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Quantities of Interest

A Quantity of Interest (QoI) is a function of the output that determines the network output behavior that the attributions describe.

The quantity of interest lets us specify what we want to explain. Often, this is the output of the network corresponding to a particular class, addressing, e.g., "Why did the model classify a given image as a car?" However, we could also consider various combinations of outputs, allowing us to ask more specific questions, such as, "Why did the model classify a given image as a sedan and not a convertible?" The former may highlight general “car features,” such as tires, while the latter (called a comparative explanation) might focus on the roof of the car, a “car feature” not shared by convertibles.

ClassQoI

Quantity of interest for attributing output towards a specified class.

__init__(self, cl) special

Parameters:

Name Type Description Default
cl int

The index of the class the QoI is for.

required
Source code in trulens/nn/quantities.py
def __init__(self, cl: int):
    """
    Parameters:
        cl:
            The index of the class the QoI is for.
    """
    self.cl = cl

ClassSeqQoI

Quantity of interest for attributing output towards a sequence of classes for each input.

__init__(self, seq_labels) special

Parameters:

Name Type Description Default
seq_labels List[int]

A sequence of classes corresponding to each input.

required
Source code in trulens/nn/quantities.py
def __init__(self, seq_labels: List[int]):
    """
    Parameters:
        seq_labels:
            A sequence of classes corresponding to each input.
    """
    self.seq_labels = seq_labels

ComparativeQoI

Quantity of interest for attributing network output towards a given class, relative to another.

__init__(self, cl1, cl2) special

Parameters:

Name Type Description Default
cl1 int

The index of the class the QoI is for.

required
cl2 int

The index of the class to compare against.

required
Source code in trulens/nn/quantities.py
def __init__(self, cl1: int, cl2: int):
    """
    Parameters:
        cl1:
            The index of the class the QoI is for.
        cl2:
            The index of the class to compare against.
    """
    self.cl1 = cl1
    self.cl2 = cl2

InternalChannelQoI

Quantity of interest for attributing output towards the output of an internal convolutional layer channel, aggregating using a specified operation.

Also works for non-convolutional dense layers, where the given neuron's activation is returned.

__init__(self, channel, channel_axis=None, agg_fn=None) special

Parameters:

Name Type Description Default
channel Union[int, List[int]]

Channel to return. If a list is provided, then the quantity sums over each of the channels in the list.

required
channel_axis Optional[int]

Channel dimension index, if relevant, e.g., for 2D convolutional layers. If channel_axis is None, then the channel axis of the relevant backend will be used. This argument is not used when the channels are scalars, e.g., for dense layers.

None
agg_fn Optional[Callable]

Function with which to aggregate the remaining dimensions (except the batch dimension) in order to get a single scalar value for each channel. If agg_fn is None then a sum over each neuron in the channel will be taken. This argument is not used when the channels are scalars, e.g., for dense layers.

None
Source code in trulens/nn/quantities.py
def __init__(
        self,
        channel: Union[int, List[int]],
        channel_axis: Optional[int] = None,
        agg_fn: Optional[Callable] = None):
    """
    Parameters:
        channel:
            Channel to return. If a list is provided, then the quantity sums 
            over each of the channels in the list.

        channel_axis:
            Channel dimension index, if relevant, e.g., for 2D convolutional
            layers. If `channel_axis` is `None`, then the channel axis of 
            the relevant backend will be used. This argument is not used 
            when the channels are scalars, e.g., for dense layers.

        agg_fn:
            Function with which to aggregate the remaining dimensions 
            (except the batch dimension) in order to get a single scalar 
            value for each channel. If `agg_fn` is `None` then a sum over 
            each neuron in the channel will be taken. This argument is not 
            used when the channels are scalars, e.g., for dense layers.
    """
    if channel_axis is None:
        channel_axis = get_backend().channel_axis
    if agg_fn is None:
        agg_fn = InternalChannelQoI._batch_sum

    self._channel_ax = channel_axis
    self._agg_fn = agg_fn
    self._channels = channel if isinstance(channel, list) else [channel]

LambdaQoI

Generic quantity of interest allowing the user to specify a function of the model's output as the QoI.

__init__(self, function) special

Parameters:

Name Type Description Default
function Callable

A callable that takes a single argument representing the model's tensor output and returns a differentiable batched scalar tensor representing the QoI.

required
Source code in trulens/nn/quantities.py
def __init__(self, function: Callable):
    """
    Parameters:
        function:
            A callable that takes a single argument representing the model's 
            tensor output and returns a differentiable batched scalar tensor 
            representing the QoI.
    """
    if len(signature(function).parameters) != 1:
        raise ValueError(
            'QoI function must take exactly 1 argument, but provided '
            'function takes {} arguments'.format(
                len(signature(function).parameters)))

    self.function = function

MaxClassQoI

Quantity of interest for attributing output towards the maximum-predicted class.

__init__(self, axis=1, activation=None) special

Parameters:

Name Type Description Default
axis int

Output dimension over which max operation is taken.

1
activation Optional[Union[Callable, str]]

Activation function to be applied to the output before taking the max. If activation is a string, use the corresponding named activation function implemented by the backend. The following strings are currently supported as shorthands for the respective standard activation functions:

  • 'sigmoid'
  • 'softmax'

If activation is None, no activation function is applied to the input.

None
Source code in trulens/nn/quantities.py
def __init__(
        self, axis: int = 1, activation: Union[Callable, str, None] = None):
    """
    Parameters:
        axis:
            Output dimension over which max operation is taken.

        activation:
            Activation function to be applied to the output before taking 
            the max. If `activation` is a string, use the corresponding 
            named activation function implemented by the backend. The 
            following strings are currently supported as shorthands for the
            respective standard activation functions:

            - `'sigmoid'` 
            - `'softmax'` 

            If `activation` is `None`, no activation function is applied to
            the input.
    """
    self._axis = axis
    self.activation = activation

QoI

Interface for quantities of interest. The Quantity of Interest (QoI) is a function of the output specified by the slice that determines the network output behavior that the attributions describe.

__call__(self, y) special

Computes the distribution of interest from an initial point.

Parameters:

Name Type Description Default
y Union[Any, List[Any]]

Output point from which the quantity is derived. Must be a differentiable tensor.

required

Returns:

Type Description
Union[Any, List[Any]]

A differentiable batched scalar tensor representing the QoI.

Source code in trulens/nn/quantities.py
@abstractmethod
def __call__(self, y: TensorLike) -> TensorLike:
    """
    Computes the distribution of interest from an initial point.

    Parameters:
        y:
            Output point from which the quantity is derived. Must be a
            differentiable tensor.

    Returns:
        A differentiable batched scalar tensor representing the QoI.
    """
    raise NotImplementedError

QoiCutSupportError

Exception raised if the quantity of interest is called on a cut whose output is not supported by the quantity of interest.

ThresholdQoI

Quantity of interest for attributing network output toward the difference between two regions seperated by a given threshold. I.e., the quantity of interest is the "high" elements minus the "low" elements, where the high elements have activations above the threshold and the low elements have activations below the threshold.

Use case: bianry segmentation.

__init__(self, threshold, low_minus_high=False, activation=None) special

Parameters:

Name Type Description Default
threshold float

A threshold to determine the element-wise sign of the input tensor. The elements with activations higher than the threshold will retain their sign, while the elements with activations lower than the threshold will have their sign flipped (or vice versa if low_minus_high is set to True).

required
low_minus_high bool

If True, substract the output with activations above the threshold from the output with activations below the threshold. If False, substract the output with activations below the threshold from the output with activations above the threshold.

False
activation Optional[Union[Callable, str]]

str or function, optional Activation function to be applied to the quantity before taking the threshold. If activation is a string, use the corresponding activation function implemented by the backend (currently supported: 'sigmoid' and 'softmax'). Otherwise, if activation is not None, it will be treated as a callable. If activation is None, do not apply an activation function to the quantity.

None
Source code in trulens/nn/quantities.py
def __init__(
        self,
        threshold: float,
        low_minus_high: bool = False,
        activation: Union[Callable, str, None] = None):
    """
    Parameters:
        threshold:
            A threshold to determine the element-wise sign of the input 
            tensor. The elements with activations higher than the threshold 
            will retain their sign, while the elements with activations 
            lower than the threshold will have their sign flipped (or vice 
            versa if `low_minus_high` is set to `True`).
        low_minus_high:
            If `True`, substract the output with activations above the 
            threshold from the output with activations below the threshold. 
            If `False`, substract the output with activations below the 
            threshold from the output with activations above the threshold.
        activation: str or function, optional
            Activation function to be applied to the quantity before taking
            the threshold. If `activation` is a string, use the 
            corresponding activation function implemented by the backend 
            (currently supported: `'sigmoid'` and `'softmax'`). Otherwise, 
            if `activation` is not `None`, it will be treated as a callable.
            If `activation` is `None`, do not apply an activation function 
            to the quantity.
    """
    # TODO(klas):should this support an aggregation function? By default
    #   this is a sum, but it could, for example, subtract the greatest
    #   positive element from the least negative element.
    self.threshold = threshold
    self.low_minus_high = low_minus_high
    self.activation = activation