Sparse Symbol API¶

Overview¶

This document lists the routines of the sparse symbolic expression package:

mxnet.symbol.sparse Sparse Symbol API of MXNet.

The Sparse Symbol API, defined in the symbol.sparse package, provides sparse neural network graphs and auto-differentiation.

The storage type of a variable is speficied by the stype attribute of the variable. The storage type of a symbolic expression is inferred based on the storage types of the variables and the operators.

>>> a = mx.sym.Variable('a', stype='csr')
>>> b = mx.sym.Variable('b')
>>> c = mx.sym.dot(a, b, transpose_a=True)
>>> type(c)

>>> e = c.bind(mx.cpu(), {'a': mx.nd.array([[1,0,0]]).tostype('csr'), 'b':mx.nd.ones((1,2))})
>>> y = e.forward()
# the result storage type of dot(csr.T, dense) is inferred to be `row_sparse`
>>> y
[]
>>> y[0].asnumpy()
array([ 1.,  1.],
      [ 0.,  0.],
      [ 0.,  0.]], dtype=float32)

Note

most operators provided in mxnet.symbol.sparse are similar to those in mxnet.symbol although there are few differences:

Only a subset of operators in mxnet.symbol have efficient sparse implementations in mxnet.symbol.sparse.
If an operator do not occur in the mxnet.symbol.sparse namespace, that means the operator does not have an efficient sparse implementation yet. If sparse inputs are passed to such an operator, it will convert inputs to the dense format and fallback to the already available dense implementation.
The storage types (stype) of sparse operators’ outputs depend on the storage types of inputs. By default the operators not available in mxnet.symbol.sparse infer “default” (dense) storage type for outputs. Please refer to the API reference section for further details on specific operators.

In the rest of this document, we list sparse related routines provided by the symbol.sparse package.

Symbol creation routines¶

`zeros_like`	Return an array of zeros with the same shape, type and storage type as the input array.
`mxnet.symbol.var`	Creates a symbolic variable with specified name.

Symbol manipulation routines¶

Changing symbol storage type¶

cast_storage Casts tensor storage type to the new type.

Joining arrays¶

concat Joins input arrays along a given axis.

Indexing routines¶

`slice`	Slices a region of the array.
`retain`	pick rows specified by user input index array from a row sparse matrix

Mathematical functions¶

Arithmetic operations¶

`elemwise_add`	Adds arguments element-wise.
`elemwise_sub`	Subtracts arguments element-wise.
`elemwise_mul`	Multiplies arguments element-wise.
`broadcast_add`	Returns element-wise sum of the input arrays with broadcasting.
`broadcast_sub`	Returns element-wise difference of the input arrays with broadcasting.
`broadcast_mul`	Returns element-wise product of the input arrays with broadcasting.
`broadcast_div`	Returns element-wise division of the input arrays with broadcasting.
`negative`	Numerical negative of the argument, element-wise.
`dot`	Dot product of two arrays.
`add_n`	Adds all input arguments element-wise.

Trigonometric functions¶

`sin`	Computes the element-wise sine of the input array.
`tan`	Computes the element-wise tangent of the input array.
`arcsin`	Returns element-wise inverse sine of the input array.
`arctan`	Returns element-wise inverse tangent of the input array.
`degrees`	Converts each element of the input array from radians to degrees.
`radians`	Converts each element of the input array from degrees to radians.

Hyperbolic functions¶

`sinh`	Returns the hyperbolic sine of the input array, computed element-wise.
`tanh`	Returns the hyperbolic tangent of the input array, computed element-wise.
`arcsinh`	Returns the element-wise inverse hyperbolic sine of the input array, computed element-wise.
`arctanh`	Returns the element-wise inverse hyperbolic tangent of the input array, computed element-wise.

Reduce functions¶

`sum`	Computes the sum of array elements over given axes.
`mean`	Computes the mean of array elements over given axes.

Rounding¶

`round`	Returns element-wise rounded value to the nearest integer of the input.
`rint`	Returns element-wise rounded value to the nearest integer of the input.
`fix`	Returns element-wise rounded value to the nearest integer towards zero of the input.
`floor`	Returns element-wise floor of the input.
`ceil`	Returns element-wise ceiling of the input.
`trunc`	Return the element-wise truncated value of the input.

Exponents and logarithms¶

`expm1`	Returns `exp(x) - 1` computed element-wise on the input.
`log1p`	Returns element-wise `log(1 + x)` value of the input.

Powers¶

`sqrt`	Returns element-wise square-root value of the input.
`square`	Returns element-wise squared value of the input.

Miscellaneous¶

`clip`	Clips (limits) the values in an array.
`abs`	Returns element-wise absolute value of the input.
`sign`	Returns element-wise sign of the input.

Neural network¶

More¶

`make_loss`	Make your own loss function in network construction.
`stop_gradient`	Stops gradient computation.
`Embedding`	Maps integer indices to vector representations (embeddings).
`LinearRegressionOutput`	Computes and optimizes for squared loss during backward propagation.
`LogisticRegressionOutput`	Applies a logistic function to the input.

API Reference¶

Sparse Symbol API of MXNet.

mxnet.symbol.sparse.ElementWiseSum(*args, **kwargs)¶

Adds all input arguments element-wise.

\[add\_n(a_1, a_2, ..., a_n) = a_1 + a_2 + ... + a_n\]

add_n is potentially more efficient than calling add by n times.

The storage type of add_n output depends on storage types of inputs

add_n(row_sparse, row_sparse, ..) = row_sparse
add_n(default, csr, default) = default
add_n(any input combinations longer than 4 (>4) with at least one default type) = default
otherwise, add_n falls all inputs back to default storage and generates default storage

Defined in src/operator/tensor/elemwise_sum.cc:L156 This function support variable length of positional input.

Parameters:	args (Symbol[]) – Positional input arguments name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.Embedding(data=None, weight=None, input_dim=_Null, output_dim=_Null, dtype=_Null, sparse_grad=_Null, name=None, attr=None, out=None, **kwargs)¶

Maps integer indices to vector representations (embeddings).

This operator maps words to real-valued vectors in a high-dimensional space, called word embeddings. These embeddings can capture semantic and syntactic properties of the words. For example, it has been noted that in the learned embedding spaces, similar words tend to be close to each other and dissimilar words far apart.

For an input array of shape (d1, ..., dK), the shape of an output array is (d1, ..., dK, output_dim). All the input values should be integers in the range [0, input_dim).

If the input_dim is ip0 and output_dim is op0, then shape of the embedding weight matrix must be (ip0, op0).

By default, if any index mentioned is too large, it is replaced by the index that addresses the last vector in an embedding matrix.

Examples:

input_dim = 4
output_dim = 5

// Each row in weight matrix y represents a word. So, y = (w0,w1,w2,w3)
y = [[  0.,   1.,   2.,   3.,   4.],
     [  5.,   6.,   7.,   8.,   9.],
     [ 10.,  11.,  12.,  13.,  14.],
     [ 15.,  16.,  17.,  18.,  19.]]

// Input array x represents n-grams(2-gram). So, x = [(w1,w3), (w0,w2)]
x = [[ 1.,  3.],
     [ 0.,  2.]]

// Mapped input x to its vector representation y.
Embedding(x, y, 4, 5) = [[[  5.,   6.,   7.,   8.,   9.],
                          [ 15.,  16.,  17.,  18.,  19.]],

                         [[  0.,   1.,   2.,   3.,   4.],
                          [ 10.,  11.,  12.,  13.,  14.]]]

The storage type of weight can be either row_sparse or default.

Note

If “sparse_grad” is set to True, the storage type of gradient w.r.t weights will be “row_sparse”. Only a subset of optimizers support sparse gradients, including SGD, AdaGrad and Adam. Note that by default lazy updates is turned on, which may perform differently from standard updates. For more details, please check the Optimization API at: /api/python/optimization/optimization.html

Defined in src/operator/tensor/indexing_op.cc:L267

Parameters:	data (Symbol) – The input array to the embedding operator. weight (Symbol) – The embedding weight matrix. input_dim (int, required) – Vocabulary size of the input indices. output_dim (int, required) – Dimension of the embedding vectors. dtype ({'float16', 'float32', 'float64', 'int32', 'int64', 'int8', 'uint8'},optional, default='float32') – Data type of weight. sparse_grad (boolean, optional, default=0) – Compute row sparse gradient in the backward calculation. If set to True, the grad’s storage type is row_sparse. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.FullyConnected(data=None, weight=None, bias=None, num_hidden=_Null, no_bias=_Null, flatten=_Null, name=None, attr=None, out=None, **kwargs)¶

Applies a linear transformation: \(Y = XW^T + b\).

If flatten is set to be true, then the shapes are:

data: (batch_size, x1, x2, ..., xn)
weight: (num_hidden, x1 * x2 * ... * xn)
bias: (num_hidden,)
out: (batch_size, num_hidden)

If flatten is set to be false, then the shapes are:

data: (x1, x2, ..., xn, input_dim)
weight: (num_hidden, input_dim)
bias: (num_hidden,)
out: (x1, x2, ..., xn, num_hidden)

The learnable parameters include both weight and bias.

If no_bias is set to be true, then the bias term is ignored.

Note

The sparse support for FullyConnected is limited to forward evaluation with row_sparse weight and bias, where the length of weight.indices and bias.indices must be equal to num_hidden. This could be useful for model inference with row_sparse weights trained with importance sampling or noise contrastive estimation.

To compute linear transformation with ‘csr’ sparse data, sparse.dot is recommended instead of sparse.FullyConnected.

Defined in src/operator/nn/fully_connected.cc:L272

Parameters:	data (Symbol) – Input data. weight (Symbol) – Weight matrix. bias (Symbol) – Bias parameter. num_hidden (int, required) – Number of hidden nodes of the output. no_bias (boolean, optional, default=0) – Whether to disable bias parameter. flatten (boolean, optional, default=1) – Whether to collapse all but the first axis of the input data tensor. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.LinearRegressionOutput(data=None, label=None, grad_scale=_Null, name=None, attr=None, out=None, **kwargs)¶

Computes and optimizes for squared loss during backward propagation. Just outputs data during forward propagation.

If \(\hat{y}_i\) is the predicted value of the i-th sample, and \(y_i\) is the corresponding target value, then the squared loss estimated over \(n\) samples is defined as

\(\text{SquaredLoss}(\textbf{Y}, \hat{\textbf{Y}} ) = \frac{1}{n} \sum_{i=0}^{n-1} \lVert \textbf{y}_i - \hat{\textbf{y}}_i \rVert_2\)

Note

Use the LinearRegressionOutput as the final output layer of a net.

The storage type of label can be default or csr

LinearRegressionOutput(default, default) = default
LinearRegressionOutput(default, csr) = default

By default, gradients of this loss function are scaled by factor 1/m, where m is the number of regression outputs of a training example. The parameter grad_scale can be used to change this scale to grad_scale/m.

Defined in src/operator/regression_output.cc:L92

Parameters:	data (Symbol) – Input data to the function. label (Symbol) – Input label to the function. grad_scale (float, optional, default=1) – Scale the gradient by a float factor name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.LogisticRegressionOutput(data=None, label=None, grad_scale=_Null, name=None, attr=None, out=None, **kwargs)¶

Applies a logistic function to the input.

The logistic function, also known as the sigmoid function, is computed as \(\frac{1}{1+exp(-\textbf{x})}\).

Commonly, the sigmoid is used to squash the real-valued output of a linear model \(wTx+b\) into the [0,1] range so that it can be interpreted as a probability. It is suitable for binary classification or probability prediction tasks.

Note

Use the LogisticRegressionOutput as the final output layer of a net.

The storage type of label can be default or csr

LogisticRegressionOutput(default, default) = default
LogisticRegressionOutput(default, csr) = default

The loss function used is the Binary Cross Entropy Loss:

\(-{(y\log(p) + (1 - y)\log(1 - p))}\)

Where y is the ground truth probability of positive outcome for a given example, and p the probability predicted by the model. By default, gradients of this loss function are scaled by factor 1/m, where m is the number of regression outputs of a training example. The parameter grad_scale can be used to change this scale to grad_scale/m.

Defined in src/operator/regression_output.cc:L152

Parameters:	data (Symbol) – Input data to the function. label (Symbol) – Input label to the function. grad_scale (float, optional, default=1) – Scale the gradient by a float factor name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.MAERegressionOutput(data=None, label=None, grad_scale=_Null, name=None, attr=None, out=None, **kwargs)¶

Computes mean absolute error of the input.

MAE is a risk metric corresponding to the expected value of the absolute error.

If \(\hat{y}_i\) is the predicted value of the i-th sample, and \(y_i\) is the corresponding target value, then the mean absolute error (MAE) estimated over \(n\) samples is defined as

\(\text{MAE}(\textbf{Y}, \hat{\textbf{Y}} ) = \frac{1}{n} \sum_{i=0}^{n-1} \lVert \textbf{y}_i - \hat{\textbf{y}}_i \rVert_1\)

Note

Use the MAERegressionOutput as the final output layer of a net.

The storage type of label can be default or csr

MAERegressionOutput(default, default) = default
MAERegressionOutput(default, csr) = default

By default, gradients of this loss function are scaled by factor 1/m, where m is the number of regression outputs of a training example. The parameter grad_scale can be used to change this scale to grad_scale/m.

Defined in src/operator/regression_output.cc:L120

Parameters:	data (Symbol) – Input data to the function. label (Symbol) – Input label to the function. grad_scale (float, optional, default=1) – Scale the gradient by a float factor name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.abs(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise absolute value of the input.

Example:

abs([-2, 0, 3]) = [2, 0, 3]

The storage type of abs output depends upon the input storage type:

abs(default) = default

abs(row_sparse) = row_sparse

abs(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L668

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.adagrad_update(weight=None, grad=None, history=None, lr=_Null, epsilon=_Null, wd=_Null, rescale_grad=_Null, clip_gradient=_Null, name=None, attr=None, out=None, **kwargs)¶

Update function for AdaGrad optimizer.

Referenced from Adaptive Subgradient Methods for Online Learning and Stochastic Optimization, and available at http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf.

Updates are applied by:

rescaled_grad = clip(grad * rescale_grad, clip_gradient)
history = history + square(rescaled_grad)
w = w - learning_rate * rescaled_grad / sqrt(history + epsilon)

Note that non-zero values for the weight decay option are not supported.

Defined in src/operator/optimizer_op.cc:L665

Parameters:	weight (Symbol) – Weight grad (Symbol) – Gradient history (Symbol) – History lr (float, required) – Learning rate epsilon (float, optional, default=1e-07) – epsilon wd (float, optional, default=0) – weight decay rescale_grad (float, optional, default=1) – Rescale gradient to grad = rescale_gradgrad. clip_gradient* (float, optional, default=-1) – Clip gradient to the range of [-clip_gradient, clip_gradient] If clip_gradient <= 0, gradient clipping is turned off. grad = max(min(grad, clip_gradient), -clip_gradient). name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.adam_update(weight=None, grad=None, mean=None, var=None, lr=_Null, beta1=_Null, beta2=_Null, epsilon=_Null, wd=_Null, rescale_grad=_Null, clip_gradient=_Null, lazy_update=_Null, name=None, attr=None, out=None, **kwargs)¶

Update function for Adam optimizer. Adam is seen as a generalization of AdaGrad.

Adam update consists of the following steps, where g represents gradient and m, v are 1st and 2nd order moment estimates (mean and variance).

\[\begin{split}g_t = \nabla J(W_{t-1})\\ m_t = \beta_1 m_{t-1} + (1 - \beta_1) g_t\\ v_t = \beta_2 v_{t-1} + (1 - \beta_2) g_t^2\\ W_t = W_{t-1} - \alpha \frac{ m_t }{ \sqrt{ v_t } + \epsilon }\end{split}\]

It updates the weights using:

m = beta1*m + (1-beta1)*grad
v = beta2*v + (1-beta2)*(grad**2)
w += - learning_rate * m / (sqrt(v) + epsilon)

However, if grad’s storage type is row_sparse, lazy_update is True and the storage type of weight is the same as those of m and v, only the row slices whose indices appear in grad.indices are updated (for w, m and v):

for row in grad.indices:
    m[row] = beta1*m[row] + (1-beta1)*grad[row]
    v[row] = beta2*v[row] + (1-beta2)*(grad[row]**2)
    w[row] += - learning_rate * m[row] / (sqrt(v[row]) + epsilon)

Defined in src/operator/optimizer_op.cc:L495

Parameters:	weight (Symbol) – Weight grad (Symbol) – Gradient mean (Symbol) – Moving mean var (Symbol) – Moving variance lr (float, required) – Learning rate beta1 (float, optional, default=0.9) – The decay rate for the 1st moment estimates. beta2 (float, optional, default=0.999) – The decay rate for the 2nd moment estimates. epsilon (float, optional, default=1e-08) – A small constant for numerical stability. wd (float, optional, default=0) – Weight decay augments the objective function with a regularization term that penalizes large weights. The penalty scales with the square of the magnitude of each weight. rescale_grad (float, optional, default=1) – Rescale gradient to grad = rescale_gradgrad. clip_gradient* (float, optional, default=-1) – Clip gradient to the range of [-clip_gradient, clip_gradient] If clip_gradient <= 0, gradient clipping is turned off. grad = max(min(grad, clip_gradient), -clip_gradient). lazy_update (boolean, optional, default=1) – If true, lazy updates are applied if gradient’s stype is row_sparse and all of w, m and v have the same stype name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.add_n(*args, **kwargs)¶

Adds all input arguments element-wise.

\[add\_n(a_1, a_2, ..., a_n) = a_1 + a_2 + ... + a_n\]

add_n is potentially more efficient than calling add by n times.

The storage type of add_n output depends on storage types of inputs

add_n(row_sparse, row_sparse, ..) = row_sparse
add_n(default, csr, default) = default
add_n(any input combinations longer than 4 (>4) with at least one default type) = default
otherwise, add_n falls all inputs back to default storage and generates default storage

Defined in src/operator/tensor/elemwise_sum.cc:L156 This function support variable length of positional input.

Parameters:	args (Symbol[]) – Positional input arguments name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.arccos(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise inverse cosine of the input array.

The input should be in range [-1, 1]. The output is in the closed interval \([0, \pi]\)

\[arccos([-1, -.707, 0, .707, 1]) = [\pi, 3\pi/4, \pi/2, \pi/4, 0]\]

The storage type of arccos output is always dense

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L123

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.arccosh(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the element-wise inverse hyperbolic cosine of the input array, computed element-wise.

The storage type of arccosh output is always dense

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L264

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.arcsin(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise inverse sine of the input array.

The input should be in the range [-1, 1]. The output is in the closed interval of [\(-\pi/2\), \(\pi/2\)].

\[arcsin([-1, -.707, 0, .707, 1]) = [-\pi/2, -\pi/4, 0, \pi/4, \pi/2]\]

The storage type of arcsin output depends upon the input storage type:

arcsin(default) = default

arcsin(row_sparse) = row_sparse

arcsin(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L104

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.arcsinh(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the element-wise inverse hyperbolic sine of the input array, computed element-wise.

The storage type of arcsinh output depends upon the input storage type:

arcsinh(default) = default

arcsinh(row_sparse) = row_sparse

arcsinh(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L250

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.arctan(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise inverse tangent of the input array.

The output is in the closed interval \([-\pi/2, \pi/2]\)

\[arctan([-1, 0, 1]) = [-\pi/4, 0, \pi/4]\]

The storage type of arctan output depends upon the input storage type:

arctan(default) = default

arctan(row_sparse) = row_sparse

arctan(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L144

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.arctanh(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the element-wise inverse hyperbolic tangent of the input array, computed element-wise.

The storage type of arctanh output depends upon the input storage type:

arctanh(default) = default

arctanh(row_sparse) = row_sparse

arctanh(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L281

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.broadcast_add(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise sum of the input arrays with broadcasting.

broadcast_plus is an alias to the function broadcast_add.

Example:

x = [[ 1.,  1.,  1.],
     [ 1.,  1.,  1.]]

y = [[ 0.],
     [ 1.]]

broadcast_add(x, y) = [[ 1.,  1.,  1.],
                       [ 2.,  2.,  2.]]

broadcast_plus(x, y) = [[ 1.,  1.,  1.],
                        [ 2.,  2.,  2.]]

Supported sparse operations:

broadcast_add(csr, dense(1D)) = dense broadcast_add(dense(1D), csr) = dense

Defined in src/operator/tensor/elemwise_binary_broadcast_op_basic.cc:L58

Parameters:	lhs (Symbol) – First input to the function rhs (Symbol) – Second input to the function name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.broadcast_div(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise division of the input arrays with broadcasting.

Example:

x = [[ 6.,  6.,  6.],
     [ 6.,  6.,  6.]]

y = [[ 2.],
     [ 3.]]

broadcast_div(x, y) = [[ 3.,  3.,  3.],
                       [ 2.,  2.,  2.]]

Supported sparse operations:

broadcast_div(csr, dense(1D)) = csr

Defined in src/operator/tensor/elemwise_binary_broadcast_op_basic.cc:L187

Parameters:	lhs (Symbol) – First input to the function rhs (Symbol) – Second input to the function name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.broadcast_minus(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise difference of the input arrays with broadcasting.

broadcast_minus is an alias to the function broadcast_sub.

Example:

x = [[ 1.,  1.,  1.],
     [ 1.,  1.,  1.]]

y = [[ 0.],
     [ 1.]]

broadcast_sub(x, y) = [[ 1.,  1.,  1.],
                       [ 0.,  0.,  0.]]

broadcast_minus(x, y) = [[ 1.,  1.,  1.],
                         [ 0.,  0.,  0.]]

Supported sparse operations:

broadcast_sub/minus(csr, dense(1D)) = dense broadcast_sub/minus(dense(1D), csr) = dense

Defined in src/operator/tensor/elemwise_binary_broadcast_op_basic.cc:L106

Parameters:	lhs (Symbol) – First input to the function rhs (Symbol) – Second input to the function name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.broadcast_mul(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise product of the input arrays with broadcasting.

Example:

x = [[ 1.,  1.,  1.],
     [ 1.,  1.,  1.]]

y = [[ 0.],
     [ 1.]]

broadcast_mul(x, y) = [[ 0.,  0.,  0.],
                       [ 1.,  1.,  1.]]

Supported sparse operations:

broadcast_mul(csr, dense(1D)) = csr

Defined in src/operator/tensor/elemwise_binary_broadcast_op_basic.cc:L146

Parameters:	lhs (Symbol) – First input to the function rhs (Symbol) – Second input to the function name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.broadcast_plus(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise sum of the input arrays with broadcasting.

broadcast_plus is an alias to the function broadcast_add.

Example:

x = [[ 1.,  1.,  1.],
     [ 1.,  1.,  1.]]

y = [[ 0.],
     [ 1.]]

broadcast_add(x, y) = [[ 1.,  1.,  1.],
                       [ 2.,  2.,  2.]]

broadcast_plus(x, y) = [[ 1.,  1.,  1.],
                        [ 2.,  2.,  2.]]

Supported sparse operations:

broadcast_add(csr, dense(1D)) = dense broadcast_add(dense(1D), csr) = dense

Defined in src/operator/tensor/elemwise_binary_broadcast_op_basic.cc:L58

Parameters:	lhs (Symbol) – First input to the function rhs (Symbol) – Second input to the function name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.broadcast_sub(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise difference of the input arrays with broadcasting.

broadcast_minus is an alias to the function broadcast_sub.

Example:

x = [[ 1.,  1.,  1.],
     [ 1.,  1.,  1.]]

y = [[ 0.],
     [ 1.]]

broadcast_sub(x, y) = [[ 1.,  1.,  1.],
                       [ 0.,  0.,  0.]]

broadcast_minus(x, y) = [[ 1.,  1.,  1.],
                         [ 0.,  0.,  0.]]

Supported sparse operations:

broadcast_sub/minus(csr, dense(1D)) = dense broadcast_sub/minus(dense(1D), csr) = dense

Defined in src/operator/tensor/elemwise_binary_broadcast_op_basic.cc:L106

Parameters:	lhs (Symbol) – First input to the function rhs (Symbol) – Second input to the function name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.cast_storage(data=None, stype=_Null, name=None, attr=None, out=None, **kwargs)¶

Casts tensor storage type to the new type.

When an NDArray with default storage type is cast to csr or row_sparse storage, the result is compact, which means:

for csr, zero values will not be retained
for row_sparse, row slices of all zeros will not be retained

The storage type of cast_storage output depends on stype parameter:

cast_storage(csr, ‘default’) = default
cast_storage(row_sparse, ‘default’) = default
cast_storage(default, ‘csr’) = csr
cast_storage(default, ‘row_sparse’) = row_sparse
cast_storage(csr, ‘csr’) = csr
cast_storage(row_sparse, ‘row_sparse’) = row_sparse

Example:

dense = [[ 0.,  1.,  0.],
         [ 2.,  0.,  3.],
         [ 0.,  0.,  0.],
         [ 0.,  0.,  0.]]

# cast to row_sparse storage type
rsp = cast_storage(dense, 'row_sparse')
rsp.indices = [0, 1]
rsp.values = [[ 0.,  1.,  0.],
              [ 2.,  0.,  3.]]

# cast to csr storage type
csr = cast_storage(dense, 'csr')
csr.indices = [1, 0, 2]
csr.values = [ 1.,  2.,  3.]
csr.indptr = [0, 1, 3, 3, 3]

Defined in src/operator/tensor/cast_storage.cc:L71

Parameters:	data (Symbol) – The input. stype ({'csr', 'default', 'row_sparse'}, required) – Output storage type. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.cbrt(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise cube-root value of the input.

\[cbrt(x) = \sqrt[3]{x}\]

Example:

cbrt([1, 8, -125]) = [1, 2, -5]

The storage type of cbrt output depends upon the input storage type:

cbrt(default) = default

cbrt(row_sparse) = row_sparse

cbrt(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L889

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.ceil(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise ceiling of the input.

The ceil of the scalar x is the smallest integer i, such that i >= x.

Example:

ceil([-2.1, -1.9, 1.5, 1.9, 2.1]) = [-2., -1.,  2.,  2.,  3.]

The storage type of ceil output depends upon the input storage type:

ceil(default) = default

ceil(row_sparse) = row_sparse

ceil(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L746

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.clip(data=None, a_min=_Null, a_max=_Null, name=None, attr=None, out=None, **kwargs)¶

Clips (limits) the values in an array.

Given an interval, values outside the interval are clipped to the interval edges. Clipping x between a_min and a_x would be:

clip(x, a_min, a_max) = max(min(x, a_max), a_min))

Example:

x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

clip(x,1,8) = [ 1.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  8.]

The storage type of clip output depends on storage types of inputs and the a_min, a_max parameter values:

clip(default) = default

clip(row_sparse, a_min <= 0, a_max >= 0) = row_sparse

clip(csr, a_min <= 0, a_max >= 0) = csr

clip(row_sparse, a_min < 0, a_max < 0) = default

clip(row_sparse, a_min > 0, a_max > 0) = default

clip(csr, a_min < 0, a_max < 0) = csr

clip(csr, a_min > 0, a_max > 0) = csr

Defined in src/operator/tensor/matrix_op.cc:L618

Parameters:	data (Symbol) – Input array. a_min (float, required) – Minimum value a_max (float, required) – Maximum value name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.concat(*data, **kwargs)¶

Joins input arrays along a given axis.

Note

Concat is deprecated. Use concat instead.

The dimensions of the input arrays should be the same except the axis along which they will be concatenated. The dimension of the output array along the concatenated axis will be equal to the sum of the corresponding dimensions of the input arrays.

The storage type of concat output depends on storage types of inputs

concat(csr, csr, ..., csr, dim=0) = csr
otherwise, concat generates output with default storage

Example:

x = [[1,1],[2,2]]
y = [[3,3],[4,4],[5,5]]
z = [[6,6], [7,7],[8,8]]

concat(x,y,z,dim=0) = [[ 1.,  1.],
                       [ 2.,  2.],
                       [ 3.,  3.],
                       [ 4.,  4.],
                       [ 5.,  5.],
                       [ 6.,  6.],
                       [ 7.,  7.],
                       [ 8.,  8.]]

Note that you cannot concat x,y,z along dimension 1 since dimension
0 is not the same for all the input arrays.

concat(y,z,dim=1) = [[ 3.,  3.,  6.,  6.],
                      [ 4.,  4.,  7.,  7.],
                      [ 5.,  5.,  8.,  8.]]

Defined in src/operator/nn/concat.cc:L368 This function support variable length of positional input.

Parameters:	data (Symbol[]) – List of arrays to concatenate dim (int, optional, default='1') – the dimension to be concated. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.cos(data=None, name=None, attr=None, out=None, **kwargs)¶

Computes the element-wise cosine of the input array.

The input should be in radians (\(2\pi\) rad equals 360 degrees).

\[cos([0, \pi/4, \pi/2]) = [1, 0.707, 0]\]

The storage type of cos output is always dense

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L63

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.cosh(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the hyperbolic cosine of the input array, computed element-wise.

\[cosh(x) = 0.5\times(exp(x) + exp(-x))\]

The storage type of cosh output is always dense

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L216

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.degrees(data=None, name=None, attr=None, out=None, **kwargs)¶

Converts each element of the input array from radians to degrees.

\[degrees([0, \pi/2, \pi, 3\pi/2, 2\pi]) = [0, 90, 180, 270, 360]\]

The storage type of degrees output depends upon the input storage type:

degrees(default) = default

degrees(row_sparse) = row_sparse

degrees(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L163

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.dot(lhs=None, rhs=None, transpose_a=_Null, transpose_b=_Null, forward_stype=_Null, name=None, attr=None, out=None, **kwargs)¶

Dot product of two arrays.

dot‘s behavior depends on the input array dimensions:

1-D arrays: inner product of vectors
2-D arrays: matrix multiplication
N-D arrays: a sum product over the last axis of the first input and the first axis of the second input

For example, given 3-D x with shape (n,m,k) and y with shape (k,r,s), the result array will have shape (n,m,r,s). It is computed by:
```
dot(x,y)[i,j,a,b] = sum(x[i,j,:]*y[:,a,b])
```
Example:
```
x = reshape([0,1,2,3,4,5,6,7], shape=(2,2,2))
y = reshape([7,6,5,4,3,2,1,0], shape=(2,2,2))
dot(x,y)[0,0,1,1] = 0
sum(x[0,0,:]*y[:,1,1]) = 0
```

The storage type of dot output depends on storage types of inputs, transpose option and forward_stype option for output storage type. Implemented sparse operations include:

dot(default, default, transpose_a=True/False, transpose_b=True/False) = default
dot(csr, default, transpose_a=True) = default
dot(csr, default, transpose_a=True) = row_sparse
dot(csr, default) = default
dot(csr, row_sparse) = default
dot(default, csr) = csr (CPU only)
dot(default, csr, forward_stype=’default’) = default
dot(default, csr, transpose_b=True, forward_stype=’default’) = default

If the combination of input storage types and forward_stype does not match any of the above patterns, dot will fallback and generate output with default storage.

Note

If the storage type of the lhs is “csr”, the storage type of gradient w.r.t rhs will be “row_sparse”. Only a subset of optimizers support sparse gradients, including SGD, AdaGrad and Adam. Note that by default lazy updates is turned on, which may perform differently from standard updates. For more details, please check the Optimization API at: /api/python/optimization/optimization.html

Defined in src/operator/tensor/dot.cc:L77

Parameters:	lhs (Symbol) – The first input rhs (Symbol) – The second input transpose_a (boolean, optional, default=0) – If true then transpose the first input before dot. transpose_b (boolean, optional, default=0) – If true then transpose the second input before dot. forward_stype ({None, 'csr', 'default', 'row_sparse'},optional, default='None') – The desired storage type of the forward output given by user, if thecombination of input storage types and this hint does not matchany implemented ones, the dot operator will perform fallback operationand still produce an output of the desired storage type. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.elemwise_add(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Adds arguments element-wise.

The storage type of elemwise_add output depends on storage types of inputs

elemwise_add(row_sparse, row_sparse) = row_sparse

elemwise_add(csr, csr) = csr

elemwise_add(default, csr) = default

elemwise_add(csr, default) = default

elemwise_add(default, rsp) = default

elemwise_add(rsp, default) = default

otherwise, elemwise_add generates output with default storage

Parameters:	lhs (Symbol) – first input rhs (Symbol) – second input name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.elemwise_div(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Divides arguments element-wise.

The storage type of elemwise_div output is always dense

Parameters:	lhs (Symbol) – first input rhs (Symbol) – second input name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.elemwise_mul(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Multiplies arguments element-wise.

The storage type of elemwise_mul output depends on storage types of inputs

elemwise_mul(default, default) = default

elemwise_mul(row_sparse, row_sparse) = row_sparse

elemwise_mul(default, row_sparse) = row_sparse

elemwise_mul(row_sparse, default) = row_sparse

elemwise_mul(csr, csr) = csr

otherwise, elemwise_mul generates output with default storage

Parameters:	lhs (Symbol) – first input rhs (Symbol) – second input name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.elemwise_sub(lhs=None, rhs=None, name=None, attr=None, out=None, **kwargs)¶

Subtracts arguments element-wise.

The storage type of elemwise_sub output depends on storage types of inputs

elemwise_sub(row_sparse, row_sparse) = row_sparse

elemwise_sub(csr, csr) = csr

elemwise_sub(default, csr) = default

elemwise_sub(csr, default) = default

elemwise_sub(default, rsp) = default

elemwise_sub(rsp, default) = default

otherwise, elemwise_sub generates output with default storage

Parameters:	lhs (Symbol) – first input rhs (Symbol) – second input name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.exp(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise exponential value of the input.

\[exp(x) = e^x \approx 2.718^x\]

Example:

exp([0, 1, 2]) = [1., 2.71828175, 7.38905621]

The storage type of exp output is always dense

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L929

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.expm1(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns exp(x) - 1 computed element-wise on the input.

This function provides greater precision than exp(x) - 1 for small values of x.

The storage type of expm1 output depends upon the input storage type:

expm1(default) = default

expm1(row_sparse) = row_sparse

expm1(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L1008

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.fix(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise rounded value to the nearest integer towards zero of the input.

Example:

fix([-2.1, -1.9, 1.9, 2.1]) = [-2., -1.,  1., 2.]

The storage type of fix output depends upon the input storage type:

fix(default) = default

fix(row_sparse) = row_sparse

fix(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L803

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.floor(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise floor of the input.

The floor of the scalar x is the largest integer i, such that i <= x.

Example:

floor([-2.1, -1.9, 1.5, 1.9, 2.1]) = [-3., -2.,  1.,  1.,  2.]

The storage type of floor output depends upon the input storage type:

floor(default) = default

floor(row_sparse) = row_sparse

floor(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L765

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.ftrl_update(weight=None, grad=None, z=None, n=None, lr=_Null, lamda1=_Null, beta=_Null, wd=_Null, rescale_grad=_Null, clip_gradient=_Null, name=None, attr=None, out=None, **kwargs)¶

Update function for Ftrl optimizer. Referenced from Ad Click Prediction: a View from the Trenches, available at http://dl.acm.org/citation.cfm?id=2488200.

It updates the weights using:

rescaled_grad = clip(grad * rescale_grad, clip_gradient)
z += rescaled_grad - (sqrt(n + rescaled_grad**2) - sqrt(n)) * weight / learning_rate
n += rescaled_grad**2
w = (sign(z) * lamda1 - z) / ((beta + sqrt(n)) / learning_rate + wd) * (abs(z) > lamda1)

If w, z and n are all of row_sparse storage type, only the row slices whose indices appear in grad.indices are updated (for w, z and n):

for row in grad.indices:
    rescaled_grad[row] = clip(grad[row] * rescale_grad, clip_gradient)
    z[row] += rescaled_grad[row] - (sqrt(n[row] + rescaled_grad[row]**2) - sqrt(n[row])) * weight[row] / learning_rate
    n[row] += rescaled_grad[row]**2
    w[row] = (sign(z[row]) * lamda1 - z[row]) / ((beta + sqrt(n[row])) / learning_rate + wd) * (abs(z[row]) > lamda1)

Defined in src/operator/optimizer_op.cc:L632

Parameters:	weight (Symbol) – Weight grad (Symbol) – Gradient z (Symbol) – z n (Symbol) – Square of grad lr (float, required) – Learning rate lamda1 (float, optional, default=0.01) – The L1 regularization coefficient. beta (float, optional, default=1) – Per-Coordinate Learning Rate beta. wd (float, optional, default=0) – Weight decay augments the objective function with a regularization term that penalizes large weights. The penalty scales with the square of the magnitude of each weight. rescale_grad (float, optional, default=1) – Rescale gradient to grad = rescale_gradgrad. clip_gradient* (float, optional, default=-1) – Clip gradient to the range of [-clip_gradient, clip_gradient] If clip_gradient <= 0, gradient clipping is turned off. grad = max(min(grad, clip_gradient), -clip_gradient). name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.gamma(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the gamma function (extension of the factorial function to the reals), computed element-wise on the input array.

The storage type of gamma output is always dense

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.gammaln(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise log of the absolute value of the gamma function of the input.

The storage type of gammaln output is always dense

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.log(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise Natural logarithmic value of the input.

The natural logarithm is logarithm in base e, so that log(exp(x)) = x

The storage type of log output is always dense

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L941

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.log10(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise Base-10 logarithmic value of the input.

10**log10(x) = x

The storage type of log10 output is always dense

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L953

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.log1p(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise log(1 + x) value of the input.

This function is more accurate than log(1 + x) for small x so that \(1+x\approx 1\)

The storage type of log1p output depends upon the input storage type:

log1p(default) = default

log1p(row_sparse) = row_sparse

log1p(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L990

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.log2(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise Base-2 logarithmic value of the input.

2**log2(x) = x

The storage type of log2 output is always dense

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L965

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.make_loss(data=None, name=None, attr=None, out=None, **kwargs)¶

Make your own loss function in network construction.

This operator accepts a customized loss function symbol as a terminal loss and the symbol should be an operator with no backward dependency. The output of this function is the gradient of loss with respect to the input data.

For example, if you are a making a cross entropy loss function. Assume out is the predicted output and label is the true label, then the cross entropy can be defined as:

cross_entropy = label * log(out) + (1 - label) * log(1 - out)
loss = make_loss(cross_entropy)

We will need to use make_loss when we are creating our own loss function or we want to combine multiple loss functions. Also we may want to stop some variables’ gradients from backpropagation. See more detail in BlockGrad or stop_gradient.

The storage type of make_loss output depends upon the input storage type:

make_loss(default) = default

make_loss(row_sparse) = row_sparse

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L298

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.mean(data=None, axis=_Null, keepdims=_Null, exclude=_Null, name=None, attr=None, out=None, **kwargs)¶

Computes the mean of array elements over given axes.

Defined in src/operator/tensor/broadcast_reduce_op_value.cc:L131

Parameters:	data (Symbol) – The input axis (Shape or None, optional, default=None) – The axis or axes along which to perform the reduction. The default, axis=(), will compute over all elements into a scalar array with shape (1,). If axis is int, a reduction is performed on a particular axis. If axis is a tuple of ints, a reduction is performed on all the axes specified in the tuple. If exclude is true, reduction will be performed on the axes that are NOT in axis instead. Negative values means indexing from right to left. keepdims (boolean, optional, default=0) – If this is set to True, the reduced axes are left in the result as dimension with size one. exclude (boolean, optional, default=0) – Whether to perform reduction on axis that are NOT in axis instead. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.negative(data=None, name=None, attr=None, out=None, **kwargs)¶

Numerical negative of the argument, element-wise.

The storage type of negative output depends upon the input storage type:

negative(default) = default

negative(row_sparse) = row_sparse

negative(csr) = csr

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.norm(data=None, ord=_Null, axis=_Null, keepdims=_Null, name=None, attr=None, out=None, **kwargs)¶

Computes the norm on an NDArray.

This operator computes the norm on an NDArray with the specified axis, depending on the value of the ord parameter. By default, it computes the L2 norm on the entire array. Currently only ord=2 supports sparse ndarrays.

Examples:

x = [[[1, 2],
      [3, 4]],
     [[2, 2],
      [5, 6]]]

norm(x, ord=2, axis=1) = [[3.1622777 4.472136 ]
                          [5.3851647 6.3245554]]

norm(x, ord=1, axis=1) = [[4., 6.],
                          [7., 8.]]

rsp = x.cast_storage('row_sparse')

norm(rsp) = [5.47722578]

csr = x.cast_storage('csr')

norm(csr) = [5.47722578]

Defined in src/operator/tensor/broadcast_reduce_op_value.cc:L345

Parameters:	data (Symbol) – The input ord (int, optional, default='2') – Order of the norm. Currently ord=1 and ord=2 is supported. axis (Shape or None, optional, default=None) – The axis or axes along which to perform the reduction. The default, axis=(), will compute over all elements into a scalar array with shape (1,). If axis is int, a reduction is performed on a particular axis. If axis is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. keepdims (boolean, optional, default=0) – If this is set to True, the reduced axis is left in the result as dimension with size one. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.radians(data=None, name=None, attr=None, out=None, **kwargs)¶

Converts each element of the input array from degrees to radians.

\[radians([0, 90, 180, 270, 360]) = [0, \pi/2, \pi, 3\pi/2, 2\pi]\]

The storage type of radians output depends upon the input storage type:

radians(default) = default

radians(row_sparse) = row_sparse

radians(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L182

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.relu(data=None, name=None, attr=None, out=None, **kwargs)¶

Computes rectified linear.

\[max(features, 0)\]

The storage type of relu output depends upon the input storage type:

relu(default) = default

relu(row_sparse) = row_sparse

relu(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L85

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.retain(data=None, indices=None, name=None, attr=None, out=None, **kwargs)¶

pick rows specified by user input index array from a row sparse matrix and save them in the output sparse matrix.

Example:

data = [[1, 2], [3, 4], [5, 6]]
indices = [0, 1, 3]
shape = (4, 2)
rsp_in = row_sparse(data, indices)
to_retain = [0, 3]
rsp_out = retain(rsp_in, to_retain)
rsp_out.values = [[1, 2], [5, 6]]
rsp_out.indices = [0, 3]

The storage type of retain output depends on storage types of inputs

retain(row_sparse, default) = row_sparse
otherwise, retain is not supported

Defined in src/operator/tensor/sparse_retain.cc:L53

Parameters:	data (Symbol) – The input array for sparse_retain operator. indices (Symbol) – The index array of rows ids that will be retained. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.rint(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise rounded value to the nearest integer of the input.

Note

For input n.5 rint returns n while round returns n+1.
For input -n.5 both rint and round returns -n-1.

Example:

rint([-1.5, 1.5, -1.9, 1.9, 2.1]) = [-2.,  1., -2.,  2.,  2.]

The storage type of rint output depends upon the input storage type:

rint(default) = default

rint(row_sparse) = row_sparse

rint(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L727

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.round(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise rounded value to the nearest integer of the input.

Example:

round([-1.5, 1.5, -1.9, 1.9, 2.1]) = [-2.,  2., -2.,  2.,  2.]

The storage type of round output depends upon the input storage type:

round(default) = default

round(row_sparse) = row_sparse

round(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L706

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.rsqrt(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise inverse square-root value of the input.

\[rsqrt(x) = 1/\sqrt{x}\]

Example:

rsqrt([4,9,16]) = [0.5, 0.33333334, 0.25]

The storage type of rsqrt output is always dense

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L866

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sgd_mom_update(weight=None, grad=None, mom=None, lr=_Null, momentum=_Null, wd=_Null, rescale_grad=_Null, clip_gradient=_Null, lazy_update=_Null, name=None, attr=None, out=None, **kwargs)¶

Momentum update function for Stochastic Gradient Descent (SGD) optimizer.

Momentum update has better convergence rates on neural networks. Mathematically it looks like below:

\[\begin{split}v_1 = \alpha * \nabla J(W_0)\\ v_t = \gamma v_{t-1} - \alpha * \nabla J(W_{t-1})\\ W_t = W_{t-1} + v_t\end{split}\]

It updates the weights using:

v = momentum * v - learning_rate * gradient
weight += v

Where the parameter momentum is the decay rate of momentum estimates at each epoch.

However, if grad’s storage type is row_sparse, lazy_update is True and weight’s storage type is the same as momentum’s storage type, only the row slices whose indices appear in grad.indices are updated (for both weight and momentum):

for row in gradient.indices:
    v[row] = momentum[row] * v[row] - learning_rate * gradient[row]
    weight[row] += v[row]

Defined in src/operator/optimizer_op.cc:L372

Parameters:	weight (Symbol) – Weight grad (Symbol) – Gradient mom (Symbol) – Momentum lr (float, required) – Learning rate momentum (float, optional, default=0) – The decay rate of momentum estimates at each epoch. wd (float, optional, default=0) – Weight decay augments the objective function with a regularization term that penalizes large weights. The penalty scales with the square of the magnitude of each weight. rescale_grad (float, optional, default=1) – Rescale gradient to grad = rescale_gradgrad. clip_gradient* (float, optional, default=-1) – Clip gradient to the range of [-clip_gradient, clip_gradient] If clip_gradient <= 0, gradient clipping is turned off. grad = max(min(grad, clip_gradient), -clip_gradient). lazy_update (boolean, optional, default=1) – If true, lazy updates are applied if gradient’s stype is row_sparse and both weight and momentum have the same stype name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sgd_update(weight=None, grad=None, lr=_Null, wd=_Null, rescale_grad=_Null, clip_gradient=_Null, lazy_update=_Null, name=None, attr=None, out=None, **kwargs)¶

Update function for Stochastic Gradient Descent (SDG) optimizer.

It updates the weights using:

weight = weight - learning_rate * (gradient + wd * weight)

However, if gradient is of row_sparse storage type and lazy_update is True, only the row slices whose indices appear in grad.indices are updated:

for row in gradient.indices:
    weight[row] = weight[row] - learning_rate * (gradient[row] + wd * weight[row])

Defined in src/operator/optimizer_op.cc:L331

Parameters:	weight (Symbol) – Weight grad (Symbol) – Gradient lr (float, required) – Learning rate wd (float, optional, default=0) – Weight decay augments the objective function with a regularization term that penalizes large weights. The penalty scales with the square of the magnitude of each weight. rescale_grad (float, optional, default=1) – Rescale gradient to grad = rescale_gradgrad. clip_gradient* (float, optional, default=-1) – Clip gradient to the range of [-clip_gradient, clip_gradient] If clip_gradient <= 0, gradient clipping is turned off. grad = max(min(grad, clip_gradient), -clip_gradient). lazy_update (boolean, optional, default=1) – If true, lazy updates are applied if gradient’s stype is row_sparse. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sigmoid(data=None, name=None, attr=None, out=None, **kwargs)¶

Computes sigmoid of x element-wise.

\[y = 1 / (1 + exp(-x))\]

The storage type of sigmoid output is always dense

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L101

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sign(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise sign of the input.

Example:

sign([-2, 0, 3]) = [-1, 0, 1]

The storage type of sign output depends upon the input storage type:

sign(default) = default

sign(row_sparse) = row_sparse

sign(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L687

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sin(data=None, name=None, attr=None, out=None, **kwargs)¶

Computes the element-wise sine of the input array.

The input should be in radians (\(2\pi\) rad equals 360 degrees).

\[sin([0, \pi/4, \pi/2]) = [0, 0.707, 1]\]

The storage type of sin output depends upon the input storage type:

sin(default) = default

sin(row_sparse) = row_sparse

sin(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L46

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sinh(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the hyperbolic sine of the input array, computed element-wise.

\[sinh(x) = 0.5\times(exp(x) - exp(-x))\]

The storage type of sinh output depends upon the input storage type:

sinh(default) = default

sinh(row_sparse) = row_sparse

sinh(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L201

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.slice(data=None, begin=_Null, end=_Null, step=_Null, name=None, attr=None, out=None, **kwargs)¶

Slices a region of the array.

Note

crop is deprecated. Use slice instead.

This function returns a sliced array between the indices given by begin and end with the corresponding step.

For an input array of shape=(d_0, d_1, ..., d_n-1), slice operation with begin=(b_0, b_1...b_m-1), end=(e_0, e_1, ..., e_m-1), and step=(s_0, s_1, ..., s_m-1), where m <= n, results in an array with the shape (|e_0-b_0|/|s_0|, ..., |e_m-1-b_m-1|/|s_m-1|, d_m, ..., d_n-1).

The resulting array’s k-th dimension contains elements from the k-th dimension of the input array starting from index b_k (inclusive) with step s_k until reaching e_k (exclusive).

If the k-th elements are None in the sequence of begin, end, and step, the following rule will be used to set default values. If s_k is None, set s_k=1. If s_k > 0, set b_k=0, e_k=d_k; else, set b_k=d_k-1, e_k=-1.

The storage type of slice output depends on storage types of inputs

slice(csr) = csr
otherwise, slice generates output with default storage

Note

When input data storage type is csr, it only supports

step=(), or step=(None,), or step=(1,) to generate a csr output. For other step parameter values, it falls back to slicing a dense tensor.

Example:

x = [[  1.,   2.,   3.,   4.],
     [  5.,   6.,   7.,   8.],
     [  9.,  10.,  11.,  12.]]

slice(x, begin=(0,1), end=(2,4)) = [[ 2.,  3.,  4.],
                                   [ 6.,  7.,  8.]]
slice(x, begin=(None, 0), end=(None, 3), step=(-1, 2)) = [[9., 11.],
                                                          [5.,  7.],
                                                          [1.,  3.]]

Defined in src/operator/tensor/matrix_op.cc:L413

Parameters:	data (Symbol) – Source input begin (Shape(tuple), required) – starting indices for the slice operation, supports negative indices. end (Shape(tuple), required) – ending indices for the slice operation, supports negative indices. step (Shape(tuple), optional, default=[]) – step for the slice operation, supports negative values. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sqrt(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise square-root value of the input.

\[\textrm{sqrt}(x) = \sqrt{x}\]

Example:

sqrt([4, 9, 16]) = [2, 3, 4]

The storage type of sqrt output depends upon the input storage type:

sqrt(default) = default

sqrt(row_sparse) = row_sparse

sqrt(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L846

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.square(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns element-wise squared value of the input.

\[square(x) = x^2\]

Example:

square([2, 3, 4]) = [4, 9, 16]

The storage type of square output depends upon the input storage type:

square(default) = default

square(row_sparse) = row_sparse

square(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L823

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.stop_gradient(data=None, name=None, attr=None, out=None, **kwargs)¶

Stops gradient computation.

Stops the accumulated gradient of the inputs from flowing through this operator in the backward direction. In other words, this operator prevents the contribution of its inputs to be taken into account for computing gradients.

Example:

v1 = [1, 2]
v2 = [0, 1]
a = Variable('a')
b = Variable('b')
b_stop_grad = stop_gradient(3 * b)
loss = MakeLoss(b_stop_grad + a)

executor = loss.simple_bind(ctx=cpu(), a=(1,2), b=(1,2))
executor.forward(is_train=True, a=v1, b=v2)
executor.outputs
[ 1.  5.]

executor.backward()
executor.grad_arrays
[ 0.  0.]
[ 1.  1.]

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L265

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.sum(data=None, axis=_Null, keepdims=_Null, exclude=_Null, name=None, attr=None, out=None, **kwargs)¶

Computes the sum of array elements over given axes.

Note

sum and sum_axis are equivalent. For ndarray of csr storage type summation along axis 0 and axis 1 is supported. Setting keepdims or exclude to True will cause a fallback to dense operator.

Example:

data = [[[1, 2], [2, 3], [1, 3]],
        [[1, 4], [4, 3], [5, 2]],
        [[7, 1], [7, 2], [7, 3]]]

sum(data, axis=1)
[[  4.   8.]
 [ 10.   9.]
 [ 21.   6.]]

sum(data, axis=[1,2])
[ 12.  19.  27.]

data = [[1, 2, 0],
        [3, 0, 1],
        [4, 1, 0]]

csr = cast_storage(data, 'csr')

sum(csr, axis=0)
[ 8.  3.  1.]

sum(csr, axis=1)
[ 3.  4.  5.]

Defined in src/operator/tensor/broadcast_reduce_op_value.cc:L115

Parameters:	data (Symbol) – The input axis (Shape or None, optional, default=None) – The axis or axes along which to perform the reduction. The default, axis=(), will compute over all elements into a scalar array with shape (1,). If axis is int, a reduction is performed on a particular axis. If axis is a tuple of ints, a reduction is performed on all the axes specified in the tuple. If exclude is true, reduction will be performed on the axes that are NOT in axis instead. Negative values means indexing from right to left. keepdims (boolean, optional, default=0) – If this is set to True, the reduced axes are left in the result as dimension with size one. exclude (boolean, optional, default=0) – Whether to perform reduction on axis that are NOT in axis instead. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.tan(data=None, name=None, attr=None, out=None, **kwargs)¶

Computes the element-wise tangent of the input array.

The input should be in radians (\(2\pi\) rad equals 360 degrees).

\[tan([0, \pi/4, \pi/2]) = [0, 1, -inf]\]

The storage type of tan output depends upon the input storage type:

tan(default) = default

tan(row_sparse) = row_sparse

tan(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L83

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.tanh(data=None, name=None, attr=None, out=None, **kwargs)¶

Returns the hyperbolic tangent of the input array, computed element-wise.

\[tanh(x) = sinh(x) / cosh(x)\]

The storage type of tanh output depends upon the input storage type:

tanh(default) = default

tanh(row_sparse) = row_sparse

tanh(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_trig.cc:L234

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.trunc(data=None, name=None, attr=None, out=None, **kwargs)¶

Return the element-wise truncated value of the input.

The truncated value of the scalar x is the nearest integer i which is closer to zero than x is. In short, the fractional part of the signed number x is discarded.

Example:

trunc([-2.1, -1.9, 1.5, 1.9, 2.1]) = [-2., -1.,  1.,  1.,  2.]

The storage type of trunc output depends upon the input storage type:

trunc(default) = default

trunc(row_sparse) = row_sparse

trunc(csr) = csr

Defined in src/operator/tensor/elemwise_unary_op_basic.cc:L785

Parameters:	data (Symbol) – The input array. name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.where(condition=None, x=None, y=None, name=None, attr=None, out=None, **kwargs)¶

Return the elements, either from x or y, depending on the condition.

Given three ndarrays, condition, x, and y, return an ndarray with the elements from x or y, depending on the elements from condition are true or false. x and y must have the same shape. If condition has the same shape as x, each element in the output array is from x if the corresponding element in the condition is true, and from y if false.

If condition does not have the same shape as x, it must be a 1D array whose size is the same as x’s first dimension size. Each row of the output array is from x’s row if the corresponding element from condition is true, and from y’s row if false.

Note that all non-zero values are interpreted as True in condition.

Examples:

x = [[1, 2], [3, 4]]
y = [[5, 6], [7, 8]]
cond = [[0, 1], [-1, 0]]

where(cond, x, y) = [[5, 2], [3, 8]]

csr_cond = cast_storage(cond, 'csr')

where(csr_cond, x, y) = [[5, 2], [3, 8]]

Defined in src/operator/tensor/control_flow_op.cc:L57

Parameters:	condition (Symbol) – condition array x (Symbol) – y (Symbol) – name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol

mxnet.symbol.sparse.zeros_like(data=None, name=None, attr=None, out=None, **kwargs)¶

Return an array of zeros with the same shape, type and storage type as the input array.

The storage type of zeros_like output depends on the storage type of the input

zeros_like(row_sparse) = row_sparse
zeros_like(csr) = csr
zeros_like(default) = default

Examples:

x = [[ 1.,  1.,  1.],
     [ 1.,  1.,  1.]]

zeros_like(x) = [[ 0.,  0.,  0.],
                 [ 0.,  0.,  0.]]

Parameters:	data (Symbol) – The input name (string, optional.) – Name of the resulting symbol.
Returns:	The result symbol.
Return type:	Symbol