Note

Click here to download the full example code

# 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 )
```

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()
```

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)