# Pairwise distances with ONNX (pdist)#

Function pdist computes pairwise distances between observations in n-dimensional space. It is not that difficult to convert that into ONNX when the dimension of the input is always the same. What if not?

from jyquickhelper import add_notebook_menu

%load_ext mlprodict


## Function pdist#

The function pdist distances. Let’s denote a list of vectors , function pdist returns the matrix where .

import numpy
from scipy.spatial.distance import pdist, squareform

M = numpy.array([[0, 1],
[1, 2],
[0.1, 1.1],
[2, 2]], dtype=float)

d1 = squareform(pdist(M, metric='sqeuclidean'))
d1

array([[0.  , 2.  , 0.02, 5.  ],
[2.  , 0.  , 1.62, 1.  ],
[0.02, 1.62, 0.  , 4.42],
[5.  , 1.  , 4.42, 0.  ]])


The two following functions are implemented to reduce the number of allocations the algorithm requires.

def custom_pdist(M):
n = M.shape[0]
res = numpy.zeros((n, n))
buffer = numpy.empty(M.shape)
for i in range(n):
numpy.subtract(M, M[i], out=buffer)  # broadcasted substraction
numpy.square(buffer, out=buffer)
res[i, :] = numpy.sum(buffer, axis=1)
return res

d2 = custom_pdist(M)
d2

array([[0.  , 2.  , 0.02, 5.  ],
[2.  , 0.  , 1.62, 1.  ],
[0.02, 1.62, 0.  , 4.42],
[5.  , 1.  , 4.42, 0.  ]])


This function computes distances wheres only are necessary since the final matrix is symmetric. Let’s change the implementation to reflect that.

def custom_pdist_lower(M):
n = M.shape[0]
res = numpy.zeros((n, n))
buffer = numpy.empty((M.shape[0]-1, M.shape[1]))
a = numpy.empty(M.shape[0])
for i in range(1, n):
numpy.subtract(M[:i], M[i], out=buffer[:i])  # broadcasted substraction
numpy.square(buffer[:i], out=buffer[:i])
numpy.sum(buffer[:i], axis=1, out=a[:i])
res[:i, i] = a[:i]
res[i, :i] = a[:i]
return res

d3 = custom_pdist_lower(M)
d3

array([[0.  , 2.  , 0.02, 5.  ],
[2.  , 0.  , 1.62, 1.  ],
[0.02, 1.62, 0.  , 4.42],
[5.  , 1.  , 4.42, 0.  ]])


## Loop mechanism in ONNX#

Operator Loop seems appropriate but it is just a loop wheras Scan holds accumulator. The first graph is what is repeated inside the loop.

from skl2onnx.algebra.onnx_ops import OnnxAdd, OnnxIdentity, OnnxScan
from skl2onnx.common.data_types import FloatTensorType

initial = numpy.array([0, 0]).astype(numpy.float32).reshape((2,))
x = numpy.array([1, 2, 3, 4, 5, 6]).astype(numpy.float32).reshape((3, 2))

scan_body = id_node.to_onnx(
{'sum_in': initial, 'next': initial},
outputs=[('sum_out', FloatTensorType()),
('scan_out', FloatTensorType())])

# add -l 1 if nothing shows up
%onnxview scan_body


The operator Scan repeats this graph a couple of times. sum_in is an accumulator, next is the iterated row from the input matrix.

node = OnnxScan('initial', 'x', output_names=['y', 'z'],
num_scan_inputs=1, body=scan_body.graph)

model_def = node.to_onnx(
{'initial': initial, 'x': x},
outputs=[('y', FloatTensorType()),
('z', FloatTensorType())])

# add -l 1 if nothing shows up
%onnxview model_def


All together in the same graph.

# add -l 1 if nothing shows up
%onnxview model_def -r 1

from mlprodict.onnxrt import OnnxInference
oinf = OnnxInference(model_def)
res = oinf.run({'initial': initial, 'x': x})
res['y']

array([ 9., 12.], dtype=float32)

res['z']

array([[ 1.,  2.],
[ 4.,  6.],
[ 9., 12.]], dtype=float32)


## Back to pdist#

sklearn-onnx implements function pdist with ONNX operators. The parameter inputs=[('x', FloatTensorType()) tels the method to_onnx that the dimension of the inputs is not fixed and should not be checked.

# from skl2onnx.algebra.complex_functions import squareform_pdist_

from collections import OrderedDict
from skl2onnx.algebra.onnx_ops import (
OnnxSub, OnnxReduceSumSquare, OnnxSqueeze,
OnnxIdentity, OnnxScan)
from skl2onnx.common.data_types import FloatTensorType

def squareform_pdist(X, op_version=17, **kwargs):
"""Returns the ONNX graph which computes
squareform(pdist(X, metric='sqeuclidean')."""

# The subgraph executed at every iteration.
opv = op_version
diff = OnnxSub('next_in', 'next', output_names=['diff'], op_version=opv)
id_next = OnnxIdentity('next_in', output_names=['next_out'], op_version=opv)
norm = OnnxReduceSumSquare(diff, output_names=['norm'], axes=[1], op_version=opv)
flat = OnnxSqueeze(norm, numpy.array([1], dtype=numpy.int64),
output_names=['scan_out'], op_version=opv)
scan_body = id_next.to_onnx(
OrderedDict([('next_in', FloatTensorType()),
('next', FloatTensorType())]),
# Size must be empty otherwise onnxruntime fails
# at execution time if it receives a matrix
# with a different shape. With 'None', the same ONNX graph
# can compute pairwise distance for any shape.
outputs=[('next_out', FloatTensorType([None, None])),
('scan_out', FloatTensorType([None]))],
other_outputs=[flat])

# The loop.
# 'scan0_{idself}' means the variable name will include
# id(OnnxScan), this is needed if squareform_pdist is used
# twice in the same graph.
node = OnnxScan(X, X, output_names=['scan0_{idself}', 'scan1_{idself}'],
num_scan_inputs=1, body=scan_body.graph, op_version=opv,
**kwargs)
return node[1]

opv = 17
onnx_fct = OnnxIdentity(squareform_pdist('x', op_version=opv),
output_names='Y', op_version=opv)
model_def = onnx_fct.to_onnx(inputs=[('x', FloatTensorType())],
target_opset=opv)

# add -l 1 if nothing shows up
%onnxview model_def

from collections import OrderedDict
from skl2onnx.algebra.onnx_ops import (
OnnxSub, OnnxReduceSumSquare, OnnxSqueeze,
OnnxIdentity, OnnxScan)
from skl2onnx.common.data_types import FloatTensorType

def squareform_pdist(X, op_version=17, **kwargs):
# The subgraph executed at every iteration.
opv = op_version
diff = OnnxSub('next_in', 'next', output_names=['diff'], op_version=opv)
id_next = OnnxIdentity('next_in', output_names=['next_out'], op_version=opv)
norm = OnnxReduceSumSquare(diff, output_names=['norm'], axes=[1], op_version=opv)
flat = OnnxSqueeze(norm, numpy.array([1], dtype=numpy.int64),
output_names=['scan_out'], op_version=opv)
scan_body = id_next.to_onnx(
OrderedDict([('next_in', FloatTensorType()),
('next', FloatTensorType())]),
outputs=[('next_out', FloatTensorType([None, None])),
('scan_out', FloatTensorType([None]))],
other_outputs=[flat])

# The loop.
node = OnnxScan(X, X, output_names=['scan0_{idself}', 'scan1_{idself}'],
num_scan_inputs=1, body=scan_body.graph, op_version=opv,
**kwargs)
return node[1]

opv = 17
onnx_fct = OnnxIdentity(squareform_pdist('x', op_version=opv),
output_names='Y', op_version=opv)
model_def = onnx_fct.to_onnx(inputs=[('x', FloatTensorType())])


Notice the double arrow. Input x is used twice, once as an permanent state involved in broacasted substract, another time to iterator rows. On the other side, the first output of operator Scan is a permanent state equal to the input, the second one is an aggregation of results produced at each iteration. Each of those produces a row of a final matrix.

oinf = OnnxInference(model_def)
body = oinf['Sc_Scan', 'body']

# add -l 1 if nothing shows up
%onnxview body.g


All together.

# add -l 1 if nothing shows up
%onnxview model_def -r 1


Let’s now execute the graph and compare it with the original graph.

d1 = squareform(pdist(M, metric='sqeuclidean'))
d1

array([[0.  , 2.  , 0.02, 5.  ],
[2.  , 0.  , 1.62, 1.  ],
[0.02, 1.62, 0.  , 4.42],
[5.  , 1.  , 4.42, 0.  ]])

oinf.run({'x': M})['Y']

array([[0.  , 2.  , 0.02, 5.  ],
[2.  , 0.  , 1.62, 1.  ],
[0.02, 1.62, 0.  , 4.42],
[5.  , 1.  , 4.42, 0.  ]])

%timeit squareform(pdist(M, metric='sqeuclidean'))

9.57 µs ± 166 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)

%timeit custom_pdist(M)

36.7 µs ± 2.28 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

%timeit custom_pdist_lower(M)

35.7 µs ± 646 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

%timeit oinf.run({'x': M})['Y']

206 µs ± 17.3 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)

M32 = M.astype(numpy.float32)

model_def.ir_version = 8

oinfrt = OnnxInference(model_def, runtime="onnxruntime1")
oinfrt.run({'x': M32})['Y']

array([[0.        , 2.        , 0.02000001, 5.        ],
[2.        , 0.        , 1.6199999 , 1.        ],
[0.02000001, 1.6199999 , 0.        , 4.42      ],
[5.        , 1.        , 4.42      , 0.        ]], dtype=float32)

%timeit oinfrt.run({'x': M32})['Y']

47.2 µs ± 295 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)


## Benchmark#

from timeit import Timer

def measure_time(name, stmt, context, repeat=10, number=10):
tim = Timer(stmt, globals=context)
res = numpy.array(tim.repeat(repeat=repeat, number=number))
res /= number
mean = numpy.mean(res)
dev = numpy.mean(res ** 2)
dev = (dev - mean**2) ** 0.5
return dict(average=mean, deviation=dev, min_exec=numpy.min(res),
max_exec=numpy.max(res), repeat=repeat, number=number,
nrows=context['M'].shape[0], ncols=context['M'].shape[1],
name=name)

measure_time("scipy", "squareform(pdist(M, metric='sqeuclidean'))",
context={'squareform': squareform, 'M': M,
'pdist': pdist})

{'average': 4.610000061802566e-05,
'deviation': 8.591571512763607e-06,
'min_exec': 3.507999936118722e-05,
'max_exec': 6.718999939039349e-05,
'repeat': 10,
'number': 10,
'nrows': 4,
'ncols': 2,
'name': 'scipy'}

from tqdm import trange

def generator():
for feat in [5, 10, 50, 100]:
for n in [5, 10, 20, 50, 100, 400, 1000]:
if n <= 500 or feat <= 10:
yield feat, n

all_values = list(generator())

rows = []

with trange(len(all_values)) as t:
for i in t:
feat, n = all_values[i]
t.set_description("feat=%d n=%d" % (feat, n))
M = numpy.random.rand(n, feat)

context = {'squareform': squareform, 'M': M, 'pdist': pdist}
res = measure_time("scipy", "squareform(pdist(M, metric='sqeuclidean'))", context=context)
res['dimres'] = squareform(pdist(M, metric='sqeuclidean')).shape[0]
rows.append(res)

context = {'M': M, 'custom_pdist': custom_pdist}
res = measure_time("numpy", "custom_pdist(M)", context=context)
res['dimres'] = custom_pdist(M).shape[0]
rows.append(res)

context = {'M': M, 'custom_pdist_lower': custom_pdist_lower}
res = measure_time("numpy-lower", "custom_pdist_lower(M)", context=context)
res['dimres'] = custom_pdist_lower(M).shape[0]
rows.append(res)

context = {'oinf': oinf, 'M': M}
res = measure_time("onnx-py", "oinf.run({'x': M})['Y']", context=context)
res['dimres'] = oinf.run({'x': M})['Y'].shape[0]
rows.append(res)

M32 = M.astype(numpy.float32)
context = {'oinfrt': oinfrt, 'M': M32}
res = measure_time("onnx-rt", "oinfrt.run({'x': M})['Y']", context=context)
res['dimres'] = oinfrt.run({'x': M32})['Y'].shape[0]
rows.append(res)

from pandas import DataFrame
df = DataFrame(rows)

feat=100 n=400: 100%|██████████| 26/26 [01:32<00:00,  3.56s/it]

average deviation min_exec max_exec repeat number nrows ncols name dimres
0 0.000045 7.066342e-06 0.000040 0.000065 10 10 5 5 scipy 5
1 0.000121 3.075137e-05 0.000084 0.000189 10 10 5 5 numpy 5
2 0.000046 9.946988e-07 0.000045 0.000049 10 10 5 5 numpy-lower 5
3 0.000400 1.665463e-04 0.000224 0.000716 10 10 5 5 onnx-py 5
4 0.000055 3.251956e-06 0.000051 0.000063 10 10 5 5 onnx-rt 5
from pandas import pivot_table
piv = pivot_table(df, index=["nrows"], columns= ['ncols', 'name'], values='average')

nrows 5 10 20 50 100
ncols name
5 numpy 0.000121 0.000159 0.000248 0.000542 0.001130
numpy-lower 0.000046 0.000116 0.000239 0.000662 0.001294
onnx-py 0.000400 0.000501 0.000886 0.002395 0.004488
onnx-rt 0.000055 0.000092 0.000148 0.000382 0.000956
scipy 0.000045 0.000021 0.000027 0.000027 0.000070
10 numpy 0.000063 0.000087 0.000211 0.001672 0.001868
numpy-lower 0.000061 0.000117 0.000280 0.001676 0.001954
onnx-py 0.000253 0.000530 0.000950 0.007125 0.004770
onnx-rt 0.000068 0.000110 0.000178 0.000951 0.001071
scipy 0.000032 0.000019 0.000018 0.000048 0.000096
50 numpy 0.000051 0.000098 0.000220 0.000664 0.001796
numpy-lower 0.000048 0.000112 0.000248 0.000702 0.001657
onnx-py 0.000283 0.000507 0.000910 0.002543 0.005693
onnx-rt 0.000072 0.000109 0.000187 0.000573 0.001665
scipy 0.000030 0.000029 0.000034 0.000068 0.000167
100 numpy 0.000066 0.000106 0.000234 0.000870 0.002847
numpy-lower 0.000068 0.000119 0.000262 0.000798 0.002007
onnx-py 0.000303 0.000465 0.000963 0.002710 0.007020
onnx-rt 0.000076 0.000104 0.000218 0.000786 0.002568
scipy 0.000028 0.000027 0.000034 0.000088 0.000305
%matplotlib inline

import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 3, figsize=(14, 3))
for i, ncol in enumerate([10, 50, 100]):
piv = df[df.ncols==ncol].pivot(index="nrows", columns="name", values="average")
piv.plot(ax=ax[i], logy=True, logx=True)
ax[i].set_title("ncol=%d" % ncol)
ax;


Curves are not linear and rather difficult to interpret. The algorithm numpy-lower and scipy should be close as the cost of both algorithm are similar. However, scipy reduces the number of trips between C and python. The C implementation of the distance is here: sqeuclidean_distance_double. The final cost is a combination of computation, multithreading, allocations…

from pyquickhelper.pycode.profiling import profile
M = numpy.random.rand(100, 10)

pr1, df1 = profile(lambda: [squareform(pdist(M, metric='sqeuclidean'))
for i in range(0, 1000)],
as_df=True)
pr2, df2 = profile(lambda: [custom_pdist_lower(M)
for i in range(0, 1000)],
as_df=True)

C:xavierdupre__home_GitHubpyquickhelpersrcpyquickhelperpycodeprofiling.py:541: FutureWarning: The default value of numeric_only in DataFrameGroupBy.sum is deprecated. In a future version, numeric_only will default to False. Either specify numeric_only or select only columns which should be valid for the function.
df = df.groupby(['namefct', 'file'], as_index=False).sum().sort_values(
C:xavierdupre__home_GitHubpyquickhelpersrcpyquickhelperpycodeprofiling.py:541: FutureWarning: The default value of numeric_only in DataFrameGroupBy.sum is deprecated. In a future version, numeric_only will default to False. Either specify numeric_only or select only columns which should be valid for the function.
df = df.groupby(['namefct', 'file'], as_index=False).sum().sort_values(
ax = df1[['namefct', 'cum_tall']].head(n=15).set_index('namefct').plot(
kind='bar', figsize=(8, 3), rot=30)
ax.set_title("scipy")
for la in ax.get_xticklabels():
la.set_horizontalalignment('right')

ax = df2[['namefct', 'cum_tall']].head(n=15).set_index('namefct').plot(
kind='bar', figsize=(8, 3), rot=30)
ax.set_title("numpy-lower")
for la in ax.get_xticklabels():
la.set_horizontalalignment('right');


Universal function do not seem to be very efficient in our case. The last graph shows time ratio between implementations of pdist and the baseline scipy.

fig, ax = plt.subplots(1, 3, figsize=(14, 3))
for i, ncol in enumerate([10, 50, 100]):
piv = df[df.ncols==ncol].pivot(index="nrows", columns="name", values="average")
piv['numpy / scipy'] = piv['numpy'] / piv['scipy']
piv['numpy-lower / scipy'] = piv['numpy-lower'] / piv['scipy']
piv['onnx-py / scipy'] = piv['onnx-py'] / piv['scipy']
piv['onnx-rt / scipy'] = piv['onnx-rt'] / piv['scipy']
piv = piv[['numpy / scipy', 'numpy-lower / scipy',
'onnx-py / scipy', 'onnx-rt / scipy']]
piv.plot(ax=ax[i], logy=True, logx=True)
ax[i].plot([0, max(piv.index)], [1, 1], '--', color='black')
ax[i].plot([0, max(piv.index)], [10, 10], '--', color='black')
ax[i].set_title("ncol=%d" % ncol)
ax;


## Test with a new operator CDist#

The final question is: should we introduce a new operator intoONNX specifications? The function pdist is not necessarily often used for a big number of observations as the square matrix it produces will even bigger. It seems reasonable. We showed that a python runtime based on numpy would not help, the implementation must be done in C++ or directly used the scipy version. The experiment was done with a GaussianProcessRegressor. The following section tests with and without a new operator CDist reusing scipy implementation.

import numpy
from sklearn.model_selection import train_test_split
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import ExpSineSquared
from mlprodict.onnx_conv import to_onnx
from mlprodict.onnxrt import OnnxInference

X, y = iris.data, iris.target
X_train, X_test, y_train, __ = train_test_split(X, y, random_state=12)
clr = GaussianProcessRegressor(ExpSineSquared(), alpha=20.)
clr.fit(X_train, y_train)

model_def = to_onnx(clr, X_train)

%onnxview model_def -r 1

model_def_cdist = to_onnx(clr, X_train,
options={GaussianProcessRegressor: {'optim': 'cdist'}})
%onnxview model_def_cdist

oinf = OnnxInference(model_def)
oinf_cdist = OnnxInference(model_def_cdist)

%timeit oinf.run({'X': X_test})

5.02 ms ± 237 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

%timeit oinf_cdist.run({'X': X_test})

396 µs ± 12.9 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)

oinfrt = OnnxInference(model_def, runtime="onnxruntime1")
oinfrt_cdist = OnnxInference(model_def_cdist)

%timeit oinfrt_cdist.run({'X': X_test})

396 µs ± 13.3 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)


It is 10 times faster for this dataset so it is worth it. For bigger datasets, we should expect a lower gain but still significant.