LinearOperator

LinearOperator is a PyTorch package for abstracting away the linear algebra routines needed for structured matrices (or operators).

This package is in beta. Currently, most of the functionality only supports positive semi-definite and triangular matrices. Package development TODOs:

• [x] Support PSD operators
• [x] Support triangular operators
• [ ] Interface to specify structure (i.e. symmetric, triangular, PSD, etc.)
• [ ] Add algebraic routines for symmetric operators
• [ ] Add algebraic routines for generic square operators
• [ ] Add algebraic routines for generic rectangular operators
• [ ] Add sparse operators

To get started, run either

pip install linear_operator
# or
conda install linear_operator -c gpytorch

or see below for more detailed instructions.

Why LinearOperator

Before describing what linear operators are and why they make a useful abstraction, it's easiest to see an example. Let's say you wanted to compute a matrix solve:

$$\boldsymbol A^{-1} \boldsymbol b.$$

If you didn't know anything about the matrix $\boldsymbol A$, the simplest (and best) way to accomplish this in code is:

# A = torch.randn(1000, 1000)
# b = torch.randn(1000)
torch.linalg.solve(A, b)  # computes A^{-1} b

While this is easy, the solve routine is $\mathcal O(N^3)$, which gets very slow as $N$ grows large.

However, let's imagine that we knew that $\boldsymbol A$ was equal to a low rank matrix plus a diagonal (i.e. $\boldsymbol A = \boldsymbol C \boldsymbol C^\top + \boldsymbol D$ for some skinny matrix $\boldsymbol C$ and some diagonal matrix $\boldsymbol D$.) There's now a very efficient $\boldsymbol O(N)$ routine to compute $\boldsymbol A^{-1}$ (the Woodbury formula). In general, if we know that $\boldsymbol A$ has structure, we want to use efficient linear algebra routines - rather than the general routines - that exploit this structure.

Without LinearOperator

Implementing the efficient solve that exploits $\boldsymbol A$'s low-rank-plus-diagonal structure would look something like this:

def low_rank_plus_diagonal_solve(C, d, b):
    # A = C C^T + diag(d)
    # A^{-1} b = D^{-1} b - D^{-1} C (I + C^T D^{-1} C)^{-1} C^T D^{-1} b
    #   where D = diag(d)

    D_inv_b = b / d
    D_inv_C = C / d.unsqueeze(-1)
    eye = torch.eye(C.size(-2))
    return (
        D_inv_b - D_inv_C @ torch.cholesky_solve(
            C.mT @ D_inv_b,
            torch.linalg.cholesky(eye + C.mT @ D_inv_C, upper=False),
            upper=False
        )
    )


# C = torch.randn(1000, 20)
# d = torch.randn(1000)
# b = torch.randn(1000)
low_rank_plus_diagonal_solve(C, d, b)  # computes A^{-1} b in O(N) time, instead of O(N^3)

While this is efficient code, it's not ideal for a number of reasons:

1. It's a lot more complicated than torch.linalg.solve(A, b).
2. There's no object that represents $\boldsymbol A$. To perform any math with $\boldsymbol A$, we have to pass around the matrix C and the vector d.

With LinearOperator

The LinearOperator package offers the best of both worlds:

from linear_operator.operators import DiagLinearOperator, LowRankRootLinearOperator
# C = torch.randn(1000, 20)
# d = torch.randn(1000)
# b = torch.randn(1000)
A = LowRankRootLinearOperator(C) + DiagLinearOperator(d)  # represents C C^T + diag(d)

it provides an interface that lets us treat $\boldsymbol A$ as if it were a generic tensor, using the standard PyTorch API:

torch.linalg.solve(A, b)  # computes A^{-1} b efficiently!

Under-the-hood, the LinearOperator object keeps track of the algebraic structure of $\boldsymbol A$ (low rank plus diagonal) and determines the most efficient routine to use (the Woodbury formula). This way, we can get a efficient $\mathcal O(N)$ solve while abstracting away all of the details.

Crucially, $\boldsymbol A$ is never explicitly instantiated as a matrix, which makes it possible to scale to very large operators without running out of memory:

# C = torch.randn(10000000, 20)
# d = torch.randn(10000000)
# b = torch.randn(10000000)
A = LowRankRootLinearOperator(C) + DiagLinearOperator(d)  # represents a 10M x 10M matrix!
torch.linalg.solve(A, b)  # computes A^{-1} b efficiently!

What is a Linear Operator?

A linear operator is a generalization of a matrix. It is a linear function that is defined in by its application to a vector. The most common linear operators are (potentially structured) matrices, where the function applying them to a vector are (potentially efficient) matrix-vector multiplication routines.

In code, a LinearOperator is a class that

1. specifies the tensor(s) needed to define the LinearOperator,
2. specifies a _matmul function (how the LinearOperator is applied to a vector),
3. specifies a _size function (how big is the LinearOperator if it is represented as a matrix, or batch of matrices), and
4. specifies a _transpose_nonbatch function (the adjoint of the LinearOperator).
5. (optionally) defines other functions (e.g. logdet, eigh, etc.) to accelerate computations for which efficient sturcture-exploiting routines exist.

For example:

class DiagLinearOperator(linear_operator.LinearOperator):
    r"""
    A LinearOperator representing a diagonal matrix.
    """
    def __init__(self, diag):
        # diag: the vector that defines the diagonal of the matrix
        self.diag = diag

    def _matmul(self, v):
        return self.diag.unsqueeze(-1) * v

    def _size(self):
        return torch.Size([*self.diag.shape, self.diag.size(-1)])

    def _transpose_nonbatch(self):
        return self  # Diagonal matrices are symmetric

    # this function is optional, but it will accelerate computation
    def logdet(self):
        return self.diag.log().sum(dim=-1)
# ...

D = DiagLinearOperator(torch.tensor([1., 2., 3.])
# Represents the matrix
#   [[1., 0., 0.],
#    [0., 2., 0.],
#    [0., 0., 3.]]
torch.matmul(D, torch.tensor([4., 5., 6.])
# Returns [4., 10., 18.]

While _matmul, _size, and _transpose_nonbatch might seem like a limited set of functions, it turns out that most functions on the torch and torch.linalg namespaces can be efficiently implemented using only these three primitative functions.

Moreover, because _matmul is a linear function, it is very easy to compose linear operators in various ways. For example: adding two linear operators (SumLinearOperator) just requires adding the output of their _matmul functions. This makes it possible to define very complex compositional structures that still yield efficient linear algebraic routines.

Finally, LinearOperator objects can be composed with one another, yielding new LinearOperator objects and automatically keeping track of algebraic structure after each computation. As a result, users never need to reason about what efficient linear algebra routines to use (so long as the input elements defined by the user encode known input structure).

See the using LinearOperator objects section for more details.

Use Cases

There are several use cases for the LinearOperator package. Here we highlight two general themes:

Modular Code for Structured Matrices

For example, let's say that you have a generative model that involves sampling from a high-dimensional multivariate Gaussian. This sampling operation will require storing and manipulating a large covariance matrix, so to speed things up you might want to experiment with different structured approximations of that covariance matrix. This is easy with the LinearOperator package.

from gpytorch.distributions import MultivariateNormal

# variance = torch.randn(10000)
cov = DiagLinearOperator(variance)
# or
# cov = LowRankRootLinearOperator(...) + DiagLinearOperator(...)
# or
# cov = KroneckerProductLinearOperator(...)
# or
# cov = ToeplitzLinearOperator(...)
# or
# ...

mvn = MultivariateNormal(torch.zeros(cov.size(-1), cov) # 10000-dimensional MVN
mvn.rsample()  # returns a 10000-dimensional vector

Efficient Routines for Complex Operators

Many of the efficient linear algebra routines in LinearOperator are iterative algorithms based on matrix-vector multiplication. Since matrix-vector multiplication obeys many nice compositional properties it is possible to obtain efficient routines for extremely complex compositional LienarOperators:

from linear_operator.operators import KroneckerProductLinearOperator, RootLinearOperator, ToeplitzLinearOperator

# mat1 = 200 x 200 PSD matrix
# mat2 = 100 x 100 PSD matrix
# vec3 = 20000 vector

A = KroneckerProductLinearOperator(mat1, mat2) + RootLinearOperator(ToeplitzLinearOperator(vec3))
# represents a 20000 x 20000 matrix

torch.linalg.solve(A, torch.randn(20000))  # Sub O(N^3) routine!

Using LinearOperator Objects

LinearOperator objects share (mostly) the same API as torch.Tensor objects. Under the hood, these objects use __torch_function__ to dispatch all efficient linear algebra operations to the torch and torch.linalg namespaces. This includes

• torch.add
• torch.cat
• torch.clone
• torch.diagonal
• torch.dim
• torch.div
• torch.expand
• torch.logdet
• torch.matmul
• torch.numel
• torch.permute
• torch.prod
• torch.squeeze
• torch.sub
• torch.sum
• torch.transpose
• torch.unsqueeze
• torch.linalg.cholesky
• torch.linalg.eigh
• torch.linalg.eigvalsh
• torch.linalg.solve
• torch.linalg.svd

Each of these functions will either return a torch.Tensor, or a new LinearOperator object, depending on the function. For example:

# A = RootLinearOperator(...)
# B = ToeplitzLinearOperator(...)
# d = vec

C = torch.matmul(A, B)  # A new LienearOperator representing the product of A and B
torch.linalg.solve(C, d)  # A torch.Tensor

For more examples, see the examples folder.

Batch Support and Broadcasting

LinearOperator objects operate naturally in batch mode. For example, to represent a batch of 3 100 x 100 diagonal matrices:

# d = torch.randn(3, 100)
D = DiagLinearOperator(d)  # Reprents an operator of size 3 x 100 x 100

These objects fully support broadcasted operations:

D @ torch.randn(100, 2)  # Returns a tensor of size 3 x 100 x 2

D2 = DiagLinearOperator(torch.randn([2, 1, 100]))  # Represents an operator of size 2 x 1 x 100 x 100
D2 + D  # Represents an operator of size 2 x 3 x 100 x 100

Indexing

LinearOperator objects can be indexed in ways similar to torch Tensors. This includes:

• Integer indexing (get a row, column, or batch)
• Slice indexing (get a subset of rows, columns, or batches)
• LongTensor indexing (get a set of individual entries by index)
• Ellipses (support indexing operations with arbitrary batch dimensions)

D = DiagLinearOperator(torch.randn(2, 3, 100))  # Represents an operator of size 2 x 3 x 100 x 100
D[-1]  # Returns a 3 x 100 x 100 operator
D[..., :10, -5:]  # Returns a 2 x 3 x 10 x 5 operator
D[..., torch.LongTensor([0, 1, 2, 3]), torch.LongTensor([0, 1, 2, 3])]  # Returns a 2 x 3 x 4 tensor

Composition and Decoration

LinearOperators can be composed with one another in various ways. This includes

• Addition (LinearOpA + LinearOpB)
• Matrix multiplication (LinearOpA @ LinearOpB)
• Concatenation (torch.cat([LinearOpA, LinearOpB], dim=-2))
• Kronecker product (torch.kron(LinearOpA, LinearOpB))

In addition, there are many ways to "decorate" LinearOperator objects. This includes:

• Elementwise multiplying by constants (torch.mul(2., LinearOpA))
• Summing over batches (torch.sum(LinearOpA, dim=-3))
• Elementwise multiplying over batches (torch.prod(LinearOpA, dim=-3))

See the documentation for a full list of supported composition and decoration operations.

Installation

LinearOperator requires Python >= 3.8.

Standard Installation (Most Recent Stable Version)

We recommend installing via pip or Anaconda:

pip install linear_operator
# or
conda install linear_operator -c gpytorch

The installation requires the following packages:

• PyTorch >= 1.11
• Scipy

You can customize your PyTorch installation (i.e. CUDA version, CPU only option) by following the PyTorch installation instructions.

Installing from the main Branch (Latest Unsable Version)

To install what is currently on the main branch (potentially buggy and unstable):

pip install --upgrade git+https://github.com/cornellius-gp/linear_operator.git

Development Installation

If you are contributing a pull request, it is best to perform a manual installation:

git clone https://github.com/cornellius-gp/linear_operator.git
cd linear_operator
pip install -e ".[dev,docs,test]"

Contributing

See the contributing guidelines CONTRIBUTING.md for information on submitting issues and pull requests.

License

LinearOperator is MIT licensed.