tensor.nlinalg – Linear Algebra Ops Using Numpy

Note

This module is not imported by default. You need to import it to use it.

API

class theano.tensor.nlinalg.AllocDiag[source]

Allocates a square matrix with the given vector as its diagonal.

grad(inputs, g_outputs)[source]

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Parameters:
  • inputs (list of Variable) – The input variables.
  • output_grads (list of Variable) – The gradients of the output variables.
Returns:

grads – The gradients with respect to each Variable in inputs.

Return type:

list of Variable

make_node(_x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.Det[source]

Matrix determinant. Input should be a square matrix.

grad(inputs, g_outputs)[source]

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Parameters:
  • inputs (list of Variable) – The input variables.
  • output_grads (list of Variable) – The gradients of the output variables.
Returns:

grads – The gradients with respect to each Variable in inputs.

Return type:

list of Variable

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.Eig[source]

Compute the eigenvalues and right eigenvectors of a square array.

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.Eigh(UPLO='L')[source]

Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.

grad(inputs, g_outputs)[source]

The gradient function should return

\sum_n\left(W_n\frac{\partial\,w_n}
      {\partial a_{ij}} +
\sum_k V_{nk}\frac{\partial\,v_{nk}}
      {\partial a_{ij}}\right),

where [W, V] corresponds to g_outputs, a to inputs, and (w, v)=\mbox{eig}(a).

Analytic formulae for eigensystem gradients are well-known in perturbation theory:

\frac{\partial\,w_n}
{\partial a_{ij}} = v_{in}\,v_{jn}

\frac{\partial\,v_{kn}}
          {\partial a_{ij}} =
\sum_{m\ne n}\frac{v_{km}v_{jn}}{w_n-w_m}

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.EighGrad(UPLO='L')[source]

Gradient of an eigensystem of a Hermitian matrix.

make_node(x, w, v, gw, gv)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Implements the “reverse-mode” gradient for the eigensystem of a square matrix.

class theano.tensor.nlinalg.MatrixInverse[source]

Computes the inverse of a matrix A.

Given a square matrix A, matrix_inverse returns a square matrix A_{inv} such that the dot product A \cdot A_{inv} and A_{inv} \cdot A equals the identity matrix I.

Notes

When possible, the call to this op will be optimized to the call of solve.

R_op(inputs, eval_points)[source]

The gradient function should return

\frac{\partial X^{-1}}{\partial X}V,

where V corresponds to g_outputs and X to inputs. Using the matrix cookbook, one can deduce that the relation corresponds to

X^{-1} \cdot V \cdot X^{-1}.

grad(inputs, g_outputs)[source]

The gradient function should return

V\frac{\partial X^{-1}}{\partial X},

where V corresponds to g_outputs and X to inputs. Using the matrix cookbook, one can deduce that the relation corresponds to

(X^{-1} \cdot V^{T} \cdot X^{-1})^T.

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.MatrixPinv[source]

Computes the pseudo-inverse of a matrix A.

The pseudo-inverse of a matrix A, denoted A^+, is defined as: “the matrix that ‘solves’ [the least-squares problem] Ax = b,” i.e., if \bar{x} is said solution, then A^+ is that matrix such that \bar{x} = A^+b.

Note that Ax=AA^+b, so AA^+ is close to the identity matrix. This method is not faster than matrix_inverse. Its strength comes from that it works for non-square matrices. If you have a square matrix though, matrix_inverse can be both more exact and faster to compute. Also this op does not get optimized into a solve op.

L_op(inputs, outputs, g_outputs)[source]

The gradient function should return

V\frac{\partial X^+}{\partial X},

where V corresponds to g_outputs and X to inputs. According to Wikipedia, this corresponds to

(-X^+ V^T X^+ + X^+ X^{+T} V (I - X X^+) + (I - X^+ X) V X^{+T} X^+)^T.

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.QRFull(mode)[source]

Full QR Decomposition.

Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.QRIncomplete(mode)[source]

Incomplete QR Decomposition.

Computes the QR decomposition of a matrix. Factor the matrix a as qr and return a single matrix R.

make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.SVD(full_matrices=True, compute_uv=True)[source]
Parameters:
  • full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).
  • compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.
make_node(x)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.TensorInv(ind=2)[source]

Class wrapper for tensorinv() function; Theano utilization of numpy.linalg.tensorinv;

make_node(a)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class theano.tensor.nlinalg.TensorSolve(axes=None)[source]

Theano utilization of numpy.linalg.tensorsolve Class wrapper for tensorsolve function.

make_node(a, b)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

theano.tensor.nlinalg.diag(x)[source]

Numpy-compatibility method If x is a matrix, return its diagonal. If x is a vector return a matrix with it as its diagonal.

  • This method does not support the k argument that numpy supports.
class theano.tensor.nlinalg.lstsq[source]
make_node(x, y, rcond)[source]

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:node – The constructed Apply node.
Return type:Apply
perform(node, inputs, outputs)[source]

Calculate the function on the inputs and put the variables in the output storage.

Parameters:
  • node (Apply) – The symbolic Apply node that represents this computation.
  • inputs (Sequence) – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.
  • output_storage (list of list) – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.
  • params (tuple) – A tuple containing the values of each entry in __props__.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform; they could’ve been allocated by another Op’s perform method. A Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

theano.tensor.nlinalg.matrix_dot(*args)[source]

Shorthand for product between several dots.

Given N matrices A_0, A_1, .., A_N, matrix_dot will generate the matrix product between all in the given order, namely A_0 \cdot A_1 \cdot A_2 \cdot .. \cdot A_N.

theano.tensor.nlinalg.matrix_power(M, n)[source]

Raise a square matrix, M, to the (integer) power n.

This implementation uses exponentiation by squaring which is significantly faster than the naive implementation. The time complexity for exponentiation by squaring is :math: mathcal{O}((n log M)^k)

Parameters:
theano.tensor.nlinalg.qr(a, mode='reduced')[source]

Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

Parameters:
  • a (array_like, shape (M, N)) – Matrix to be factored.
  • mode ({'reduced', 'complete', 'r', 'raw'}, optional) –

    If K = min(M, N), then

    ’reduced’
    returns q, r with dimensions (M, K), (K, N)
    ’complete’
    returns q, r with dimensions (M, M), (M, N)
    ’r’
    returns r only with dimensions (K, N)
    ’raw’
    returns h, tau with dimensions (N, M), (K,)

    Note that array h returned in ‘raw’ mode is transposed for calling Fortran.

    Default mode is ‘reduced’

Returns:

  • q (matrix of float or complex, optional) – A matrix with orthonormal columns. When mode = ‘complete’ the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/- 1 in that case.
  • r (matrix of float or complex, optional) – The upper-triangular matrix.

theano.tensor.nlinalg.svd(a, full_matrices=1, compute_uv=1)[source]

This function performs the SVD on CPU.

Parameters:
  • full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).
  • compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.
Returns:

U, V, D

Return type:

matrices

theano.tensor.nlinalg.tensorinv(a, ind=2)[source]

Does not run on GPU; Theano utilization of numpy.linalg.tensorinv;

Compute the ‘inverse’ of an N-dimensional array. The result is an inverse for a relative to the tensordot operation tensordot(a, b, ind), i. e., up to floating-point accuracy, tensordot(tensorinv(a), a, ind) is the “identity” tensor for the tensordot operation.

Parameters:
  • a (array_like) – Tensor to ‘invert’. Its shape must be ‘square’, i. e., prod(a.shape[:ind]) == prod(a.shape[ind:]).
  • ind (int, optional) – Number of first indices that are involved in the inverse sum. Must be a positive integer, default is 2.
Returns:

ba’s tensordot inverse, shape a.shape[ind:] + a.shape[:ind].

Return type:

ndarray

Raises:

LinAlgError – If a is singular or not ‘square’ (in the above sense).

theano.tensor.nlinalg.tensorsolve(a, b, axes=None)[source]

Theano utilization of numpy.linalg.tensorsolve. Does not run on GPU!

Solve the tensor equation a x = b for x. It is assumed that all indices of x are summed over in the product, together with the rightmost indices of a, as is done in, for example, tensordot(a, x, axes=len(b.shape)).

Parameters:
  • a (array_like) – Coefficient tensor, of shape b.shape + Q. Q, a tuple, equals the shape of that sub-tensor of a consisting of the appropriate number of its rightmost indices, and must be such that prod(Q) == prod(b.shape) (in which sense a is said to be ‘square’).
  • b (array_like) – Right-hand tensor, which can be of any shape.
  • axes (tuple of ints, optional) – Axes in a to reorder to the right, before inversion. If None (default), no reordering is done.
Returns:

x

Return type:

ndarray, shape Q

Raises:

LinAlgError – If a is singular or not ‘square’ (in the above sense).

theano.tensor.nlinalg.trace(X)[source]

Returns the sum of diagonal elements of matrix X.

Notes

Works on GPU since 0.6rc4.