# Generic solver¶

## Overview¶

On top of the Genred interface, KeOps provides a simple conjugate gradient solver which can be used to solve large-scale Kriging/regression problems on the GPU: the KernelSolve module. Depending on your framework, you may import it using either:

from pykeops.numpy import KernelSolve  # for NumPy users, or...
from pykeops.torch import KernelSolve  # for PyTorch users.


In both cases, KernelSolve is a class with no methods: its instantiation simply returns a numerical function that can be called on arbitrary input tensors.

1. Instantiation: KernelSolve(...) takes as input a bunch of strings that specify the desired computation. It returns a python function or PyTorch layer, callable on numpy arrays or torch tensors. The syntax is:

K_inv = KernelSolve(formula, aliases, varinvalias, alpha=1e-10, axis=0, dtype='float32')

1. Call: The variable K_inv now refers to a callable object wrapped around a set of custom Cuda routines. It may be used on any set of arrays (either NumPy arrays or Torch tensors) with the correct shapes, as described in the aliases argument:

a = K_inv( arg_1, arg_2, ..., arg_p, backend='auto', device_id=-1, ranges=None)


## Documentation¶

See the numpy.KernelSolve or torch.KernelSolve API documentation for the syntax at instantiation and call times.

## An example¶

Using the generic syntax, solving a ridge regression problem with a Cauchy kernel

$\text{for } i = 1, \cdots, 10 000, \quad b_i = 0.1 \cdot a_i +\sum_{j=1}^{10 000} \frac{1}{1+\|x_i-x_j\|^2}\cdot a_j$

with respect to the $$a_i$$’s can be done with:

import torch
from pykeops.torch import Genred, KernelSolve

formula = "Inv(IntCst(1) + SqDist(X,Y)) * A"  # Positive definite Cauchy kernel
aliases = ["X = Vi(3)",  # 1st arg: one point per  line,  in dimension 3
"Y = Vj(3)",  # 2nd arg: one point per column, in dimension 3
"A = Vj(1)"]  # 3rd arg: one scalar per column

K = Genred(formula, aliases, axis=1)        # Sum with respect to "j"
K_inv = KernelSolve(formula, aliases, "A",  # Solve with respect to "A"
axis=1, alpha=.1)       # Add 0.1 to the diagonal of the kernel matrix

# Generate the data as PyTorch tensors:
x = torch.randn(10000, 3)
b = torch.randn(10000, 1)

a = K_inv(x, x, b)  # N.B.: a.shape == [10000, 1]
mean_squared_error = ((K(x, x, a) + .1*a - b)**2).sum().sqrt() / len(x)


More examples can be found in the examples , tutorials and benchmark.