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In artificial neural networks, a convolutional layer is a type of network layer that applies a convolution operation to the input. Convolutional layers are some of the primary building blocks of convolutional neural networks (CNNs), a class of neural network most commonly applied to images, video, audio, and other data that have the property of uniform translational symmetry.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain).
It would be calculated, for example, as: [(input width 227 - kernel width 11) / stride 4] + 1 = [(227 - 11) / 4] + 1 = 55. Since the kernel output is the same length as width, its area is 55×55.) LeNet has several common motifs of modern convolutional neural networks, such as convolutional layer, pooling layer and full connection layer.
The problem with polynomials may be removed by allowing the outputs of the hidden layers to be multiplied together (the "pi-sigma networks"), yielding the generalization: [38] Universal approximation theorem for pi-sigma networks — With any nonconstant activation function, a one-hidden-layer pi-sigma network is a universal approximator.
A worked example of performing a convolution. The convolution has stride 1, zero-padding, with kernel size 3-by-3. The convolution kernel is a discrete Laplacian operator. The convolutional layer is the core building block of a CNN.
For example, when = and =, Eq.3 equals , whereas direct evaluation of Eq.1 would require up to complex multiplications per output sample, the worst case being when both and are complex-valued. Also note that for any given M , {\displaystyle M,} Eq.3 has a minimum with respect to N . {\displaystyle N.} Figure 2 is a graph of the values of N ...
The convolution of f and g exists if f and g are both Lebesgue integrable functions in L 1 (R d), and in this case f∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L 1, under the discrete convolution, or more generally for the convolution on any group.
As an example, a single 5×5 convolution can be factored into 3×3 stacked on top of another 3×3. Both has a receptive field of size 5×5. The 5×5 convolution kernel has 25 parameters, compared to just 18 in the factorized version. Thus, the 5×5 convolution is strictly more powerful than the factorized version.