Convolutional Feature Maps, Kernels, and Receptive Fields
Summary
A convolutional layer applies shared kernels across spatial locations to produce output feature maps. Kernel size, stride, dilation, padding, input channels, and output channels determine spatial dimensions, parameter count, and the theoretical receptive field. A (1\times1) convolution mixes channels at each location, whereas a (3\times3) convolution also combines neighboring spatial positions.
Scope
Covered
- Two-dimensional convolution and feature-map channels.
- Kernel size, stride, padding, dilation, and output shape.
- Parameter counts for dense convolutional kernels.
- Theoretical receptive fields and stacked convolutions.
Not covered
- The operation vocabulary of a particular architecture space.
- A claim that one kernel size is universally more accurate.
- A procedure for selecting or fitting neural architectures.
Key concepts and notation
| Symbol | Meaning |
|---|---|
| (H,W) | Input height and width |
| (C_{\mathrm{in}},C_{\mathrm{out}}) | Input and output channel counts |
| (k) | Square kernel width |
| (s,p,d) | Stride, padding, and dilation |
| feature map | Spatial array associated with one channel |
| receptive field | Input region capable of affecting an output unit |
Core knowledge
Multi-channel convolution
For an input tensor (X), a convolutional output can be written
[ Y_{i,j,c_o} = b_{c_o}
- \sum_{u,v,c_i} K_{u,v,c_i,c_o}, X_{i s+u,,j s+v,,c_i}, ]
with indexing adjusted for padding and dilation. Weight sharing means the same kernel coefficients are applied at different spatial positions. This gives translation-equivariant linear processing away from boundary and sampling effects [1,2].
A dense (k\times k) convolution with bias has
[ k^2 C_{\mathrm{in}}C_{\mathrm{out}}+C_{\mathrm{out}} ]
trainable scalar parameters. Computation also depends on output spatial size, so parameter count and operation count are distinct quantities.
Spatial output dimensions
For one spatial dimension, the common output-size formula is
[ H_{\mathrm{out}}
\left\lfloor \frac{H+2p-d(k-1)-1}{s}+1 \right\rfloor. ]
The same relationship applies to width. Padding can preserve spatial extent, stride can subsample it, and dilation spaces kernel elements farther apart [1]. Framework conventions determine asymmetric padding and rounding details.
One-by-one convolution
A (1\times1) convolution applies a learned linear transformation across channels independently at every spatial position. It does not expand the spatial receptive field when stride is one, but it can change channel count, combine channel information, and introduce a new nonlinearity when followed by an activation [3].
Three-by-three convolution
A (3\times3) convolution combines each location with a local spatial neighborhood as well as mixing channels. With unit stride and suitable padding, it preserves spatial dimensions. Relative to a (1\times1) convolution at equal channel counts, it has nine times as many kernel weights.
Stacking two unit-stride (3\times3) convolutions gives a theoretical (5\times5) receptive field; a third gives (7\times7), assuming no dilation and ignoring boundaries. Nonlinearities between layers make the stack different from one larger linear convolution.
Theoretical and effective receptive fields
The theoretical receptive field is determined by connectivity, kernel sizes, strides, and dilation. It states which input positions can affect an output. The effective influence of those positions after training can be highly nonuniform and is not specified by theoretical size alone.
Conditions, limitations, and uncertainty
- Output shape depends on the precise padding and rounding convention.
- Boundary positions do not have the same input neighborhood as interior positions when padding is used.
- Parameter count does not determine accuracy, latency, memory traffic, or optimization difficulty by itself.
- A larger theoretical receptive field does not guarantee that distant input pixels materially affect a trained output.
- Convolutional layers also depend on normalization, activation, initialization, and the surrounding graph.
Related knowledge resources
pooling_branching_and_feature_aggregation: non-convolutional spatial aggregation and graph merges.cell_based_convolutional_neural_networks: convolutional operations inside reusable cells.
References
- Dumoulin V, Visin F. A guide to convolution arithmetic for deep learning. arXiv. 2016. https://arxiv.org/abs/1603.07285 [Technical guide]
- Goodfellow I, Bengio Y, Courville A. Convolutional networks. In: Deep Learning. MIT Press; 2016. https://www.deeplearningbook.org/contents/convnets.html [Textbook]
- Lin M, Chen Q, Yan S. Network in network. International Conference on Learning Representations. 2014. https://arxiv.org/abs/1312.4400 [Primary research]