Initial prototype for Deep Learning

Representing tensor and images in SystemML

In this prototype, we represent a tensor as a matrix stored in a row-major format, where first dimension of tensor and matrix are exactly the same. For example, a tensor (with all zeros) of shape [3, 2, 4, 5] can be instantiated by following DML statement:

A = matrix(0, rows=3, cols=2*4*5) 

Tensor functions:

Element-wise arithmetic operators:

Following operators work out-of-the box when both tensors X and Y have same shape:

• Element-wise exponentiation: X ^ Y
• Element-wise unary minus: -X
• Element-wise integer division: X %/% Y
• Element-wise modulus operation: X %% Y
• Element-wise multiplication: X * Y
• Element-wise division: X / Y
• Element-wise addition: X + Y
• Element-wise subtraction: X - Y

SystemML does not support implicit broadcast for above tensor operations, however one can write a DML-bodied function to do so. For example: to perform the above operations with broadcasting on second dimensions, one can use the below rep(Z, n) function:

rep = function(matrix[double] Z, int C) return (matrix[double] ret) {
	ret = Z
	for(i in 2:C) {
		ret = cbind(ret, Z)
	}
}

Using the above rep(Z, n) function, we can realize the element-wise arithmetic operation with broadcasting. Here are some examples:

• X of shape [N, C, H, W] and Y of shape [1, C, H, W]: X + Y (Note: SystemML does implicit broadcasting in this case because of the way it represents the tensor)
• X of shape [1, C, H, W] and Y of shape [N, C, H, W]: X + Y (Note: SystemML does implicit broadcasting in this case because of the way it represents the tensor)
• X of shape [N, C, H, W] and Y of shape [N, 1, H, W]: X + rep(Y, C)
• X of shape [N, C, H, W] and Y of shape [1, 1, H, W]: X + rep(Y, C)
• X of shape [N, 1, H, W] and Y of shape [N, C, H, W]: rep(X, C) + Y
• X of shape [1, 1, H, W] and Y of shape [N, C, H, W]: rep(X, C) + Y

TODO: Map the NumPy tensor calls to DML expressions.

Representing images in SystemML

The images are assumed to be stored NCHW format, where N = batch size, C = #channels, H = height of image and W = width of image. Hence, the images are internally represented as a matrix with dimension (N, C * H * W).

Convolution and Pooling built-in functions

This prototype also contains initial implementation of forward/backward functions for 2D convolution and pooling:

• conv2d(x, w, ...)
• conv2d_backward_filter(x, dout, ...) and conv2d_backward_data(w, dout, ...)
• max_pool(x, ...) and max_pool_backward(x, dout, ...)

The required arguments for all above functions are:

• stride=[stride_h, stride_w]
• padding=[pad_h, pad_w]
• input_shape=[numImages, numChannels, height_image, width_image]

The additional required argument for conv2d/conv2d_backward_filter/conv2d_backward_data functions is:

• filter_shape=[numFilters, numChannels, height_filter, width_filter]

The additional required argument for max_pool/avg_pool functions is:

• pool_size=[height_pool, width_pool]

The results of these functions are consistent with Nvidia's CuDNN library.

Border mode:

• To perform valid padding, use padding = (input_shape-filter_shape)*(stride-1)/ 2. (Hint: for stride length of 1, padding = [0, 0] performs valid padding).

• To perform full padding, use padding = ((stride-1)*input_shape + (stride+1)*filter_shape - 2*stride) / 2. (Hint: for stride length of 1, padding = [filter_h-1, filter_w-1] performs full padding).

• To perform same padding, use padding = (input_shape*(stride-1) + filter_shape - stride)/2. (Hint: for stride length of 1, padding = [(filter_h-1)/2, (filter_w-1)/2] performs same padding).

Explanation of backward functions for conv2d

Consider one-channel 3 X 3 image =

x1	x2	x3
x4	x5	x6
x7	x8	x9

and one 2 X 2 filter:

w1	w2
w3	w4

Then, conv2d(x, w, stride=[1, 1], padding=[0, 0], input_shape=[1, 1, 3, 3], filter_shape=[1, 1, 2, 2]) produces following tensor of shape [1, 1, 2, 2], which is represented as 1 X 4 matrix in NCHW format:

w1*x1 + w2*x2 + w3*x4 + w4*x5	w1*x2 + w2*x3 + w3*x5 + w4*x6	w1*x4 + w2*x5 + w3*x7 + w4*x8	w1*x5 + w2*x6 + w3*x8 + w4*x9

Let the error propagated from above layer is

y1	y2	y3	y4

Then conv2d_backward_filter(x, y, stride=[1, 1], padding=[0, 0], input_shape=[1, 1, 3, 3], filter_shape=[1, 1, 2, 2]) produces following updates for the filter:

y1*x1 + y2*x2 + y3*x4 + y4*x5	y1*x2 + y2*x3 + y3*x5 + y4*x6
y1*x4 + y2*x5 + y3*x7 + y4*x8	y1*x5 + y2*x6 + y3*x8 + y4*x9

Note: since the above update is a tensor of shape [1, 1, 2, 2], it will be represented as matrix of dimension [1, 4].

Similarly, conv2d_backward_data(w, y, stride=[1, 1], padding=[0, 0], input_shape=[1, 1, 3, 3], filter_shape=[1, 1, 2, 2]) produces following updates for the image:

w1*y1	w2*y1 + w1*y2	w2*y2
w3*y1 + w1*y3	w4*y1 + w3*y2 + w2*y3 + w1*y4	w4*y2 + w2*y4
w3*y3	w4*y3 + w3*y4	w4*y4