Hopfield Networks - Updated

	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S
1	Hopfield Networks
2	Plus and Minus Sign Recognition
3	M. Ali Yousuf
4	First version: circa 1998. Last updated: June 17, 2025
5	https://www.nobelprize.org/prizes/physics/2024/hopfield/facts/
6
7	Let's look at two simple patterns (a plus sign and a minus sign)
8
9		-1	1	-1
10	V+ =	1	1	1		V+ =	-1	1	-1	1	1	1	-1	1	-1
11		-1	1	-1
12
13		-1	-1	-1
14	V- =	1	1	1		V- =	-1	-1	-1	1	1	1	-1	-1	-1
15		-1	-1	-1
16
17	So
18			=	-1
19	and
20
21			=	1
22
23	Claim: The following matrix can recall these two patterns, even if they are noisy
24
25
26			2	0	2	-2	-2	-2	2	0	2
27			0	2	0	0	0	0	0	2	0
28			2	0	2	-2	-2	-2	2	0	2
29		M =	-2	0	-2	2	2	2	-2	0	-2
30			-2	0	-2	2	2	2	-2	0	-2
31			-2	0	-2	2	2	2	-2	0	-2
32			2	0	2	-2	-2	-2	2	0	2
33			0	2	0	0	0	0	0	2	0
34			2	0	2	-2	-2	-2	2	0	2
35
36
37	Now lets try a new (noisy) vector and see if we can recover the true pattern.
38
39		1	-1	-1
40		1	1	1		T =	1	-1	-1	1	1	1	-1	-1	-1
41		-1	-1	-1
42
43
44	We are going to calculate the matrix product: MT
45	(The Threshold is 0, anything greater than 0 is one)
46
47	2	0	2	-2	-2	-2	2	0	2		1		-10		-1
48	0	2	0	0	0	0	0	2	0		-1		-4		-1
49	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
50	-2	0	-2	2	2	2	-2	0	-2	X	1		10		1		-1	-1	-1
51	-2	0	-2	2	2	2	-2	0	-2		1	=	10	-->	1	-->	1	1	1
52	-2	0	-2	2	2	2	-2	0	-2		1		10		1		-1	-1	-1
53	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
54	0	2	0	0	0	0	0	2	0		-1		-4		-1
55	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
56
57	i.e., a minus sign
58
59	It is easy to check that we can recover the original vectors also:
60	V+
61	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
62	0	2	0	0	0	0	0	2	0		1		4		1
63	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
64	-2	0	-2	2	2	2	-2	0	-2		1		14		1		-1	1	-1
65	-2	0	-2	2	2	2	-2	0	-2		1	=	14	-->	1		1	1	1
66	-2	0	-2	2	2	2	-2	0	-2		1		14		1		-1	1	-1
67	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
68	0	2	0	0	0	0	0	2	0		1		4		1
69	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
70
71	and	V-
72	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
73	0	2	0	0	0	0	0	2	0		-1		-4		-1
74	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
75	-2	0	-2	2	2	2	-2	0	-2		1		14		1		-1	-1	-1
76	-2	0	-2	2	2	2	-2	0	-2		1	=	14	-->	1		1	1	1
77	-2	0	-2	2	2	2	-2	0	-2		1		14		1		-1	-1	-1
78	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
79	0	2	0	0	0	0	0	2	0		-1		-4		-1
80	2	0	2	-2	-2	-2	2	0	2		-1		-14		-1
81
82	Try a distorted minus sign
83
84		-1	-1	-1
85		1	1	-1		T =	-1	-1	-1	1	1	-1	-1	-1	-1
86		-1	-1	-1
87
88	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
89	0	2	0	0	0	0	0	2	0		-1		-4		-1
90	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
91	-2	0	-2	2	2	2	-2	0	-2		1		10		1		-1	-1	-1
92	-2	0	-2	2	2	2	-2	0	-2		1	=	10	-->	1		1	1	1
93	-2	0	-2	2	2	2	-2	0	-2		-1		10		1		-1	-1	-1
94	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
95	0	2	0	0	0	0	0	2	0		-1		-4		-1
96	2	0	2	-2	-2	-2	2	0	2		-1		-10		-1
97
98
99	Lets try the inverse of a minus sign
100