Contents

# Two-Dimensional Locally-connected Backward Layer

The forward two-dimensional (2D) locally-connected layer computes the value tensor
Y
by applying a set of
nKernels
2D kernels
K
of size
m
1
x
m
2
to the input argument
x
. The library supports four-dimensional input tensors
X
R
n
1
x
n
2
x
n
3
x
n
4
. Therefore, the following formula applies:
where
i
+
a
<
n
1
,
j
+
b
<
n
2
, and
r
is the kernel index.
A set of kernels is specific to the selected dimensions of the input argument
x
.
For more details, see Forward 2D Locally-connected Layer.
The backward 2D locally-connected layer computes the derivatives of the objective function
E
with respect to the input argument, weights, and biases.

## Problem Statement

Without loss of generality, let's assume that convolution kernels are applied to the last two dimensions.
Given:
• Four-dimensional tensor
G
R
n
1
x
nKernels
x
l
3
x
l
4
with the gradient from the preceding layer
• Four-dimensional tensor
X
R
n
1
x
n
2
x
n
3
x
n
4
with input data of the forward layer
• Six-dimensional tensor
K
R
nKernels
x
l
3
x
l
4
x
m
2
x
m
3
x
m
4
with kernel parameters/weights
• Three-dimensional tensor
B
R
nKernels
x
l
3
x
l
4
with the bias of each kernel.
For the above tensors:
• and
p
i
is the respective padding.
• nGroups
is defined as follows: let's assume that
n
2
is the group dimension in the input tensor for the forward 2D locally-connected layer. The output gradient tensor is split along this dimension into
nGroups
groups, and the input gradient tensor and weights tensor are split into
nGroups
groups along the
nKernels
dimension.
nKernels
and
n
2
must be multiples of
nGroups
.
The problem is to compute:
• Four-dimensional tensor
Z
R
n
1
x
n
2
x
n
3
x
n
4
such that:
• Values:
In the above formulas:
• s
3
and
s
4
are strides

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Notice revision #20110804