# Function Definitions¶

## Neural Network Layer¶

### Affine¶

Affine layer, also called as the fully connected layer. It calculates:

${\mathbf y} = {\mathbf A} {\mathbf x} + {\mathbf b}.$

where $${\mathbf x}$$ is the input and $${\mathbf y}$$ is the output.

• Input(s)
 Name Description Options x Input N-D array with shape ($$M_0 \times ... \times M_{B-1} \times D_B \times ... \times D_N$$). Dimensions before and after base_axis are flattened as if it is a matrix. weight Weight matrix with shape ($$(D_B \times ... \times D_N) \times L$$) Parameter bias Bias vector ($$L$$) Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Base axis of Affine operation. Dimensions up to base_axis is treated as sample dimension.
• Output(s)
 Name Description Options y $$(B + 1)$$-D array. ($$M_0 \times ... \times M_{B-1} \times L$$)

### Convolution¶

N-D Convolution with bias.

See references for dilated convolution (a.k.a. atrous convolution).

References:

• Input(s)
 Name Description Options x $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C \times L_1 \times ... \times L_N$$). weight $$(2 + N)$$-D array ($$C' \times C \times K_1 \times ... \times K_N$$). Parameter bias Bias vector ($$C'$$). Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 base axis $$B$$. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. group int64 1 Number of groups of channels. This makes the connection across channels sparser, by grouping connections along the mapping direction.
• Output(s)
 Name Description Options y $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C' \times L'_1 \times ... \times L'_N$$).

### DepthwiseConvolution¶

N-D Depthwise Convolution with bias.

References:

• Input(s)
 Name Description Options x $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C \times L_1 \times ... \times L_N$$). weight $$(1 + N)$$-D array ($$C \times K_1 \times ... \times K_N$$). Parameter bias Bias vector ($$C$$). Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 base axis $$B$$. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. multiplier int64 1 Number of output feature maps per input feature map.
• Output(s)
 Name Description Options y $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C \times L'_1 \times ... \times L'_N$$).

### Deconvolution¶

N-D deconvolution, also known as transposed convolution, with bias operates backward convolution (derivative of the output w.r.t. the input) plus channel-wise learned bias.

The weights are specified in the same manner as convolution() , as if it was an ordinary convolution function. The forward operation of deconvolution() will then be operationally equivalent to the backward pass of convolution() . Therefore, the number of input channels (can be seen as output channels of forward convolution) is specified in the first dimension, and the number of the output channels divided by the number of groups is specified in the second dimension.

• Input(s)
 Name Description Options x $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C \times L_1 \times ... \times L_N$$). weight $$(2 + N)$$-D array ($$C' \times C \times K_1 \times ... \times K_N$$). Parameter bias Bias vector ($$C'$$). Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 base axis $$B$$. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. group int64 1 Number of groups of channels. This makes the connection across channels sparser, by grouping connections along the mapping direction.
• Output(s)
 Name Description Options y $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C' \times L'_1 \times ... \times L'_N$$).

### DepthwiseDeconvolution¶

Depthwise deconvolution computes the transposed depthwise convolution with bias for one-dimensional and two-dimensional input data.

• Input(s)
 Name Description Options x $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C \times L_1 \times ... \times L_N$$). weight $$(1 + N)$$-D array ($$C \times K_1 \times ... \times K_N$$). Parameter bias Bias vector ($$C$$). Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 base axis $$B$$. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. divisor int64 1 Number of input feature maps per output feature map.
• Output(s)
 Name Description Options y $$(B + 1 + N)$$-D array ($$M_1 \times ... \times M_B \times C \times L'_1 \times ... \times L'_N$$).

### MaxPooling¶

Max pooling. It pools the maximum values inside the scanning kernel:

$y_{i_1, i_2} = \max_{k_1, k_2 \in K} (x_{i_1 + k_1, i_2 + k_2})$

where $$x_{i_1 + k_1, i_2 + k_2}$$ is the input and $$y_{i_1, i_2}$$ is the output.

• Input(s)
 Name Description Options x Input variable.
• Argument(s)
 Name Type Default Description kernel Shape Kernel sizes for each spatial axis. stride Shape kernel Subsampling factors for each spatial axis. ignore_border bool True If false, kernels covering borders are also considered for the output. pad Shape (0,) * len(kernel) Border padding values for each spatial axis. Padding will be added both sides of the dimension.
• Output(s)
 Name Description Options y Maximum values variable

### AveragePooling¶

Average pooling. It pools the averaged values inside the scanning kernel:

$y_{i_1, i_2} = \frac{1}{K_1 K_2} \sum_{k1} \sum_{k2} x_{i_1 + k_1, i_2 + k_2}$

where $$x_{i_1 + k_1, i_2 + k_2}$$ is the input and $$y_{i_1, i_2}$$ is the output.

• Input(s)
 Name Description Options x Input variable.
• Argument(s)
 Name Type Default Description kernel Shape Kernel sizes for each spatial axis. stride Shape kernel Subsampling factors for each spatial axis. ignore_border bool True If false, kernels covering borders are also considered for the output. pad Shape (0,) * len(kernel) Border padding values for each spatial axis. Padding will be added both sides of the dimension. including_pad bool True If true, border padding values are considered for the output.
• Output(s)
 Name Description Options y Average values variable

### GlobalAveragePooling¶

Global average pooling. It pools an averaged value from the whole image

• Input(s)
 Name Description Options x Input variable.
• Output(s)
 Name Description Options y Average values variable

### SumPooling¶

Sum pooling. It pools the summed values inside the scanning kernel:

$y_{i_1, i_2} = \sum_{k1} \sum_{k2} x_{i_1 + k_1, i_2 + k_2}$

where $$x_{i_1 + k_1, i_2 + k_2}$$ is the input and $$y_{i_1, i_2}$$ is the output.

• Input(s)
 Name Description Options x Input variable.
• Argument(s)
 Name Type Default Description kernel Shape Kernel sizes for each spatial axis. stride Shape kernel Subsampling factors for each spatial axis. ignore_border bool True If false, kernels covering borders are also considered for the output. pad Shape (0,) * len(kernel) Border padding values for each spatial axis. Padding will be added both sides of the dimension.
• Output(s)
 Name Description Options y Summed values variable

### Unpooling¶

Inverse operation of pooling. It spreads the input values:

$y_{k_1 i_1 + j_1, k_2 i_2 + j_2} = x_{i_1, i_2}$

where $$_{i_1, i_2}$$ is the input and $$y_{k_1 i_1 + j_1, k_2 i_2 + j_2}$$ is the output.

• Input(s)
 Name Description Options x Input variable.
• Argument(s)
 Name Type Default Description kernel Shape Kernel sizes for each spatial axis.
• Output(s)
 Name Description Options y Spread values variable

### Embed¶

Embed slices of a matrix/tensor with indexing array/tensor.

• Input(s)
 Name Description Options x0 Indices with shape $$(I_0, ..., I_N)$$ Integer x1 Weights with shape $$(W_0, ..., W_M)$$
• Output(s)
 Name Description Options y Output with shape $$(I_0, ..., I_N, W_1, ..., W_M)$$

## Neural Network Activation Functions¶

### Sigmoid¶

Element-wise sigmoid function.

$f(x) = \frac{1}{1 + \exp(-x)},$
• Input(s)
 Name Description Options x Input
• Output(s)
 Name Description Options y Output

### Swish¶

Element-wise swish function, by Ramachandran et al. (2017).

$y_i = \frac{x_i}{1 + \exp(-x_i)},$
References:
• Input(s)
 Name Description Options x Input
• Output(s)
 Name Description Options y Output

### Tanh¶

Element-wise hyperbolic tangent (tanh) function.

$y_i = \tanh (x_i)$
• Input(s)
 Name Description Options x N-D array
• Output(s)
 Name Description Options y N-D array with the same shape as x

### ReLU¶

Element-wise Rectified Linear Unit (ReLU) function.

$y_i = \max (0, x_i)$
• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description inplace bool False The output array is shared with the input array if True.
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LeakyReLU¶

Element-wise Leaky Rectified Linear Unit (ReLU) function.

It is defined as:

$y_i = \alpha * \min(0, x_i) + \max (0, x_i)$
• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description alpha float 0.1 The slope value multiplied to negative numbers. $$\alpha$$ in the definition.
• Output(s)
 Name Description Options y N-D array with the same shape as x

### Softmax¶

Softmax normalization. Calculates

$y_i = \frac{\exp(x_i)}{\sum_j \exp(x_j)}$

along the dimension specified by axis, where $$y_i$$ is the input and $$x_i$$ is the output.

• Input(s)
 Name Description Options x N-D array. Typically indicates a score.
• Argument(s)
 Name Type Default Description axis int64 len(x.shape) - 1 Axis normalization is taken.
• Output(s)
 Name Description Options y N-D array with the same shape as x

### ELU¶

Element-wise Exponential Linear Unit (ELU) function.

$\begin{split}y_i= \left\{ \begin{array}{ll} x_i & (x > 0)\\ \alpha (\exp(x_i) - 1) & (x \leq 0) \end{array} \right..\end{split}$
References:
• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description alpha double 1.0 Coefficient for negative outputs. $$\alpha$$ in definition
• Output(s)
 Name Description Options y N-D array with the same shape as x

### SELU¶

Element-wise Scaled Exponential Linear Unit (SELU) function by Klambauer et al. (2017).

$\begin{split}y_i= \lambda \left\{ \begin{array}{ll} x_i & (x > 0)\\ \alpha (\exp(x_i) - 1) & (x \leq 0) \end{array} \right..\end{split}$

The coefficients $$\lambda$$ and $$\alpha$$ default to the following values $$\lambda_{01}$$ and $$\alpha_{01}$$, respectively, provided by Klambauer et al. (2017):

$\begin{split}\begin{array}{lll} \lambda_{01} &=& \left( 1 - \operatorname{erfc}\left( \frac{1}{\sqrt{2}} \right) \sqrt{e} \right) \sqrt{2 \pi} \\ && \left( 2 \operatorname{erfc} \left( \sqrt{2} \right) e^2 + \pi \operatorname{erfc}\left( \frac{1}{\sqrt{2}} \right)^2 e \right. \\ && \left. - 2(2 + \pi) \operatorname{erfc} \left( \frac{1}{\sqrt{2}} \right) \sqrt{e} + \pi + 2 \right)^{-1/2} \\ &\approx& 1.0507 \\ \alpha_{01} &=& - \frac {\sqrt {\frac {2}{\pi}}} {\operatorname{erfc} \left( \frac{1}{\sqrt{2}} \right) \exp \left(\frac {1} {2} \right) - 1} \\ &\approx& 1.67326 \end{array}\end{split}$
References:
• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description scale double 1.050700987355480 The coefficient $$\lambda$$ in the definition. alpha double 1.673263242354377 The coefficient $$\alpha$$ in the definition.
• Output(s)
 Name Description Options y N-D array with the same shape as x

### CReLU¶

Element-wise Concatenated Rectified Linear Unit (CReLU) function. This function calculates the ReLU of $$x$$ and $$-x$$ , then concatenates the results together at a specified axis, and returns the resulting array.

References:
• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description axis int64 1 The ReLU activations of positive inputs and negative inputs are concatenated at axis.
• Output(s)
 Name Description Options y N-D array where axis dimension is doubled by concatenating.

### CELU¶

Element-wise Concatenated Exponential Linear Unit (CELU) function. Concatenates ELU outputs of positive and negative inputs together at specified axis.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description alpha double 1.0 Coefficient for negative outputs. $$\alpha$$ in definition. axis int64 1 The ELU activations of positive inputs and negative inputs are concatenated at axis.
• Output(s)
 Name Description Options y N-D array where axis dimension is doubled by concatenating.

### PReLU¶

Element-wise Parametrized Rectified Linear Unit function. Calculates:

$y_i = \max(0, x_i) + w_i \min(0, -x_i)$

where negative slope $$w$$ is learned and can vary across channels (an axis specified with base_axis).

• Input(s)
 Name Description Options x0 (N-D array) Input x1 (N-D array) Weights
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension.
• Output(s)
 Name Description Options y N-D array.

## Normalization¶

### BatchNormalization¶

Batch normalization.

$\begin{split}\begin{eqnarray} \mu &=& \frac{1}{M} \sum x_i \\ \sigma^2 &=& \frac{1}{M} \sum \left(x_i - \mu\right)^2 \\ \hat{x}_i &=& \frac{x_i - \mu}{\sqrt{\sigma^2 + \epsilon}} \\ y_i &=& \hat{x}_i \gamma + \beta. \end{eqnarray}\end{split}$

At testing time, the mean and variance values used are those that were computed during training by moving average.

References:

• Input(s)
 Name Description Options x N-D array of input. beta N-D array of beta which is learned. Parameter gamma N-D array of gamma which is learned. Parameter mean N-D array of running mean (modified during forward execution). Parameter variance N-D array of running variance (modified during forward execution). Parameter
• Argument(s)
 Name Type Default Description axes repeated int64 (1, ) Axes mean and variance are taken. decay_rate float 0.9 Decay rate of running mean and variance. eps float 1e-5 Tiny value to avoid zero division by std. batch_stat bool True Use mini-batch statistics rather than running ones.
• Output(s)
 Name Description Options y N-D array

### MeanSubtraction¶

It subtracts the mean of the elements of the input array, and normalizes it to $$0$$. Preprocessing arrays with this function has the effect of improving accuracy in various tasks such as image classification.

At training time, this function is defined as

$\begin{split}\begin{eqnarray} \mu &=& \frac{1}{M} \sum x_i \\ rm &=& ({\rm decay\_rate}) rm + (1 - {\rm decay\_rate}) \mu \\ y_i &=& x_i - rm \end{eqnarray}\end{split}$

At validation time, it is defined as

$y_i = x_i - rm$
Note:
The backward performs an approximated differentiation that takes into account only the latest mini-batch.
• Input(s)
 Name Description Options x N-D array of input. rmean N-D array of running mean (modified during forward execution). t Scalar of num of iteration of running mean (modified during forward execution).
• Argument(s)
 Name Type Default Description base_axis int64 1 Base axis of Mean Subtraction operation. Dimensions up to base_axis is treated as sample dimension. update_running_mean bool True Update running mean during forward execution.
• Output(s)
 Name Description Options y N-D array.

In forward pass, the function behaves as the identity.

In backward pass,

$\begin{split}g_x = \begin{cases} max & (g_y > max) \\ g_y & (otherwise) \\ min & (g_y < min) \end{cases}.\end{split}$

In the forward pass, the function behaves like the identity.

In the backward pass,

$g_x = clip\_norm \times \frac{g_y}{\|g_y\|_2}$

where $$g_x$$ is the gradient w.r.t the input, $$g_y$$ is the gradient w.r.t. the output, and $$clip\_norm$$ is the norm of $$g_y$$. this is the case that axes is not set. When axes is set, the norm is computed over axes.

• Input(s)
 Name Description Options x N-D array of input. min N-D array of minimum input value by which the gradients of the x are clipped. max N-D array of maximum input value by which the gradients of the x are clipped.
• Argument(s)
 Name Type Default Description clip_norm float 1.0 Clip to the norm of input to clip_norm in the forward pass. axes repeated int64 range(x.ndim) Axes to be reduced. If empty list is given, all dimensions are reduced to scalar. This is used in the forward pass.
• Output(s)
 Name Description Options y N-D array.

## Reduction¶

### Sum¶

Reduces a matrix along a specified axis with the sum function.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description axes repeated int64 range(x.ndim) Axes to be reduced. If empty list is given, all dimensions are reduced to scalar. keep_dims bool False Flag whether the reduced axis is kept.
• Output(s)
 Name Description Options y N-D array

### Mean¶

Reduces a matrix along a specified axis with the mean function.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description axes repeated int64 range(x.ndim) Axes to be reduced. keep_dims bool False Flag whether the reduced axis is kept.
• Output(s)
 Name Description Options y N-D array

### Max¶

Reduction along axis or axes with max operation.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description axes repeated int64 range(x.ndim) Axes to be reduced. keep_dims bool False Flag whether the reduced axis is kept.
• Output(s)
 Name Description Options y N-D array

### Min¶

Reduction along axis or axes with min operation.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description axes repeated int64 range(x.ndim) Axes to be reduced. keep_dims bool False Flag whether the reduced axis is kept.
• Output(s)
 Name Description Options y N-D array

### Prod¶

Reduction along axis or axes with product operation.

Note:
Backward computation is not accurate in a zero value input.
• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description axes repeated int64 range(x.ndim) Axes to be reduced. keep_dims bool False Flag whether the reduced axis is kept.
• Output(s)
 Name Description Options y N-D array

### ReduceSum¶

Reduction along an axis with sum operation.

Note:
This is deprecated. Use sum instead.
• Input(s)
 Name Description Options x N-D array.
• Output(s)
 Name Description Options y N-D array

### ReduceMean¶

Reduction by mean along an axis.

Note:
This is deprecated. Use mean instead.
• Input(s)
 Name Description Options x N-D array
• Output(s)
 Name Description Options y N-D array

## Arithmetic¶

$y_i = x^{(0)}_i + x^{(1)}_i$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Argument(s)
 Name Type Default Description inplace bool False The output array is shared with the 1st input array if True.
• Output(s)
 Name Description Options y N-D array

Note: This shouldn’t be called by users.

• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array

### Sub2¶

Element-wise subtraction.

$y_i = x^{(0)}_i - x^{(1)}_i$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array

### Mul2¶

Element-wise multiplication.

$y_i = x^{(0)}_i x^{(1)}_i$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array

### Div2¶

Element-wise division.

$y_i = \frac{x^{(0)}_i} {x^{(1)}_i}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array

### Pow2¶

Element-wise power function.

$y_i = {(x^{(0)}_i)} ^ {x^{(1)}_i}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array

$y_i = x_i + v$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### MulScalar¶

Element-wise scalar multiplication.

$y_i = v x_i$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### PowScalar¶

Element-wise scalar power function.

$y_i = (x_i) ^ v$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### RSubScalar¶

Element-wise scalar subtraction.

$y_i = v - x_i$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### RDivScalar¶

Element-wise scalar division.

$y_i = \frac{v}{x_i}$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### RPowScalar¶

Element-wise scalar power function.

$y_i = v ^ {x_i}$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

## Logical¶

### Sign¶

Element-wise sign function.

In the forward pass, it is defined as

$\begin{split}f(x) = \begin{cases} 1 & (x > 0) \\ -1 & (x < 0) \\ \alpha & (x = 0) \end{cases}.\end{split}$

In the backward pass, it is defined as

$\frac{\partial f(x)}{\partial x} = 1,$

or in other words, it behaves as the identity function for the gradient in the backward pass.

• Input(s)
 Name Description Options x Input
• Argument(s)
 Name Type Default Description alpha float 0.0 Value in case of $$x = 0$$.
• Output(s)
 Name Description Options y N-D array with the same shape as x

### Minimum2¶

Element-wise minimum.

$y_i = \min(x^{(0)}_i, x^{(1)}_i)$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array of min value

### Maximum2¶

Element-wise maximum.

$y_i = \max(x^{(0)}_i, x^{(1)}_i)$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y N-D array of max value

### MinimumScalar¶

Element-wise scalar minimum.

$y_i = \min(x_i, v)$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1.0 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### MaximumScalar¶

Element-wise scalar maximum.

$y_i = \max (x_i, v)$
• Input(s)
 Name Description Options x Input variable
• Argument(s)
 Name Type Default Description val double 1.0 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LogicalAnd¶

Elementwise logical AND.

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i \neq 0 \;\&\; x^{(1)}_i \neq 0) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### LogicalOr¶

Elementwise logical OR.

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 0 & (x^{(0)}_i = 0 \;\&\; x^{(1)}_i = 0) \\ 1 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### LogicalXor¶

Elementwise logical XOR.

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i = 0 \;\&\; x^{(1)}_i = 0) \\ 1 & (x^{(0)}_i \neq 0 \;\&\; x^{(1)}_i \neq 0) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### Equal¶

Element wise ‘equal’

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i = x^{(1)}_i) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### NotEqual¶

Element wise ‘not equal’

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 0 & (x^{(0)}_i = x^{(1)}_i) \\ 1 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### GreaterEqual¶

Element wise comparison. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i \geq x^{(1)}_i) \\ 0 & (x^{(0)}_i < x^{(1)}_i) \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### Greater¶

Element wise comparison. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i > x^{(1)}_i) \\ 0 & (x^{(0)}_i \leq x^{(1)}_i) \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### LessEqual¶

Element wise comparison. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i \leq x^{(1)}_i) \\ 0 & (x^{(0)}_i > x^{(1)}_i) \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### Less¶

Element wise comparison. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,x^{(1)}_i) = \begin{cases} 1 & (x^{(0)}_i < x^{(1)}_i) \\ 0 & (x^{(0)}_i \geq x^{(1)}_i) \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 N-D array x1 N-D array
• Output(s)
 Name Description Options y

### LogicalAndScalar¶

Elementwise logical AND with scalar.

$\begin{split}f(x_i,v) = \begin{cases} 1 & (x_i \neq 0 \;\&\; v \neq 0) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val bool
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LogicalOrScalar¶

Elementwise logical OR with scalar.

$\begin{split}f(x_i,v) = \begin{cases} 0 & (x_i = 0 \;\&\; v = 0) \\ 1 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val bool
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LogicalXorScalar¶

Elementwise logical XOR with scalar.

$\begin{split}f(x_i,v) = \begin{cases} 1 & (x_i = 0 \;\&\; v = 0) \\ 1 & (x_i \neq 0 \;\&\; v \neq 0) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val bool
• Output(s)
 Name Description Options y N-D array with the same shape as x

### EqualScalar¶

Element wise ‘equal’ with a scalar

$\begin{split}f(x_i,v) = \begin{cases} 1 & (x_i = v) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x
• inputs * Variable x0
• outputs * Variable output
• params * double val

### NotEqualScalar¶

Element wise ‘not equal’ with a scalar

$\begin{split}f(x_i,v) = \begin{cases} 0 & (x_i = v) \\ 1 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x
• inputs * Variable x0
• outputs * Variable output
• params * double val

### GreaterEqualScalar¶

Element wise comparison with a scalar. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,v) = \begin{cases} 1 & (x^{(0)}_i \geq v \\ 0 & (x^{(0)}_i < v \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### GreaterScalar¶

Element wise comparison with a scalar. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,v) = \begin{cases} 1 & (x^{(0)}_i > v \\ 0 & (x^{(0)}_i \leq v \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LessEqualScalar¶

Element wise comparison with a scalar. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,v) = \begin{cases} 1 & (x^{(0)}_i \leq v) \\ 0 & (x^{(0)}_i > v) \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LessScalar¶

Element wise comparison with a scalar. The $$i^{th}$$ element of the output is:

$\begin{split}f(x^{(0)}_i,v) = \begin{cases} 1 & (x^{(0)}_i < v) \\ 0 & (x^{(0)}_i \geq v) \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Argument(s)
 Name Type Default Description val double 1 Value of the scalar
• Output(s)
 Name Description Options y N-D array with the same shape as x

### LogicalNot¶

Element-wise logical NOT operation

$\begin{split}f(x_i) = \begin{cases} 1 & (x_i = 0) \\ 0 & otherwise \end{cases}.\end{split}$
• Input(s)
 Name Description Options x0 Input variable
• Output(s)
 Name Description Options y N-D array with the same shape as x

## Math¶

### Constant¶

Generate a constant-valued array.

• Input(s)
 Name Description Options
• Argument(s)
 Name Type Default Description val float 0 Constant value. shape Shape [] Shape of the output array.
• Output(s)
 Name Description Options y N-D array where all values are the specified constant.

### Abs¶

Element-wise absolute value function.

$y_i = |x_i|$
• Input(s)
 Name Description Options x Input variable
• Output(s)
 Name Description Options y Element-wise absolute variable

### Exp¶

Element-wise natural exponential function.

$y_i = \exp(x_i).$
• Input(s)
 Name Description Options x Input variable
• Output(s)
 Name Description Options y Element-wise exp variable

### Log¶

Element-wise natural logarithm function.

$y_i = \ln(x_i).$
• Input(s)
 Name Description Options x Input variable
• Output(s)
 Name Description Options y Element-wise log variable

### Identity¶

Identity function.

$y = x$
• Input(s)
 Name Description Options x N-D array.
• Output(s)
 Name Description Options y N-D array

### BatchMatmul¶

Batch matrix multiplication.

Two of batches of matrices are multiplied for each sample in a batch. A batch of matrices is composed as […, P, Q] where the last two dimensions compose matrix dimensions, and the first dimensions up to the third last dimension are considered as batch samples.

• Input(s)
 Name Description Options a N-D array with >= 2-dim. The last two dimensions will be treated as a matrix. b N-D array with >= 2-dim. The last two dimensions will be treated as a matrix. The product of the size of 0-th dimension through the size of the third last dimension must be same as that of the input a.
• Argument(s)
 Name Type Default Description transpose_a bool False Transpose the last two axes of a in matrix multiplication. transpose_b bool False Transpose the last two axes of b in matrix multiplication.
• Output(s)
 Name Description Options y Output of sample-wise matrix multiplication in a batch. When a is of a shape of [N, P, Q], b is of a shape of [N, Q, R], and transpose options are all False, the output will be a shape of [N, P, R].

### Round¶

Element-wise round function.

In the forward pass, this function simply computes round to the nearest integer value.

$y_i = round(x_i).$

In the backward pass, the simple Straight-Through Estimator (STE) is applied,

$\frac{\partial y_i}{\partial x_i} = 1.$
• Input(s)
 Name Description Options x Input variable
• Output(s)
 Name Description Options y N-D array with the same shape as x

## Array Manipulation¶

### Concatenate¶

Concatenate a variable number of input arrays along the specified axis.

• Input(s)
 Name Description Options x N-D arrays. Variadic Parameter
• Argument(s)
 Name Type Default Description axis int64 len(x[0].shape) - 1 Axis
• Output(s)
 Name Description Options y Concatenate variable

### Split¶

Split arrays at the specified axis.

note:
This function should not be called directly when constructing models. Instead, use nnabla.functions.split() which automatically sets n_output from the input’s shape and axis.
• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description axis int64 0 Axis
• Output(s)
 Name Description Options y list of N-D arrays Variadic Parameter

### Stack¶

Joins two or more arrays on a new axis.

Note:
Unlike nnabla.functions.concatenate() , which joins arrays on an existing axis, Stack joins arrays on a new axis.
• Input(s)
 Name Description Options x N-D arrays. The sizes of all the arrays to be stacked must be the same. Variadic Parameter
• Argument(s)
 Name Type Default Description axis int64 0 The axis on which to concatenate arrays. Axis indices take on values 0, 1, 2, and so on from the left. For example, to stack four (3,28,28) inputs on the second axis, specify 1. In this case, the output size will be (3,4,28,28).
• Output(s)
 Name Description Options y Output

### Slice¶

Slice arrays along specified axis.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description start repeated int64 (0,) * len(x.shape) Start indices for each axis stop repeated int64 tuple(x.shape) Stop indices for each axis step repeated int64 (1,) * len(x.shape) Step indices for each axis
• Output(s)
 Name Description Options y Sliced N-D array

### Transpose¶

Transposes tensor dimensions.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description axes repeated int64 Source axis indices for each axis.
• Output(s)
 Name Description Options y Transposed N-D array.

Broadcasting ND-array to the specified shape.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description shape Shape Shape broadcasted to. The size must be the same in axis where x’s shape is not 1.
• Output(s)
 Name Description Options y Broadcasted N-D array

Broadcasting ND-array to the specified buffer

• Input(s)
 Name Description Options x N-D array y N-D array
• Argument(s)
 Name Type Default Description axis int64 -1 Target axis to start broadcasting. If this is not set, broadcast will try to fit y to x starting from the last dimension
• Output(s)
 Name Description Options z Broadcasted N-D array

### OneHot¶

OneHot creates one-hot vector based on input indices.

• Input(s)
 Name Description Options x N-D array Integer
• Argument(s)
 Name Type Default Description shape Shape
• Output(s)
 Name Description Options output N-D array

### Flip¶

Reverses the order of elements of the specified dimension of an array.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description axes repeated int64 [len(x.shape) - 1] The index of the dimension to reverse the order of the elements. Axis indices take on values 0, 1, 2, and so on from the left. For example, to flip a 32 (W) by 24 (H) 100 RGB image (100,3,24,32) vertically and horizontally, specify (2,3).
• Output(s)
 Name Description Options y N-D array

### Shift¶

Shifts the array elements by the specified amount.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description shifts repeated int64 (0,) * len(x.shape) The amount to shift elements. For example, to shift image data to the right by 2 pixels and up 3 pixels, specify (-3,2). border_mode string (“nearest” or “reflect”) “nearest” Specify how to process the ends of arrays whose values will be undetermined as a result of shifting. nearest: The data at the ends of the original array is copied and used. reflect: Original data reflected at the ends of the original array is used.
• Output(s)
 Name Description Options y N-D array.

### Reshape¶

Reshapes the input variable in-place. It does not create a copy of the variable. The output variable (y) has a new shape but points to the same data as the input variable (x). This means that if the data in the output variable (y) is modified, the data in the input variable (x) also gets modified since the reshape was done in-place.

Note:
This function has the same behavior as the nnabla.Variable.reshape() method.
• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description shape Shape Dimensions for each axis
• Output(s)
 Name Description Options y Reshaped N-D array

### MatrixDiag¶

Returns an array where the last two dimensions consist of the diagonal matrix.

• Input(s)
 Name Description Options x N-D array with shape ($$M_0 \times \ldots \times M_N$$).
• Output(s)

### MatrixDiagPart¶

Returns an array in which the values of the last dimension consist of the diagonal elements of the last two dimensions of an input array.

• Input(s)
• Output(s)
 Name Description Options y N-D array with shape ($$M_0 \times \ldots \times M_N$$).

## Stochasticity¶

### Dropout¶

Dropout. Samples a number $$u$$ from a uniform distribution in $$[0, 1]$$ , and ignores the input if $$u > p$$.

$\begin{split}y = \left\{ \begin{array}{ll} \frac{x}{1 - p} & (u > p) \\ 0 & ({\rm otherwise}) \end{array} \right.\end{split}$
Note:

Usually dropout only applied during training as below (except Bayesian dropout).

h = PF.affine(x, num_hidden)
if train:
h = F.dropout(h, 0.5)

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description p double 0.5 $$p$$ in definition. seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y N-D array with the same shape as x

### TopKData¶

Select the k largest values from each sample in x to propagate unmodified and set all other values to 0. If abs is True, the k largest values are selected by magnitude. If reduce is True (the default), all feature dimensions are reduced to a single dimension of size k that propagates only the k largest values. Otherwise, if reduce is False, input and output dimensions are identical. Dimensions before base_axis are treated as number of sample dimensions and k values get selected from all elements of a sample (dimensions from base_axis) regardless of shape.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description k int64 Number of largest data values to propagate. abs bool False Determine largest data values by magnitude. reduce bool True Reduce feature size to one dimension of size k. base_axis int64 1 First dimension of the sample shape.
• Output(s)
 Name Description Options y N-D array.

Select the k largest gradients for each sample in x to back-propagate unmodified and set all other gradients to 0. If abs is True, the k largest gradients are selected by magnitude. Dimensions before base_axis are treated as number of sample dimensions and k gradients get selected from all gradients of a sample (dimensions from base_axis) regardless of shape.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description k int64 Number of largest data values to propagate. abs bool False Determine largest data values by magnitude. base_axis int64 1 First dimension of the sample shape.
• Output(s)
 Name Description Options y N-D array with same shape and data as x.

### Rand¶

Samples numbers from a uniform distribution $$x \sim U(low, high)$$ given lowest value $$low$$, upper bound $$high$$, and shape of the returned Variable.

• Input(s)
 Name Description Options
• Argument(s)
 Name Type Default Description low float 0 $$low$$ in definition. high float 1 $$high$$ in definition. shape Shape [] Shape of returned variable. seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y Variable with the shape specified in the argument.

### Randint¶

Samples integer numbers from a uniform distribution $$x \sim U(low, high)$$ given lowest value $$low$$, upper bound $$high$$, and shape of the returned Variable.

• Input(s)
 Name Description Options
• Argument(s)
 Name Type Default Description low int64 0 $$low$$ in definition. high int64 1 $$high$$ in definition. shape Shape [] Shape of returned variable. seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y Variable with the shape specified in the argument. The dtype is int32. Integer

### Randn¶

Samples numbers from a normal distribution $$x \sim N(\mu, \sigma)$$ given mean $$\mu$$, standard deviation $$\sigma$$, and shape of the returned Variable.

• Input(s)
 Name Description Options
• Argument(s)
 Name Type Default Description mu float 0 $$\mu$$ in definition. sigma float 1 $$\sigm$$ in definition. shape Shape [] Shape of returned variable. seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y Variable with the shape specified in the argument.

### RandomCrop¶

RandomCrop randomly extracts a portion of an array.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description shape Shape x.shape The data size to extract. For example, to randomly extract a portion of the image (3,48,48) from a 3,64,64 image, specify (3,48,48). base_axis int64 1 seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y N-D array

### RandomFlip¶

Reverses the order of elements of the specified dimension of an array at 50% probability.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description axes repeated int64 [len(x.shape) - 1] The index of the axis to reverse the order of the elements. Axis indices take on values 0, 1, 2, and so on from the left. For example, to flip a 32 (W) by 24 (H) 100 RGB images (100, 3,24,32) vertically and horizontally at random, specify (2,3). base_axis int64 1 seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y N-D array

### RandomShift¶

Randomly shifts the array elements within the specified range.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
 Name Type Default Description shifts repeated int64 (0,) * len(x.shape) Max absolute amount to shift elements. For example, to shift image data horizontally by $$\pm 2$$ pixels and vertically by $$\pm 3$$ pixels, specify (3,2). border_mode string (“nearest” or “reflect”) “nearest” Specify how to process the ends of arrays whose values will be undetermined as a result of shifting. nearest: The data at the ends of the original array is copied and used. reflect: Original data reflected at the ends of the original array is used. base_axis int64 1 seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y N-D array.

### ImageAugmentation¶

ImageAugmentation randomly alters the input image.

• Input(s)
 Name Description Options x N-D array.
• Argument(s)
• Output(s)
 Name Description Options y N-D array.

## Loss Functions¶

### SigmoidCrossEntropy¶

Element-wise cross entropy between x and the target variables, passed to a sigmoid function.

$y_i = - \left(x^{(1)}_i \ln \left(\sigma \left(x^{(0)}_i \right)\right) + \ \left(1 - x^{(1)}_i\right) \ln \left(1 - \sigma \left(x^{(0)}_i \ \right)\right)\right)$

where $$\sigma(s)=\frac{1}{1+\exp(-s)}$$.

Note:
SigmoidCrossEntropy is equivalent to Sigmoid+BinaryCrossEntropy, but computing them at once has the effect of reducing computational error.
• Input(s)
 Name Description Options x N-D array. Typically indicates a score. The value lies in $$[-\infty, \infty]$$ Parameter target N-D array of labels. Only 0 or 1 value is allowed. Integer Parameter
• Output(s)
 Name Description Options y N-D array of element-wise losses.

### BinaryCrossEntropy¶

Element-wise cross entropy between x and the target variables.

$y_i = - \left(x^{(1)}_i * \ln \left(x^{(0)}_i\right) + \left(1 - \ x^{(1)}_i\right) * \ln \left(1 - x^{(0)}_i\right)\right).$
• Input(s)
 Name Description Options x Probabilities N-D array. $$-\infty$$ to $$\infty$$. target N-D array of labels. Usually set as 0 or 1, but, unlike SigmoidCrossEntropy, it allows probability (0 to 1) as inputs and backpropagation can be done.
• Output(s)
 Name Description Options y N-D array of element-wise losses.

### SoftmaxCrossEntropy¶

Element-wise cross entropy between the variables and the variables of a label given by a category index with Softmax normalization.

$y_{j} = -\ln \left(\frac{\exp(x_{j,t_j})}{\sum_{i'} \exp(x_{j,i'})}\right)$

along dimension specified by axis ($$i$$ is the axis where normalization is performed on).

Note:
SoftmaxCrossEntropy is equivalent to Softmax+CategoricalCrossEntropy, but computing them at once has the effect of reducing computational error.
• Input(s)
 Name Description Options x N-D array. Typically indicates a score. $$(D_1 \times ... \times D_i \times ... \times D_N)$$ Parameter target N-D array of labels. $$(D_1 \times ... \times 1 \times ... \times D_N)$$ Integer Parameter
• Argument(s)
 Name Type Default Description axis int64 len(x.shape) - 1 Axis normalization is taken.
• Output(s)
 Name Description Options y N-D array of element-wise losses. $$(D_1 \times ... \times 1 \times ... \times D_N)$$

### CategoricalCrossEntropy¶

Element-wise cross entropy between x and the target t where targets are given by a category index.

$y_{j} = -\ln \left( x_{j, t_j} \right)$

along dimension specified by axis ($$i$$ is the axis where normalization is performed on).

• Input(s)
 Name Description Options x N-D array. Typically indicates a score. $$(D_1 \times ... \times D_i \times ... \times D_N)$$ Parameter target N-D array of labels. $$(D_1 \times ... \times 1 \times ... \times D_N)$$ Integer Parameter
• Argument(s)
 Name Type Default Description axis int64 len(x.shape) - 1 Axis normalization is taken.
• Output(s)
 Name Description Options y N-D array of element-wise losses. $$(D_1 \times ... \times 1 \times ... \times D_N)$$

### SquaredError¶

Element-wise squared error

$y_i = \left(x^{(0)}_i - x^{(1)}_i\right)^2.$
• Input(s)
 Name Description Options x0 N-D array. x1 N-D array.
• Output(s)
 Name Description Options y N-D array.

### AbsoluteError¶

Element-wise absolute error

$y_i = | x^{(0)}_i - x^{(1)}_i |.$
• Input(s)
 Name Description Options x0 N-D array. x1 N-D array.
• Output(s)
 Name Description Options y N-D array.

### HuberLoss¶

Element-wise Huber loss

$\begin{split}y_i= \left\{ \begin{array}{ll} d^2 & (|d| < \delta)\\ \delta (2 |d| - \delta) & ({\rm otherwise}) \end{array} \right.\end{split}$

where $$d = x^{(0)}_i - x^{(1)}_i$$

• Input(s)
 Name Description Options x0 N-D array. x1 N-D array.
• Argument(s)
 Name Type Default Description delta float 1.0 Delta
• Output(s)
 Name Description Options y N-D array of element-wise losses.

### EpsilonInsensitiveLoss¶

Element-wise Epsilon Insensitive Loss

$\begin{split}y_i= \left\{ \begin{array}{ll} | x^{(0)}_i - x^{(1)}_i | - \epsilon & if \ \ | x^{(0)}_i - x^{(1)}_i | > \epsilon \\ 0 & otherwise \end{array} \right.\end{split}$
• Input(s)
 Name Description Options x0 N-D array. x1 N-D array.
• Argument(s)
 Name Type Default Description epsilon float Insensitive parameter.
• Output(s)
 Name Description Options y N-D array of element-wise losses.

### KLMultinomial¶

The Kullback Leibler Divergence for multinomial distributions.

$D = \sum_i p_i \log \left( \frac{p_i}{q_i} \right)$
• Input(s)
 Name Description Options p N-D array of the source categorical probabilities q N-D array of the target categorical probabilities
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension.
• Output(s)
 Name Description Options D Kullback Leibler divergence $$KL(p \parallel q)$$.

## Quantization Neural Network Layers¶

### BinarySigmoid¶

Element-wise binary sigmoid function. In the forward pass, it computes

$\begin{split}f(x) = \begin{cases} 1 & (x > 0) \\ 0 & ({\rm otherwise})\end{cases},\end{split}$

but in the backward pass, a straight-through approximation of the gradient is used, i.e.,

$\begin{split}\frac{\partial f(x)}{\partial x} = \begin{cases} 0 & (|x| \geq 1) \\ \frac{1}{2} & ({\rm otherwise}) \end{cases}.\end{split}$

References:

• Input(s)
 Name Description Options x Input .
• Output(s)
 Name Description Options y Output.

### BinaryTanh¶

Element-wise binary tanh function. In the forward pass, it computes

$\begin{split}f(x) = \begin{cases} 1 & (x > 0) \\ -1 & ({\rm otherwise}) \end{cases},\end{split}$

but in the backward pass, a straight-through approximation of the gradient is used, i.e.,

$\begin{split}\frac{\partial f(x)}{\partial x} = \begin{cases} 0 & (|x| \geq 1) \\ 1 & ({\rm otherwise}) \end{cases}.\end{split}$

References:

• Input(s)
 Name Description Options x Input .
• Output(s)
 Name Description Options y Output.

### BinaryConnectAffine¶

This function provides a BinaryConnect affine layer. It computes in the forward pass

$y_j = \sum_{i} sign(w_{j,i}) x_i,$

i.e., the weights $$w_{j,i}$$ are binarized to $$sign(w_{j,i})$$ and, hence, each weight is in $$\{-1,\,1\}$$. By this weight binarization, the inner product computations do not require any multiplications anymore as they turn into additions/subtractions.

This function should be used together with batch_normalization().

Note

1) If you would like to share the binary weights between other layers, please use the standard, floating value weights (weight) and not the binary weights (binary_weight).

2) The weights and the binary weights become in sync only after a call to forward(), and not after a call to backward(). If you wish to store the parameters of the network, remember to call forward(), once before doing so, otherwise the weights and the binary weights will not be in sync.

3) CPU and GPU implementations now use floating values for binary_weight, since this function is for simulation purposes.

References:

• Input(s)
 Name Description Options x Input . weight Weight . Parameter binary_weight Binarized weight . Parameter bias Bias. Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension.
• Output(s)
 Name Description Options y Output.

### BinaryConnectConvolution¶

This function provides a BinaryConnect convolution layer. It computes in the forward pass

$y_{n, a, b} = \sum_{m} \sum_{i} \sum_{j} sign(w_{n, m, i, j}) x_{m, a + i, b + j},$

i.e., the weights $$w_{n, m, i, j}$$ are binarized to $$sign(w_{n, m, i, j})$$ and, hence, each weight is in $$\{-1,\,1\}$$. By this weight binarization, the inner product computations do not require any multiplications anymore as they turn into additions/subtractions.

This function should be used together with batch_normalization().

Reference

Note

1) If you would like to share the binary weights between other layers, please use the standard, floating value weights (weight) and not the binary weights (binary_weight).

2) The weights and the binary weights become in sync only after a call to forward(), and not after a call to backward(). If you wish to store the parameters of the network, remember to call forward(), once before doing so, otherwise the weights and the binary weights will not be in sync.

3) CPU and GPU implementations now use floating values for binary_weight, since this function is for simulation purposes.

• Input(s)
 Name Description Options x Input. weight Weight. Parameter binary_weight Binarized weight. Parameter bias Bias. Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. group int64 1 Number of groups of channels. This makes the connection across channels sparser, by grouping connections along the mapping direction.
• Output(s)
 Name Description Options y Output

### BinaryWeightAffine¶

This function provides a Binary Weight Network affine layer. It computes in the forward pass

$y_j = \frac{1}{\|\mathbf{w}_j\|_{\ell_1}} \sum_{i} sign(w_{j,i}) x_i$

i.e., the weights $$w_{j,i}$$ are binarized to $$sign(w_{j,i})$$ and, hence, each weight is in $$\{-1,\,1\}$$. By this weight binarization, the inner product computations turn into additions/subtractions which are followed by multiplication with the scaling factor $$\alpha_j = \frac{1}{\|\mathbf{w}_j\|_{\ell_1}}$$.

Reference

Note

1) If you would like to share the binary weights with other layers, please use the standard, floating value weights (weight) and not the binary weights (binary_weight).

2) The weights and the binary weights become in sync only after a call to forward(), and not after a call to backward(). If you wish to store the parameters of the network, remember to call forward(), once before doing so, otherwise the weights and the binary weights will not be in sync.

3) CPU and GPU implementations now use floating values for binary_weight, since this function is for simulation purposes.

• Input(s)
 Name Description Options x Input . weight Weight. Parameter binary_weight Binarized weight. Parameter alpha Alpha. Parameter bias Bias. Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension.
• Output(s)
 Name Description Options y Output.

### BinaryWeightConvolution¶

This function provides a Binary Weight Network convolution layer. It computes in the forward pass

$y_{n, a, b} = \frac{1}{\|\mathbf{w}_n\|_{\ell_1}} \sum_{m} \sum_{i} \sum_{j} sign(w_{n, m, i, j}) x_{m, a + i, b + j}.$

i.e., the weights $$w_{n, m, i, j}$$ are binarized to $$sign(w_{n, m, i, j})$$ and, hence, each weight is in $$\{-1,\,1\}$$. By this weight binarization, the inner product computations turn into additions/subtractions which are followed by multiplication with the scaling factor $$\alpha_n = \frac{1}{\|\mathbf{w}_n\|_{\ell_1}}$$.

Reference

Note

1) If you would like to share the binary weights between other standard layers, please use the standard, floating value weights (weight) and not the binary weights (binary_weight).

2) The weights and the binary weights become in sync only after a call to forward(), and not after a call to backward(). If you wish to store the parameters of the network, remember to call forward(), once before doing so, otherwise the weights and the binary weights will not be in sync.

3) CPU and GPU implementations now use floating values for binary_weight, since this function is for simulation purposes.

• Input(s)
 Name Description Options x Input. weight Weight. Parameter binary_weight Binarized weight. Parameter alpha Alpha. Parameter bias Bias. Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. group int64 1 Number of groups of channels. This makes the connection across channels sparser, by grouping connections along the mapping direction.
• Output(s)
 Name Description Options y Output

### INQAffine¶

This function provides a INQ affine layer. It computes in the forward pass

$y_j = \sum_{i} w_{j,i} x_i,$

where the weights $$w_{j,i}$$ are quantized sequentially during training to power-of-two numbers. In the backward pass, only the non-fixed (i.e., learnable) weights are updated.

References:

• Input(s)
 Name Description Options x Input . weight Weight . Parameter indicator_fixedweights Indicates which weights are already fixed (0 = not fixed, 1 = fixed) . Integer Parameter bias Bias. Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension. num_bits int64 4 Number of bits per weight. Needs to be >= 2 as two bits are used to code zero and sign of weight. inq_iterations repeated int64 () List which specifies after how many forward passes we fix 50% of the learnable weights. If we have done as many iterations as specified in the last element of inq_iterations, then all weights are fixed. selection_algorithm string (“largest_abs” or “random”) “largest_abs” Chooses algorithm that we use for selecting the weights to fix (“largest_abs” … fix weights with largest absolute value, “random” … fix weights randomly) seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y Output.

### INQConvolution¶

This function provides a INQ convolution layer. It computes in the forward pass

$y_{n, a, b} = \sum_{m} \sum_{i} \sum_{j} w_{n, m, i, j} x_{m, a + i, b + j},$

where the weights $$w_{j,i}$$ are quantized sequentially during training to power-of-two numbers. In the backward pass, only the non-fixed (i.e., learnable) weights are updated.

Reference

• Input(s)
 Name Description Options x Input. weight Weight. Parameter indicator_fixedweights Indicates which weights are already fixed (0 = not fixed, 1 = fixed) . Integer Parameter bias Bias. Optional Parameter
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension. pad Shape (0,) * (len(x.shape) - (base_axis+1)) Padding sizes for dimensions. stride Shape (1,) * (len(x.shape) - (base_axis+1)) Stride sizes for dimensions. dilation Shape (1,) * (len(x.shape) - (base_axis+1)) Dilation sizes for dimensions. group int64 1 Number of groups of channels. This makes the connection across channels sparser, by grouping connections along the mapping direction. num_bits int64 4 Number of bits per weight. Needs to be >= 2 as two bits are used to code zero and sign of weight. inq_iterations repeated int64 () List which specifies after how many forward passes we fix 50% of the learnable weights. If we have done as many iterations as specified in the last element of inq_iterations, then all weights are fixed. selection_algorithm string (“largest_abs” or “random”) “largest_abs” Chooses algorithm that we use for selecting the weights to fix (“largest_abs” … fix weights with largest absolute value, “random” … fix weights randomly) seed int64 -1 Random seed. When -1, seed is sampled from global random number generator.
• Output(s)
 Name Description Options y Output

### FixedPointQuantize¶

This function uniformly quantizes values in fixed-point number representation.

In the forward pass,

$\begin{split}q_i= \left\{ \begin{array}{ll} max & if \ \ \ x_i > max \\ sign(x_i) \times floor(|x_i| \delta^{-1} + 2^{-1}) \times \delta & if \ \ min \le x_i \le max \\ min & if \ \ x_i < min \\ \end{array} \right.,\end{split}$

where $$\delta$$ is the step size, $$(min, max) :=(- (2^{n-1} - 1)\delta, (2^{n-1} - 1)\delta)$$ if $$sign$$ is true, $$(min, max) := (0, (2^n - 1) \delta)$$ otherwise, and $$n$$ is the total bit-width used.

In the backward pass when using ste_fine_grained as false,

$\frac{\partial q_i}{\partial x_i} = 1.$

In the backward pass when using ste_fine_grained as true,

$\begin{split}\frac{\partial q_i}{\partial x_i}= \left\{ \begin{array}{ll} 0 & if \ \ \ x_i > max \\ 1 & if \ \ min \le x_i \le max \\ 0 & if \ \ x_i < min \\ \end{array} \right..\end{split}$

Note

Quantized values are stored as floating point number, since this function is for simulation purposes.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description sign bool True Indicate the signed number or the unsigned number. Default is true. n int64 8 Bit width used. Note that sign consumes one bit. $$n-1$$ is used for number representation in signed case. delta float 2**-4 Step size. ste_fine_grained bool True Straight Through Estimator is fine-grained or not.
• Output(s)
 Name Description Options y N-D array.

### Pow2Quantize¶

This function quantizes values in the power of 2 number representation, in other words, it is linear (uniform) quantization in $$log_2$$ domain.

In the forward pass of signed case,

$\begin{split}q_i= \left\{ \begin{array}{ll} max_{+} & if \ \ \overline{q_i} > max_{+} \\ \overline{q_i} & if \ \ min_{+} \le \overline{q_i} \le max_{+} \\ min_{+} & if \ \ 0 \le \overline{q_i} < min_{+} \\ min_{-} & if \ \ min_{-} < \overline{q_i} < 0 \\ \overline{q_i} & if \ \ max_{-} \le \overline{q_i} \le min_{-}\\ max_{-} & if \ \ \overline{q_i} < max_{-} \\ \end{array} \right.,\end{split}$

where

$\begin{split}&& max_{+} = 2^{m}, min_{+} = 2^{m - (2^{n-1} - 1)},\\ && max_{-} = -2^{m}, min_{-} = -2^{m - (2^{n-1} - 1)},\\ && \overline{q_i} = sign(x_i) \times 2^{round(\log_2 |x_i|)}.\end{split}$

This quantization uses the geometric mean between two power-of-two numbers as quantization threshold.

In the forward pass of unsigned case,

$\begin{split}q_i= \left\{ \begin{array}{ll} max & if \ \ \overline{q_i} > max \\ \overline{q_i} & if \ \ min \le \overline{q_i} \le max \\ min & if \ \ 0 < \overline{q_i} < min \\ \end{array} \right.,\end{split}$

where

$\begin{split}&& max = 2^{m}, min = 2^{m - (2^{n} - 1)},\\ && \overline{q_i} = 2^{int(\log_2 |x_i|)}.\end{split}$

When using with_zero as true, a pruning threshold is used to round an input to 0 or $$min$$. The pruning threshold is defined in this function as the following,

$pruning\ threshold = min \times 2^{-\frac{1}{2}}.$

If an absolute value of the input is lesser than this value, the input is rounded to 0, otherwise $$min$$.

In the backward pass when using ste_fine_grained as false,

$\frac{\partial q_i}{\partial x_i} = 1.$

In the backward pass when using ste_fine_grained as true,

$\begin{split}\frac{\partial q_i}{\partial x_i}= \left\{ \begin{array}{ll} 0 & if \ \ \overline{q_i} > max_{+} \\ 1 & if \ \ otherwise \\ 0 & if \ \ \overline{q_i} < max_{-} \\ \end{array} \right..\end{split}$

There are some literatures using pow2 quantization in their proposed methods.

References:

Note

Quantized values are stored as floating point number, since this function is for simulation purposes.

• Input(s)
 Name Description Options x N-D array
• Argument(s)
 Name Type Default Description sign bool True Indicate the signed number or the unsigned number. Default is true. with_zero bool True Indicate using zero as a quantized value. Default is true. Note that zero consumes one bit. n int64 8 Bit width used, Note that sign consumes one bit. $$n-1$$ is used for number representation in signed case. Default is 8. m int64 1 $$2^m$$ is the upper bound of the dynamic range and $$-2^m$$ is the lower bound, $$m \in \mathcal{Z}$$. Default is 1. ste_fine_grained bool True Straight Through Estimator is fine-grained or not.
• Output(s)
 Name Description Options y N-D array.

## Validation¶

### TopNError¶

Top N error along the dimension specified by the axis, the element of outputs is

$\begin{split}y_i = \left \{ \begin{array}{l} 1 \ (x_i \ is \ not \ within \ N-th \ place) \\ 0 \ (x_i \ is \ within \ N-th \ place) \end{array} \right.\end{split}$
• Input(s)
 Name Description Options x Probabilities N-D array. $$D_1 \times ... \times D_i \times ... \times D_N$$ target N-D array of labels. $$D_1 \times ... \times 1 \times ... \times D_N$$ Integer
• Argument(s)
 Name Type Default Description axis int64 len(x.shape) - 1 Axis on which the top N error is calculated. n int64 1 top N
• Output(s)
 Name Description Options output Element-wise error N-D array. ($$D_1 \times ... \times 1 \times ... \times D_N$$)

### BinaryError¶

Elementwise binary error.

$\begin{split}y_i = \left \{ \begin{array}{l} 0 ((x^{(0)} \geq 0.5) = (x^{(1)} \geq 0.5)) \\ 1 ((x^{(0)} \geq 0.5) \neq (x^{(1)} \geq 0.5)) \end{array} \right.\end{split}$
• Input(s)
 Name Description Options x Probabilities N-D array. f$-inftyf$ to f$inftyf$. target Labels N-D array. Usually set as 0 or 1, but, it allows probability (0 to 1) as inputs.
• Output(s)
 Name Description Options output Element-wise errors N-D array.

### ConfusionMatrix¶

Confusion matrix. The return value is already summed over samples.

• Input(s)
 Name Description Options x Probabilities N-D array. (f$D_1 times … times D_i times … times D_Nf$) target Labels N-D array. (f$D_1 times … times 1 times … times D_Nf$) Integer
• Argument(s)
 Name Type Default Description axis int64 len(x.shape) - 1 Axis on which the confusion matrix is calculated.
• Output(s)
 Name Description Options output Confusion matrix 2-D array. Col index is estimated class. Row index is label class.

## Unsupported, Special Use¶

### VATNoise¶

This layer is a special layer for GUI network designing, specialized for getting the noise of virtual adversarial training.

In the backward process, the weight parameter will be replaced with the gradient.

Forward

$y_i = \frac{\epsilon x_i}{\sqrt{\sum_k x_k^2 + c}}$

Backward

$\delta x_i = 0$
$w_i = \epsilon \delta y_i$
Note:
This layer is a special layer for GUI network designing.
References:
• Input(s)
 Name Description Options x N-D array of noise input. Noise is standard Gaussian noise initially, but the next step, fed back gradient variable. w N-D array for keep gradient values.
• Argument(s)
 Name Type Default Description base_axis int64 1 Dimensions up to base_axis is treated as sample dimension. eps float 1.0 Noise norm (l2) factor.
• Output(s)
 Name Description Options y N-D array

### Sink¶

Creates a dummy variable used to call forward or backward function of multiple variables at one place.

This takes any numbers of input variables with any shape, and creates a single 0-shape outputs. The forward pass does nothing. The backward pass set ones to the input grads if one_input_grad is set as true.

• Input(s)
 Name Description Options x Any number of inputs with any shape. Variadic
• Argument(s)
 Name Type Default Description one_input_grad bool True Set grads of inputs as one during backward. It is useful to set false if you want to set external gradients to the input variables.
• Output(s)
 Name Description Options y Dummy variable.