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Introduction to RKeOps

Ghislain Durif

2024-02-13

Source: ci/vignettes/introduction_to_rkeops.Rmd
introduction_to_rkeops.Rmd

Authors

Please contact us for any bug report, question or feature request by filing a report on our GitHub issue tracker!

Core library - KeOps, PyKeOps, KeOpsLab:

R bindings - RKeOps:

  • Amelie Vernay, from the University of Montpellier.
  • Chloe Serre-Combe, from the University of Montpellier.
  • Ghislain Durif, from CNRS.

Contributors:

Beyond explicit code contributions, KeOps has grown out of numerous discussions with applied mathematicians and machine learning experts. We would especially like to thank Alain Trouvé, Stanley Durrleman, Gabriel Peyré and Michael Bronstein for their valuable suggestions and financial support.

Citation

If you use this code in a research paper, please cite our original publication:

Charlier, B., Feydy, J., Glaunès, J. A., Collin, F.-D. & Durif, G. Kernel Operations on the GPU, with Autodiff, without Memory Overflows. Journal of Machine Learning Research 22, 1–6 (2021).

@article{JMLR:v22:20-275,
  author  = {Benjamin Charlier and Jean Feydy and Joan Alexis Glaunès and François-David Collin and Ghislain Durif},
  title   = {Kernel Operations on the GPU, with Autodiff, without Memory Overflows},
  journal = {Journal of Machine Learning Research},
  year    = {2021},
  volume  = {22},
  number  = {74},
  pages   = {1-6},
  url     = {https://jmlr.org/papers/v22/20-275.html}
}

For applications to geometric (deep) learning, you may also consider our NeurIPS 2020 paper:

@article{feydy2020fast,
    title={Fast geometric learning with symbolic matrices},
    author={Feydy, Jean and Glaun{\`e}s, Joan and Charlier, Benjamin and Bronstein, Michael},
    journal={Advances in Neural Information Processing Systems},
    volume={33},
    year={2020}
}

What is RKeOps?

RKeOps is the R package interfacing the KeOps library.

KeOps

Seamless Kernel Operations on GPU (or CPU), with auto-differentiation and without memory overflows

The KeOps library (http://www.kernel-operations.io) provides routines to compute generic reductions of large 2d arrays whose entries are given by a mathematical formula. Using a C++/CUDA-based implementation with GPU support, it combines a tiled reduction scheme with an automatic differentiation engine. Relying on online map-reduce schemes, it is perfectly suited to the scalable computation of kernel dot products and the associated gradients, even when the full kernel matrix does not fit into the GPU memory.

KeOps is all about breaking through this memory bottleneck and making GPU power available for seamless standard mathematical routine computations. As of 2019, this effort has been mostly restricted to the operations needed to implement Convolutional Neural Networks: linear algebra routines and convolutions on grids, images and volumes. KeOps provides CPU and GPU support without the cost of developing a specific CUDA implementation of your custom mathematical operators.

To ensure its versatility, KeOps can be used through Matlab, Python (NumPy or PyTorch) and R back-ends.

RKeOps

RKeOps is a library that can

  • Compute generic reduction (row-wise or column-wise) of very large array/matrices, i.e. \[\sum_{i=1}^M a_{ij} \ \ \ \ \text{or}\ \ \ \ \sum_{j=1}^N a_{ij}\] for some matrix \(A = [a_{ij}]_{M \times N}\) with \(M\) rows and \(N\) columns, whose entries \(a_{ij}\) can be defined with basic math formulae or matrix operators.

  • Compute kernel dot products, i.e. \[\sum_{i=1}^M K(\mathbf x_i, \mathbf y_j)\ \ \ \ \text{or}\ \ \ \ \sum_{j=1}^N K(\mathbf x_i, \mathbf y_j)\] for a kernel function \(K\) and some vectors \(\mathbf x_i\), \(\mathbf y_j\in \mathbb{R}^D\) that are generally rows of some data matrices \(\mathbf X = [x_{ik}]_{M \times D}\) and \(\mathbf Y = [y_{jk}]_{N \times D}\) respectively.

  • Compute the associated gradients

Applications: RKeOps can be used to implement a wide range of problems encountered in machine learning, statistics and more: such as \(k\)-nearest neighbor classification, \(k\)-means clustering, Gaussian-kernel-based problems (e.g. linear system with Ridge regularization), etc.

Why using RKeOps?

RKeOps provides

  • a symbolic computation framework to seamlessly process very large arrays writing R-base-matrix-like operations based on lazy evaluations.

  • an API to create user-defined operators based on generic mathematical formulae, that can be applied to data matrices such as \(\mathbf X = [x_{ik}]_{M \times D}\) and \(\mathbf Y = [y_{jk}]_{N \times D}\).

  • fast computation on GPU without memory overflow, especially to process very large dimensions \(M\) and \(N\) (e.g. \(\approx 10^4\) or \(10^6\)) over indexes \(i\) and \(j\).

  • automatic differentiation and gradient computations for user-defined operators.

Installing and using RKeOps 

# install rkeops
install.packages("rkeops")
# load rkeops
library(rkeops)
# create a dedicated Python environment with reticulate (to be done only once)
reticulate::virtualenv_create("rkeops")
# activate the dedicated Python environment
reticulate::use_virtualenv(virtualenv = "rkeops", required = TRUE)
# install rkeops requirements (to be done only once)
install_rkeops()

For more details, see the specific “Using RKeOps” article or the corresponding vignette:

vignette("using_rkeops", package = "rkeops")

Examples in R

Example in R with LazyTensors

Lazy evaluation allows to write intermediate computations as symbolic operations that are not directly evaluated. The real evaluation is only made on final computation.

To do so, you can use LazyTensors. These are objects wrapped around R matrices or vectors used to create symbolic formulas for the KeOps reduction operations. A typical use case is the following:

Let us say that we want to compute (for all \(j=1,\dots,N\)):

\[ a_j = \sum_{i=1}^{M} \exp\left(-\frac{\Vert \mathbf x_i - \mathbf y_j\Vert^2}{s^2}\right) \]

with

  • parameter: \(s \in\mathbb R\)

  • \(i\)-indexed variables \(\mathbf X = [\mathbf x_i]_{i=1,...,M} \in\mathbb R^{M\times 3}\)

  • \(j\)-indexed variables \(\mathbf Y = [\mathbf y_j]_{j=1,...,N} \in\mathbb R^{N\times 3}\)

The associated code would be:

# to run computation on CPU (default mode)
rkeops_use_cpu()
# OR
# to run computations on GPU (to be used only if relevant)
rkeops_use_gpu()

# Data
M <- 10000
N <- 15000
x <- matrix(runif(N * 3), nrow = M, ncol = 3) # arbitrary R matrix representing 
                                              # 10000 data points in R^3
y <- matrix(runif(M * 3), nrow = N, ncol = 3) # arbitrary R matrix representing 
                                              # 15000 data points in R^3
s <- 0.1                                      # scale parameter

# Turn our Tensors into KeOps symbolic variables:
x_i <- LazyTensor(x, "i")     # symbolic object representing an arbitrary row of x, 
                              # indexed by the letter "i"
y_j <- LazyTensor(y, "j")     # symbolic object representing an arbitrary row of y, 
                              # indexed by the letter "j"

# Perform large-scale computations, without memory overflows:
D_ij <- sum((x_i - y_j)^2)    # symbolic matrix of pairwise squared distances, 
                              # with 10000 rows and 15000 columns

K_ij <- exp(- D_ij / s^2)     # symbolic matrix, 10000 rows and 15000 columns

# D_ij and K_ij are only symbolic at that point, no computation is done

# Computing the result without storing D_ij and K_ij:
a_j <- sum(K_ij, index = "i") # actual R matrix (in fact a row vector of 
                              # length 15000 here)
                              # containing the column sums of K_ij
                              # (i.e. the sums over the "i" index, for each 
                              # "j" index)

More in the dedicated “RKeOps LazyTensor” article or in the corresponding vignette:

vignette("LazyTensor_rkeops", package = "rkeops")

Example in R with generic reductions

We want to implement with RKeOps the following mathematical formula \[\sum_{j=1}^{N} \exp\Big(-\sigma || \mathbf x_i - \mathbf y_j ||_2^{\,2}\Big)\,\mathbf b_j\] with

  • parameter: \(\sigma\in\mathbb R\)

  • \(i\)-indexed variables \(\mathbf X = [\mathbf x_i]_{i=1,...,M} \in\mathbb R^{M\times 3}\)

  • \(j\)-indexed variables \(\mathbf Y = [\mathbf y_j]_{j=1,...,N} \in\mathbb R^{N\times 3}\) and \(\mathbf B = [\mathbf b_j]_{j=1,...,N} \in\mathbb R^{N\times 6}\)

In R, we can define the corresponding KeOps formula as a simple text string:

formula = "Sum_Reduction(Exp(-s * SqNorm2(x - y)) * b, 0)"
  • SqNorm2 = squared \(\ell_2\) norm
  • Exp = exponential
  • Sum_reduction(..., 0) = sum reduction over the dimension 0 i.e. sum on the \(j\)’s (1 to sum over the \(i\)’s)

and the corresponding arguments of the formula, i.e. parameters or variables indexed by \(i\) or \(j\) with their corresponding inner dimensions:

args = c("x = Vi(3)",      # vector indexed by i (of dim 3)
         "y = Vj(3)",      # vector indexed by j (of dim 3)
         "b = Vj(6)",      # vector indexed by j (of dim 6)
         "s = Pm(1)")      # parameter (scalar)

Then we just compile the corresponding operator and apply it to some data

# compilation
op <- keops_kernel(formula, args)
# data and parameter values
nx <- 100
ny <- 150
X <- matrix(runif(nx*3), nrow=nx)   # matrix 100 x 3
Y <- matrix(runif(ny*3), nrow=ny)   # matrix 150 x 3
B <- matrix(runif(ny*6), nrow=ny)   # matrix 150 x 6
s <- 0.2
# computation (order of the input arguments should be similar to `args`)
res <- op(list(X, Y, B, s))

CPU and GPU computing

Based on your LazyTensor-based operations or formulae, RKeOps compiles on the fly operators that can be used to run the corresponding computations on CPU or GPU, it uses a tiling scheme to decompose the data and avoid (i) useless and costly memory transfers between host and GPU (performance gain) and (ii) memory overflow.

Note: You can use the same code (i.e. define the same operators) for CPU or GPU computing. The only difference will be the compiler used for the compilation of your operators (upon the availability of CUDA on your system).

To use CPU computing mode, you can call rkeops_use_cpu() (with an optional argument ncore specifying the number of cores used to run parallel computations).

To use GPU computing mode, you can call rkeops_use_gpu() (with an optional argument device to choose a specific GPU id to run computations).

Matrix reduction and kernel operator

The general framework of RKeOps (and KeOps) is to provide fast and scalable matrix operations on GPU, in particular kernel-based computations of the form \[\underset{i=1,...,M}{\text{reduction}}\ G(\boldsymbol{\sigma}, \mathbf x_i, \mathbf y_j) \ \ \ \ \text{or}\ \ \ \ \underset{j=1,...,N}{\text{reduction}}\ G(\boldsymbol{\sigma}, \mathbf x_i, \mathbf y_j)\] where

  • \(\boldsymbol{\sigma}\in\mathbb R^L\) is a vector of parameters

  • \(\mathbf x_i\in \mathbb{R}^D\) and \(\mathbf y_j\in \mathbb{R}^{D'}\) are two vectors of data (potentially with different dimensions)

  • \(G: \mathbb R^L \times \mathbb R^D \times \mathbb R^{D'} \to \mathbb R\) is a function of the data and the parameters, that can be expressed through a composition of generic operators

  • \(\text{reduction}\) is a generic reduction operation over the index \(i\) or \(j\) (e.g. sum)

RKeOps creates (and compiles on the fly) an operator implementing your formula. You can apply it to your data, or compute its gradient regarding some data points.

Note: You can use a wide range of reduction such as sum, min, argmin, max, argmax, etc.

What you need to do

To use RKeOps you only need to express your computations as a formula with the previous form.

RKeOps allows to use a wide range of mathematical functions to define your operators (see https://www.kernel-operations.io/keops/api/math-operations.html).

You can use two type of input matrices with RKeOps:

  • ones whose rows (or columns) are indexed by \(i=1,...,M\) such as \(\mathbf X = [x_{ik}]_{M \times D}\)

  • others whose rows (or columns) are indexed by \(j=1,...,N\) such as \(\mathbf Y = [y_{ik'}]_{N \times D'}\)

More details about input matrices (size, storage order) are given in the “Using RKeOps” article or the corresponding vignette::

vignette("using_rkeops", package = "rkeops")

Generic kernel function

With RKeOps, you can define kernel functions \(K: \mathbb R^D \times \mathbb R^D \to \mathbb R\) such as, for some vectors \(\mathbf x_i\), \(\mathbf y_j\in \mathbb{R}^D\)

  • the linear kernel (standard scalar product) \(K(\mathbf x_i, \mathbf y_j) = \big\langle \mathbf x_i \, , \, \mathbf y_j \big\rangle\)

  • the Gaussian kernel \(K(\mathbf x_i, \mathbf y_j) = \exp\left(-\frac{1}{2\sigma^2} \Vert \mathbf x_i - \mathbf y_j \Vert_2^{\,2}\right)\) with \(\sigma>0\)

  • and more…

Then you can compute reductions based on such functions, especially when the \(M \times N\) matrix \(\mathbf K = [K(\mathbf x_i, \mathbf y_j)]\) is too large to fit into memory, such as

  • Kernel reduction: \[\sum_{i=1}^M K(\mathbf x_i, \mathbf y_j)\ \ \ \ \text{or}\ \ \ \ \sum_{j=1}^N K(\mathbf x_i, \mathbf y_j)\]

  • Convolution-like operations: \[\sum_{i=1}^M K(\mathbf x_i, \mathbf y_j)\boldsymbol\beta_j\ \ \ \ \text{or}\ \ \ \ \sum_{j=1}^N K(\mathbf x_i, \mathbf y_j)\boldsymbol\beta_j\] for some vectors \((\boldsymbol\beta_j)_{j=1,...,N} \in \mathbb R^{N\times D}\)

  • More complex operations: \[\sum_{i=1}^{M}\, K_1(\mathbf x_i, \mathbf y_j)\, K_2(\mathbf u_i, \mathbf v_j)\,\langle \boldsymbol\alpha_i\, ,\,\boldsymbol\beta_j\rangle \ \ \ \ \text{or}\ \ \ \ \sum_{j=1}^{N}\, K_1(\mathbf x_i, \mathbf y_j)\, K_2(\mathbf u_i, \mathbf v_j)\,\langle \boldsymbol\alpha_i\, ,\,\boldsymbol\beta_j\rangle\] for some kernels \(K_1\) and \(K_2\), and some \(D\)-vectors \((\mathbf x_i)_{i=1,...,M}, (\mathbf u_i)_{i=1,...,M}, (\boldsymbol\alpha_i)_{i=1,...,M} \in \mathbb R^{M\times D}\) and \((\mathbf y_j)_{j=1,...,N}, (\mathbf v_j)_{j=1,...,N}, (\boldsymbol\beta_j)_{j=1,...,N} \in \mathbb R^{N\times D}\).

Examples/Tutorials/Benchmarks

Using RKeOps

See this article or the corresponding vignette:

vignette("using_rkeops", package = "rkeops")

LazyTensors

See this article or the corresponding vignette:

vignette("LazyTensor_rkeops", package = "rkeops")

Kernel interpolation

See this article or the corresponding vignette:

vignette("kernel_interpolation_rkeops", package = "rkeops")

Benchmarks

See the corresponding directory rkeops/benchmarks in the project source repository.