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tfp.experimental.distributions.marginal_fns.ps.where

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Returns the indices of non-zero elements, or multiplexes x and y.

This operation has two modes:

  1. Return the indices of non-zero elements - When only condition is provided the result is an int64 tensor where each row is the index of a non-zero element of condition. The result's shape is [tf.math.count_nonzero(condition), tf.rank(condition)].
  2. Multiplex x and y - When both x and y are provided the result has the shape of x, y, and condition broadcast together. The result is taken from x where condition is non-zero or y where condition is zero.

1. Return the indices of non-zero elements

If x and y are not provided (both are None):

tf.where will return the indices of condition that are non-zero, in the form of a 2-D tensor with shape [n, d], where n is the number of non-zero elements in condition (tf.count_nonzero(condition)), and d is the number of axes of condition (tf.rank(condition)).

Indices are output in row-major order. The condition can have a dtype of tf.bool, or any numeric dtype.

Here condition is a 1-axis bool tensor with 2 True values. The result has a shape of [2,1]

tf.where([True, False, False, True]).numpy()
array([[0],
       [3]])

Here condition is a 2-axis integer tensor, with 3 non-zero values. The result has a shape of [3, 2].

tf.where([[1, 0, 0], [1, 0, 1]]).numpy()
array([[0, 0],
       [1, 0],
       [1, 2]])

Here condition is a 3-axis float tensor, with 5 non-zero values. The output shape is [5, 3].

float_tensor = [[[0.1, 0], [0, 2.2], [3.5, 1e6]],
                [[0,   0], [0,   0], [99,    0]]]
tf.where(float_tensor).numpy()
array([[0, 0, 0],
       [0, 1, 1],
       [0, 2, 0],
       [0, 2, 1],
       [1, 2, 0]])

These indices are the same that tf.sparse.SparseTensor would use to represent the condition tensor:

sparse = tf.sparse.from_dense(float_tensor)
sparse.indices.numpy()
array([[0, 0, 0],
       [0, 1, 1],
       [0, 2, 0],
       [0, 2, 1],
       [1, 2, 0]])

A complex number is considered non-zero if either the real or imaginary component is non-zero:

tf.where([complex(0.), complex(1.), 0+1j, 1+1j]).numpy()
array([[1],
       [2],
       [3]])

2. Multiplex x and y

If x and y are also provided (both have non-None values) the condition tensor acts as a mask that chooses whether the corresponding element / row in the output should be taken from x (if the element in condition is True) or y (if it is False).

The shape of the result is formed by broadcasting together the shapes of condition, x, and y.

When all three inputs have the same size, each is handled element-wise.

tf.where([True, False, False, True],
         [1, 2, 3, 4],
         [100, 200, 300, 400]).numpy()
array([  1, 200, 300,   4], dtype=int32)

There are two main rules for broadcasting:

  1. If a tensor has fewer axes than the others, length-1 axes are added to the left of the shape.
  2. Axes with length-1 are streched to match the coresponding axes of the other tensors.

A length-1 vector is streched to match the other vectors:

tf.where([True, False, False, True], [1, 2, 3, 4], [100]).numpy()
array([  1, 100, 100,   4], dtype=int32)

A scalar is expanded to match the other arguments:

tf.where([[True, False], [False, True]], [[1, 2], [3, 4]], 100).numpy()
array([[  1, 100], [100,   4]], dtype=int32)
tf.where([[True, False], [False, True]], 1, 100).numpy()
array([[  1, 100], [100,   1]], dtype=int32)

A scalar condition returns the complete x or y tensor, with broadcasting applied.

tf.where(True, [1, 2, 3, 4], 100).numpy()
array([1, 2, 3, 4], dtype=int32)
tf.where(False, [1, 2, 3, 4], 100).numpy()
array([100, 100, 100, 100], dtype=int32)

For a non-trivial example of broadcasting, here condition has a shape of [3], x has a shape of [3,3], and y has a shape of [3,1]. Broadcasting first expands the shape of condition to [1,3]. The final broadcast shape is [3,3]. condition will select columns from x and y. Since y only has one column, all columns from y will be identical.

tf.where([True, False, True],
         x=[[1, 2, 3],
            [4, 5, 6],
            [7, 8, 9]],
         y=[[100],
            [200],
            [300]]
).numpy()
array([[ 1, 100, 3],
       [ 4, 200, 6],
       [ 7, 300, 9]], dtype=int32)

Note that if the gradient of either branch of the tf.where generates a NaN, then the gradient of the entire tf.where will be NaN. This is because the gradient calculation for tf.where combines the two branches, for performance reasons.

A workaround is to use an inner tf.where to ensure the function has no asymptote, and to avoid computing a value whose gradient is NaN by replacing dangerous inputs with safe inputs.

Instead of this,

x = tf.constant(0., dtype=tf.float32)
with tf.GradientTape() as tape:
  tape.watch(x)
  y = tf.where(x < 1., 0., 1. / x)
print(tape.gradient(y, x))
tf.Tensor(nan, shape=(), dtype=float32)

Although, the 1. / x values are never used, its gradient is a NaN when x = 0. Instead, we should guard that with another tf.where

x = tf.constant(0., dtype=tf.float32)
with tf.GradientTape() as tape:
  tape.watch(x)
  safe_x = tf.where(tf.equal(x, 0.), 1., x)
  y = tf.where(x < 1., 0., 1. / safe_x)
print(tape.gradient(y, x))
tf.Tensor(0.0, shape=(), dtype=float32)

See also:

condition A tf.Tensor of dtype bool, or any numeric dtype. condition must have dtype bool when x and y are provided.
x If provided, a Tensor which is of the same type as y, and has a shape broadcastable with condition and y.
y If provided, a Tensor which is of the same type as x, and has a shape broadcastable with condition and x.
name A name of the operation (optional).

If x and y are provided: A Tensor with the same type as x and y, and shape that is broadcast from condition, x, and y. Otherwise, a Tensor with shape [tf.math.count_nonzero(condition), tf.rank(condition)].

ValueError When exactly one of x or y is non-None, or the shapes are not all broadcastable.