Scaling up Gaussian convolutions on 3D point clouds

Let’s compare the performances of PyTorch and KeOps on simple Gaussian RBF kernel products, as the number of samples grows from 100 to 1,000,000.

Note

In this demo, we use exact bruteforce computations (tensorized for PyTorch and online for KeOps), without leveraging any multiscale or low-rank (multipole) decomposition of the Kernel matrix. Please visit the documentation of the GeomLoss package for a discussion of clever, scalable schemes.

Setup

import importlib
import os
import time

import numpy as np
import torch
from matplotlib import pyplot as plt

use_cuda = torch.cuda.is_available()

Benchmark specifications:

D  = 3        # Let's do this in 3D
MAXTIME = 10 if use_cuda else 1   # Max number of seconds before we break the loop
REDTIME = 2  if use_cuda else .2  # Decrease the number of runs if computations take longer than 2s...

# Number of samples that we'll loop upon
NS = [100, 200, 500,
      1000, 2000, 5000,
      10000, 20000, 50000,
      100000, 200000, 500000,
      1000000]

Synthetic dataset. Feel free to use a Stanford Bunny, or whatever!

def generate_samples(N, device, lang, batchsize=None):
    """Create point clouds sampled non-uniformly on a sphere of diameter 1."""

    B = () if batchsize is None else (batchsize,)

    if lang == 'torch':
        if device == 'cuda':
            torch.cuda.manual_seed_all(1234)
        else:
            torch.manual_seed(1234)

        x = torch.randn(B + (N, D), device=device)
        x[:,0] += 1
        x  = x / (2*x.norm(dim=1,keepdim=True))

        y = torch.randn(B + (N, D), device=device)
        y[:,1] += 2
        y  = y / (2*y.norm(dim=1,keepdim=True))

        # Draw a random source signal:
        b = torch.randn(B + (N, 1), device=device)

    else:
        np.random.seed(1234)

        x = np.random.rand(*(B + (N, D))).astype('float32')
        y = np.random.rand(*(B + (N, D))).astype('float32')
        b = np.random.randn(*(B + (N,))).astype('float32')

    return x, y, b

Define a simple Gaussian RBF product, using a tensorized implementation:

def gaussianconv_numpy(x, y, b):
    K_xy = np.exp( - np.sum( (x[:,None,:] - y[None,:,:]) **2, axis=2) )

    return K_xy@b


def gaussianconv_pytorch(x, y, b):
    D_xx = (x*x).sum(-1).unsqueeze(1)         # (N,1)
    D_xy = torch.matmul( x, y.permute(1,0) )  # (N,D) @ (D,M) = (N,M)
    D_yy = (y*y).sum(-1).unsqueeze(0)         # (1,M)
    D_xy = D_xx - 2*D_xy + D_yy
    K_xy = (-D_xy).exp()

    return K_xy @ b

Define a simple Gaussian RBF product, using an online implementation:

from pykeops.torch import generic_sum

gaussianconv_keops = generic_sum("Exp(-SqDist(X,Y)) * B",  # Formula
                                 "A = Vi(1)",              # Output
                                 "X = Vi({})".format(D),   # 1st argument
                                 "Y = Vj({})".format(D),   # 2nd argument
                                 "B = Vj(1)" )             # 3rd argument

Finally, perform the same operation with our high-level pykeops.torch.LazyTensor wrapper:

from pykeops.torch import LazyTensor


def gaussianconv_lazytensor(x, y, b):
    nbatchdims = len(x.shape) - 2
    x_i = LazyTensor(x.unsqueeze(-2))  # (B, M, 1, D)
    y_j = LazyTensor(y.unsqueeze(-3))  # (B, 1, N, D)
    D_ij = ((x_i - y_j) ** 2).sum(-1)  # (B, M, N, 1)
    K_ij = (- D_ij).exp()  # (B, M, N, 1)
    S_ij = K_ij * b.unsqueeze(-3)  # (B, M, N, 1) * (B, 1, N, 1)
    return S_ij.sum(dim=nbatchdims + 1)

Benchmarking loops

def benchmark(routine_batchsize, dev, N, loops=10, lang='torch'):
    """Times a convolution on an N-by-N problem."""

    if isinstance(routine_batchsize, tuple):
        Routine, B = routine_batchsize
    else:
        Routine, B = routine_batchsize, None

    importlib.reload(torch)  # In case we had a memory overflow just before...
    device = torch.device(dev)
    x, y, b = generate_samples(N, device, lang, batchsize=B)

    # We simply benchmark a convolution
    code = "a = Routine( x, y, b ) "
    exec( code, locals() ) # Warmup run, to compile and load everything

    t_0 = time.perf_counter()  # Actual benchmark --------------------
    if use_cuda: torch.cuda.synchronize()
    for i in range(loops):
        exec( code, locals() )
    if use_cuda: torch.cuda.synchronize()
    elapsed = time.perf_counter() - t_0  # ---------------------------

    if B is None:
        print("{:3} NxN convolution, with N ={:7}: {:3}x{:3.6f}s".format(loops, N, loops, elapsed / loops))
        return elapsed / loops
    else:
        print("{:3}x{:3} NxN convolution, with N ={:7}: {:3}x{:3}x{:3.6f}s".format(
            B, loops, N, B, loops, elapsed / (B * loops)))
        return elapsed / (B * loops)


def bench_config(Routine, backend, dev, l) :
    """Times a convolution for an increasing number of samples."""

    print("Backend : {}, Device : {} -------------".format(backend, dev))

    times = []
    try :
        Nloops = [100, 10, 1]
        nloops = Nloops.pop(0)
        for n in NS :
            elapsed = benchmark(Routine, dev, n, loops=nloops, lang=l)

            times.append( elapsed )
            if (nloops * elapsed > MAXTIME) \
            or (nloops * elapsed > REDTIME/10 and len(Nloops) > 0 ) :
                nloops = Nloops.pop(0)

    except RuntimeError :
        print("**\nMemory overflow !")
    except IndexError :
        print("**\nToo slow !")

    return times + (len(NS)-len(times)) * [np.nan]


def full_bench(title, routines) :
    """Benchmarks the varied backends of a geometric loss function."""

    backends = [ backend for (_, backend, _) in routines ]

    print("Benchmarking : {} ===============================".format(title))

    lines  = [ NS ]
    for routine, backend, lang in routines :
        lines.append( bench_config(routine, backend, "cuda" if use_cuda else "cpu", lang) )

    benches = np.array(lines).T

    # Creates a pyplot figure:
    plt.figure(figsize=(12,8))
    linestyles = ["o-", "s-", "^-"]
    for i, backend in enumerate(backends):
        plt.plot( benches[:,0], benches[:,i+1], linestyles[i],
                  linewidth=2, label='backend = "{}"'.format(backend) )

        for (j, val) in enumerate( benches[:,i+1] ):
            if np.isnan(val) and j > 0:
                x, y = benches[j-1,0], benches[j-1,i+1]
                plt.annotate('Memory overflow!',
                    xy=(x, 1.05*y),
                    horizontalalignment='center',
                    verticalalignment='bottom')
                break

    plt.title('Runtimes for {} in dimension {}'.format(title, D))
    plt.xlabel('Number of samples')
    plt.ylabel('Seconds')
    plt.yscale('log') ; plt.xscale('log')
    plt.legend(loc='upper left')
    plt.grid(True, which="major", linestyle="-")
    plt.grid(True, which="minor", linestyle="dotted")
    plt.axis([NS[0], NS[-1], 1e-5, MAXTIME])
    plt.tight_layout()

    # Save as a .csv to put a nice Tikz figure in the papers:
    header = "Npoints " + " ".join(backends)
    os.makedirs("output", exist_ok=True)
    np.savetxt("output/benchmark_convolutions_3D.csv", benches,
               fmt='%-9.5f', header=header, comments='')

NumPy vs. PyTorch vs. KeOps

routines = [ (gaussianconv_numpy,   "Numpy",   "numpy"),
             (gaussianconv_pytorch, "PyTorch", "torch"),
             (gaussianconv_keops,   "KeOps",   "torch"), ]
full_bench( "Gaussian Matrix-Vector products", routines )
../_images/sphx_glr_plot_benchmarks_convolutions_3D_001.png

Out:

Benchmarking : Gaussian Matrix-Vector products ===============================
Backend : Numpy, Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000293s
100 NxN convolution, with N =    200: 100x0.001627s
100 NxN convolution, with N =    500: 100x0.007978s
 10 NxN convolution, with N =   1000:  10x0.035831s
  1 NxN convolution, with N =   2000:   1x0.143121s
  1 NxN convolution, with N =   5000:   1x0.867392s
  1 NxN convolution, with N =  10000:   1x3.352783s
  1 NxN convolution, with N =  20000:   1x13.611010s
**
Too slow !
Backend : PyTorch, Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000241s
100 NxN convolution, with N =    200: 100x0.000229s
100 NxN convolution, with N =    500: 100x0.000229s
100 NxN convolution, with N =   1000: 100x0.000238s
100 NxN convolution, with N =   2000: 100x0.000423s
100 NxN convolution, with N =   5000: 100x0.002336s
 10 NxN convolution, with N =  10000:  10x0.010855s
 10 NxN convolution, with N =  20000:  10x0.039532s
**
Memory overflow !
Backend : KeOps, Device : cuda -------------
Compiling libKeOpstorch9451ba29f0 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch9451ba29f0:
       formula: Sum_Reduction(Exp(-SqDist(X,Y)) * B,0)
       aliases: X = Vi(0,3); Y = Vj(1,3); B = Vj(2,1);
       dtype  : float32
... Done.
100 NxN convolution, with N =    100: 100x0.000166s
100 NxN convolution, with N =    200: 100x0.000138s
100 NxN convolution, with N =    500: 100x0.000150s
100 NxN convolution, with N =   1000: 100x0.000161s
100 NxN convolution, with N =   2000: 100x0.000195s
100 NxN convolution, with N =   5000: 100x0.000301s
100 NxN convolution, with N =  10000: 100x0.000469s
100 NxN convolution, with N =  20000: 100x0.001180s
100 NxN convolution, with N =  50000: 100x0.004697s
 10 NxN convolution, with N = 100000:  10x0.017843s
 10 NxN convolution, with N = 200000:  10x0.070646s
  1 NxN convolution, with N = 500000:   1x0.428023s
  1 NxN convolution, with N =1000000:   1x1.709376s

Genred vs. LazyTensor vs. batched LazyTensor

routines = [(gaussianconv_keops, "KeOps (Genred)", "torch"),
            (gaussianconv_lazytensor, "KeOps (LazyTensor)", "torch"),
            ((gaussianconv_lazytensor, 10), "KeOps (LazyTensor, batchsize=10)", "torch"), ]
full_bench( "Gaussian Matrix-Vector products", routines )

plt.show()
../_images/sphx_glr_plot_benchmarks_convolutions_3D_002.png

Out:

Benchmarking : Gaussian Matrix-Vector products ===============================
Backend : KeOps (Genred), Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000127s
100 NxN convolution, with N =    200: 100x0.000130s
100 NxN convolution, with N =    500: 100x0.000138s
100 NxN convolution, with N =   1000: 100x0.000150s
100 NxN convolution, with N =   2000: 100x0.000178s
100 NxN convolution, with N =   5000: 100x0.000268s
100 NxN convolution, with N =  10000: 100x0.000404s
100 NxN convolution, with N =  20000: 100x0.001037s
100 NxN convolution, with N =  50000: 100x0.004816s
 10 NxN convolution, with N = 100000:  10x0.018006s
 10 NxN convolution, with N = 200000:  10x0.071372s
  1 NxN convolution, with N = 500000:   1x0.432676s
  1 NxN convolution, with N =1000000:   1x1.729578s
Backend : KeOps (LazyTensor), Device : cuda -------------
Compiling libKeOpstorch4cd7af70c0 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch4cd7af70c0:
       formula: Sum_Reduction((Exp(Minus(Sum(Square((Var(0,3,0) - Var(1,3,1)))))) * Var(2,1,1)),0)
       aliases: Var(0,3,0); Var(1,3,1); Var(2,1,1);
       dtype  : float32
... Done.
100 NxN convolution, with N =    100: 100x0.000331s
100 NxN convolution, with N =    200: 100x0.000314s
100 NxN convolution, with N =    500: 100x0.000316s
100 NxN convolution, with N =   1000: 100x0.000329s
100 NxN convolution, with N =   2000: 100x0.000363s
100 NxN convolution, with N =   5000: 100x0.000468s
100 NxN convolution, with N =  10000: 100x0.000640s
100 NxN convolution, with N =  20000: 100x0.001355s
100 NxN convolution, with N =  50000: 100x0.004976s
 10 NxN convolution, with N = 100000:  10x0.018273s
 10 NxN convolution, with N = 200000:  10x0.071754s
  1 NxN convolution, with N = 500000:   1x0.436207s
  1 NxN convolution, with N =1000000:   1x1.744026s
Backend : KeOps (LazyTensor, batchsize=10), Device : cuda -------------
 10x100 NxN convolution, with N =    100:  10x100x0.000075s
 10x100 NxN convolution, with N =    200:  10x100x0.000073s
 10x100 NxN convolution, with N =    500:  10x100x0.000073s
 10x100 NxN convolution, with N =   1000:  10x100x0.000075s
 10x100 NxN convolution, with N =   2000:  10x100x0.000081s
 10x100 NxN convolution, with N =   5000:  10x100x0.000119s
 10x100 NxN convolution, with N =  10000:  10x100x0.000253s
 10x100 NxN convolution, with N =  20000:  10x100x0.000789s
 10x100 NxN convolution, with N =  50000:  10x100x0.004729s
 10x 10 NxN convolution, with N = 100000:  10x 10x0.026386s
 10x  1 NxN convolution, with N = 200000:  10x  1x0.140907s
 10x  1 NxN convolution, with N = 500000:  10x  1x1.586358s
 10x  1 NxN convolution, with N =1000000:  10x  1x8.246210s
/home/bcharlier/tmp/libkeops/pykeops/benchmarks/plot_benchmarks_convolutions_3D.py:258: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.show()

Total running time of the script: ( 4 minutes 50.098 seconds)

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