log(dh), log(dw))]
    target_class_ids: [batch, num_rois]. Integer class IDs.
    pred_bbox: [batch, num_rois, num_classes, (dy, dx, log(dh), log(dw))]
    """
    # Reshape to merge batch and roi dimensions for simplicity.
    target_class_ids = K.reshape(target_class_ids, (-1,))
    target_bbox = K.reshape(target_bbox, (-1, 4))
    pred_bbox = K.reshape(pred_bbox, (-1, K.int_shape(pred_bbox)[2], 4))

    # Only positive ROIs contribute to the loss. And only
    # the right class_id of each ROI. Get their indicies.
    positive_roi_ix = tf.where(target_class_ids > 0)[:, 0]
    positive_roi_class_ids = tf.cast(tf.gather(target_class_ids, positive_roi_ix), tf.int64)
    indices = tf.stack([positive_roi_ix, positive_roi_class_ids], axis=1)

    # Gather the deltas (predicted and true) that contribute to loss
    target_bbox = tf.gather(target_bbox, positive_roi_ix)
    pred_bbox = tf.gather_nd(pred_bbox, indices)

    # Smooth-L1 Loss
    loss = K.switch(tf.size(target_bbox) > 0,
                    smooth_l1_loss(y_true=target_bbox, y_pred=pred_bbox),
                    tf.constant(0.0))
    loss = K.mean(loss)
    loss = K.reshape(loss, [1, 1])
    return loss 

def rpn_bbox_loss_graph(config, target_bbox, rpn_match, rpn_bbox):
    """Return the RPN bounding box loss graph.

    config: the model config object.
    target_bbox: [batch, max positive anchors, (dy, dx, log(dh), log(dw))].
        Uses 0 padding to fill in unsed bbox deltas.
    rpn_match: [batch, anchors, 1]. Anchor match type. 1=positive,
               -1=negative, 0=neutral anchor.
    rpn_bbox: [batch, anchors, (dy, dx, log(dh), log(dw))]
    """
    # Positive anchors contribute to the loss, but negative and
    # neutral anchors (match value of 0 or -1) don't.
    rpn_match = K.squeeze(rpn_match, -1)
    indices = tf.where(K.equal(rpn_match, 1))

    # Pick bbox deltas that contribute to the loss
    rpn_bbox = tf.gather_nd(rpn_bbox, indices)

    # Trim target bounding box deltas to the same length as rpn_bbox.
    batch_counts = K.sum(K.cast(K.equal(rpn_match, 1), tf.int32), axis=1)
    target_bbox = batch_pack_graph(target_bbox, batch_counts,
                                   config.IMAGES_PER_GPU)

    # TODO: use smooth_l1_loss() rather than reimplementing here
    #       to reduce code duplication
    diff = K.abs(target_bbox - rpn_bbox)
    less_than_one = K.cast(K.less(diff, 1.0), "float32")
    loss = (less_than_one * 0.5 * diff ** 2) + (1 - less_than_one) * (diff - 0.5)

    loss = K.switch(tf.size(loss) > 0, K.mean(loss), tf.constant(0.0))
    return loss 

def smooth_l1_loss(y_true, y_pred):
    """Implements Smooth-L1 loss.
    y_true and y_pred are typicallly: [N, 4], but could be any shape.
    """
    diff = K.abs(y_true - y_pred)
    less_than_one = K.cast(K.less(diff, 1.0), "float32")
    loss = (less_than_one * 0.5 * diff ** 2) + (1 - less_than_one) * (diff - 0.5)
    return loss 

def pair_loss(y_true, y_pred):
    y_true = tf.cast(y_true, tf.int32)
    parts = tf.dynamic_partition(y_pred, y_true, 2)
    y_pos = parts[1]
    y_neg = parts[0]
    y_pos = tf.expand_dims(y_pos, 0)
    y_neg = tf.expand_dims(y_neg, -1)
    out = K.sigmoid(y_neg - y_pos)
    return K.mean(out) 

def viterbi_decode(x, U, b_start=None, b_end=None, mask=None):
    '''Computes the best tag sequence y for a given input x, i.e. the one that
    maximizes the value of path_energy.'''
    x = add_boundary_energy(x, b_start, b_end, mask)

    alpha_0 = x[:, 0, :]
    gamma_0 = K.zeros_like(alpha_0)
    initial_states = [gamma_0, alpha_0]
    _, gamma = _forward(x,
                        lambda B: [K.cast(K.argmax(B, axis=1), K.floatx()), K.max(B, axis=1)],
                        initial_states,
                        U,
                        mask)
    y = _backward(gamma, mask)
    return y 

def accuracy(y_true, y_pred):
    '''Compute classification accuracy with a fixed threshold on distances.
    '''
    return K.mean(K.equal(y_true, K.cast(y_pred < 0.5, y_true.dtype))) 

def path_energy0(y, x, U, mask=None):
    '''Path energy without boundary potential handling.'''
    n_classes = K.shape(x)[2]
    y_one_hot = K.one_hot(y, n_classes)

    # Tag path energy
    energy = K.sum(x * y_one_hot, 2)
    energy = K.sum(energy, 1)

    # Transition energy
    y_t = y[:, :-1]
    y_tp1 = y[:, 1:]
    U_flat = K.reshape(U, [-1])
    # Convert 2-dim indices (y_t, y_tp1) of U to 1-dim indices of U_flat:
    flat_indices = y_t * n_classes + y_tp1
    U_y_t_tp1 = K.gather(U_flat, flat_indices)

    if mask is not None:
        mask = K.cast(mask, K.floatx())
        y_t_mask = mask[:, :-1]
        y_tp1_mask = mask[:, 1:]
        U_y_t_tp1 *= y_t_mask * y_tp1_mask

    energy += K.sum(U_y_t_tp1, axis=1)

    return energy