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+# Design Doc: Selected Rows
+
+`SelectedRows` is a kind of sparse tensor data type, which is designed to support `embedding` operators. The gradient of embedding table is a sparse tensor. Only a few rows are non-zero values in that tensor. It is straightforward to represent the sparse tensor by the following sparse tensor data structure:
+
+```cpp
+class SelectedRows {
+ private:
+  vector<int> rows_;
+  Tensor value_;
+  int height_;
+};
+```
+
+The field `height_` shows the first dimension of `SelectedRows`. The `rows` are the indices of which rows of `SelectedRows` are non-zeros. The `value_` field is an N-dim tensor and shape is `[rows.size() /* NUM_ROWS */, ...]`, which supplies values for each row. The dimension of `SelectedRows` satisfies `[height_] + value_.shape[1:]`.
+
+Suppose that a SelectedRows-typed variable `x` has many rows, but only two of them have values -- row 73 is `[1, 2]` and row 84 is `[3, 4]`, the `SelectedRows` representation would be:
+
+```
+x = SelectedRow {
+  rows = [73, 84],
+  value = [[1, 2], [3,4]]
+}
+```
+
+
+## SelectedRows in Protobuf
+
+`SelectedRows` is a kind of `Variable`. `VarDesc` in protobuf should describe the `SelectedRows` information. Only the tensor dimension of a `SelectedRows` will be described in compile-time since the `rows_` and `value_` are related to training data. 
+So we use `TensorDesc` to unify `data_type` and `dims`. A LodTensorDesc contains a `TensorDesc` and `lod_level`. The description of `SelectedRows` is a Tensor description.
+
+```proto
+message TensorDesc {
+  required DataType data_type = 1;
+  repeated int64 dims = 2; // [UNK, 640, 480] is saved as [-1, 640, 480]
+}
+
+message LodTensorDesc {
+  required TensorDesc tensor = 1;
+  optional int lod_level = 2;
+}
+
+message VarDesc {
+  required string name = 1;
+  enum VarType { 
+    LOD_TENSOR = 0;
+    SELECTED_ROWS = 1;
+  }
+  required VarType type = 2;
+  optional LodTensorDesc lod_desc = 3;
+  optional TensorDesc selected_rows_desc = 4;
+  optional bool persistable = 5 [ default = false ];
+}
+```
+
+## InferShape for Selected Rows
+
+Just like `LoD` information, `InferShape` method will inference output tensor type as well. The operator should decide whether its output is a `SelectedRows` or `Dense` tensor.
+
+For example, the gradient operator of `TableLookup` will always generate `SelectedRows`. Its `InferShape` method should be like following
+
+```cpp
+void TableLookupGrad::InferShape(context) {
+  ...
+  context.SetDataType("Embedding.Grad", kSelectedRows);
+}
+```
+
+
+## Sparse Operators
+
+There are several operators should be written to support `SelectedRows`. They are:
+
+1. Operators which generates `SelectedRows` gradient. e.g. Gradient of `TableLookupOp`.
+2. Optimize operators which support `SelectedRows` gradient. e.g. `SGD` or `AdaGrad` for `SelectedRows`. However, there should be only one `SGD` operator. `OpWithKernel::Run` should select a suitable kernel for both `dense` tensor or `SelectedRows`.