Signal Processing - July 2017 - 34

Furthermore, note that in the setting of multiple domains,
there is no immediate way to define a meaningful spatial pooling operation, as the number of points on different domains can
vary, and their order can be arbitrary. It is, however, possible to
pool pointwise features produced by a network by aggregating
all the local information into a single vector. One possibility
for such a pooling is computing the statistics of the pointwise
features, e.g., the mean or covariance [47]. Note that after such
a pooling, all of the spatial information is lost.
On a Euclidean domain, due to shift invariance the convolution can be thought of as passing a template at each point
of the domain and recording the correlation of the template
with the function at that point. Thinking of image filtering,
this amounts to extracting a (typically square) patch of pixels, multiplying it elementwise with a template and summing
up the results, then moving to the next position in a sliding window manner. Shift invariance implies that the very
operation of extracting the patch at each position is always
the same.
One of the major problems in applying the same paradigm to non-Euclidean domains is the lack of shift invariance,
implying that the patch operator extracting a local patch would
be position dependent. Furthermore, the typical lack of meaningful global parameterization for a graph or manifold forces
to represent the patch in some local intrinsic system of coordinates. Such a mapping can be obtained by defining a set of
weighting functions v 1 (x, $), f, v J (x, $) localized to positions

near x (see examples in Figure 3). Extracting a patch amounts
to averaging the function f at each point by these weights,
D j (x) f =

#X f (xl ) v j (x, xl ) dxl , j = 1, f, J,

providing for a spatial definition of an intrinsic equivalent of
convolution
( f * g) (x) = / g j D j (x) f,

Anisotropic
Heat Kernel
(b)

Geodesic Polar
Coordinates
(c)

(40)

j

where g denotes the template coefficients applied on the patch
extracted at each point. Overall, (39) and (40) act as a kind of
nonlinear filtering of f, and the patch operator D is specified
by defining the weighting functions v 1, f, v J . Such filters are
localized by construction, and the number of parameters is
equal to the number of weighting functions J = O (1) . Several
frameworks for non-Euclidean CNNs essentially amount to
different choices of these weights. The spectrum-free methods
(ChebNet and GCN) described in the previous section can
also be thought of in terms of local weighting functions, as it is
easy to see the analogy between (40) and (34).

Geodesic CNN
Because manifolds naturally come with a low-dimensional
tangent space associated with each point, it is natural to work
in a local system of coordinates in the tangent space [47]. In
particular, on 2-D manifolds one can create a polar system of
coordinates around x where the radial coordinate is given by
some intrinsic distance t (xl ) = d (x, xl ), and the angular coordinate i (x) is obtained by ray shooting from a point at equispaced angles. The weighting functions in this case can be
obtained as a product of Gaussians
v ij (x, xl ) = e - (t (xl ) - t i)

Diffusion
Distance
(a)

(39)

2

/2v 2t - (i (x l ) - i j) 2 /2v i2

e

,

(41)

where i = 1, f, J and j = 1, f, J l denote the indices of the
radial and angular bins, respectively. The resulting JJ l weights
are bins of width v t # v i in the polar coordinates [Figure 3(c)
and (f)].

Anisotropic CNN
(d)

(e)

(f)

FIGURE 3. (a)-(c) The examples of intrinsic weighting functions used to
construct a patch operator at the point marked in black (different colors
represent different weighting functions). (a) Diffusion distance allows to
map neighbor points according to their distance from the reference point,
thus defining a 1-D system of local intrinsic coordinates. (b) Anisotropic
heat kernels of different scale and orientations and (c) geodesic polar
weights are 2-D systems of coordinates. (d)-(f) The representation of
the weighting functions in the local polar (t, i) system of coordinates
(red curves represent the 0.5 level set).

34

We have already seen the non-Euclidean heat equation (S5),
whose heat kernel h t (x, ·) produces localized blob-like
weights around the point x [see Figure S3(a)]. Varying the diffusion time t controls the spread of the kernel. However, such
kernels are isotropic, meaning that the heat flows equally fast
in all the directions. A more general anisotropic diffusion [48]
equation on a manifold
ft (x, t) = - div (A (x) df (x, t))

(42)

involves the thermal conductivity tensor A (x) (in the case of
2-D manifolds, a 2 × 2 matrix is applied to the intrinsic gradient

IEEE SIGNAL PROCESSING MAGAZINE

|

July 2017

|



Table of Contents for the Digital Edition of Signal Processing - July 2017

Signal Processing - July 2017 - Cover1
Signal Processing - July 2017 - Cover2
Signal Processing - July 2017 - 1
Signal Processing - July 2017 - 2
Signal Processing - July 2017 - 3
Signal Processing - July 2017 - 4
Signal Processing - July 2017 - 5
Signal Processing - July 2017 - 6
Signal Processing - July 2017 - 7
Signal Processing - July 2017 - 8
Signal Processing - July 2017 - 9
Signal Processing - July 2017 - 10
Signal Processing - July 2017 - 11
Signal Processing - July 2017 - 12
Signal Processing - July 2017 - 13
Signal Processing - July 2017 - 14
Signal Processing - July 2017 - 15
Signal Processing - July 2017 - 16
Signal Processing - July 2017 - 17
Signal Processing - July 2017 - 18
Signal Processing - July 2017 - 19
Signal Processing - July 2017 - 20
Signal Processing - July 2017 - 21
Signal Processing - July 2017 - 22
Signal Processing - July 2017 - 23
Signal Processing - July 2017 - 24
Signal Processing - July 2017 - 25
Signal Processing - July 2017 - 26
Signal Processing - July 2017 - 27
Signal Processing - July 2017 - 28
Signal Processing - July 2017 - 29
Signal Processing - July 2017 - 30
Signal Processing - July 2017 - 31
Signal Processing - July 2017 - 32
Signal Processing - July 2017 - 33
Signal Processing - July 2017 - 34
Signal Processing - July 2017 - 35
Signal Processing - July 2017 - 36
Signal Processing - July 2017 - 37
Signal Processing - July 2017 - 38
Signal Processing - July 2017 - 39
Signal Processing - July 2017 - 40
Signal Processing - July 2017 - 41
Signal Processing - July 2017 - 42
Signal Processing - July 2017 - 43
Signal Processing - July 2017 - 44
Signal Processing - July 2017 - 45
Signal Processing - July 2017 - 46
Signal Processing - July 2017 - 47
Signal Processing - July 2017 - 48
Signal Processing - July 2017 - 49
Signal Processing - July 2017 - 50
Signal Processing - July 2017 - 51
Signal Processing - July 2017 - 52
Signal Processing - July 2017 - 53
Signal Processing - July 2017 - 54
Signal Processing - July 2017 - 55
Signal Processing - July 2017 - 56
Signal Processing - July 2017 - 57
Signal Processing - July 2017 - 58
Signal Processing - July 2017 - 59
Signal Processing - July 2017 - 60
Signal Processing - July 2017 - 61
Signal Processing - July 2017 - 62
Signal Processing - July 2017 - 63
Signal Processing - July 2017 - 64
Signal Processing - July 2017 - 65
Signal Processing - July 2017 - 66
Signal Processing - July 2017 - 67
Signal Processing - July 2017 - 68
Signal Processing - July 2017 - 69
Signal Processing - July 2017 - 70
Signal Processing - July 2017 - 71
Signal Processing - July 2017 - 72
Signal Processing - July 2017 - 73
Signal Processing - July 2017 - 74
Signal Processing - July 2017 - 75
Signal Processing - July 2017 - 76
Signal Processing - July 2017 - 77
Signal Processing - July 2017 - 78
Signal Processing - July 2017 - 79
Signal Processing - July 2017 - 80
Signal Processing - July 2017 - 81
Signal Processing - July 2017 - 82
Signal Processing - July 2017 - 83
Signal Processing - July 2017 - 84
Signal Processing - July 2017 - 85
Signal Processing - July 2017 - 86
Signal Processing - July 2017 - 87
Signal Processing - July 2017 - 88
Signal Processing - July 2017 - 89
Signal Processing - July 2017 - 90
Signal Processing - July 2017 - 91
Signal Processing - July 2017 - 92
Signal Processing - July 2017 - 93
Signal Processing - July 2017 - 94
Signal Processing - July 2017 - 95
Signal Processing - July 2017 - 96
Signal Processing - July 2017 - 97
Signal Processing - July 2017 - 98
Signal Processing - July 2017 - 99
Signal Processing - July 2017 - 100
Signal Processing - July 2017 - 101
Signal Processing - July 2017 - 102
Signal Processing - July 2017 - 103
Signal Processing - July 2017 - 104
Signal Processing - July 2017 - 105
Signal Processing - July 2017 - 106
Signal Processing - July 2017 - 107
Signal Processing - July 2017 - 108
Signal Processing - July 2017 - 109
Signal Processing - July 2017 - 110
Signal Processing - July 2017 - 111
Signal Processing - July 2017 - 112
Signal Processing - July 2017 - 113
Signal Processing - July 2017 - 114
Signal Processing - July 2017 - 115
Signal Processing - July 2017 - 116
Signal Processing - July 2017 - 117
Signal Processing - July 2017 - 118
Signal Processing - July 2017 - 119
Signal Processing - July 2017 - 120
Signal Processing - July 2017 - 121
Signal Processing - July 2017 - 122
Signal Processing - July 2017 - 123
Signal Processing - July 2017 - 124
Signal Processing - July 2017 - 125
Signal Processing - July 2017 - 126
Signal Processing - July 2017 - 127
Signal Processing - July 2017 - 128
Signal Processing - July 2017 - 129
Signal Processing - July 2017 - 130
Signal Processing - July 2017 - 131
Signal Processing - July 2017 - 132
Signal Processing - July 2017 - 133
Signal Processing - July 2017 - 134
Signal Processing - July 2017 - 135
Signal Processing - July 2017 - 136
Signal Processing - July 2017 - 137
Signal Processing - July 2017 - 138
Signal Processing - July 2017 - 139
Signal Processing - July 2017 - 140
Signal Processing - July 2017 - 141
Signal Processing - July 2017 - 142
Signal Processing - July 2017 - 143
Signal Processing - July 2017 - 144
Signal Processing - July 2017 - 145
Signal Processing - July 2017 - 146
Signal Processing - July 2017 - 147
Signal Processing - July 2017 - 148
Signal Processing - July 2017 - 149
Signal Processing - July 2017 - 150
Signal Processing - July 2017 - 151
Signal Processing - July 2017 - 152
Signal Processing - July 2017 - 153
Signal Processing - July 2017 - 154
Signal Processing - July 2017 - 155
Signal Processing - July 2017 - 156
Signal Processing - July 2017 - 157
Signal Processing - July 2017 - 158
Signal Processing - July 2017 - 159
Signal Processing - July 2017 - 160
Signal Processing - July 2017 - 161
Signal Processing - July 2017 - 162
Signal Processing - July 2017 - 163
Signal Processing - July 2017 - 164
Signal Processing - July 2017 - 165
Signal Processing - July 2017 - 166
Signal Processing - July 2017 - 167
Signal Processing - July 2017 - 168
Signal Processing - July 2017 - 169
Signal Processing - July 2017 - 170
Signal Processing - July 2017 - 171
Signal Processing - July 2017 - 172
Signal Processing - July 2017 - 173
Signal Processing - July 2017 - 174
Signal Processing - July 2017 - 175
Signal Processing - July 2017 - 176
Signal Processing - July 2017 - 177
Signal Processing - July 2017 - 178
Signal Processing - July 2017 - 179
Signal Processing - July 2017 - 180
Signal Processing - July 2017 - 181
Signal Processing - July 2017 - 182
Signal Processing - July 2017 - 183
Signal Processing - July 2017 - 184
Signal Processing - July 2017 - 185
Signal Processing - July 2017 - 186
Signal Processing - July 2017 - 187
Signal Processing - July 2017 - 188
Signal Processing - July 2017 - 189
Signal Processing - July 2017 - 190
Signal Processing - July 2017 - 191
Signal Processing - July 2017 - 192
Signal Processing - July 2017 - 193
Signal Processing - July 2017 - 194
Signal Processing - July 2017 - 195
Signal Processing - July 2017 - 196
Signal Processing - July 2017 - Cover3
Signal Processing - July 2017 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201809
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201807
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201805
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201803
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201801
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0917
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0717
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0517
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0317
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0916
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0716
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0516
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0316
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0915
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0715
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0515
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0315
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0914
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0714
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0514
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0314
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0913
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0713
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0513
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0313
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0912
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0712
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0512
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0312
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0911
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0711
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0511
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0311
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0910
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0710
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0510
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0310
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0909
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0709
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0509
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0309
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1108
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0908
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0708
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0508
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0308
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0108
https://www.nxtbookmedia.com