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Remove breaks to improve readability
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José Valim
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1 changed files with 0 additions and 52 deletions
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@ -39,8 +39,6 @@ Systems of equations are a central theme in numerical computing.
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These equations are often expressed and solved with multidimensional
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arrays. For example, this is a two dimensional array:
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<!-- livebook:{"break_markdown":true} -->
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$$
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\begin{bmatrix}
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1 & 2 \\
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@ -48,8 +46,6 @@ $$
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\end{bmatrix}
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$$
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<!-- livebook:{"break_markdown":true} -->
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Elixir programmers typically express a similar data structure using
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a list of lists, like this:
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@ -77,8 +73,6 @@ than `Enum.map/2` and `Enum.reduce/3`.
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In this section, we'll look at some of the various tools for
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creating and interacting with tensors.
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<!-- livebook:{"break_markdown":true} -->
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To get started, we'll first need to include the Nx dependency.
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In Livebook, we can simply use Mix install. At the time of this writing,
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there's no Hex package for Nx, so we'll use a git
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@ -102,8 +96,6 @@ Now, everything is set up, so we're ready to create some tensors.
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### Creating tensors
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<!-- livebook:{"break_markdown":true} -->
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Start out by getting a feel for Nx through its documentation.
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Do so through the IEx helpers, like this:
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@ -138,8 +130,6 @@ The result shows all of the major fields that make up a tensor:
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* The shape of the tensor, going left to right, with the outside dimensions listed first.
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* The names of each dimension.
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<!-- livebook:{"break_markdown":true} -->
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We can easily convert it to a binary:
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```elixir
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@ -168,8 +158,6 @@ because they must often be processed. Elixir binaries make quick work
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of dealing with numerical data structured for platforms other than
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Elixir.
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<!-- livebook:{"break_markdown":true} -->
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We can get any cell of the tensor:
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```elixir
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@ -200,8 +188,6 @@ Now,
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* return a `{2, 2}` tensor containing the first two columns
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of the first two rows
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<!-- livebook:{"break_markdown":true} -->
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We can get information about this most recent term with
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the IEx helper `i`, like this:
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@ -223,8 +209,6 @@ to get more documentation about tensors. We could also open an Elixir
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cell, type Nx.tensor, and hover the cursor over the word `tensor`
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to see the help about that function.
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<!-- livebook:{"break_markdown":true} -->
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We can get the shape of the tensor with `Nx.shape/1`:
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```elixir
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@ -242,8 +226,6 @@ changes the metadata, so it has no notable cost.
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The new tensor has the same type, but a new shape.
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<!-- livebook:{"break_markdown":true} -->
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Now, reshape the tensor to contain three dimensions with
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one batch, one row, and four columns.
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@ -263,8 +245,6 @@ Nx.tensor([[1, 2, 3]], names: [:rows, :cols], type: {:u, 8})
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We created a tensor of the shape `{1, 3}`, with the type `u8`,
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the values `[1, 2, 3]`, and two axes named `rows` and `cols`.
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<!-- livebook:{"break_markdown":true} -->
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Now we know how to create tensors, so it's time to do something with them.
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## Tensor aware functions
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@ -324,8 +304,6 @@ Nx.sum(tensor, axes: [:x])
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Nx sums the values across the `x` dimension: `1 + 2` in the first row
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and `3 + 4` in the second row.
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<!-- livebook:{"break_markdown":true} -->
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Now,
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* create a `{2, 2, 2}` tensor
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@ -341,8 +319,6 @@ Sometimes, we need to combine two tensors together with an
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operator. Let's say we wanted to subtract one tensor from
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another. Mathematically, the expression looks like this:
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<!-- livebook:{"break_markdown":true} -->
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$$
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\begin{bmatrix}
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5 & 6 \\
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@ -358,8 +334,6 @@ $$
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\end{bmatrix}
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$$
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<!-- livebook:{"break_markdown":true} -->
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To solve this problem, add each integer on the left with the
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corresponding integer on the right. Unfortunately, we cannot
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use Elixir's built-in subtraction operator as it is not tensor-aware.
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@ -383,8 +357,6 @@ Often, the dimensions of tensors in an operator don't match.
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For example, you might want to subtract a `1` from every
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element of a `{2, 2}` tensor, like this:
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<!-- livebook:{"break_markdown":true} -->
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$$
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\begin{bmatrix}
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1 & 2 \\
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@ -396,8 +368,6 @@ $$
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\end{bmatrix}
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$$
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<!-- livebook:{"break_markdown":true} -->
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Mathematically, it's the same as this:
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$$
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@ -415,8 +385,6 @@ $$
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\end{bmatrix}
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$$
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<!-- livebook:{"break_markdown":true} -->
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That means we need a way to convert `1` to a `{2, 2}` tensor.
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`Nx.broadcast/2` solves that problem. This function takes
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a tensor or a scalar and a shape.
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@ -451,8 +419,6 @@ Nx.subtract(10, tensor)
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Or subtract a row or column. Mathematically, it would look like this:
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<!-- livebook:{"break_markdown":true} -->
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$$
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\begin{bmatrix}
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1 & 2 \\
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@ -467,12 +433,8 @@ $$
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\end{bmatrix}
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$$
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<!-- livebook:{"break_markdown":true} -->
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which is the same as this:
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<!-- livebook:{"break_markdown":true} -->
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$$
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\begin{bmatrix}
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1 & 2 \\
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@ -488,8 +450,6 @@ $$
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\end{bmatrix}
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$$
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<!-- livebook:{"break_markdown":true} -->
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This rewrite happens in Nx too, also through a broadcast. We want to
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broadcast the tensor `[1, 2]` to match the `{2, 2}` shape, like this:
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@ -531,8 +491,6 @@ h Nx.broadcast
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Much of the time, you won't have to broadcast yourself. Many of
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the functions and operators Nx supports will do so automatically.
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<!-- livebook:{"break_markdown":true} -->
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We can use tensor-aware operators via various `Nx` functions and
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many of them implicitly broadcast tensors.
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@ -556,8 +514,6 @@ over classic Elixir functions.
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individual operations and using a just-in-time (JIT) compiler to emit
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highly specialized native code for the desired computation unit.
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<!-- livebook:{"break_markdown":true} -->
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We don't have to do anything special to get access to
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get tensor awareness beyond importing `Nx.Defn` and writing
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our code within a `defn` block.
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@ -612,8 +568,6 @@ defn subtract(a, b) do
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end
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```
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<!-- livebook:{"break_markdown":true} -->
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Now, it's your turn. Add a `defn` to `TensorMath`
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that accepts two tensors representing the lengths of sides of a
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right triangle and uses the pythagorean theorem to return the
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@ -647,20 +601,14 @@ a function returning the gradient. We have two functions: `poly/1`
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is a simple numerical definitin, and `poly_slope_at/1` returns
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its gradient:
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<!-- livebook:{"break_markdown":true} -->
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$$
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poly: f(x) = 3x^2 + 2x + 1 \\
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$$
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<!-- livebook:{"break_markdown":true} -->
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$$
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polySlopeAt: g(x) = 6x + 2
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$$
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<!-- livebook:{"break_markdown":true} -->
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Here's the Elixir equivalent of those functions:
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```elixir
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