# Linear Operators¶

Finite dimensional linear operators allow matrix algebra without explicitly constructing a full matrix representation. Instead it suffices to define a matrix-vector product and a shape attribute. This avoids unnecessary memory usage and can often be more convenient to derive.

:

# Make inline plots vector graphics instead of raster graphics
%matplotlib inline
from IPython.display import set_matplotlib_formats

set_matplotlib_formats("pdf", "svg")

# Plotting
import matplotlib.pyplot as plt

plt.style.use("../../probnum.mplstyle")


## Sparsity and Operator Arithmetic¶

We begin by sampling a random sparse matrix, represent it as a LinearOperator and perform some simple arithmetic.

:

import numpy as np
import scipy.sparse
from probnum import linops

# Linear operator from sparse matrix
n = 50
mat = scipy.sparse.rand(m=n, n=n, density=0.05, random_state=42)
A = linops.Matrix(A=mat)

# Linear operator arithmetic
Id = linops.Identity(shape=n)
B = A + 1.5 * Id

:

# Plot
matdict = {"$A$": A.todense(), "$B$": B.todense()}
vmin = np.min([np.min(mat) for mat in list(matdict.values())])
vmax = np.max([np.max(mat) for mat in list(matdict.values())])

fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(6, 3.5))
for i, (title, mat) in enumerate(matdict.items()):
axes[i].imshow(mat, vmin=vmin, vmax=vmax)
axes[i].set_axis_off()
axes[i].title.set_text(title)
fig.tight_layout() ## The Kronecker Product¶

A frequently occuring linear operator is the Kronecker product of two matrices. Example application areas are matrix-variate distributions, signal processing, image processing, semi-definite programming and deep learning. We will illustrate some basic properties here.

:

# Kronecker Product
A = np.array([[4, 1, 4], [2, 3, 2]])
B = np.array([, ])
W = linops.Kronecker(A, B)
V = linops.Kronecker(B, A)

:

# Plot
matdict = {
"$A$": A,
"$B$": B,
"$A \otimes B$": W.todense(),
"$B \otimes A$": V.todense(),
}
vmin = np.min([np.min(mat) for mat in list(matdict.values())])
vmax = np.max([np.max(mat) for mat in list(matdict.values())])

fig, axes = plt.subplots(
nrows=1, ncols=4, figsize=(6, 3.5), gridspec_kw={"width_ratios": [3, 1, 3, 3]}
)
for i, (title, mat) in enumerate(matdict.items()):
axes[i].imshow(mat, vmin=vmin, vmax=vmax)
axes[i].set_axis_off()
axes[i].title.set_text(title)
fig.tight_layout() The elements of the Kronecker product $$A \otimes B$$ consist of all possible products of an entry of $$B \in \mathbb{R}^{m_2 \times n_2}$$ with an entry of $$A \in \mathbb{R}^{m_1 \times n_1}$$. If this was represented as a full matrix instead of a linear operator, one would have to store and multiply with a matrix of dimensions $$m_1m_2 \times n_1n_2$$. However, by just defining the matrix-vector product instead, it suffices to store the matrices $$A$$ and $$B$$.

A useful relative of the Kronecker product is the symmetric Kronecker product. As we will see later it can be used to define a distribution over symmetric matrices.

:

# Symmetric Kronecker Product
C = np.array([[5, 1], [2, 10]])
D = np.array([[1, 2], [3, 4]])
Ws = linops.SymmetricKronecker(C, D)
Vs = linops.SymmetricKronecker(D, C)

:

# Plot
matdict = {
"$C$": C,
"$D$": D,
"$C \otimes_s D$": Ws.todense(),
"$D \otimes_s C$": Vs.todense(),
}
vmin = np.min([np.min(mat) for mat in list(matdict.values())])
vmax = np.max([np.max(mat) for mat in list(matdict.values())])

fig, axes = plt.subplots(
nrows=1, ncols=4, figsize=(6, 3.5), gridspec_kw={"width_ratios": [2, 2, 3, 3]}
)
for i, (title, mat) in enumerate(matdict.items()):
axes[i].imshow(mat, vmin=vmin, vmax=vmax)
axes[i].set_axis_off()
axes[i].title.set_text(title)
fig.tight_layout() Note, that the symmetric Kronecker product generally does not have a symmetric matrix representation. If its arguments are symmetric, then so is their product. We also observe another property of the symmetric Kronecker product, namely that $$C \otimes_s D = D \otimes_s C$$. This is not the case for the Kronecker product as we saw above.

## Linear Operators and Random Variables¶

When defining distributions over $$m \times n$$ matrices, the covariance has dimension $$(mn)^2$$, which quickly becomes prohibitively large. This can be somewhat remedied by using linear operators as parameters, since for most computations with matrix-variate random variables the full covariance matrix never needs to be formed explicitly. An example is the matrix-variate normal distribution, where one often assumes Kronecker-structured covariances. One can use the symmetric Kronecker product in this context to define a distribution over symmetric matrices.

:

import scipy.sparse
from probnum import randvars

import probnum.problems.zoo.linalg

# Linear operators as random variable parameters
n = 15
rng = np.random.default_rng(seed=42)
mat = probnum.problems.zoo.linalg.random_spd_matrix(rng=rng, dim=n)

opmean = linops.Matrix(A=mat)
opcov = linops.SymmetricKronecker(2 * linops.Identity(n))

Y = randvars.Normal(mean=opmean, cov=opcov)
print(Y)

# Draw samples
Ysamples = Y.sample(rng=rng, size=3)

<Normal with shape=(15, 15), dtype=float64>

:

# Plot
rvdict = {
"$\mathbb{E}(\mathsf{Y})$": Y.mean.todense(),
"$\mathsf{Y}_1$": Ysamples,
"$\mathsf{Y}_2$": Ysamples,
"$\mathsf{Y}_3$": Ysamples,
}
vmin = np.min([np.min(mat) for mat in list(rvdict.values())])
vmax = np.max([np.max(mat) for mat in list(rvdict.values())])

fig, axes = plt.subplots(nrows=1, ncols=4, figsize=(10, 5), sharey=True)
for i, (title, rv) in enumerate(rvdict.items()):
axes[i].imshow(rv, vmin=vmin, vmax=vmax)
axes[i].set_axis_off()
axes[i].title.set_text(title)
plt.tight_layout() [ ]: