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Structured Matrices

Structured matrices

Requirements and Installation

See the instructions here. Then simply

pip install backends-matrix

Example

>>> import lab as B

>>> from matrix import Diagonal

>>> d = Diagonal(B.rand(2, 3))  # A batch of diagonal marices

>>> d
<diagonal matrix: batch=(2,), shape=(3, 3), dtype=float64
 diag=[[0.427 0.912 0.622]
       [0.777 0.048 0.808]]>

>>> 2 * d
<diagonal matrix: batch=(2,), shape=(3, 3), dtype=float64
 diag=[[0.854 1.824 1.243]
       [1.553 0.096 1.616]]>
  
>>> 2 * d + 1
<Woodbury matrix: batch=(2,), shape=(3, 3), dtype=float64
 diag=<diagonal matrix: batch=(2,), shape=(3, 3), dtype=float64
       diag=[[0.854 1.824 1.243]
             [1.553 0.096 1.616]]>
 lr=<low-rank matrix: batch=(), shape=(3, 3), dtype=int64, rank=1
     left=[[1]
           [1]
           [1]]
     middle=<diagonal matrix: batch=(), shape=(1, 1), dtype=int64
             diag=[1]>>>

>>> B.inv(2 * d + 1)
<Woodbury matrix: batch=(2,), shape=(3, 3), dtype=float64
 diag=<diagonal matrix: batch=(2,), shape=(3, 3), dtype=float64
       diag=[[ 1.171  0.548  0.804]
             [ 0.644 10.386  0.619]]>
 lr=<low-rank matrix: batch=(2,), shape=(3, 3), dtype=float64, rank=1
     left=<dense matrix: batch=(2,), shape=(3, 1), dtype=float64
           mat=[[[ 1.171]
                 [ 0.548]
                 [ 0.804]]

                [[ 0.644]
                 [10.386]
                 [ 0.619]]]>
     middle=<dense matrix: batch=(2,), shape=(1, 1), dtype=float64
             mat=[[[-0.284]]

                  [[-0.079]]]>
     right=<dense matrix: batch=(2,), shape=(3, 1), dtype=float64
            mat=[[[ 1.171]
                  [ 0.548]
                  [ 0.804]]

                 [[ 0.644]
                  [10.386]
                  [ 0.619]]]>>>

>>> B.inv(B.inv(2 * d + 1))
<Woodbury matrix: batch=(2,), shape=(3, 3), dtype=float64
 diag=<diagonal matrix: batch=(2,), shape=(3, 3), dtype=float64
       diag=[[0.854 1.824 1.243]
             [1.553 0.096 1.616]]>
 lr=<low-rank matrix: batch=(2,), shape=(3, 3), dtype=float64, rank=1
     left=<dense matrix: batch=(2,), shape=(3, 1), dtype=float64
           mat=[[[1.]
                 [1.]
                 [1.]]

                [[1.]
                 [1.]
                 [1.]]]>
     middle=<dense matrix: batch=(2,), shape=(1, 1), dtype=float64
             mat=[[[1.]]

                  [[1.]]]>
     right=<dense matrix: batch=(2,), shape=(3, 1), dtype=float64
            mat=[[[1.]
                  [1.]
                  [1.]]

                 [[1.]
                  [1.]
                  [1.]]]>>>

>>> B.inv(B.inv(2 * d + 1)) - 1
<diagonal matrix: batch=(2,), shape=(3, 3), dtype=float64
 diag=[[0.854 1.824 1.243]
       [1.553 0.096 1.616]]>

Matrix Types

All matrix types are subclasses of AbstractMatrix.

The following base types are provided:

Zero
Dense
Diagonal
Constant
LowerTriangular
UpperTriangular

The following composite types are provided:

LowRank
Woodbury
Kronecker
TiledBlocks

Functions

The following functions are added to LAB. They can be accessed with B.<function> where import lab as B.

shape_broadcast(*elements)
shape_batch(a, *indices)
shape_batch_broadcast(*elements)
shape_matrix(a, *indices)
shape_matrix_broadcast(*elements)

broadcast_batch_to(a, *batch)

dense(a)
fill_diag(a, diag_len)
block(*rows)
block_diag(*blocks)

matmul_diag(a, b, tr_a=False, tr_b=False)

pd_inv(a)
schur(a)
pd_schur(a)
iqf(a, b, c)
iqf_diag(a, b, c)

ratio(a, c)
root(a)

sample(a, num=1)