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 os
import numpy as np
import time
from matplotlib import pyplot as plt

import importlib
import torch

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-4, 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: 100x0.000421s
100 NxN convolution, with N =    200: 100x0.001759s
100 NxN convolution, with N =    500: 100x0.010887s
 10 NxN convolution, with N =   1000:  10x0.037291s
  1 NxN convolution, with N =   2000:   1x0.151025s
  1 NxN convolution, with N =   5000:   1x0.841828s
  1 NxN convolution, with N =  10000:   1x3.242485s
  1 NxN convolution, with N =  20000:   1x12.794234s
**
Too slow !
Backend : PyTorch, Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000159s
100 NxN convolution, with N =    200: 100x0.000157s
100 NxN convolution, with N =    500: 100x0.000159s
100 NxN convolution, with N =   1000: 100x0.000170s
100 NxN convolution, with N =   2000: 100x0.000424s
100 NxN convolution, with N =   5000: 100x0.002323s
 10 NxN convolution, with N =  10000:  10x0.010904s
 10 NxN convolution, with N =  20000:  10x0.039553s
**
Memory overflow !
Backend : KeOps, Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000125s
100 NxN convolution, with N =    200: 100x0.000125s
100 NxN convolution, with N =    500: 100x0.000133s
100 NxN convolution, with N =   1000: 100x0.000146s
100 NxN convolution, with N =   2000: 100x0.000173s
100 NxN convolution, with N =   5000: 100x0.000258s
100 NxN convolution, with N =  10000: 100x0.000417s
100 NxN convolution, with N =  20000: 100x0.001067s
100 NxN convolution, with N =  50000: 100x0.004768s
 10 NxN convolution, with N = 100000:  10x0.017990s
 10 NxN convolution, with N = 200000:  10x0.071256s
  1 NxN convolution, with N = 500000:   1x0.432217s
  1 NxN convolution, with N =1000000:   1x1.731090s

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: 100x0.000119s
100 NxN convolution, with N =    200: 100x0.000124s
100 NxN convolution, with N =    500: 100x0.000134s
100 NxN convolution, with N =   1000: 100x0.000148s
100 NxN convolution, with N =   2000: 100x0.000179s
100 NxN convolution, with N =   5000: 100x0.000280s
100 NxN convolution, with N =  10000: 100x0.000454s
100 NxN convolution, with N =  20000: 100x0.001131s
100 NxN convolution, with N =  50000: 100x0.004985s
 10 NxN convolution, with N = 100000:  10x0.018807s
 10 NxN convolution, with N = 200000:  10x0.072771s
  1 NxN convolution, with N = 500000:   1x0.442033s
  1 NxN convolution, with N =1000000:   1x2.343521s
Backend : KeOps (LazyTensor), Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000298s
100 NxN convolution, with N =    200: 100x0.000301s
100 NxN convolution, with N =    500: 100x0.000310s
100 NxN convolution, with N =   1000: 100x0.000322s
100 NxN convolution, with N =   2000: 100x0.000361s
100 NxN convolution, with N =   5000: 100x0.000471s
100 NxN convolution, with N =  10000: 100x0.000658s
100 NxN convolution, with N =  20000: 100x0.001436s
100 NxN convolution, with N =  50000: 100x0.007684s
 10 NxN convolution, with N = 100000:  10x0.034021s
  1 NxN convolution, with N = 200000:   1x0.133892s
  1 NxN convolution, with N = 500000:   1x1.165402s
  1 NxN convolution, with N =1000000:   1x12.377574s
**
Too slow !
Backend : KeOps (LazyTensor, batchsize=10), Device : cuda -------------
 10x100 NxN convolution, with N =    100:  10x100x0.000074s
 10x100 NxN convolution, with N =    200:  10x100x0.000078s
 10x100 NxN convolution, with N =    500:  10x100x0.000085s
 10x100 NxN convolution, with N =   1000:  10x100x0.000098s
 10x100 NxN convolution, with N =   2000:  10x100x0.000148s
 10x100 NxN convolution, with N =   5000:  10x100x0.000420s
 10x100 NxN convolution, with N =  10000:  10x100x0.001404s
 10x100 NxN convolution, with N =  20000:  10x100x0.005176s
 10x 10 NxN convolution, with N =  50000:  10x 10x0.030826s
 10x  1 NxN convolution, with N = 100000:  10x  1x0.121361s
 10x  1 NxN convolution, with N = 200000:  10x  1x0.483594s
 10x  1 NxN convolution, with N = 500000:  10x  1x3.012165s
 10x  1 NxN convolution, with N =1000000:  10x  1x12.030773s
**
Too slow !
/home/bcharlier/keops/pykeops/benchmarks/plot_benchmarks_convolutions_3D.py:259: 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: ( 6 minutes 37.333 seconds)

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