# Standard Function implementations¶

Chainer provides basic Function implementations in the chainer.functions package. Most of them are wrapped by plain Python functions, which users should use.

Note

As of v1.5, the concept of parameterized functions are gone, and they are replaced by corresponding Link implementations. They are still put in the functions namespace for backward compatibility, though it is strongly recommended to use them via the chainer.links package.

## Activation functions¶

### clipped_relu¶

chainer.functions.clipped_relu(x, z=20.0)[source]

Clipped Rectifier Unit function.

This function is expressed as $$ClippedReLU(x, z) = \min(\max(0, x), z)$$, where $$z(>0)$$ is a clipping value.

Parameters: x (Variable) – Input variable. z (float) – Clipping value. (default = 20.0) Output variable. Variable

### crelu¶

chainer.functions.crelu(x, axis=1)[source]

Concatenated Rectified Linear Unit function.

This function is expressed as $$f(x) = (\max(0, x), \max(0, -x))$$, where two output values are concatenated along an axis.

Parameters: x (Variable) – Input variable. axis (int) – Axis that the output values are concatenated along Output variable. Variable

### elu¶

chainer.functions.elu(x, alpha=1.0)[source]

Exponential Linear Unit function.

This function is expressed as

$\begin{split}f(x) = \left \{ \begin{array}{ll} x & {\rm if}~ x \ge 0 \\ \alpha (\exp(x) - 1) & {\rm if}~ x < 0, \end{array} \right.\end{split}$

where $$\alpha$$ is a parameter. See: http://arxiv.org/abs/1511.07289

Parameters: x (Variable) – Input variable. alpha (float) – Parameter $$\alpha$$. Output variable. Variable

### hard_sigmoid¶

chainer.functions.hard_sigmoid(x)[source]

Elementwise hard-sigmoid function.

This function is defined as

$\begin{split}f(x) = \left \{ \begin{array}{ll} 0 & {\rm if}~ x < -0.25 \\ 0.2 x + 0.5 & {\rm if}~ -0.25 < x < 0.25 \\ 1 & {\rm if}~ 0.25 < x. \end{array} \right.\end{split}$
Parameters: x (Variable) – Input variable. Output variable. Variable

### leaky_relu¶

chainer.functions.leaky_relu(x, slope=0.2)[source]

Leaky Rectified Linear Unit function.

This function is expressed as $$f(x) = \max(x, ax)$$, where $$a$$ is a configurable slope value.

Parameters: x (Variable) – Input variable. slope (float) – Slope value $$a$$. Output variable. Variable

### log_softmax¶

chainer.functions.log_softmax(x, use_cudnn=True)[source]

Channelwise log-softmax function.

This function computes its logarithm of softmax along the second axis. Let $$i = (i_1, i_2, \dots, i_d)^{\top}$$ be the d dimensional index array and $$x = f(i)$$ be the corresponding d dimensional input array. For each index $$i$$ of the input array $$f(i)$$, it computes the logarithm of the probability $$\log p(x)$$ defined as

$p(i) = {\exp(f(i)) \over \sum_{i'_2} \exp(f(i'))},$

where $$i' = (i_1, i'_2, \dots, i_d)$$.

$p(x) = {\exp(f(x)) \over \sum_{x'} \exp(f(x'))}.$

This method is theoretically equivalent to log(softmax(x)) but is more stable.

Note

log(softmax(x)) may cause underflow when x is too small, because softmax(x) may returns 0. log_softmax method is more stable.

Parameters: x (Variable) – Input variable. use_cudnn (bool) – If True, cuDNN is enabled and cuDNN ver. 3 or later is used, then this function uses cuDNN as the core implementation. Output variable. Variable

### lstm¶

chainer.functions.lstm(c_prev, x)[source]

Long Short-Term Memory units as an activation function.

This function implements LSTM units with forget gates. Let the previous cell state $$c_{\text{prev}}$$ and the incoming signal $$x$$.

First, the incoming signal $$x$$ is split into four arrays $$a, i, f, o$$ of the same shapes along the second axis. It means that $$x$$ ‘s second axis must have 4 times the length of $$c_{\text{prev}}$$.

The split input signals are corresponding to:

• $$a$$ : sources of cell input
• $$i$$ : sources of input gate
• $$f$$ : sources of forget gate
• $$o$$ : sources of output gate

Second, it computes outputs as:

$\begin{split}c &= \tanh(a) \text{sigmoid}(i) + c_{\text{prev}} \text{sigmoid}(f), \\ h &= \tanh(c) \text{sigmoid}(o).\end{split}$

These are returned as a tuple of two variables.

This function supports variable length inputs. The mini-batch size of the current input must be equal to or smaller than that of the previous one. When mini-batch size of x is smaller than that of c, this function only updates c[0:len(x)] and doesn’t change the rest of c, c[len(x):]. So, please sort input sequences in descending order of lengths before applying the function.

Parameters: c_prev (Variable) – Variable that holds the previous cell state. The cell state should be a zero array or the output of the previous call of LSTM. x (Variable) – Variable that holds the incoming signal. It must have the second dimension four times of that of the cell state, Two Variable objects c and h. c is the updated cell state. h indicates the outgoing signal. tuple

See the original paper proposing LSTM with forget gates: Long Short-Term Memory in Recurrent Neural Networks.

Example

Assuming y is the current input signal, c is the previous cell state, and h is the previous output signal from an lstm function. Each of y, c and h has n_units channels. Most typical preparation of x is:

>>> import chainer, chainer.functions as F
>>> n_units = 100
>>> y = chainer.Variable(numpy.zeros((1, n_units), 'f'))
>>> h = chainer.Variable(numpy.zeros((1, n_units), 'f'))
>>> c = chainer.Variable(numpy.zeros((1, n_units), 'f'))
>>> model = chainer.Chain(w=F.Linear(n_units, 4 * n_units),
...                       v=F.Linear(n_units, 4 * n_units),)
>>> x = model.w(y) + model.v(h)
>>> c, h = F.lstm(c, x)


It corresponds to calculate the input sources $$a, i, f, o$$ from the current input y and the previous output h. Different parameters are used for different kind of input sources.

### maxout¶

chainer.functions.maxout(x, pool_size, axis=1)[source]

Maxout activation function.

It accepts an input tensor x, reshapes the axis dimension (say the size being M * pool_size) into two dimensions (M, pool_size), and takes maximum along the axis dimension. The output of this function is same as x except that axis dimension is transformed from M * pool_size to M.

Typically, x is the output of a linear layer or a convolution layer. The following is the example where we use maxout() in combination with a Linear link.

>>> import numpy, chainer, chainer.links as L
>>> in_size, out_size, pool_size = 100, 100, 100
>>> l = L.Linear(in_size, out_size * pool_size)
>>> x = chainer.Variable(numpy.zeros((1, in_size), 'f'))  # prepare data
>>> x = l(x)
>>> y = maxout(x, pool_size)

Parameters: x (Variable) – Input variable. Its first dimension is assumed to be the minibatch dimension. The other dimensions are treated as one concatenated dimension. Output variable. Variable

### prelu¶

chainer.functions.prelu(x, W)[source]

Parametric ReLU function.

It accepts two arguments: an input x and a weight array W and computes the output as $$PReLU(x) = \max(x, W*x)$$, where $$*$$ is an elementwise multiplication for each sample in the batch.

When the PReLU function is combined with two-dimensional convolution, the elements of parameter $$a$$ are typically shared across the same filter of different pixels. In order to support such usage, this function supports the shape of parameter array that indicates leading dimensions of input arrays except the batch dimension.

For example $$W$$ has the shape of $$(2, 3, 4)$$, $$x$$ must have the shape of $$(B, 2, 3, 4, S1, ..., SN)$$ where B is batch size and the number of trailing S’s is arbitrary non-negative integer.

Parameters: x (Variable) – Input variable. Its first argument is assumed to be the minibatch dimension. W (Variable) – Weight variable. Output variable Variable

### relu¶

chainer.functions.relu(x, use_cudnn=True)[source]

Rectified Linear Unit function $$f(x)=\max(0, x)$$.

Parameters: x (Variable) – Input variable. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. Variable

### sigmoid¶

chainer.functions.sigmoid(x, use_cudnn=True)[source]

Elementwise sigmoid logistic function $$f(x)=(1 + \exp(-x))^{-1}$$.

Parameters: x (Variable) – Input variable. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. Variable

### slstm¶

chainer.functions.slstm(c_prev1, c_prev2, x1, x2)[source]

S-LSTM units as an activation function.

This function implements S-LSTM unit. It is an extension of LSTM unit applied to tree structures. The function is applied to binary trees. Each node has two child nodes. It gets four arguments, previous cell states $$c_1$$ and $$c_2$$, and incoming signals $$x_1$$ and $$x_2$$.

First both input signals $$x_1$$ and $$x_2$$ are split into eight arrays $$a_1, i_1, f_1, o_1$$, and $$a_2, i_2, f_2, o_2$$. They have the same shape along the second axis. It means that $$x_1$$ and $$x_2$$ ‘s second axis must have 4 times the length of $$c_{1 \text{prev}}$$ and $$c_{2 \text{prev}}$$.

The split input signals are corresponding to:

• $$a_i$$ : sources of cell input
• $$i_i$$ : sources of input gate
• $$f_i$$ : sources of forget gate
• $$o_i$$ : sources of output gate

It computes outputs as:

$\begin{split}c &= \tanh(a_1 + a_2) \sigma(i_1 + i_2) + c_{1 \text{prev}} \sigma(f_1) + c_{2 \text{prev}} \sigma(f_2), \\ h &= \tanh(c) \sigma(o_1 + o_2),\end{split}$

where $$\sigma$$ is the elementwise sigmoid function. The function returns $$c$$ and $$h$$ as a tuple.

Parameters: c_prev1 (Variable) – Variable that holds the previous cell state of the first child node. The cell state should be a zero array or the output of the previous call of LSTM. c_prev2 (Variable) – Variable that holds the previous cell state of the second child node. x1 (Variable) – Variable that holds the incoming signal from the first child node. It must have the second dimension four times of that of the cell state, x2 (Variable) – Variable that holds the incoming signal from the second child node. Two Variable objects c and h. c is the cell state. h indicates the outgoing signal. tuple

See detail in paper: Long Short-Term Memory Over Tree Structures.

### softmax¶

chainer.functions.softmax(x, use_cudnn=True)[source]

Channelwise softmax function.

This function computes its softmax along the second axis. Let $$x = (x_1, x_2, \dots, x_d)^{\top}$$ be the d dimensional index array and $$f(x)$$ be the d dimensional input array. For each index $$x$$ of the input array $$f(x)$$, it computes the probability $$p(x)$$ defined as $$p(x) = {\exp(f(x)) \over \sum_{x_2} \exp(f(x))}$$.

Parameters: x (Variable) – Input variable. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. Variable

### softplus¶

chainer.functions.softplus(x, beta=1.0)[source]

Elementwise softplus function.

This function is expressed as $$f(x) = \frac{1}{\beta}\log(1 + \exp(\beta x))$$, where $$\beta$$ is a parameter.

Parameters: x (Variable) – Input variable. beta (float) – Parameter $$\beta$$. Output variable. Variable

### tanh¶

chainer.functions.tanh(x, use_cudnn=True)[source]

Elementwise hyperbolic tangent function.

Parameters: x (Variable) – Input variable. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. Variable

## Array manipulations¶

chainer.functions.broadcast(*args)[source]

Parameters: args (Variables) – Variables to be broadcasted. Tuple of Variable objects which are broadcasted from given arguments. tuple

chainer.functions.broadcast_to(x, shape)[source]

Broadcast a given variable to a given shape.

Parameters: x (Variable) – Variable to be broadcasted. shape (tuple of int) – The shape of the output variable. Output variable broadcasted to the given shape. Variable

### cast¶

chainer.functions.cast(x, typ)[source]

Cast an input variable to a given type.

Parameters: x (Variable) – Input variable. typ (str of dtype) – Typecode or data type to cast. Variable holding a casted array. Variable

### concat¶

chainer.functions.concat(xs, axis=1)[source]

Concatenates given variables along an axis.

Parameters: xs (tuple of Variables) – Variables to be concatenated. axis (int) – Axis that the input arrays are concatenated along. Output variable. Variable

### copy¶

chainer.functions.copy(x, dst)[source]

Copies the input variable onto the specified device.

This function copies the array of input variable onto the device specified by dst. When dst == -1, it copies the array onto the host memory. This function supports copies from host to device, from device to device and from device to host.

Parameters: x (Variable) – Variable to be copied. dst – Target device specifier. Output variable. Variable

### expand_dims¶

chainer.functions.expand_dims(x, axis)[source]

Expands dimensions of an input variable without copy.

Parameters: x (Variable) – Input variable. axis (int) – Position where new axis is to be inserted. Variable that holds a expanded input. Variable

### flatten¶

chainer.functions.flatten(x)[source]

Flatten a given array.

Parameters: x (Varaiable) – Input variable. Output variable. Variable

### get_item¶

chainer.functions.get_item(x, slices)[source]

Extract elements from array with specified shape, axes and offsets.

Parameters: x (tuple of Variables) – Variable to be sliced. slices (int, slice, None or Ellipsis or tuple of them) – Basic slicing to slice a variable. It supports int, slice, newaxis (equivalent to None) and Ellipsis. Variable object which contains sliced array of x. Variable

Note

See NumPy document for details of indexing.

### hstack¶

chainer.functions.hstack(xs)[source]

Concatenate variables horizontally (column wise).

Parameters: xs (list of chainer.Variable) – Variables to be concatenated. Output variable. Variable

### permutate¶

chainer.functions.permutate(x, indices, axis=0, inv=False)[source]

Permutates a given variable along an axis.

This function permutate x with given indices. That means y[i] = x[indices[i]] for all i. Note that this result is same as y = x.take(indices). indices must be a permutation of [0, 1, ..., len(x) - 1].

When inv is True, indices is treated as its inverse. That means y[indices[i]] = x[i].

Parameters: x (Variable) – Variable to permutate. indices (Variable) – Indices to extract from the variable. axis (int) – Axis that the input array is permutate along. inv (bool) – If True, indices is treated as its inverse. Output variable. Variable

### reshape¶

chainer.functions.reshape(x, shape)[source]

Reshapes an input variable without copy.

Parameters: x (Variable) – Input variable. shape (tuple of ints) – Target shape. Variable that holds a reshaped version of the input variable. Variable

### rollaxis¶

chainer.functions.rollaxis(x, axis, start=0)[source]

Roll the axis backwards to the given position.

Parameters: x (Variable) – Input variable. axis (int) – The axis to roll backwards. start (int) – The place to which the axis is moved. Variable whose axis is rolled. Variable

### select_item¶

chainer.functions.select_item(x, t)[source]

Select elements stored in given indices.

This function returns t.choose(x.T), that means y[i] == x[i, t[i]] for all i.

Parameters: x (Variable) – Variable storing arrays. t (Variable) – Variable storing index numbers. Variable that holds t-th element of x. Variable

### separate¶

chainer.functions.separate(x, axis=0)[source]

Separates an array along a given axis.

This function separates an array along a given axis. For example, shape of an array is (2, 3, 4). When it separates the array with axis=1, it returns three (2, 4) arrays.

This function is an inverse of chainer.functions.stack().

Parameters: x (chainer.Variable) – Variable to be separated. axis (int) – Axis along which variables are separated. Output variables. tuple of chainer.Variable

### split_axis¶

chainer.functions.split_axis(x, indices_or_sections, axis, force_tuple=False)[source]

Splits given variables along an axis.

Parameters: x (tuple of Variables) – Variables to be split. indices_or_sections (int or 1-D array) – If this argument is an integer, N, the array will be divided into N equal arrays along axis. If it is a 1-D array of sorted integers, it indicates the positions where the array is split. axis (int) – Axis that the input array is split along. force_tuple (bool) – If True, this method returns a tuple even when the number of outputs is one. Tuple of Variable objects if the number of outputs is more than 1 or Variable otherwise. When force_tuple is True, returned value is always a tuple regardless of the number of outputs. tuple or Variable

Note

This function raises ValueError if at least one of the outputs is split to zero-size (i.e. axis-th value of its shape is zero).

### stack¶

chainer.functions.stack(xs, axis=0)[source]

Concatenate variables along a new axis.

Parameters: xs (list of chainer.Variable) – Variables to be concatenated. axis (int) – Axis of result along which variables are stacked. Output variable. Variable

### swapaxes¶

chainer.functions.swapaxes(x, axis1, axis2)[source]

Swap two axes of a variable.

Parameters: x (Variable) – Input variable. axis1 (int) – The first axis to swap. axis2 (int) – The second axis to swap. Variable whose axes are swapped. Variable

### transpose¶

chainer.functions.transpose(x, axes=None)[source]

Permute the dimensions of an input variable without copy.

Parameters: x (Variable) – Input variable. axes (tuple of ints) – By default, reverse the dimensions, otherwise permute the axes according to the values given. Variable whose axes are permuted. Variable

### transpose_sequence¶

chainer.functions.transpose_sequence(xs)[source]

Transpose a list of Variables.

This function transposes a list of Variable s and returns a list of Variable s. For example a user gives [(0, 1, 2, 3), (4, 5), (6)], the function returns [(0, 4, 6), (1, 5), (2), (3)]. Note that a given list needs to be sorted by each length of Variable.

Parameters: xs (list of ~chainer.Variable) – Variables to transpose. Transposed list. tuple or Variable

### vstack¶

chainer.functions.vstack(xs)[source]

Concatenate variables vertically (row wise).

Parameters: xs (list of chainer.Variable) – Variables to be concatenated. Output variable. Variable

### where¶

chainer.functions.where(condition, x, y)[source]

Choose elements depending on condition.

This function choose values depending on a given condition. All condition, x, and y must have the same shape.

Parameters: condition (Variable) – Variable containing the condition. Only boolean array is permitted. x (Variable) – Variable chosen when condition is True. y (Variable) – Variable chosen when condition is False. Variable containing chosen values. Variable

## Neural network connections¶

### bilinear¶

chainer.functions.bilinear(e1, e2, W, V1=None, V2=None, b=None)[source]

Applies a bilinear function based on given parameters.

This is a building block of Neural Tensor Network (see the reference paper below). It takes two input variables and one or four parameters, and outputs one variable.

To be precise, denote six input arrays mathematically by $$e^1\in \mathbb{R}^{I\cdot J}$$, $$e^2\in \mathbb{R}^{I\cdot K}$$, $$W\in \mathbb{R}^{J \cdot K \cdot L}$$, $$V^1\in \mathbb{R}^{J \cdot L}$$, $$V^2\in \mathbb{R}^{K \cdot L}$$, and $$b\in \mathbb{R}^{L}$$, where $$I$$ is mini-batch size. In this document, we call $$V^1$$, $$V^2$$, and $$b$$ linear parameters.

The output of forward propagation is calculated as

$y_{il} = \sum_{jk} e^1_{ij} e^2_{ik} W_{jkl} + \ \sum_{j} e^1_{ij} V^1_{jl} + \sum_{k} e^2_{ik} V^2_{kl} + b_{l}.$

Note that V1, V2, b are optional. If these are not given, then this function omits the last three terms in the above equation.

Note

This function accepts an input variable e1 or e2 of a non-matrix array. In this case, the leading dimension is treated as the batch dimension, and the other dimensions are reduced to one dimension.

Note

In the original paper, $$J$$ and $$K$$ must be equal and the author denotes $$[V^1 V^2]$$ (concatenation of matrices) by $$V$$.

Parameters: e1 (Variable) – Left input variable. e2 (Variable) – Right input variable. W (Variable) – Quadratic weight variable. V1 (Variable) – Left coefficient variable. V2 (Variable) – Right coefficient variable. b (Variable) – Bias variable. Output variable. Variable
See:
Reasoning With Neural Tensor Networks for Knowledge Base Completion [Socher+, NIPS2013].

### convolution_2d¶

chainer.functions.convolution_2d(x, W, b=None, stride=1, pad=0, use_cudnn=True, cover_all=False)[source]

Two-dimensional convolution function.

This is an implementation of two-dimensional convolution in ConvNets. It takes three variables: the input image x, the filter weight W, and the bias vector b.

Notation: here is a notation for dimensionalities.

• $$n$$ is the batch size.
• $$c_I$$ and $$c_O$$ are the number of the input and output, respectively.
• $$h$$ and $$w$$ are the height and width of the input image, respectively.
• $$k_H$$ and $$k_W$$ are the height and width of the filters, respectively.
Parameters: x (Variable) – Input variable of shape $$(n, c_I, h, w)$$. W (Variable) – Weight variable of shape $$(c_O, c_I, k_H, k_W)$$. b (Variable) – Bias variable of length $$c_O$$ (optional). stride (int or pair of ints) – Stride of filter applications. stride=s and stride=(s, s) are equivalent. pad (int or pair of ints) – Spatial padding width for input arrays. pad=p and pad=(p, p) are equivalent. use_cudnn (bool) – If True, then this function uses cuDNN if available. cover_all (bool) – If True, all spatial locations are convoluted into some output pixels. It may make the output size larger. Output variable. Variable

The two-dimensional convolution function is defined as follows. Then the Convolution2D function computes correlations between filters and patches of size $$(k_H, k_W)$$ in x. Note that correlation here is equivalent to the inner product between expanded vectors. Patches are extracted at positions shifted by multiples of stride from the first position -pad for each spatial axis. The right-most (or bottom-most) patches do not run over the padded spatial size.

Let $$(s_Y, s_X)$$ be the stride of filter application, and $$(p_H, p_W)$$ the spatial padding size. Then, the output size $$(h_O, w_O)$$ is determined by the following equations:

$\begin{split}h_O &= (h + 2p_H - k_H) / s_Y + 1,\\ w_O &= (w + 2p_W - k_W) / s_X + 1.\end{split}$

If the bias vector is given, then it is added to all spatial locations of the output of convolution.

Convolution2D

### convolution_nd¶

chainer.functions.convolution_nd(x, W, b=None, stride=1, pad=0, use_cudnn=True, cover_all=False)[source]

N-dimensional convolution function.

This is an implementation of N-dimensional convolution which is generalized two-dimensional convolution in ConvNets. It takes three variables: the input x, the filter weight W and the bias vector b.

Notation: here is a notation for dimensionalities.

• $$N$$ is the number of spatial dimensions.
• $$n$$ is the batch size.
• $$c_I$$ and $$c_O$$ are the number of the input and output channels, respectively.
• $$d_1, d_2, ..., d_N$$ are the size of each axis of the input’s spatial dimensions, respectively.
• $$k_1, k_2, ..., k_N$$ are the size of each axis of the filters, respectively.
Parameters: x (Variable) – Input variable of shape $$(n, c_I, d_1, d_2, ..., d_N)$$. W (Variable) – Weight variable of shape $$(c_O, c_I, k_1, k_2, ..., k_N)$$. b (Variable) – One-dimensional bias variable with length $$c_O$$ (optional). stride (int or tuple of ints) – Stride of filter applications $$(s_1, s_2, ..., s_N)$$. stride=s is equivalent to (s, s, ..., s). pad (int or tuple of ints) – Spatial padding width for input arrays $$(p_1, p_2, ..., p_N)$$. pad=p is equivalent to (p, p, ..., p). use_cudnn (bool) – If True, then this function uses cuDNN if available. See below for the excact conditions. cover_all (bool) – If True, all spatial locations are convoluted into some output pixels. It may make the output size larger. cover_all needs to be False if you want to use cuDNN. Output variable. Variable

This function uses cuDNN implementation for its forward and backward computation if ALL of the following conditions are satisfied:

• cuda.cudnn_enabled is True
• use_cudnn is True
• The number of spatial dimensions is more than one.
• cover_all is False
• The input’s dtype is equal to the filter weight’s.
• The dtype is FP32, FP64 or FP16(cuDNN version is equal to or greater than v3)

ConvolutionND, convolution_2d()

### deconvolution_2d¶

chainer.functions.deconvolution_2d(x, W, b=None, stride=1, pad=0, outsize=None, use_cudnn=True)[source]

Two dimensional deconvolution function.

This is an implementation of two-dimensional deconvolution. It takes three variables: input image x, the filter weight W, and the bias vector b.

Parameters: x (Variable) – Input variable of shape $$(n, c_I, h, w)$$. W (Variable) – Weight variable of shape $$(c_I, c_O, k_H, k_W)$$. b (Variable) – Bias variable of length $$c_O$$ (optional). stride (int or pair of ints) – Stride of filter applications. stride=s and stride=(s, s) are equivalent. pad (int or pair of ints) – Spatial padding width for input arrays. pad=p and pad=(p, p) are equivalent. outsize (tuple) – Expected output size of deconvolutional operation. It should be pair of height and width $$(out_H, out_W)$$. Default value is None and the outsize is estimated by input size, stride and pad. use_cudnn (bool) – If True, then this function uses cuDNN if available.

The filter weight has four dimensions $$(c_I, c_O, k_H, k_W)$$ which indicate the number of the number of input channels, output channels, height and width of the kernels, respectively.

The bias vector is of size $$c_O$$.

Let $$X$$ be the input tensor of dimensions $$(n, c_I, h, w)$$, $$(s_Y, s_X)$$ the stride of filter application, and $$(p_H, p_W)$$ the spatial padding size. Then, the output size $$(h_O, w_O)$$ is determined by the following equations:

$\begin{split}h_O &= s_Y (h - 1) + k_H - 2p_H,\\ w_O &= s_X (w - 1) + k_W - 2p_W.\end{split}$

### embed_id¶

chainer.functions.embed_id(x, W, ignore_label=None)[source]

Efficient linear function for one-hot input.

This function implements so called word embedding. It takes two arguments: a set of IDs (words) x in $$B$$ dimensional integer vector, and a set of all ID (word) embeddings W in $$V \times d$$ float32 matrix. It outputs $$B \times d$$ matrix whose i-th column is the x[i]-th column of W.

This function is only differentiable on the input W.

Parameters: x (Variable) – Batch vectors of IDs. W (Variable) – Representation of each ID (a.k.a. word embeddings). ignore_label (int or None) – If ignore_label is an int value, i-th column of return value is filled with 0. Output variable. Variable

EmbedID

### linear¶

chainer.functions.linear(x, W, b=None)[source]

Linear function, or affine transformation.

It accepts two or three arguments: an input minibatch x, a weight matrix W, and optionally a bias vector b. It computes $$Y = xW^\top + b$$.

Parameters: x (Variable) – Input variable. Its first dimension is assumed to be the minibatch dimension. The other dimensions are treated as concatenated one dimension whose size must be N. W (Variable) – Weight variable of shape (M, N). b (Variable) – Bias variable (optional) of shape (M,). Output variable. Variable

## Evaluation functions¶

### accuracy¶

chainer.functions.accuracy(y, t, ignore_label=None)[source]

Computes muticlass classification accuracy of the minibatch.

Parameters: y (Variable) – Variable holding a matrix whose (i, j)-th element indicates the score of the class j at the i-th example. t (Variable) – Variable holding an int32 vector of ground truth labels. ignore_label (int or None) – Skip calculating accuracy if the true label is ignore_label. A variable holding a scalar array of the accuracy. Variable

Note

This function is non-differentiable.

## Loss functions¶

### bernoulli_nll¶

chainer.functions.bernoulli_nll(x, y)[source]

Computes the negative log-likelihood of a Bernoulli distribution.

This function calculates the negative log-likelihood of a Bernoulli distribution.

$-B(x; p) = -\sum_i {x_i \log(p_i) + (1 - x_i)\log(1 - p_i)},$

where $$p = \sigma(y)$$, and $$\sigma(\cdot)$$ is a sigmoid function.

Note

As this function uses a sigmoid function, you can pass a result of fully-connected layer (that means Linear) to this function directly.

Parameters: x (Variable) – Input variable. y (Variable) – A variable representing the parameter of Bernoulli distribution. A variable representing negative log-likelihood. Variable

### connectionist_temporal_classification¶

chainer.functions.connectionist_temporal_classification(x, t, blank_symbol, input_length=None, label_length=None)[source]

Connectionist Temporal Classification loss function.

Connectionist Temporal Classification(CTC) [Graves2006] is a loss function of sequence labeling where the alignment between the inputs and target is unknown. See also [Graves2012]

Parameters: x (sequence of Variable) – RNN output at each time. x must be a list of Variable s. Each element of x, x[i] is a Variable representing output of RNN at time i. t (Variable) – Expected label sequence. blank_symbol (int) – Index of blank_symbol. This value must be non-negative. input_length (Variable) – Length of valid sequence for each of mini batch x (optional). If input_length is skipped, It regards that all of x is valid input. label_length (Variable) – Length of valid sequence for each of mini batch t (optional). If label_length is skipped, It regards that all of t is valid input. A variable holding a scalar value of the CTC loss. Variable

Note

You need to input x without applying to activation functions(e.g. softmax function), because this function applies softmax functions to x before calculating CTC loss to avoid numerical limitations. You also need to apply softmax function to forwarded values before you decode it.

Note

This function is differentiable only by x.

Note

This function supports (batch, sequence, 1-dimensional input)-data.

 [Graves2006] Alex Graves, Santiago Fernandez, Faustino Gomez, Jurgen Schmidhuber, Connectionist Temporal Classification: Labelling Unsegmented Sequence Data with Recurrent Neural Networks

### contrastive¶

chainer.functions.contrastive(x0, x1, y, margin=1)[source]

Computes contrastive loss.

It takes a pair of variables and a label as inputs. The label is 1 when those two input variables are similar, or 0 when they are dissimilar. Let $$N$$ and $$K$$ denote mini-batch size and the dimension of input variables, respectively. The shape of both input variables should be (N, K).

$L = \frac{1}{2N} \left( \sum_{n=1}^N y_n d_n^2 + (1 - y_n) \max ({\rm margin} - d_n, 0)^2 \right)$

where $$d_n = \| {\bf x_0}_n - {\bf x_1}_n \|_2$$. $$N$$ denotes the mini-batch size. Input variables, x0 and x1, have $$N$$ vectors, and each vector is K-dimensional. Therefore, $${\bf x_0}_n$$ and $${\bf x_1}_n$$ are $$n$$-th K-dimensional vectors of x0 and x1.

Parameters: x0 (Variable) – The first input variable. The shape should be (N, K), where N denotes the mini-batch size, and K denotes the dimension of x0. x1 (Variable) – The second input variable. The shape should be the same as x0. y (Variable) – Labels. All values should be 0 or 1. The shape should be (N,), where N denotes the mini-batch size. margin (float) – A parameter for contrastive loss. It should be positive value. A variable holding a scalar that is the loss value calculated by the above equation. Variable

Note

This cost can be used to train siamese networks. See Learning a Similarity Metric Discriminatively, with Application to Face Verification for details.

### crf1d¶

chainer.functions.crf1d(cost, xs, ys)[source]

Calculates negative log-likelihood of linear-chain CRF.

It takes a transition cost matrix, a sequence of costs, and a sequence of labels. Let $$c_{st}$$ be a transition cost from a label $$s$$ to a label $$t$$, $$x_{it}$$ be a cost of a label $$t$$ at position $$i$$, and $$y_i$$ be an expected label at position $$i$$. The negative log-likelihood of linear-chain CRF is defined as

$L = -\left( \sum_{i=1}^l x_{iy_i} + \ \sum_{i=1}^{l-1} c_{y_i y_{i+1}} - {\log(Z)} \right) ,$

where $$l$$ is the length of the input sequence and $$Z$$ is the normalizing constant called partition function.

Parameters: cost (Variable) – A $$K \times K$$ matrix which holds transition cost between two labels, where $$K$$ is the number of labels. xs (list of Variable) – Input feature vector for each label. Each Variable holds a $$B \times K$$ matrix, where $$B$$ is mini-batch size, $$K$$ is the number of labels. ys (list of Variable) – Expected output labels. Each Variable holds a $$B$$ integer vector. A variable holding the average negative log-likelihood of the input sequences. Variable

### cross_covariance¶

chainer.functions.cross_covariance(y, z)[source]

Computes the sum-squared cross-covariance penalty between y and z

Parameters: y (Variable) – Variable holding a matrix where the first dimension corresponds to the batches z (Variable) – Variable holding a matrix where the first dimension corresponds to the batches A variable holding a scalar of the cross covariance loss. Variable

Note

This cost can be used to disentangle variables. See http://arxiv.org/abs/1412.6583v3 for details.

### gaussian_kl_divergence¶

chainer.functions.gaussian_kl_divergence(mean, ln_var)[source]

Computes the KL-divergence of Gaussian variables from the standard one.

Given two variable mean representing $$\mu$$ and ln_var representing $$\log(\sigma^2)$$, this function returns a variable representing the KL-divergence between the given multi-dimensional Gaussian $$N(\mu, S)$$ and the standard Gaussian $$N(0, I)$$

$D_{\mathbf{KL}}(N(\mu, S) \| N(0, I)),$

where $$S$$ is a diagonal matrix such that $$S_{ii} = \sigma_i^2$$ and $$I$$ is an identity matrix.

Parameters: mean (Variable) – A variable representing mean of given gaussian distribution, $$\mu$$. ln_var (Variable) – A variable representing logarithm of variance of given gaussian distribution, $$\log(\sigma^2)$$. A variable representing KL-divergence between given gaussian distribution and the standard gaussian. Variable

### gaussian_nll¶

chainer.functions.gaussian_nll(x, mean, ln_var)[source]

Computes the negative log-likelihood of a Gaussian distribution.

Given two variable mean representing $$\mu$$ and ln_var representing $$\log(\sigma^2)$$, this function returns the negative log-likelihood of $$x$$ on a Gaussian distribution $$N(\mu, S)$$,

$-\log N(x; \mu, \sigma^2) = \log\left(\sqrt{(2\pi)^D |S|}\right) + \frac{1}{2}(x - \mu)^\top S^{-1}(x - \mu),$

where $$D$$ is a dimension of $$x$$ and $$S$$ is a diagonal matrix where $$S_{ii} = \sigma_i^2$$.

Parameters: x (Variable) – Input variable. mean (Variable) – A variable representing mean of a Gaussian distribution, $$\mu$$. ln_var (Variable) – A variable representing logarithm of variance of a Gaussian distribution, $$\log(\sigma^2)$$. A variable representing the negative log-likelihood. Variable

### hinge¶

chainer.functions.hinge(x, t, norm='L1')[source]

Computes the hinge loss for a one-of-many classification task.

$L = \frac{1}{N} \sum_{n=1}^N \sum_{k=1}^K \left[ \max(0, 1 - \delta\{l_n = k\} t_{nk}) \right]^p$

where $$N$$ denotes the batch size, $$K$$ is the number of classes of interest,

$\begin{split}\delta \{ {\rm condition} \} = \left \{ \begin{array}{cc} 1 & {\rm if~condition} \\ -1 & {\rm otherwise,} \end{array} \right.\end{split}$

and

$\begin{split}p = \left \{ \begin{array}{cc} 1 & {\rm if~norm} = {\rm 'L1'} \\ 2 & {\rm if~norm} = {\rm 'L2'.} \end{array} \right.\end{split}$
Parameters: x (Variable) – Input variable. The shape of x should be ($$N$$, $$K$$). t (Variable) – The $$N$$-dimensional label vector $${\bf l}$$ with values $$l_n \in \{0, 1, 2, \dots, K-1\}$$. The shape of t should be ($$N$$,). norm (string) – Specifies norm type. Only either ‘L1’ or ‘L2’ is acceptable. A variable object holding a scalar array of the hinge loss $$L$$. Variable

### huber_loss¶

chainer.functions.huber_loss(x, t, delta)[source]

Loss function which is less sensitive to outliers in data than MSE.

$a = x - t$

and

$\begin{split}L_{\delta}(a) = \left \{ \begin{array}{cc} \frac{1}{2} a^2 & {\rm if~|a| \leq \delta} \\ \delta (|a| - \frac{1}{2} \delta) & {\rm otherwise,} \end{array} \right.\end{split}$
Parameters: x (Variable) – Input variable. The shape of x should be ($$N$$, $$K$$). t (Variable) – Target variable for regression. The shape of t should be ($$N$$, $$K$$). delta (float) – Constant variable for huber loss function as used in definition. A variable object holding a scalar array of the huber loss $$L_{\delta}$$. Variable
See:
Huber loss - Wikipedia.

### mean_squared_error¶

chainer.functions.mean_squared_error(x0, x1)[source]

Mean squared error function.

This function computes mean squared error between two variables. The mean is taken over the minibatch. Note that the error is not scaled by 1/2.

### negative_sampling¶

chainer.functions.negative_sampling(x, t, W, sampler, sample_size)[source]

Negative sampling loss function.

In natural language processing, especially language modeling, the number of words in a vocabulary can be very large. Therefore, you need to spend a lot of time calculating the gradient of the embedding matrix.

By using the negative sampling trick you only need to calculate the gradient for a few sampled negative examples.

The objective function is below:

$f(x, p) = \log \sigma(x^\top w_p) + \ k E_{i \sim P(i)}[\log \sigma(- x^\top w_i)],$

where $$\sigma(\cdot)$$ is a sigmoid function, $$w_i$$ is the weight vector for the word $$i$$, and $$p$$ is a positive example. It is approximated with $$k$$ examples $$N$$ sampled from probability $$P(i)$$, like this:

$f(x, p) \approx \log \sigma(x^\top w_p) + \ \sum_{n \in N} \log \sigma(-x^\top w_n).$

Each sample of $$N$$ is drawn from the word distribution $$P(w)$$. This is calculated as $$P(w) = \frac{1}{Z} c(w)^\alpha$$, where $$c(w)$$ is the unigram count of the word $$w$$, $$\alpha$$ is a hyper-parameter, and $$Z$$ is the normalization constant.

Parameters: x (Variable) – Batch of input vectors. t (Variable) – Vector of ground truth labels. W (Variable) – Weight matrix. sampler (FunctionType) – Sampling function. It takes a shape and returns an integer array of the shape. Each element of this array is a sample from the word distribution. A WalkerAlias object built with the power distribution of word frequency is recommended. sample_size (int) – Number of samples.

### sigmoid_cross_entropy¶

chainer.functions.sigmoid_cross_entropy(x, t, use_cudnn=True, normalize=True)[source]

Computes cross entropy loss for pre-sigmoid activations.

Parameters: x (Variable) – A variable object holding a matrix whose (i, j)-th element indicates the unnormalized log probability of the j-th unit at the i-th example. t (Variable) – Variable holding an int32 vector of ground truth labels. If t[i] == -1, corresponding x[i] is ignored. Loss is zero if all ground truth labels are -1. normalize (bool) – Variable holding a boolean value which determines the normalization constant. If true, this function normalizes the cross entropy loss across all instances. If else, it only normalizes along a batch size. A variable object holding a scalar array of the cross entropy loss. Variable

Note

This function is differentiable only by x.

### softmax_cross_entropy¶

chainer.functions.softmax_cross_entropy(x, t, use_cudnn=True, normalize=True, cache_score=True)[source]

Computes cross entropy loss for pre-softmax activations.

Parameters: x (Variable) – Variable holding a multidimensional array whose element indicates unnormalized log probability: the first axis of the variable represents the number of samples, and the second axis represents the number of classes. While this function computes a usual softmax cross entropy if the number of dimensions is equal to 2, it computes a cross entropy of the replicated softmax if the number of dimensions is greater than 2. t (Variable) – Variable holding an int32 vector of ground truth labels. If t[i] == -1, corresponding x[i] is ignored. normalize (bool) – If true, this function normalizes the cross entropy loss across all instances. If false, it only normalizes along a batch size. cache_score (bool) – When it is True, the function stores result of forward computation to use it on backward computation. It reduces computational cost though consumes more memory. A variable holding a scalar array of the cross entropy loss. Variable

Note

This function is differentiable only by x.

### triplet¶

chainer.functions.triplet(anchor, positive, negative, margin=0.2)[source]

Computes triplet loss.

It takes a triplet of variables as inputs, $$a$$, $$p$$ and $$n$$: anchor, positive example and negative example respectively. The triplet defines a relative similarity between samples. Let $$N$$ and $$K$$ denote mini-batch size and the dimension of input variables, respectively. The shape of all input variables should be $$(N, K)$$.

$L(a, p, n) = \frac{1}{N} \left( \sum_{i=1}^N \max \{d(a_i, p_i) - d(a_i, n_i) + {\rm margin}, 0\} \right)$

where $$d(x_i, y_i) = \| {\bf x}_i - {\bf y}_i \|_2^2$$.

Parameters: anchor (Variable) – The anchor example variable. The shape should be $$(N, K)$$, where $$N$$ denotes the minibatch size, and $$K$$ denotes the dimension of the anchor. positive (Variable) – The positive example variable. The shape should be the same as anchor. negative (Variable) – The negative example variable. The shape should be the same as anchor. margin (float) – A parameter for triplet loss. It should be a positive value. A variable holding a scalar that is the loss value calculated by the above equation. Variable

Note

This cost can be used to train triplet networks. See Learning Fine-grained Image Similarity with Deep Ranking for details.

## Mathematical functions¶

### argmax¶

chainer.functions.argmax(x, axis=None)[source]

Returns index which holds maximum of array elements over a given axis.

Parameters: x (Variable) – Array to find maximum elements. axis (None or int) – Axis over which a max is performed. The default (axis = None) is perform a max over all the dimensions of the input array. Output variable. Variable

### argmin¶

chainer.functions.argmin(x, axis=None)[source]

Returns index which holds minimum of array elements over a given axis.

Parameters: x (Variable) – Array to find minimum elements. axis (None or int) – Axis over which a min is performed. The default (axis = None) is perform a min over all the dimensions of the input array. Output variable. Variable

### batch_inv¶

chainer.functions.batch_inv(a)[source]

Computes the inverse of a batch of square matrices.

Parameters: a (Variable) – Input array to compute the determinant for. Shape of the array should be (m, n, n) where m is the number of matrices in the batch, and n is the dimensionality of a square matrix. Inverse of every matrix in the batch of matrices. Variable

### batch_l2_norm_squared¶

chainer.functions.batch_l2_norm_squared(x)[source]

L2 norm (a.k.a. Euclidean norm) squared.

This function implements the square of L2 norm on a vector. No reduction along batch axis is done.

Parameters: x (Variable) – Input variable. The first dimension is assumed to be the minibatch dimension. If x has more than two dimensions all but the first dimension are flattened to one dimension. Two dimensional output variable. Variable

### batch_matmul¶

chainer.functions.batch_matmul(a, b, transa=False, transb=False)[source]

Computes the batch matrix multiplications of two sets of arrays.

Parameters: a (Variable) – The left operand of the batch matrix multiplications. A 2-D array of shape (B, N) is considered as B $$N \times 1$$ matrices. A 3-D array of shape (B, M, N) is considered as B $$M \times N$$ matrices. b (Variable) – The right operand of the batch matrix multiplications. Its array is treated as matrices in the same way as a‘s array. transa (bool) – If True, transpose each matrix in a. transb (bool) – If True, transpose each matrix in b. The result of the batch matrix multiplications as a 3-D array. Variable

### bias¶

chainer.functions.bias(x, y, axis=1)[source]

Elementwise summation with broadcasting.

Computes a elementwise summation of two input variables, with the shape of the latter variable broadcasted to match the shape of the former. axis is the first axis of the first variable along which the second variable is applied.

The term “broadcasting” here comes from Caffe’s bias layer so the “broadcasting” with the following arguments:

   x : 100 x 3 x 40 x 60
y : 3 x 40
axis : 1


is equivalent to the following numpy broadcasting:

x : 100 x 3 x 40 x 60
y :   1 x 3 x 40 x 1


Note that how the axis indicates to which axis of x we apply y.

Parameters: x (Variable) – Input variable to be summed. y (Variable) – Input variable to sum, broadcasted. axis (int) – The first axis of x along which y is applied. Output variable. Variable

### clip¶

chainer.functions.clip(x, x_min, x_max)[source]

Clips (limits) elements of input variable.

Given an interval [x_min, xmax], elements outside the interval are clipped to the interval edges.

Parameters: x (Variable) – Input variable to be clipped. x_min (float) – Minimum value. x_max (float) – Maximum value. Output variable. Variable

### cos¶

chainer.functions.cos(x)[source]

Elementwise cos function.

### cosh¶

chainer.functions.cosh(x)[source]

Elementwise hyperbolic cosine function.

$y_i = \cosh x_i.$
Parameters: x (Variable) – Input variable. Output variable. Variable

### exp¶

chainer.functions.exp(x)[source]

Elementwise exponential function.

### identity¶

chainer.functions.identity(*inputs)[source]

Just returns input variables.

### inv¶

chainer.functions.inv(a)[source]

Computes the inverse of of square matrix.

Parameters: a (Variable) – Input array to compute the determinant for. Shape of the array should be (n, n) where n is the dimensionality of a square matrix. Matrix inverse of a. Variable

### linear_interpolate¶

chainer.functions.linear_interpolate(p, x, y)[source]

Elementwise linear-interpolation function.

This function is defined as

$f(p, x, y) = p x + (1 - p) y.$
Parameters: p (Variable) – Input variable. x (Variable) – Input variable. y (Variable) – Input variable. Output variable. Variable

### log¶

chainer.functions.log(x)[source]

Elementwise natural logarithm function.

### log10¶

chainer.functions.log10(x)[source]

Elementwise logarithm function to the base 10.

$y_i = \log_10 x_i.$
Parameters: x (Variable) – Input variable. Output variable. Variable

### log1p¶

chainer.functions.log1p(x)[source]

Elementwise natural logarithm plus one function.

### log2¶

chainer.functions.log2(x)[source]

Elementwise logarithm function to the base 2.

$y_i = \log_2 x_i.$
Parameters: x (Variable) – Input variable. Output variable. Variable

### logsumexp¶

chainer.functions.logsumexp(x, axis=None)[source]

Log-sum-exp of array elements over a given axis.

This function calculates logarithm of sum of exponential of array elements.

$y_i = \log\left(\sum_j \exp(x_{ij})\right)$
Parameters: x (Variable) – Elements to log-sum-exp. axis (None, int, or tuple of int) – Axis which a sum is performed. The default (axis = None) is perform a sum over all the dimensions of the input array. Output variable. Variable

### matmul¶

chainer.functions.matmul(a, b, transa=False, transb=False)[source]

Computes the matrix multiplication of two arrays.

Parameters: a (Variable) – The left operand of the matrix multiplication. A 1-D array of shape (N,) is considered as an $$N \times 1$$ matrix. A 2-D array of shape (M, N) is considered as an $$M \times N$$ matrix. b (Variable) – The right operand of the matrix multiplication. Its array is treated as a matrix in the same way as a‘s array. transa (bool) – If True, transpose a. transb (bool) – If True, transpose b. The result of the matrix multiplication as a 2-D array. Variable

### max¶

chainer.functions.max(x, axis=None, keepdims=False)[source]

Maximum of array elements over a given axis.

Parameters: x (Variable) – Array to be maximized. axis (None, int, or tuple of int) – Axis over which a max is performed. The default (axis = None) is perform a max over all the dimensions of the input array. Output variable. Variable

### maximum¶

chainer.functions.maximum(x1, x2)[source]

Element-wise maximum of input variables.

Parameters: x1 (Variable) – Input variables to be compared. x2 (Variable) – Input variables to be compared. Output variable. Variable

### min¶

chainer.functions.min(x, axis=None, keepdims=False)[source]

Minimum of array elements over a given axis.

Parameters: x (Variable) – Array to be minimized. axis (None, int, or tuple of int) – Axis over which a min is performed. The default (axis = None) is perform a min over all the dimensions of the input array. Output variable. Variable

### minimum¶

chainer.functions.minimum(x1, x2)[source]

Element-wise minimum of input variables.

Parameters: x1 (Variable) – Input variables to be compared. x2 (Variable) – Input variables to be compared. Output variable. Variable

### rsqrt¶

chainer.functions.rsqrt(x)[source]

Computes elementwise reciprocal of square root of input $$x_i$$.

$y_i = {1 \over \sqrt x_i}.$
Parameters: x (Variable) – Input variable. Output variable. Variable

### scale¶

chainer.functions.scale(x, y, axis=1)[source]

Elementwise product with broadcasting.

Computes a elementwise product of two input variables, with the shape of the latter variable broadcasted to match the shape of the former. axis is the first axis of the first variable along which the second variable is applied.

The term “broadcasting” here comes from Caffe’s scale layer so the “broadcasting” with the following arguments:

   x : 100 x 3 x 40 x 60
y : 3 x 40
axis : 1


is equivalent to the following numpy broadcasting:

x : 100 x 3 x 40 x 60
y :   1 x 3 x 40 x 1


Note that how the axis indicates to which axis of x we apply y.

Parameters: x (Variable) – Input variable to be scaled. y (Variable) – Input variable to scale, broadcasted. axis (int) – The first axis of x along which y is applied. Output variable. Variable

### sin¶

chainer.functions.sin(x)[source]

Elementwise sin function.

### sinh¶

chainer.functions.sinh(x)[source]

Elementwise hyperbolic sine function.

$y_i = \sinh x_i.$
Parameters: x (Variable) – Input variable. Output variable. Variable

### sqrt¶

chainer.functions.sqrt(x)[source]

Elementwise square root function.

$y_i = \sqrt x_i.$

If the value of $$x_i$$ is negative, it returns Nan for $$y_i$$ respect to underlying numpy and cupy specification.

Parameters: x (Variable) – Input variable. Output variable. Variable

### sum¶

chainer.functions.sum(x, axis=None)[source]

Sum of array elements over a given axis.

Parameters: x (Variable) – Elements to sum. axis (None, int, or tuple of int) – Axis which a sum is performed. The default (axis = None) is perform a sum over all the dimensions of the input array. Output variable. Variable

### tanh¶

Hyperbolic tangent function is described in “Activation functions” section.

### tan¶

chainer.functions.tan(x)[source]

Elementwise tan function.

## Noise injections¶

### dropout¶

chainer.functions.dropout(x, ratio=0.5, train=True)[source]

Drops elements of input variable randomly.

This function drops input elements randomly with probability ratio and scales the remaining elements by factor 1 / (1 - ratio). In testing mode, it does nothing and just returns x.

Parameters: x (Variable) – Input variable. ratio (float) – Dropout ratio. train (bool) – If True, executes dropout. Otherwise, does nothing. Output variable. Variable

See the paper by G. Hinton: Improving neural networks by preventing co-adaptation of feature detectors.

### gaussian¶

chainer.functions.gaussian(mean, ln_var)[source]

Gaussian sampling function.

It takes mean $$\mu$$ and logarithm of variance $$\log(\sigma^2)$$ as input and output a sample drawn from gaussian $$N(\mu, \sigma)$$.

Parameters: mean (Variable) – Input variable representing mean $$\mu$$. ln_var (Variable) – Input variable representing logarithm of variance $$\log(\sigma^2)$$. Output variable. Variable

## Normalization functions¶

### batch_normalization¶

chainer.functions.batch_normalization(x, gamma, beta, eps=2e-05, running_mean=None, running_var=None, decay=0.9, use_cudnn=True)[source]

Batch normalization function.

It takes the input variable x and two parameter variables gamma and beta. The input must have the batch size and the features (or channels) as the first two dimensions of its shape. The input can have more than two dimensions, where the remaining dimensions are considered as spatial dimensions, which are considered as a part of the batch size. That is, the total batch size will be considered to be the product of all dimensions except the second dimension.

Note: If this function is called, it will not be possible to access the updated running mean and variance statistics, because they are members of the function object, which cannot be accessed by the caller. If it is desired to access the updated running statistics, it is necessary to get a new instance of the function object, call the object, and then access the running_mean and/or running_var attributes. See the corresponding Link class for an example of how to do this.

Parameters: x (Variable) – The input variable. gamma (Variable) – The scaling parameter of normalized data. beta (Variable) – The shifting parameter of scaled normalized data. eps (float) – Epsilon value for numerical stability. running_mean (array) – The running average of the mean. This is a running average of the mean over several mini-batches using the decay parameter. If None, the running average is not computed. If this is None, then runnng_var must also be None. running_var (array) – The running average of the variance. This is a running average of the variance over several mini-batches using the decay parameter. If None, the running average is not computed. If this is None, then running_mean must also be None. decay (float) – Decay rate of moving average. It is used during training. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation.

links.BatchNormalization

### fixed_batch_normalization¶

chainer.functions.fixed_batch_normalization(x, gamma, beta, mean, var, eps=2e-05, use_cudnn=True)[source]

Batch normalization function with fixed statistics.

This is a variant of batch normalization, where the mean and variance statistics are given by the caller as fixed variables. This is used on testing mode of the batch normalization layer, where batch statistics cannot be used for prediction consistency.

Parameters: x (Variable) – The input variable. gamma (Variable) – The scaling parameter of normalized data. beta (Variable) – The shifting parameter of scaled normalized data. mean (Variable) – The shifting parameter of input. var (Variable) – The square of scaling parameter of input. eps (float) – Epsilon value for numerical stability. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation.

functions.batch_normalization(), links.BatchNormalization

### local_response_normalization¶

chainer.functions.local_response_normalization(x, n=5, k=2, alpha=0.0001, beta=0.75)[source]

Local response normalization across neighboring channels.

This function implements normalization across channels. Let $$x$$ an input image with $$N$$ channels. Then, this function computes an output image $$y$$ by following formula:

$y_i = {x_i \over \left( k + \ \alpha \sum_{j=\max{1, i - n/2}}^{\min{N, i + n/2}} \ x_j^2 \right)^\beta}.$
Parameters: x (Variable) – Input variable. n (int) – Normalization window width. k (float) – Smoothing parameter. alpha (float) – Normalizer scaling parameter. beta (float) – Normalizer power parameter. Output variable. Variable

See: Section 3.3 of ImageNet Classification with Deep Convolutional Neural Networks

### normalize¶

chainer.functions.normalize(x, eps=1e-05)[source]

L2 norm squared (a.k.a. Euclidean norm).

This function implements L2 normalization on a 1D vector. No reduction is done along batch axis. Let $$x$$ be an input vector of dimension $$(N, K)$$, where $$N$$ and $$K$$ denote mini-batch size and the dimension of the input variable. Then, this function computes an output vector $$y$$ by the following equation:

$y_i = {x_i \over \| x_i \|_2}$

$$eps$$ is used to avoid division by zero when $$x_i=0$$

Parameters: x (Variable) – Two dimensional output variable. The first dimension is assumed to be the mini-batch dimension. eps (float) – Epsilon value for numerical stability. Two dimensional output variable, the same shape as $$x$$. Variable

## Spatial pooling¶

### average_pooling_2d¶

chainer.functions.average_pooling_2d(x, ksize, stride=None, pad=0, use_cudnn=True)[source]

Spatial average pooling function.

This function acts similarly to Convolution2D, but it computes the average of input spatial patch for each channel without any parameter instead of computing the inner products.

Parameters: x (Variable) – Input variable. ksize (int or pair of ints) – Size of pooling window. ksize=k and ksize=(k, k) are equivalent. stride (int or pair of ints or None) – Stride of pooling applications. stride=s and stride=(s, s) are equivalent. If None is specified, then it uses same stride as the pooling window size. pad (int or pair of ints) – Spatial padding width for the input array. pad=p and pad=(p, p) are equivalent. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. Variable

Note

This function currently does not support cover_all mode as max_pooling_2d(). Average pooling runs in non-cover-all mode.

### max_pooling_2d¶

chainer.functions.max_pooling_2d(x, ksize, stride=None, pad=0, cover_all=True, use_cudnn=True)[source]

Spatial max pooling function.

This function acts similarly to Convolution2D, but it computes the maximum of input spatial patch for each channel without any parameter instead of computing the inner products.

Parameters: x (Variable) – Input variable. ksize (int or pair of ints) – Size of pooling window. ksize=k and ksize=(k, k) are equivalent. stride (int or pair of ints or None) – Stride of pooling applications. stride=s and stride=(s, s) are equivalent. If None is specified, then it uses same stride as the pooling window size. pad (int or pair of ints) – Spatial padding width for the input array. pad=p and pad=(p, p) are equivalent. cover_all (bool) – If True, all spatial locations are pooled into some output pixels. It may make the output size larger. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. Variable

### roi_pooling_2d¶

chainer.functions.roi_pooling_2d(x, rois, outh, outw, spatial_scale)[source]

Spatial Region of Interest (ROI) pooling function.

This function acts similarly to MaxPooling2D, but it computes the maximum of input spatial patch for each channel with the region of interest.

Parameters: x (Variable) – Input variable. The shape is expected to be 4 dimentional: (n: batch, c: channel, h, height, w: width). rois (Variable) – Input roi variable. The shape is expected to be (n: data size, 5), and each datum is set as below: (batch_index, x_min, y_min, x_max, y_max). outh (int) – Height of output image after pooled. outw (int) – Width of output image after pooled. spatial_scale (float) – Scale of the roi is resized. Output variable. Variable

See the original paper proposing ROIPooling: Fast R-CNN.

### spatial_pyramid_pooling_2d¶

chainer.functions.spatial_pyramid_pooling_2d(x, pyramid_height, pooling_class, use_cudnn=True)[source]

Spatial pyramid pooling function.

It outputs a fixed-length vector regardless of input feature map size.

It performs pooling operation to the input 4D-array x with different kernel sizes and padding sizes, and then flattens all dimensions except first dimension of all pooling results, and finally concatenates them along second dimension.

At $$i$$-th pyramid level, the kernel size $$(k_h^{(i)}, k_w^{(i)})$$ and padding size $$(p_h^{(i)}, p_w^{(i)})$$ of pooling operation are calculated as below:

$\begin{split}k_h^{(i)} &= \lceil b_h / 2^i \rceil, \\ k_w^{(i)} &= \lceil b_w / 2^i \rceil, \\ p_h^{(i)} &= (2^i k_h^{(i)} - b_h) / 2, \\ p_w^{(i)} &= (2^i k_w^{(i)} - b_w) / 2,\end{split}$

where $$\lceil \cdot \rceil$$ denotes the ceiling function, and $$b_h, b_w$$ are height and width of input variable x, respectively. Note that index of pyramid level $$i$$ is zero-based.

See detail in paper: Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition.

Parameters: x (Variable) – Input variable. The shape of x should be (batchsize, # of channels, height, width). pyramid_height (int) – the number of pyramid levels pooling_class (MaxPooling2D or AveragePooling2D) – Only MaxPooling2D class can be available for now. use_cudnn (bool) – If True and cuDNN is enabled, then this function uses cuDNN as the core implementation. Output variable. The shape of the output variable will be $$(batchsize, c \sum_{h=0}^{H-1} 2^{2h}, 1, 1)$$, where $$c$$ is the number of channels of input variable x and $$H$$ is the number of pyramid levels. Variable

Note

This function uses some pooling classes as components to perform spatial pyramid pooling. Now it supports only MaxPooling2D as elemental pooling operator so far.

### unpooling_2d¶

chainer.functions.unpooling_2d(x, ksize, stride=None, pad=0, outsize=None, cover_all=True)[source]

Inverse operation of pooling for 2d array.

This function acts similarly to Deconvolution2D, but it spreads input 2d array’s value without any parameter instead of computing the inner products.

Parameters: x (Variable) – Input variable. ksize (int or pair of ints) – Size of pooling window. ksize=k and ksize=(k, k) are equivalent. stride (int, pair of ints or None) – Stride of pooling applications. stride=s and stride=(s, s) are equivalent. If None is specified, then it uses same stride as the pooling window size. pad (int or pair of ints) – Spatial padding width for the input array. pad=p and pad=(p, p) are equivalent. outsize (None or pair of ints) – Expected output size (height, width) of array after the operation. If None, the size (height or width) is estimated from the size of input array in first batch with get_deconv_outsize(). If outsize is not None, the result of outsize applied to get_conv_outsize() must be equal to the shape of the 2d array in the input batch x. cover_all (bool) – If True, the output size may be smaller than the size if cover_all is False. This flag serves to align behavior to the pooling functions which can cover all input locations, see max_pooling_2d() and convolution_2d(). Output variable. Variable