IEEE Circuits and Systems Magazine - Q1 2021 - 21

Lemma 4 shows that multiplication is distributive over
addition, and it implements a mapping that preserves distance. These are useful properties of hyperdimensional
arithmetic. By keeping the relation between the vectors,
multiplication preserves the relation between entities.
Lemma 4
Multiplication is distributive over addition and implements
a mapping that preserves distance, i.e., mapping two vectors ^ X, Y h with the same vector M in the hyperdimensional
space yields:
d ^ X M , Y M h = d ^ X, Y h

(34)

Proof. If the bipolar ^-1, 1 h representation is adopted,
the XOR operator is replaced by regular multiplication.
Since it distributes over ordinary addition, it also does
so in this bipolar case. The symbol ; ; in (35) denotes
normalization.
M ) ; X +Y ; = ;M ) X + M )Y ;

(35)

By mapping two vectors ^ X, Y h with the same vector M,
the relation between the vectors is maintained as shown
in (36):
X M ) YM = ^ M ) X h ) ^ M ) Y h = X ) Y
< X M ) Y M < & d ^ X M , Y M h = d ^ X, Y h

(36)

Y
and Lemma 4 is proved.
Permutation changes the order of vector components.
The permutation can be described as a list of integers
^1, 2, 3, f, 10000 h in the permuted order or represented

as multiplication by a special kind of matrix. This permutation matrix will have all '0's except exactly one '1' in
each row and each column. Similar to vector multiplication, when a permutation is applied to different vectors,
the distance between them is maintained. As with sets,
elements of a sequence can be represented by a single
hypervector in a process called flattening. However, if
sequences are flattened by the sum, the order of the elements will be lost. Thus, the elements must be " labeled "
according to their position in the sequence so that the
value of X appears under different variants depending
on the position in the sequence, which can be done with
permutations [70]. Sequences in time are important to
introduce memory in the systems, which can also be
represented with pointer chains or linked lists [70].
The HDC has several known applications. For example, sequence prediction based on sparse hyperdimensional coding is used for online learning and prediction
[71], which can be useful for predicting mobile phone use
patterns, including the prediction of the next launched
application and the next GPS location of a user. Another
example of an HDC application is the learning and classification of biosignals, such as electromyography (EMG),
electroencephalography (EEG), and electrocorticography (ECoG) [72]. A full set of Hyper-Dimensional (HD)
network templates comprehensively encodes body potentials and brain neural activity recorded from different electrodes within a single HD, processed as a single
entity for learning and robust classification purposes.
The diagram of a programmable processor based
on HDC for supervised classification [73] is presented
in Fig. 11. This HDC processor encompasses three main

Item Memory
Item 1

Address
In

In

Address
Out

ARGMIN

Out

Controller

HD Mapper

Item N

dH

Class X

dH

Class Y

DPU

DPU

DPU

DPU

DPU

DPU

DPU

DPU

DPU
Encoder

Associative Memory
Figure 11. The Generic HDC-based Processor: major components and dataflow [73]-Data Processing Unit (DPU).

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