[Review] XNOR-Nets: ImageNet Classification Using Binary Convolutional Neural Networks



The two models presented:

In Binary-Weight-Networks, the (convolution) filters are approximated with binary values resulting in 32 x memory saving.

In XNOR-Networks, both the filters and the input to convolutional layers are binary. … This results in 58 x faster convolutional operations…


XNOR-Nets offer the possibility of running state-of-the-art networks on CPUs (rather than GPUs) in real-time.

Binary Convolutional Neural Networks

For future discussions we use the following mathematical notation for a CNN layer:

Convolution with binary weights

In binary convolutional networks, we estimate the convolution filter weight as , where is a scalar scaling factor and . Hence, we estimate the convolution operation as follows:

To find an optimal estimation for we solve the following problem:

Going straight to the answer:


The gradients are computed as follows:

where , the estimated value of .

The gradient values are kepted as real values; they cannot be binarized due to excessive information loss. Optimization is done by either SGD with momentum or ADAM.


Convolutions are a set of dot products between a submatrix of the input and a filter. Thus we attempt to express dot products in terms of binary operations.

Binary Dot Product

For vectors and , we approximate the dot product between and as

We solve the following optimization problem:

Going straight to the answer:

Convolution with binary inputs and weights

Calculating for every submatrix in input tensor involves a large number of redundant computations. To overcome this inefficiency we first calculate

which is an average over absolute values of along its channel. Then, we convolve with a 2D filter where :

This acts as a global spatially across the submatrices. Now we can estimate our convolution with binary inputs and weights as:


A CNN block in XNOR-Net has the following structure:

[Binary Normalization] - [Binary Activation] - [Binary Convolution] - [Pool]

The BinNorm layer normalizes the input batch by its mean and variance. The BinActiv layer calculates and . We may insert a non-linear activation function between the BinConv layer and the Pool layer.


The paper implemented the AlexNet, the Residual Net, and a GoogLenet variant(Darknet) with binary convolutions. This resulted in a few percent point of accuracy decrease, but overall worked fairly well. Refer to the paper for details.


Binary convolutions were not at all entirely binary; the gradients had to be real values. It would be fascinating if even the gradient is binarizable.

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