PaddlePaddle's RNN doesn't require that all instances have the same length. To do so, we introduce an extension to Tensor, namely, LoD Tensor.
## Challenge of Variable-length Inputs
People usually represent a mini-batch by a Tensor. For example, a mini-batch of 32 images, each of size 32x32, is a 10x32x32 Tensor. So a transformation, T, of all images can be a matrix multiplication of the 32x32xO-dimensional tensor T and the 10x32x32 Tensor.
Another example is that each mini-batch contains 32 sentences, where each word is a D-dimensional one-hot vector. If all sentences have the same length L, we can represent this mini-batch by a 32xLxD tensor. However, in most cases, sentences have variable lengths, and we will need an index data structure to record these variable lengths.
## LoD as a Solution
### Mini-Batch of variable-length sentenses
Let's imagine a mini-batch of 3 variable lengths sentences, containing 3, 1, and 2 words respectively. We can represent it by a (3+1+2)xD tensor plus some index information:
```
3
3 1 2
||| | ||
```
Each `|` represents a D-dimensional word vectors. The number 3 on top indicate 3 sentences, and numbers 3, 1, and 2 on the second level represent the number of words in each sentence.
### Mini-Batch of variable-length videos
This approach generalizes to the case where elements are not words, but higher dimensional objects, like images. Suppose that a mini-batch contains videos of the same frame size 640x480. If a mini-batch contains 3 videos of 3, 1, and 2 frames respectively. The underlying tensor is of size (3+1+2)x640x480. The index information illustrates as:
```
3
3 1 2
口口口 口 口口
```
where each `口` represents an image.
### Mini-Batch of fixed-size images
Let's get back to a typical example, image classification, where each mini-batch has M fixed-sized images. The LoD Tensor representation is
```
M
1 1 1 1 1
口口口口 ... 口
```
The many 1's on the second level seem duplicated. For this particular case of 2 levels and the second level always have length 1, we can ignore the LoD index.
### Design and summarization
In summary, as long as that the essential elements (words or images) have the same size, we can represent mini-batches by a LoD Tensor:
- The underlying tensor has size LxD1xD2x..., where D1xD2... is the size of the essential elements, and
- the first dimension size L has an additon property -- a LoD index as a nested vector:
```c++
typedefstd::vector<std::vector>>LoD;
```
- The LoD index can is not necessary when there are only two levels and all elements of the second level have length 1.
## Slicing of LoD Tensor
Consider that we have a network with three levels of RNN: the top level one handles articles, the second level one handles sentences, and the basic level one handles words. This network requires that mini-batches represented by 4 level LoD Tensor, for example,
```
3
3 1 2
3 2 4 1 2 3
||| || |||| | || |||
```
To allow each level of RNN to handle its input, we define **the slicing of a LoD Tensor is defined as getting the j-th sequence on level i, or the <i,j>-slice**
For example, the <2,1>-slice of above slice is
```
2
||
```
and the <1,2>-slice of above example is
```
2
2 3
|| |||
```
Let's go on slicing this slice. Its <1,1>-slice is
```
3
|||
```
### The General Slicing Algorithm
The algorithm, with over-simplified data structure, is defined as
