Transposed convolution handles multiple input and output channels using the same fundamental principles as regular convolution. If the input tensor has $$c_i$$ channels, the transposed convolution applies a distinct $$k_h 	imes k_w$$ spatial kernel to each individual input channel. When configuring the layer to produce multiple output channels, it allocates a complete set of these $$c_i$$ kernels for every desired output channel. As a result, the complete multi-channel kernel tensor possesses a total shape of $$c_i 	imes k_h 	imes k_w$$ for each output channel, allowing the network to learn rich, multi-dimensional upsampling transformations.

Multiple Channels in Transposed Convolution

Unlike regular convolution, where padding is added to the borders of the input tensor, padding in transposed convolution is applied directly to the output. Specifying a padding value removes that number of rows and columns from the outer boundaries of the transposed convolution result. For example, if we apply a padding of $$1$$ on all sides, the first and last rows and columns are discarded from the final output tensor. In frameworks like PyTorch and MXNet, this can be implemented using the `padding=1` argument in `nn.ConvTranspose2d` or `nn.Conv2DTranspose`.

```python
# PyTorch
tconv = nn.ConvTranspose2d(1, 1, kernel_size=2, padding=1, bias=False)
tconv.weight.data = K
tconv(X)
```

```python
# MXNet
tconv = nn.Conv2DTranspose(1, kernel_size=2, padding=1)
tconv.initialize(init.Constant(K))
tconv(X)
```

Padding in Transposed Convolution

In transposed convolution, strides determine the spatial arrangement of the intermediate results rather than the sliding movement of the kernel over the input. Specifically, a stride greater than $$1$$ increases the spacing between the broadcasted elements in the intermediate tensor, which expands both the height and width of the final output. For instance, increasing the stride from $$1$$ to $$2$$ with a $$2 \times 2$$ kernel places the generated values two pixels apart in the intermediate matrix. This expansion mechanism can be implemented by setting the stride parameter to `stride=2` in PyTorch's `nn.ConvTranspose2d` or `strides=2` in MXNet's `nn.Conv2DTranspose`.

```python
# PyTorch
tconv = nn.ConvTranspose2d(1, 1, kernel_size=2, stride=2, bias=False)
tconv.weight.data = K
tconv(X)
```

```python
# MXNet
tconv = nn.Conv2DTranspose(1, kernel_size=2, strides=2)
tconv.initialize(init.Constant(K))
tconv(X)
```

Strides in Transposed Convolution

A primary application of transposed convolutional layers is to perfectly reverse the spatial dimensionality transformations introduced by standard convolutional layers. If an input tensor $$\mathsf{X}$$ is passed through a convolutional layer $$f$$ to produce an output $$\mathsf{Y} = f(\mathsf{X})$$, a transposed convolutional layer $$g$$ can be constructed to invert this shape change. By configuring $$g$$ with the exact same spatial hyperparameters (kernel size, padding, and stride) as $$f$$, and setting its output channel count to match the channels of $$\mathsf{X}$$, the resulting tensor $$g(\mathsf{Y})$$ will possess the identical shape as the original input $$\mathsf{X}$$. This property can be verified programmatically in frameworks like PyTorch and MXNet by ensuring `tconv(conv(X)).shape == X.shape` when both layers share identical parameters.

```python
# PyTorch
X = torch.rand(size=(1, 10, 16, 16))
conv = nn.Conv2d(10, 20, kernel_size=5, padding=2, stride=3)
tconv = nn.ConvTranspose2d(20, 10, kernel_size=5, padding=2, stride=3)
tconv(conv(X)).shape == X.shape
```

```python
# MXNet
X = np.random.uniform(size=(1, 10, 16, 16))
conv = nn.Conv2D(20, kernel_size=5, padding=2, strides=3)
tconv = nn.Conv2DTranspose(10, kernel_size=5, padding=2, strides=3)
conv.initialize()
tconv.initialize()
tconv(conv(X)).shape == X.shape
```

Transposed Convolution Shape Reversal

In contrast to a regular convolution that reduces input elements via a kernel, the transposed convolution broadcasts input elements via the kernel, producing an output that is generally larger than the input. For a basic transposed convolution with a stride of $$1$$ and no padding, an $$n_h 	imes n_w$$ input tensor and a $$k_h 	imes k_w$$ kernel are used to produce an $$(n_h + k_h - 1) 	imes (n_w + k_w - 1)$$ output tensor. For each element in the input tensor, it is multiplied by the entire kernel to produce a $$k_h 	imes k_w$$ tensor. This resulting tensor replaces a corresponding portion of an intermediate tensor initialized with zeros, based on the element's original position. The final output is formed by summing all these intermediate results. For example, if both the input tensor and the kernel are $$2 	imes 2$$ matrices with values $$[[0, 1], [2, 3]]$$, the basic transposed convolution yields a $$3 	imes 3$$ output tensor with values $$[[0, 0, 1], [0, 4, 6], [4, 12, 9]]$$.

Claude

The cross-correlation operation involves mapping a filter matrix (or kernel) to every possible position on an input matrix. At each position, the corresponding elements of the input and the kernel are multiplied together, and the products are summed to calculate a single entry in the resulting output matrix. While a strict mathematical convolution requires flipping the two-dimensional kernel both horizontally and vertically prior to this process, deep learning implementations typically omit this mirroring step and compute the cross-correlation directly. Despite this technical distinction, it is standard terminology in deep learning literature to refer to the cross-correlation operation simply as a convolution.

The Cross-Correlation (Convolution) Operation

Dive into Deep Learning

This node before explained how it works, and now we need to understand how to write the operation mathematically and more importantly code it efficiently. So suppose we have the input I with the shape H x H x C, where B - the batch, H - is the height and width of the image( just taking square image for the example), C - the amount of channels. One kernel k has a size of K x K x C( also taking square kernel for the sake of the example). b is the bias for that kernel

$ ( I * K)_{i,j} = \sum_{m=0}^{k-1} \sum_{n=0}^{k-1} \sum_{c=0}^{C} K_{m,n,c} I_{i + m, j + n, c} + b$
-------------------------------------------------
$ ( I * K)$ - is the output matrix for the current kernel with size of H-K+1, H-K+1. This is the variation of the Convolution with no padding and stride one. Most of the libraries call it convolution but it is actually an operation called cross-correlation. In terms of coding the formula for convolution, there are several ways to implement it. One of them might be just straight brute force with nested loops, turning everything int a big matrix multiplication, and even using Fourier transforms. While brute force solution is straight forward, other two ways are not that easy to visualize and understand


Mathematical Implementation of Forward Propagation

Here is the visualization of the convolution operation were you can play with different hyperparameters of CNN:
https://ezyang.github.io/convolution-visualizer/index.html

Convolution Visualizer

On the left, we have a 6 x 6 grayscale image, so we represent it using a 6 x 6 x 1 matrix because there are no separate RGB channels. To detect vertical edges, we construct a 3 x 3 matrix, called filter or kernel, shown in the middle. We convolve the 6 x 6 matrix with the 3 x 3 filter. The output of this convolution operator will be a 4 x 4 matrix.

To compute the upper left element of this 4 x 4 matrix, we take the 3 x 3 filter and paste it on top of the 3 x 3 region of the original input image and sum up all the 9 the element-wise products. So, the result would be 3 x 1 + 1 x 1 + 2 x 1 + 0 x 0 + 5 x 0 + 7 x 0 + 1 x (-1) + 8 x (-1) + 2 x (-1) = -5. Next, to compute the second element, we shift the pasted filter matrix one step to the right on the 6 x 6 image, do the same element-wise products, and then add them up, which will give us -4, and so on.

To compute the next rows, we shift the pasted filter matrix one step down on the 6 x 6 image. Then, we repeat the same element-wise products, and additions for the second row, and so on.

Calculating Cross-Correlation (Convolution) Operation Example

In a standard cross-correlation operation, the convolution window slides over the input tensor one element at a time. However, to increase computational efficiency or to downsample the representation, we can traverse multiple elements per slide, skipping intermediate locations. The number of rows and columns traversed per slide is called the stride. Strided convolution is particularly useful when the convolution kernel is large, as it efficiently captures a broad area of the underlying image.

Strided Convolution

A 2D convolution layer performs an entry per entry multiplication between the input and the filter, where the input and the filter are 2D matrices. 

In a 3D convolution layer, the same operation is performed on 3D inputs and the filters, such that:
- We multiply corresponding entries in the input and filter cubes and add up the results to determine the output entry.
- In addition to shifting (striding) the filter on two dimensions of the input matrix to perform the convolution operation, we also shift it in the third dimension.

For example, if we have a 4 x 4 x 4 input and a 3 x 3 x 3 filter, because we can shift the filter for only one entry per dimension, the output will be 2 x 2 x 2.

3D Convolution Layers

For example, if we have a 3 x 3 x 3 RGB filter, to perform the convolution operation, we start with placing the filter in the upper left most position of the image. We can consider this 3 x 3 x 3 filter as a cube with 27 parameters, where the first dimension is called "width," the second is called height, and the third indicates the number of channels. We multiply each of these 27 numbers with the corresponding numbers from the red, green, and blue channels of the image and sum up the results to calculate the corresponding output number. Then to compute the next output, we take this cube and slide it over by one, do the 27 multiplications, and add up the results to calculate the next output number, and so on. This way, convolving a 6 x 6 x 3 image with a 3 x 3 x 3 filter will give us a 4 x 4 output.

Convolutions Over Volumes

A convolutional layer performs a cross-correlation operation between its input and a kernel, and subsequently adds a scalar bias to the result to generate an output. During model training, the convolutional layer learns two key parameters: the kernel (weights) and the scalar bias. Prior to training, these kernel weights are generally initialized with random values, analogous to the initialization process in fully connected layers.

Convolutional Layer

To execute a two-dimensional cross-correlation, the convolution window starts at the upper-left of the input tensor and systematically translates across it, moving rightwards and downwards. At each valid position, the kernel tensor undergoes an elementwise multiplication with the currently overlaid input subtensor. The sum of these individual products produces a single scalar, which becomes the corresponding element in the resulting output tensor.

Two-Dimensional Convolution Operation Procedure

When applying a two-dimensional cross-correlation without padding, the resulting output tensor is dimensionally smaller than the input. Since the kernel must fit completely within the boundaries of the input tensor, an input of size $$n_\textrm{h} \times n_\textrm{w}$$ convolved with a kernel of size $$k_\textrm{h} \times k_\textrm{w}$$ will yield an output shape calculated by the formula: $$(n_\textrm{h}-k_\textrm{h}+1) \times (n_\textrm{w}-k_\textrm{w}+1)$$.

Computation of Convolution Output Size

A convolution kernel can act as a finite difference operator to locate pixel changes for edge detection. For example, a $$1 \times 2$$ kernel like $$[1, -1]$$ computes the difference between horizontally adjacent pixels, $$x_{i,j} - x_{(i+1),j}$$. This cross-correlation operation serves as a discrete approximation of the first derivative in the horizontal direction. Mathematically, for an image function $$f(i,j)$$, its derivative is $$-\partial_i f(i,j) = \lim_{\epsilon \to 0} \frac{f(i,j) - f(i+\epsilon,j)}{\epsilon}$$. By applying this kernel, the output is zero where adjacent pixels are identical and non-zero at boundaries, effectively detecting edges.

Convolution Kernel as a Finite Difference Operator

As an example of padding in a convolutional layer, consider a $$3 	imes 3$$ input tensor padded with zero values to increase its size to $$5 	imes 5$$. If we apply a two-dimensional cross-correlation using a $$2 	imes 2$$ kernel on this padded input, the resulting output will be a $$4 	imes 4$$ matrix. The added zero pixels do not change the sum of the products when the convolution window covers them.

Example of Two-Dimensional Cross-Correlation with Padding

In convolutional operations, there are standard baseline configurations that are assumed unless specified otherwise. By default, the padding is set to $$0$$ (meaning no extra border pixels are added) and the stride is set to $$1$$ (meaning the convolution window slides exactly one element at a time).

Default Padding and Stride Values

When the input data contains multiple channels, denoted as $$c_	extrm{i}$$, the convolution kernel must have the same number of input channels to perform cross-correlation. If the kernel's two-dimensional spatial window shape is $$k_	extrm{h} 	imes k_	extrm{w}$$, a kernel tensor of shape $$k_	extrm{h} 	imes k_	extrm{w}$$ is required for every input channel. Concatenating these $$c_	extrm{i}$$ tensors yields a convolution kernel with an overall shape of $$c_	extrm{i} 	imes k_	extrm{h} 	imes k_	extrm{w}$$.

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