Add LoD Tensor design doc

paddle/framework/lod_tensor.md

+# LoD (Level-of-Detail) Tensor
+
+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++
+  typedef std::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
+
+```c++
+typedef vector<vector<int> > LoD;
+
+struct LoDTensor {
+  LoD lod_;
+  float* tensor_;
+};
+
+LoDTensor Slice(const LoDTensor& lodt, int level, int sequence) {
+
+}
+```
+
+### Slicing the Top Level
+
+Please be aware that an RNN operator only slices the top level of a LoD Tensor to get the step inputs.
+
+```c++
+LoDTensor Slice(const LoDTensor& lodt, int sequence) {
+
+}
+```