Compositional Linear Algebra (CoLA)

CoLA is a framework for scalable linear algebra, automatically exploiting the structure often found in machine learning problems and beyond. CoLA supports both PyTorch and JAX.

Installation

pip install cola-ml

Features in CoLA

Large scale linear algebra routines for solve(A,b), eig(A), logdet(A), exp(A), trace(A), diag(A), sqrt(A).
Provides (user extendible) compositional rules to exploit structure through multiple dispatch.
Has memory-efficient autodiff rules for iterative algorithms.
Works with PyTorch or JAX, supporting GPU hardware acceleration.
Supports operators with complex numbers and low precision.
Provides linear algebra operations for both symmetric and non-symmetric matrices.

See https://cola.readthedocs.io/en/latest/ for our full documentation and many examples.

Quick start guide

LinearOperators. The core object in CoLA is the LinearOperator. You can add and subtract them +, -, multiply by constants *, /, matrix multiply them @ and combine them in other ways: kron, kronsum, block_diag etc.

import jax.numpy as jnp
import cola

A = cola.ops.Diagonal(jnp.arange(5) + .1)
B = cola.ops.Dense(jnp.array([[2., 1.], [-2., 1.1], [.01, .2]]))
C = B.T @ B
D = C + 0.01 * cola.ops.I_like(C)
E = cola.ops.Kronecker(A, cola.ops.Dense(jnp.ones((2, 2))))
F = cola.ops.BlockDiag(E, D)

v = jnp.ones(F.shape[-1])
print(F @ v)

[0.2       0.2       2.2       2.2       4.2       4.2       6.2
 6.2       8.2       8.2       7.8121004 2.062    ]

Performing Linear Algebra. With these objects we can perform linear algebra operations even when they are very big.

print(cola.linalg.trace(F))
Q = F.T @ F + 1e-3 * cola.ops.I_like(F)
b = cola.linalg.inv(Q) @ v
print(jnp.linalg.norm(Q @ b - v))
print(cola.linalg.eig(F)[0][:5])
print(cola.sqrt(A))

31.2701
0.0010193728
[ 2.0000000e-01+0.j  0.0000000e+00+0.j  2.1999998e+00+0.j
 -1.1920929e-07+0.j  4.1999998e+00+0.j]
diag([0.31622776 1.0488088  1.4491377  1.7606816  2.0248456 ])

For many of these functions, if we know additional information about the matrices we can annotate them to enable the algorithms to run faster.

Qs = cola.SelfAdjoint(Q)
%timeit cola.linalg.inv(Q) @ v
%timeit cola.linalg.inv(Qs) @ v

JAX and PyTorch. We support both ML frameworks.

import torch

A = cola.ops.Dense(torch.Tensor([[1., 2.], [3., 4.]]))
print(cola.linalg.trace(cola.kron(A, A)))

import jax.numpy as jnp
A = cola.ops.Dense(jnp.array([[1., 2.], [3., 4.]]))
print(cola.linalg.trace(cola.kron(A, A)))

tensor(25.)
25.0

and both support autograd (and jit):

from jax import grad, jit, vmap

def myloss(x):
    A = cola.ops.Dense(jnp.array([[1., 2.], [3., x]]))
    return jnp.ones(2) @ cola.linalg.inv(A) @ jnp.ones(2)


g = jit(vmap(grad(myloss)))(jnp.array([.5, 10.]))
print(g)

[-0.06611571 -0.12499995]

Citing us

If you use CoLA, please cite the following paper:

Andres Potapczynski, Marc Finzi, Geoff Pleiss, and Andrew Gordon Wilson. "CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebra." 2023.

@article{potapczynski2023cola,
  title={{CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebra}},
  author={Andres Potapczynski and Marc Finzi and Geoff Pleiss and Andrew Gordon Wilson},
  journal={arXiv preprint arXiv:2309.03060},
  year={2023}
}

Features implemented

Linear Algebra	inverse	eig	diag	trace	logdet	exp	sqrt	f(A)	SVD	pseudoinverse
Implementation	✓	✓	✓	✓	✓	✓	✓	✓

LinearOperators	Diag	BlockDiag	Kronecker	KronSum	Sparse	Jacobian	Hessian	Fisher	Concatenated	Triangular	FFT	Tridiagonal
Implementation	✓	✓	✓	✓	✓	✓	✓		✓	✓		✓

Annotations	SelfAdjoint	PSD	Unitary
Implementation	✓	✓	✓

Contributing

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

CoLA is Apache 2.0 licensed.

Support and contact

Please raise an issue if you find a bug or slow performance when using CoLA.

akshat01112001/cola