We develop, prepare, and deploy TensorFlow fashions from R. However that doesn’t imply we don’t make use of documentation, weblog posts, and examples written in Python. We glance up particular performance within the official TensorFlow API docs; we get inspiration from different individuals’s code.

Relying on how snug you might be with Python, there’s an issue. For instance: You’re presupposed to know the way *broadcasting* works. And maybe, you’d say you’re vaguely accustomed to it: So when arrays have completely different shapes, some parts get duplicated till their shapes match and … and isn’t R vectorized anyway?

Whereas such a worldwide notion may fit usually, like when skimming a weblog put up, it’s not sufficient to know, say, examples within the TensorFlow API docs. On this put up, we’ll attempt to arrive at a extra precise understanding, and examine it on concrete examples.

Talking of examples, listed below are two motivating ones.

## Broadcasting in motion

The primary makes use of TensorFlow’s `matmul`

to multiply two tensors. Would you wish to guess the outcome – not the numbers, however the way it comes about usually? Does this even run with out error – shouldn’t matrices be two-dimensional (*rank*-2 tensors, in TensorFlow communicate)?

```
a tf$fixed(keras::array_reshape(1:12, dim = c(2, 2, 3)))
a
# tf.Tensor(
# [[[ 1. 2. 3.]
# [ 4. 5. 6.]]
#
# [[ 7. 8. 9.]
# [10. 11. 12.]]], form=(2, 2, 3), dtype=float64)
b tf$fixed(keras::array_reshape(101:106, dim = c(1, 3, 2)))
b
# tf.Tensor(
# [[[101. 102.]
# [103. 104.]
# [105. 106.]]], form=(1, 3, 2), dtype=float64)
c tf$matmul(a, b)
```

Second, here’s a “actual instance” from a TensorFlow Chance (TFP) github concern. (Translated to R, however holding the semantics).

In TFP, we will have *batches* of distributions. That, per se, isn’t a surprise. However have a look at this:

```
library(tfprobability)
d tfd_normal(loc = c(0, 1), scale = matrix(1.5:4.5, ncol = 2, byrow = TRUE))
d
# tfp.distributions.Regular("Regular", batch_shape=[2, 2], event_shape=[], dtype=float64)
```

We create a batch of 4 regular distributions: every with a unique *scale* (1.5, 2.5, 3.5, 4.5). However wait: there are solely two *location* parameters given. So what are their *scales*, respectively?

Fortunately, TFP builders Brian Patton and Chris Suter defined the way it works: TFP truly does broadcasting – with distributions – identical to with tensors!

We get again to each examples on the finish of this put up. Our principal focus shall be to elucidate broadcasting as achieved in NumPy, as NumPy-style broadcasting is what quite a few different frameworks have adopted (e.g., TensorFlow).

Earlier than although, let’s shortly evaluation just a few fundamentals about NumPy arrays: index or *slice* them (indexing usually referring to single-element extraction, whereas slicing would yield – effectively – slices containing a number of parts); easy methods to parse their shapes; some terminology and associated background.

Although not sophisticated per se, these are the sorts of issues that may be complicated to rare Python customers; but they’re usually a prerequisite to efficiently making use of Python documentation.

Acknowledged upfront, we’ll actually limit ourselves to the fundamentals right here; for instance, we gained’t contact superior indexing which – identical to tons extra –, could be regarded up intimately within the NumPy documentation.

## Few info about NumPy

### Primary slicing

For simplicity, we’ll use the phrases indexing and slicing kind of synonymously any further. The essential gadget here’s a *slice*, particularly, a `begin:cease`

construction indicating, for a single dimension, which vary of parts to incorporate within the choice.

In distinction to R, Python indexing is zero-based, and the top index is unique:

And second, once we add tensors with shapes `(3, 3)`

and `(3,)`

, the 1-d tensor ought to get added to each row (not each column):

```
a tf$fixed(matrix(1:9, ncol = 3, byrow = TRUE), dtype = tf$float32)
a
# tf.Tensor(
# [[1. 2. 3.]
# [4. 5. 6.]
# [7. 8. 9.]], form=(3, 3), dtype=float32)
b tf$fixed(c(100, 200, 300))
b
# tf.Tensor([100. 200. 300.], form=(3,), dtype=float32)
a + b
# tf.Tensor(
# [[101. 202. 303.]
# [104. 205. 306.]
# [107. 208. 309.]], form=(3, 3), dtype=float32)
```

Now again to the preliminary `matmul`

instance.

## Again to the puzzles

The documentation for matmul says,

The inputs should, following any transpositions, be tensors of rank >= 2 the place the internal 2 dimensions specify legitimate matrix multiplication dimensions, and any additional outer dimensions specify matching batch dimension.

So right here (see code slightly below), the internal two dimensions look good – `(2, 3)`

and `(3, 2)`

– whereas the one (one and solely, on this case) batch dimension reveals mismatching values `2`

and `1`

, respectively.

A case for broadcasting thus: Each “batches” of `a`

get matrix-multiplied with `b`

.

```
a tf$fixed(keras::array_reshape(1:12, dim = c(2, 2, 3)))
a
# tf.Tensor(
# [[[ 1. 2. 3.]
# [ 4. 5. 6.]]
#
# [[ 7. 8. 9.]
# [10. 11. 12.]]], form=(2, 2, 3), dtype=float64)
b tf$fixed(keras::array_reshape(101:106, dim = c(1, 3, 2)))
b
# tf.Tensor(
# [[[101. 102.]
# [103. 104.]
# [105. 106.]]], form=(1, 3, 2), dtype=float64)
c tf$matmul(a, b)
c
# tf.Tensor(
# [[[ 622. 628.]
# [1549. 1564.]]
#
# [[2476. 2500.]
# [3403. 3436.]]], form=(2, 2, 2), dtype=float64)
```

Let’s shortly examine this actually is what occurs, by multiplying each batches individually:

```
tf$matmul(a[1, , ], b)
# tf.Tensor(
# [[[ 622. 628.]
# [1549. 1564.]]], form=(1, 2, 2), dtype=float64)
tf$matmul(a[2, , ], b)
# tf.Tensor(
# [[[2476. 2500.]
# [3403. 3436.]]], form=(1, 2, 2), dtype=float64)
```

Is it too bizarre to be questioning if broadcasting would additionally occur for matrix dimensions? E.g., might we attempt `matmul`

ing tensors of shapes `(2, 4, 1)`

and `(2, 3, 1)`

, the place the `4 x 1`

matrix could be broadcast to `4 x 3`

? – A fast take a look at reveals that no.

To see how actually, when coping with TensorFlow operations, it pays off overcoming one’s preliminary reluctance and really seek the advice of the documentation, let’s attempt one other one.

Within the documentation for matvec, we’re informed:

Multiplies matrix a by vector b, producing a * b.

The matrix a should, following any transpositions, be a tensor of rank >= 2, with form(a)[-1] == form(b)[-1], and form(a)[:-2] in a position to broadcast with form(b)[:-1].

In our understanding, given enter tensors of shapes `(2, 2, 3)`

and `(2, 3)`

, `matvec`

ought to carry out two matrix-vector multiplications: as soon as for every batch, as listed by every enter’s leftmost dimension. Let’s examine this – to this point, there isn’t a broadcasting concerned:

```
# two matrices
a tf$fixed(keras::array_reshape(1:12, dim = c(2, 2, 3)))
a
# tf.Tensor(
# [[[ 1. 2. 3.]
# [ 4. 5. 6.]]
#
# [[ 7. 8. 9.]
# [10. 11. 12.]]], form=(2, 2, 3), dtype=float64)
b = tf$fixed(keras::array_reshape(101:106, dim = c(2, 3)))
b
# tf.Tensor(
# [[101. 102. 103.]
# [104. 105. 106.]], form=(2, 3), dtype=float64)
c tf$linalg$matvec(a, b)
c
# tf.Tensor(
# [[ 614. 1532.]
# [2522. 3467.]], form=(2, 2), dtype=float64)
```

Doublechecking, we manually multiply the corresponding matrices and vectors, and get:

```
tf$linalg$matvec(a[1, , ], b[1, ])
# tf.Tensor([ 614. 1532.], form=(2,), dtype=float64)
tf$linalg$matvec(a[2, , ], b[2, ])
# tf.Tensor([2522. 3467.], form=(2,), dtype=float64)
```

The identical. Now, will we see broadcasting if `b`

has only a single batch?

```
b = tf$fixed(keras::array_reshape(101:103, dim = c(1, 3)))
b
# tf.Tensor([[101. 102. 103.]], form=(1, 3), dtype=float64)
c tf$linalg$matvec(a, b)
c
# tf.Tensor(
# [[ 614. 1532.]
# [2450. 3368.]], form=(2, 2), dtype=float64)
```

Multiplying each batch of `a`

with `b`

, for comparability:

```
tf$linalg$matvec(a[1, , ], b)
# tf.Tensor([ 614. 1532.], form=(2,), dtype=float64)
tf$linalg$matvec(a[2, , ], b)
# tf.Tensor([[2450. 3368.]], form=(1, 2), dtype=float64)
```

It labored!

Now, on to the opposite motivating instance, utilizing *tfprobability*.

### Broadcasting in all places

Right here once more is the setup:

```
library(tfprobability)
d tfd_normal(loc = c(0, 1), scale = matrix(1.5:4.5, ncol = 2, byrow = TRUE))
d
# tfp.distributions.Regular("Regular", batch_shape=[2, 2], event_shape=[], dtype=float64)
```

What’s going on? Let’s examine *location* and *scale* individually:

```
d$loc
# tf.Tensor([0. 1.], form=(2,), dtype=float64)
d$scale
# tf.Tensor(
# [[1.5 2.5]
# [3.5 4.5]], form=(2, 2), dtype=float64)
```

Simply specializing in these tensors and their shapes, and having been informed that there’s broadcasting occurring, we will cause like this: Aligning each shapes on the correct and lengthening `loc`

’s form by `1`

(on the left), now we have `(1, 2)`

which can be broadcast with `(2,2)`

– in matrix-speak, `loc`

is handled as a row and duplicated.

Which means: We’ve two distributions with imply (0) (one in every of scale (1.5), the opposite of scale (3.5)), and in addition two with imply (1) (corresponding scales being (2.5) and (4.5)).

Right here’s a extra direct technique to see this:

```
d$imply()
# tf.Tensor(
# [[0. 1.]
# [0. 1.]], form=(2, 2), dtype=float64)
d$stddev()
# tf.Tensor(
# [[1.5 2.5]
# [3.5 4.5]], form=(2, 2), dtype=float64)
```

Puzzle solved!

Summing up, broadcasting is easy “in principle” (its guidelines are), however may have some training to get it proper. Particularly along with the truth that features / operators do have their very own views on which elements of its inputs ought to broadcast, and which shouldn’t. Actually, there isn’t a means round wanting up the precise behaviors within the documentation.

Hopefully although, you’ve discovered this put up to be an excellent begin into the subject. Perhaps, just like the creator, you are feeling such as you would possibly see broadcasting occurring wherever on the earth now. Thanks for studying!