# magni.utils.matrices module¶

Module providing matrix emulators.

The matrix emulators of this module are wrappers of fast linear operations giving the fast linear operations the same basic interface as a numpy ndarray. Thereby allowing fast linear operations and numpy ndarrays to be used interchangably in other parts of the package.

## Routine listings¶

Matrix(magni.utils.validation.types.MatrixBase)
Wrap fast linear operations in a matrix emulator.
MatrixCollection(magni.utils.validation.types.MatrixBase)
Wrap multiple matrix emulators in a single matrix emulator.
Separable2DTransform(Matrix)
Wrap a linear 2D separable transform in a matrix emulator.
SRM(MatrixCollection)
Wrap a Structurally Random Matrix (SRM) in a matrix emulator.
SumApproximationMatrix(object)
Wrap a sum approximation in a matrix emulator.
norm(A, ord=None)
Compute a norm of a matrix.

magni.imaging._fastops
Fast linear operations.
magni.imaging._mtx1D
1D transforms matrices for use 2D separable transforms.
class magni.utils.matrices.Matrix(func, conj_trans, args, shape, is_complex=False, is_valid=True)[source]

Wrap fast linear operations in a matrix emulator.

Matrix defines a few attributes and internal methods which ensures that instances have the same basic interface as a numpy ndarray instance without explicitly forming the array. This basic interface allows instances to be multiplied with vectors, to be transposed, to be complex conjuagted, and to assume a shape. Also, instances have an attribute which explicitly forms the matrix as an ndarray.

Parameters: func (function) – The fast linear operation applied to the vector when multiplying the matrix with a vector. conj_trans (function) – The fast linear operation applied to the vector when multiplying the complex conjuated transposed matrix with a vector. args (list or tuple) – The arguments which should be passed to func and conj_trans in addition to the vector. shape (list or tuple) – The shape of the emulated matrix. is_complex (bool) – The indicator of whether or not the emulated matrix is defined for the complex numbers (the default is False, which implies that the emulated matrix is defined for the real numbers only). is_valid (bool) – The indicator of whether or not the fast linear operation corresponds to a valid matrix (se discussion below) (the default is True, which implies that the matrix is considered valid).

magni.utils.validation.types.MatrixBase
Superclass of the present class.

Notes

The is_valid indicator is an implementation detail used to control possibly missing operations in the fast linear operation. For instance, consider the Discrete Fourier Transform (DFT) and its inverse transform. The forward DFT transform is a matrix-vector product involving the corresponding DFT matrix. The inverse transform is a matrix-vetor product involving the complex conjugate transpose DFT matrix. That is, it involves complex conjugation and transposition, both of which are (individually) valid transformations of the DFT matrix. However, when implementing the DFT (and its inverse) using a Fast Fourier Transform (FFT), only the combined (complex conjugate, transpose) operation is available. Thus, when using an FFT based magni.util.matrices.Matrix, one can get, e.g., a Matrix.T object corresponding to its transpose. However, the operation of computing a matrix-vector product involving the tranpose DFT matrix is not available and the Matrix.T is, consequently, considered an invalid matrix. Only the combined Matrix.T.conj() or Matrix.conj().T is considered a valid matrix.

Warning

The tranpose operation changed in magni 1.5.0.

For complex matrices, the the T tranpose operation now yields an invalid matrix as described above. Prior to magni 1.5.0, the T would yield the “inverse” which would usually be the complex conjugated transpose for complex matrices.

Examples

For example, the negative identity matrix could be emulated as

>>> import numpy as np, magni
>>> from magni.utils.matrices import Matrix
>>> func = lambda vec: -vec
>>> matrix = Matrix(func, func, (), (3, 3))


The example matrix will have the desired shape:

>>> matrix.shape
(3, 3)


The example matrix will behave just like an explicit matrix:

>>> vec = np.float64([1, 2, 3]).reshape(3, 1)
>>> np.set_printoptions(suppress=True)
>>> matrix.dot(vec)
array([[-1.],
[-2.],
[-3.]])


If, at some point, an explicit representation of the matrix is required, this can easily be obtained:

>>> matrix.A
array([[-1., -0., -0.],
[-0., -1., -0.],
[-0., -0., -1.]])


Likewise, the transpose of the matrix can be obtained:

>>> matrix.T.A
array([[-1., -0., -0.],
[-0., -1., -0.],
[-0., -0., -1.]])

__array__()[source]

Return ndarray representation of the matrix.

matrix_state

Return a copy of the internal matrix state.

The internal matrix state consists of:

• func: The forward fast linear operator.
• conj_trans: The backward fast linear operator.
• args: The tuple of extra arguments passed to func and conj_trans.
• is_complex: The indicator of whether or not the matrix is complex.
• is_valid: The indicator of whether or not the matrix is valid.
Returns: matrix_state (dict) – A copy of the internal matrix state
A

Explicitly form the matrix.

The fast linear operations implicitly define a matrix which is usually not explicitly formed. However, some functionality might require a more advanced matrix interface than that provided by this class.

Returns: matrix (numpy.ndarray) – The explicit matrix.

Notes

The explicit matrix is formed by multiplying the matrix with the columns of an identity matrix and stacking the resulting vectors as columns in a matrix.

T

Get the transpose of the matrix.

Returns: matrix (Matrix) – The transpose of the matrix.

Notes

The fast linear operation and the fast linear transposed operation of the resulting matrix are the same as those of the current matrix except swapped. The shape is modified accordingly. This returns an invalid matrix if the entries are complex numbers as only the complex conjugate transpose is considered valid.

conj()[source]

Get the complex conjugate of the matrix.

Returns: matrix (Matrix) – The complex conjugate of the matrix.

Notes

The fast linear operation and the fast linear transposed operation of the resulting matrix are the same as those of the current matrix. This returns an invalid matrix if the entries are complex numbers as only the complex conjugate transpose is considered valid.

dot(vec)[source]

Multiply the matrix with a vector.

Parameters: vec (numpy.ndarray) – The vector which the matrix is multiplied with. vec (numpy.ndarray) – The result of the multiplication.

Notes

This method honors magni.utils.validation.enable_validate_once.

class magni.utils.matrices.MatrixCollection(matrices)[source]

Wrap multiple matrix emulators in a single matrix emulator.

MatrixCollection defines a few attributes and internal methods which ensures that instances have the same basic interface as a numpy ndarray instance without explicitly forming the ndarray. This basic interface allows instances to be multiplied with vectors, to be transposed, to be complex conjugated, and to assume a shape. Also, instances have an attribute which explicitly forms the matrix.

Parameters: matrices (list or tuple) – The collection of matrices (e.g. ndarrays or Matrix instances).

magni.utils.validation.types.MatrixBase
Superclass of the present class.
Matrix
Matrix emulator.

Examples

For example, two matrix emulators can be combined into one. That is, the matrix:

>>> import numpy as np, magni
>>> func = lambda vec: -vec
>>> negate = magni.utils.matrices.Matrix(func, func, (), (3, 3))
>>> np.set_printoptions(suppress=True)
>>> negate.A
array([[-1., -0., -0.],
[-0., -1., -0.],
[-0., -0., -1.]])


And the matrix:

>>> func = lambda vec: vec[::-1]
>>> reverse = magni.utils.matrices.Matrix(func, func, (), (3, 3))
>>> reverse.A
array([[ 0.,  0.,  1.],
[ 0.,  1.,  0.],
[ 1.,  0.,  0.]])


Can be combined into one matrix emulator using the present class:

>>> from magni.utils.matrices import MatrixCollection
>>> matrix = MatrixCollection((negate, reverse))


The example matrix will have the desired shape:

>>> matrix.shape
(3, 3)


The example matrix will behave just like an explicit matrix:

>>> vec = np.float64([1, 2, 3]).reshape(3, 1)
>>> matrix.dot(vec)
array([[-3.],
[-2.],
[-1.]])


If, at some point, an explicit representation of the matrix is required, this can easily be obtained:

>>> matrix.A
array([[-0., -0., -1.],
[-0., -1., -0.],
[-1., -0., -0.]])


Likewise, the transpose of the matrix can be obtained:

>>> matrix.T.A
array([[-0., -0., -1.],
[-0., -1., -0.],
[-1., -0., -0.]])

__array__()[source]

Return ndarray representation of the matrix.

matrix_state

Return a copy of the internal matrix state.

The internal matrix state consists of:

• matrices: The tuple of matrices in the matrix collection
Returns: matrix_state (dict) – A copy of the internal matrix state
A

Explicitly form the matrix.

The collection of matrices implicitly defines a matrix which is usually not explicitly formed. However, some functionality might require a more advanced matrix interface than that provided by this class.

Returns: matrix (numpy.ndarray) – The explicit matrix.

Notes

The explicit matrix is formed by multiplying the matrix with the columns of an identity matrix and stacking the resulting vectors as columns in a matrix.

shape

Get the shape of the matrix.

The shape of the product of a number of matrices is the number of rows of the first matrix times the number of columns of the last matrix.

Returns: shape (tuple) – The shape of the matrix.
T

Get the transpose of the matrix.

The transpose of the product of the number of matrices is the product of the transpose of the matrices in reverse order.

Returns: matrix (MatrixCollection) – The transpose of the matrix.
conj()[source]

Get the complex conjugate of the matrix.

The complex conjugate of the product of the number of matrices is the product of the complex conjugates of the matrices.

Returns: matrix (MatrixCollection) – The complex conjugate of the matrix.
dot(vec)[source]

Multiply the matrix with a vector.

Parameters: vec (numpy.ndarray) – The vector which the matrix is multiplied with. vec (numpy.ndarray) – The result of the multiplication.

Notes

This method honors magni.utils.validation.enable_validate_once.

class magni.utils.matrices.Separable2DTransform(mtx_l, mtx_r)[source]

Wrap a linear 2D separable transform in a matrix emulator.

A linear 2D separable transform is defined by the two matrices mtx_l and mtx_r in the sense that the Kronecker product kron(mtx_l, mtx_r) yields the full matrix of the linear 2D separable transform when the linear transform is implemented as a matrix-vector product. See e.g. [1] for the details.

Separable2DTransform defines a few attributes and internal methods which ensures that instances have the same basic interface as a numpy matrix instance without explicitly forming the matrix. This basic interface allows instances to be multiplied with vectors, to be transposed, and to assume a shape. Also, instances have an attribute which explicitly forms the matrix.

Parameters: mtx_l (ndarray) – The “left” matrix in the defining Kronecker product. mtx_r (ndarray) – The “right” matrix in the defining Kronecker product.

magni.utils.matrices.Matrix
Superclass of the present class.

References

Examples

For example, the transform based on a 2-by-3 matrix and a 5-by-4 matrix

>>> import numpy as np
>>> from magni.imaging import vec2mat, mat2vec
>>> from magni.utils.matrices import Separable2DTransform
>>> mtx_l = np.arange(6).reshape(2, 3)
>>> mtx_r = np.arange(40)[::2].reshape(5, 4)
>>> sep_matrix = Separable2DTransform(mtx_l, mtx_r)


The full transform matrix shape is:

>>> print(tuple(int(s) for s in sep_matrix.shape))
(10, 12)


The matrix behaves like an explicit matrix

>>> vec = np.arange(36)[::3].reshape(12, 1)
>>> sep_matrix.dot(vec)
array([[  936],
[ 3528],
[ 2712],
[10056],
[ 4488],
[16584],
[ 6264],
[23112],
[ 8040],
[29640]])


which, due to the separability of the transform, may also be computed as

>>> vec_as_mat = vec2mat(vec, (3, 4))
>>> mat2vec(mtx_l.dot(vec_as_mat.dot(mtx_r.T)))
array([[  936],
[ 3528],
[ 2712],
[10056],
[ 4488],
[16584],
[ 6264],
[23112],
[ 8040],
[29640]])


The explicit matrix is given by the kronecker product of the two matrices

>>> np.all(sep_matrix.A == np.kron(mtx_l, mtx_r))
True


The transpose of the matrix is also easily obtainable

>>> sep_matrix_transpose = sep_matrix.T
>>> sep_matrix_transpose.A
array([[  0,   0,   0,   0,   0,   0,  24,  48,  72,  96],
[  0,   0,   0,   0,   0,   6,  30,  54,  78, 102],
[  0,   0,   0,   0,   0,  12,  36,  60,  84, 108],
[  0,   0,   0,   0,   0,  18,  42,  66,  90, 114],
[  0,   8,  16,  24,  32,   0,  32,  64,  96, 128],
[  2,  10,  18,  26,  34,   8,  40,  72, 104, 136],
[  4,  12,  20,  28,  36,  16,  48,  80, 112, 144],
[  6,  14,  22,  30,  38,  24,  56,  88, 120, 152],
[  0,  16,  32,  48,  64,   0,  40,  80, 120, 160],
[  4,  20,  36,  52,  68,  10,  50,  90, 130, 170],
[  8,  24,  40,  56,  72,  20,  60, 100, 140, 180],
[ 12,  28,  44,  60,  76,  30,  70, 110, 150, 190]])

matrix_state

Return a copy of the internal matrix state.

The internal matrix state consists of:

• mtx_l: The “left” matrix in the defining Kronecker product.
• mtx_r: The “right” matrix in the defining Kronecker product.
Returns: matrix_state (dict) – A copy of the internal matrix state
A

Explicitly form the matrix.

For a linear separable 2D transform, the full explicit transform matrix is given by the Kronecker product of the two matrices that define the separable transform.

Returns: matrix (ndarray) – The explicit matrix.
T

Get the transpose of the matrix.

The transpose of a Kronekcer product of two matrices is the Kronecker product of the transposes of the two matrices.

Returns: matrix (Separable2DTransform) – The transpose of the matrix.
conj()[source]

Get the complex conjugate of the matrix.

The complex conjugate of a Kronekcer product of two matrices is the Kronecker product of the complex conjugates of the two matrices.

Returns: matrix (Separable2DTransform) – The complex conjugate of the matrix.
dot(vec)[source]

Multiply the matrix with a vector.

The matrix-vector product is efficiently computed by mtx_l.dot(V.dot(mtx_r.T)) where V is the matrix from which vec was created by stacking its columns.

Parameters: vec (ndarray) – The vector corresponding to the stacked columns of a matrix which this matrix is multiplied with. vec (ndarray) – The result of the matrix-vector multiplication.

Notes

This method honors magni.utils.validation.enable_allow_validate_once.

class magni.utils.matrices.SRM(F, D=None, l_ran_arr=None, g_ran_arr=None)[source]

Wrap a Structurally Random Matrix (SRM) in a matrix emulator.

Structurally Random Matrices are detailed in [2]. They are composed of a (row) sub-sampling matrix D, an orthogonal matrix F, and a pre-randomization matrix R such that the SRM (an m-by-n matrix) is given by sqrt(n/m)*DFR with DFR being the matrix product of D, F, and R.

This class implements are sligthly more general SRM than the one just described. Specifically, the scaling is absorbed into D and may be arbitrary, i.e. potentially different for each row. Furthermore, the F matrix is allowed to be an abitrary p-by-n matrix for D an m-by-p matrix. Finally, this class allows for either or both of local pre-randomization (sign changes on columns) or global pre-randomization (permutation of columns). If both local and global pre-randomization is used, the R matrix is composed as the matrix product R_gR_l with R_g the global pre-randomization and R_l the local.

Parameters: F (ndarray or magni.utils.validation.types.MatrixBase) – The p-by-n “base” matrix used in the SRM. D (ndarray or magni.utils.validation.types.MatrixBase) – The m-by-p sub-sampling matrix used in the SRM (the default is None, which implies that no sub-sampling matrix is used in the SRM). l_ran_arr (ndarray) – The length n, ordered 1D array of signs to apply to the n columns (the default is None, which implies that no signs are applied to the columns). g_ran_arr (ndarray) – The length n, ordered 1D array of indices (zero-indexed) defining the permutation of the n columns (the default is None, which implies that the columns are not permuted) - see example below.

magni.utils.matrices.MatrixCollection
Superclass of the present class.

References

 [2] T.T. Do, L.. Gan, N.H. Nguyen, and T.D. Tran, “Fast and Efficient Compressive Sensing Using Structurally Random Matrices”, IEEE Transactions on Signal Processing, vol. 60, no. 1, pp. 139-154, 2012

Examples

For example, a sub-sampling of a 3-by-3 matrix

>>> import numpy as np
>>> from magni.utils.matrices import SRM
>>> D = np.array([[0, 1, 0], [0, 0, 1]])
>>> F = np.arange(9).reshape(3, 3)
>>> A_1 = SRM(F, D=D)
>>> A_1.A.real
array([[ 3.,  4.,  5.],
[ 6.,  7.,  8.]])


or a local pre-randomization (sign change)

>>> signs = np.array([1, -1, 1])
>>> A_2 = SRM(F, l_ran_arr=signs)
>>> A_2.A.real
array([[ 0., -1.,  2.],
[ 3., -4.,  5.],
[ 6., -7.,  8.]])


or a global pre-randomization (permutation)

>>> permutation = np.array([2, 0, 1])
>>> A_3 = SRM(F, g_ran_arr=permutation)
>>> A_3.A.real
array([[ 2.,  0.,  1.],
[ 5.,  3.,  4.],
[ 8.,  6.,  7.]])


or everything together

>>> A_4 = SRM(F, D=D, l_ran_arr=signs, g_ran_arr=permutation)
>>> A_4.A.real
array([[ 5., -3.,  4.],
[ 8., -6.,  7.]])

matrix_state

Return a copy of the internal matrix state.

The internal matrix state consists of:

• matrices: The tuple of matrices in the SRM.
• l_ran_arr: The local pre-randomization array.
• g_ran_arr: The global pre-randomization array.
• F_norm: The special F matrix used in norm computations.
• includes: The dictionary detailing the SRM structure.
Returns: matrix_state (dict) – A copy of the internal matrix state
class magni.utils.matrices.SumApproximationMatrix(scaling)[source]

Bases: object

Wrap a sum approximation in a matrix emulator.

This simply emulates computing a scaled sum of the entries of a vector as a matrix vector product.

Parameters: scaling (int or float) – The scaling applied to the sum approximation.

Examples

For example,

>>> import numpy as np
>>> from magni.utils.matrices import SumApproximationMatrix
>>> np.random.seed(seed=6021)
>>> n = 10
>>> vec = np.random.randn(n, 1)
>>> np_printoptions = np.get_printoptions()
>>> np.set_printoptions(precision=5)
>>> np.array([vec.mean()])
array([ 0.43294])
>>> matrix = SumApproximationMatrix(1.0/n)
>>> np.array([matrix.dot(vec)])
array([ 0.43294])
>>> np.set_printoptions(**np_printoptions)

dot(vec)[source]

Form the matrix-vector product sum approximation.

Parameters: vec (ndarray) – The vector in the matrix-vector product. scaled sum (float) – The scaled sum of the input vector.

Notes

This method honors magni.utils.validation.enable_allow_validate_once.

magni.utils.matrices.norm(A, ord=None)[source]

Compute a norm of a matrix.

Efficient norm computations for the magni.utils.matrices matrices.

Parameters: A (magni.util.matrices.Matrix or magni.utils.matrices.MatrixCollection) – The matrix which norm is to be computed. ord (str) – The order of the norm to compute. norm (float) – The computed matrix norm.

numpy.linalg.norm()
Numpy’s function for computing norms of ndarrays

Notes

This function is a simple wrapper around numpy.linalg.norm. It exploits some properties of the matrix classes in magni.utils.matrices to optimize the norm computation in terms of speed and/or memory usage.

Warning

This function builds the ndarray corresponding to the A matrix and passes that on to numpy.linalg.norm if no smarter way of computing the norm has been implemented. Beware of possible “out of memory” issues.

Currently optimized norm computations are:

If the A matrix is a magni.util.matrices.SRM and A.F_norm is not None, the norm computation is accelerated using A.F_norm. If the SRM also includes sub-sampling, it is assumed that the F matrix has entries (or row norms) of the same size as is the case for an orthogonal matrix. Furthermore, it is assumed that any sub-sampling does not include a scaling, i.e. any scaling is not taken into account in computing the norm. If these assumptions about the SRM are not fulfilled, the computed Frobenius norm will only be an approximation to the true Frobernius norm.

Examples

Compute the Frobenius norm of a Separable2DTransform

>>> import numpy as np
>>> from magni.utils.matrices import Separable2DTransform, norm
>>> mtx_l = np.arange(6).reshape(2, 3)
>>> mtx_r = np.arange(40)[::2].reshape(5, 4)
>>> sep_matrix = Separable2DTransform(mtx_l, mtx_r)
>>> round(float(norm(sep_matrix, 'fro')), 2)
737.16


for comparison, the Frobenius computed using np.linalg.norm is

>>> round(float(np.linalg.norm(sep_matrix.A)), 2)
737.16