IEEE Circuits and Systems Magazine - Q2 2020 - 33

80
60

100

Z = [X 1 + X 2 + f + X n], (3)

80

0

[A + B + C] = 0 0 1 0 1 0 0 1 0 1.

0.24 0.25 0.26
0

94.4
94.2
94
93.8
93.6
0.248 0.25

40

0

(4)

n
Z
1
]
c / pi m 2H
>
] 1 + i=1
, favor 0,
] 2
n
	 Majority ( p 1, f, p n) = [
(5)
n
]
pi
/
]]> 1 i = 1 H
, favor 1.
+
n
\ 2

2) Multiplication
Pointwise multiplication, also called binding, aims to
form associations between two related hypervectors.
A and B are bound together to form X = A 5 B, which

0.2
0.4
0.6
0.8
Normalized Hamming Distance
(a)

1

Ham (A, B)
Ham (X, A )
Ham (X, B )

60

0

Generally speaking, the addition over odd number
of hypervectors has no ambiguity, whereas the addition over an even number can favor either 0 or 1 using
the majority function defined in Eq. (5). However, this
approach may lead to a biased result for adding two
hypervectors. Therefore, the bias in adding even number of hypervectors is usually reduced by adding an
extra random vector [15]. Fig. 2 illustrates addition of
10,000-dimensional random hypervectors repeated for
3,000 times. Comparing Fig. 2(b) to Fig. 2(c), we see that
specifying in favor of 0 or 1 has little impact over addition. It can be observed from Fig. 2 that the sum is nearly
equally similar to the input operands.

SECOND QUARTER 2020 		

91

20

A = 0 0 0 0 1 1 0 0 1 1,
B = 1 0 1 1 0 0 0 1 0 1,

C = 0 0 1 0 1 0 1 1 0 1,

0.252

0.2
0.4
0.6
0.8
Normalized Hamming Distance
(b)

1

100
Ham (A, B)
Ham (X, A )
Ham (X, B )

80
Probability (%)

	

92

20

Probability (%)

where [$] indicates the sum hypervector Z is thresholded and binarized to {0, 1}d based on the majority rule.
For convenience, Eq. (4) shows an example for the pointwise addition of three 10-bit binary vectors.

93

40

1) Addition
Pointwise addition, also referred to as bundling, computes a hypervector Z using Eq. (3) from the input hypervectors {X 1, X 2, f, X n}. Compared to random hypervectors, the generated Z is maximally similar to the n
inputs X 1, X 2, f, X n , i.e., Hamming distance between Z
and any of the n inputs is at a minimum.
	

Ham (X, A)
Ham (X, B )
Ham (X, C )
Ham (A, B)
Ham (A, C )
Ham (B, C )

100

Probability (%)

examples of data transformations using binary hypervectors. Without doubt, data transformation can
also be employed to non-binary hypervectors, which
is in essence similar to the manipulations over binary
hypervectors. The only difference is from the point
of hardware; for binary hypervectors, the pointwise
-multiplication can be realized by an exclusive or
(XOR) gate.

60

94.4
94.2
94
93.8
0.248 0.25

40
20
0

0

0.2
0.4
0.6
0.8
Normalized Hamming Distance
(c)

0.252

1

Figure 2. Hamming distance distribution of addition for 10,000bit hypervectors over 3000 cases. (a) Addition over odd number
of hypervectors; (b) and (c) shows the addition over even number favoring 0 and 1, respectively.
(a) A + B + C = X.
(b) A + B = X in favor of 0.
(c) A + B = X in favor of 1.

is approximately orthogonal to both A and B, where 5
represents the XOR operation. Eq. (6) shows the pointwise multiplication of two 10-bit binary vectors. In a
IEEE CIRCUITS AND SYSTEMS MAGAZINE	

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IEEE Circuits and Systems Magazine - Q2 2020

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