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: 100x0.000292s
100 NxN convolution, with N =    200: 100x0.001382s
100 NxN convolution, with N =    500: 100x0.009220s
 10 NxN convolution, with N =   1000:  10x0.031147s
  1 NxN convolution, with N =   2000:   1x0.143121s
  1 NxN convolution, with N =   5000:   1x0.867765s
  1 NxN convolution, with N =  10000:   1x3.342052s
  1 NxN convolution, with N =  20000:   1x13.256580s
**
Too slow !
Backend : PyTorch, Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000178s
100 NxN convolution, with N =    200: 100x0.000174s
100 NxN convolution, with N =    500: 100x0.000174s
100 NxN convolution, with N =   1000: 100x0.000186s
100 NxN convolution, with N =   2000: 100x0.000419s
100 NxN convolution, with N =   5000: 100x0.002320s
 10 NxN convolution, with N =  10000:  10x0.010794s
 10 NxN convolution, with N =  20000:  10x0.039737s
**
Memory overflow !
Backend : KeOps, Device : cuda -------------
100 NxN convolution, with N =    100: 100x0.000130s
100 NxN convolution, with N =    200: 100x0.000127s
100 NxN convolution, with N =    500: 100x0.000135s
100 NxN convolution, with N =   1000: 100x0.000149s
100 NxN convolution, with N =   2000: 100x0.000173s
100 NxN convolution, with N =   5000: 100x0.000254s
100 NxN convolution, with N =  10000: 100x0.000374s
100 NxN convolution, with N =  20000: 100x0.000898s
100 NxN convolution, with N =  50000: 100x0.003937s
 10 NxN convolution, with N = 100000:  10x0.014846s
 10 NxN convolution, with N = 200000:  10x0.058569s
  1 NxN convolution, with N = 500000:   1x0.353305s
  1 NxN convolution, with N =1000000:   1x1.399503s

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: 100x0.000124s
100 NxN convolution, with N =    200: 100x0.000128s
100 NxN convolution, with N =    500: 100x0.000136s
100 NxN convolution, with N =   1000: 100x0.000149s
100 NxN convolution, with N =   2000: 100x0.000175s
100 NxN convolution, with N =   5000: 100x0.000252s
100 NxN convolution, with N =  10000: 100x0.000375s
100 NxN convolution, with N =  20000: 100x0.000898s
100 NxN convolution, with N =  50000: 100x0.003969s
 10 NxN convolution, with N = 100000:  10x0.014856s
 10 NxN convolution, with N = 200000:  10x0.058777s
  1 NxN convolution, with N = 500000:   1x0.354500s
  1 NxN convolution, with N =1000000:   1x1.408210s
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: 100x0.000309s
100 NxN convolution, with N =   1000: 100x0.000322s
100 NxN convolution, with N =   2000: 100x0.000347s
100 NxN convolution, with N =   5000: 100x0.000422s
100 NxN convolution, with N =  10000: 100x0.000547s
100 NxN convolution, with N =  20000: 100x0.001063s
100 NxN convolution, with N =  50000: 100x0.004137s
 10 NxN convolution, with N = 100000:  10x0.015157s
 10 NxN convolution, with N = 200000:  10x0.059258s
  1 NxN convolution, with N = 500000:   1x0.358228s
  1 NxN convolution, with N =1000000:   1x1.416255s
Backend : KeOps (LazyTensor, batchsize=10), Device : cuda -------------
 10x100 NxN convolution, with N =    100:  10x100x0.000074s
 10x100 NxN convolution, with N =    200:  10x100x0.000074s
 10x100 NxN convolution, with N =    500:  10x100x0.000074s
 10x100 NxN convolution, with N =   1000:  10x100x0.000075s
 10x100 NxN convolution, with N =   2000:  10x100x0.000081s
 10x100 NxN convolution, with N =   5000:  10x100x0.000109s
 10x100 NxN convolution, with N =  10000:  10x100x0.000215s
 10x100 NxN convolution, with N =  20000:  10x100x0.000636s
 10x100 NxN convolution, with N =  50000:  10x100x0.003574s
 10x 10 NxN convolution, with N = 100000:  10x 10x0.013958s
 10x 10 NxN convolution, with N = 200000:  10x 10x0.055675s
 10x  1 NxN convolution, with N = 500000:  10x  1x0.348551s
 10x  1 NxN convolution, with N =1000000:  10x  1x1.414246s
/home/bcharlier/keops/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: ( 1 minutes 41.928 seconds)

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