1.
The symbiotic relationship
between neuroscience and
machine learning
Dr Emma C. Robinson
@emrobSci
https://emmarobinson01.com/
https://kclpure.kcl.ac.uk/portal/emma.robinson.html
1

2.
Overview
!2
1. How understanding of biological neural networks
inspires design of artificial intelligence systems
2. The application of machine learning to improve
understanding the human brain
3. Applying artificial neural networks to problems in
neuroscience

3.
Artificial Neurons
!3
Biological neuron

4.
Artificial Neurons
!4
Biological neuron
Artificial neuron:
http://cs231n.github.io/neural-networks-1/

5.
Artificial Neurons
!5 http://cs231n.github.io/neural-networks-1/
Linear Classifier:
E.g. f uses sigmoid function
Where
Squashes function to
range 0->1
=
1
1 + ez
<latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit><latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit><latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit><latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit>
z = !0x0 + !1x1 + !2x2 + b<latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit><latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit><latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit><latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit>
where

6.
Artificial Neurons
!5 http://cs231n.github.io/neural-networks-1/
Linear Classifier:
E.g. f uses sigmoid function
Where
Squashes function to
range 0->1
=
1
1 + ez
<latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit><latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit><latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit><latexit sha1_base64="PVzq+gtkigSOdx15V3TmmjGK7Mk=">AAAB/nicbVBNS8NAEN3Ur1q/ooIXL4tFEISSiKAehKIXjxWMLTSxbLabdunuJuxuhBpz8K948aDi1d/hzX/jts1BWx8MPN6bYWZemDCqtON8W6W5+YXFpfJyZWV1bX3D3ty6VXEqMfFwzGLZCpEijAriaaoZaSWSIB4y0gwHlyO/eU+korG40cOEBBz1BI0oRtpIHXvHV7TH0bkfSYQzN8/cQ3L3kHfsqlNzxoCzxC1IFRRodOwvvxvjlBOhMUNKtV0n0UGGpKaYkbzip4okCA9Qj7QNFYgTFWTj+3O4b5QujGJpSmg4Vn9PZIgrNeSh6eRI99W0NxL/89qpjk6DjIok1UTgyaIoZVDHcBQG7FJJsGZDQxCW1NwKcR+ZJLSJrGJCcKdfniXeUe2s5l4fV+sXRRplsAv2wAFwwQmogyvQAB7A4BE8g1fwZj1ZL9a79TFpLVnFzDb4A+vzBxwjlbc=</latexit>
z = !0x0 + !1x1 + !2x2 + b<latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit><latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit><latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit><latexit sha1_base64="crPtn9fuq+aJ/QiGd+Y58irm3j4=">AAACFnicbVBNS8MwGE79nPOr6tFLcAjCoLRDUA/C0IvHCdYNtlLSLN3C0qQkqTjL/oUX/4oXDypexZv/xmzrQTdfCDwf78ub94lSRpV23W9rYXFpeWW1tFZe39jc2rZ3dm+VyCQmPhZMyFaEFGGUE19TzUgrlQQlESPNaHA59pt3RCoq+I0epiRIUI/TmGKkjRTazsN5RySkh0IX3odutSCeIR6swoLWDK0ZGoV2xXXcScF54BWgAopqhPZXpytwlhCuMUNKtT031UGOpKaYkVG5kymSIjxAPdI2kKOEqCCf3DWCh0bpwlhI87iGE/X3RI4SpYZJZDoTpPtq1huL/3ntTMenQU55mmnC8XRRnDGoBRyHBLtUEqzZ0ACEJTV/hbiPJMLaRFk2IXizJ88Dv+acOd71caV+UaRRAvvgABwBD5yAOrgCDeADDB7BM3gFb9aT9WK9Wx/T1gWrmNkDf8r6/AHN0py8</latexit>
where

7.
Artificial Neurons
!6
Linear Classifier:
w1x1 + w2x2 + w0 = 0<latexit sha1_base64="M6eAlkt6VPk6r5STizNG9EQwvk8=">AAAB/HicbVDLSgMxFM34rPU1PnZugkUQhCEzWtouhKIblxUcW2iHIZOmbWjmQZKxraX4K25cqLj1Q9z5N6YPRK0HLvdwzr3k5gQJZ1Ih9GksLC4tr6xm1rLrG5tb2+bO7q2MU0GoS2Iei1qAJeUsoq5iitNaIigOA06rQfdy7FfvqJAsjm7UIKFeiNsRazGClZZ8c7/n233fPun5Tt93dEPnKOubOWTZp04hX4DIyjtFVHKgbaEJvkkOzFDxzY9GMyZpSCNFOJaybqNEeUMsFCOcjrKNVNIEky5u07qmEQ6p9IaT60fwSCtN2IqFrkjBifpzY4hDKQdhoCdDrDryrzcW//PqqWoVvSGLklTRiEwfaqUcqhiOo4BNJihRfKAJJoLpWyHpYIGJ0oGNQ5j78jxxHatk2ddnufLFLI0MOACH4BjYoADK4ApUgAsIuAeP4Bm8GA/Gk/FqvE1HF4zZzh74BeP9C19jk24=</latexit><latexit sha1_base64="M6eAlkt6VPk6r5STizNG9EQwvk8=">AAAB/HicbVDLSgMxFM34rPU1PnZugkUQhCEzWtouhKIblxUcW2iHIZOmbWjmQZKxraX4K25cqLj1Q9z5N6YPRK0HLvdwzr3k5gQJZ1Ih9GksLC4tr6xm1rLrG5tb2+bO7q2MU0GoS2Iei1qAJeUsoq5iitNaIigOA06rQfdy7FfvqJAsjm7UIKFeiNsRazGClZZ8c7/n233fPun5Tt93dEPnKOubOWTZp04hX4DIyjtFVHKgbaEJvkkOzFDxzY9GMyZpSCNFOJaybqNEeUMsFCOcjrKNVNIEky5u07qmEQ6p9IaT60fwSCtN2IqFrkjBifpzY4hDKQdhoCdDrDryrzcW//PqqWoVvSGLklTRiEwfaqUcqhiOo4BNJihRfKAJJoLpWyHpYIGJ0oGNQ5j78jxxHatk2ddnufLFLI0MOACH4BjYoADK4ApUgAsIuAeP4Bm8GA/Gk/FqvE1HF4zZzh74BeP9C19jk24=</latexit><latexit sha1_base64="M6eAlkt6VPk6r5STizNG9EQwvk8=">AAAB/HicbVDLSgMxFM34rPU1PnZugkUQhCEzWtouhKIblxUcW2iHIZOmbWjmQZKxraX4K25cqLj1Q9z5N6YPRK0HLvdwzr3k5gQJZ1Ih9GksLC4tr6xm1rLrG5tb2+bO7q2MU0GoS2Iei1qAJeUsoq5iitNaIigOA06rQfdy7FfvqJAsjm7UIKFeiNsRazGClZZ8c7/n233fPun5Tt93dEPnKOubOWTZp04hX4DIyjtFVHKgbaEJvkkOzFDxzY9GMyZpSCNFOJaybqNEeUMsFCOcjrKNVNIEky5u07qmEQ6p9IaT60fwSCtN2IqFrkjBifpzY4hDKQdhoCdDrDryrzcW//PqqWoVvSGLklTRiEwfaqUcqhiOo4BNJihRfKAJJoLpWyHpYIGJ0oGNQ5j78jxxHatk2ddnufLFLI0MOACH4BjYoADK4ApUgAsIuAeP4Bm8GA/Gk/FqvE1HF4zZzh74BeP9C19jk24=</latexit><latexit sha1_base64="M6eAlkt6VPk6r5STizNG9EQwvk8=">AAAB/HicbVDLSgMxFM34rPU1PnZugkUQhCEzWtouhKIblxUcW2iHIZOmbWjmQZKxraX4K25cqLj1Q9z5N6YPRK0HLvdwzr3k5gQJZ1Ih9GksLC4tr6xm1rLrG5tb2+bO7q2MU0GoS2Iei1qAJeUsoq5iitNaIigOA06rQfdy7FfvqJAsjm7UIKFeiNsRazGClZZ8c7/n233fPun5Tt93dEPnKOubOWTZp04hX4DIyjtFVHKgbaEJvkkOzFDxzY9GMyZpSCNFOJaybqNEeUMsFCOcjrKNVNIEky5u07qmEQ6p9IaT60fwSCtN2IqFrkjBifpzY4hDKQdhoCdDrDryrzcW//PqqWoVvSGLklTRiEwfaqUcqhiOo4BNJihRfKAJJoLpWyHpYIGJ0oGNQ5j78jxxHatk2ddnufLFLI0MOACH4BjYoADK4ApUgAsIuAeP4Bm8GA/Gk/FqvE1HF4zZzh74BeP9C19jk24=</latexit>
+
+
+ +
+
-
-
-
-
- - -
w1x1 + w2x2 + w0 > 0<latexit sha1_base64="YSO/gynGP/X0WLp/5vtFOjEa/F0=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpgrqRohuXFYwttCFMppN26OTBzMS2luKvuHGh4tYPceffOGmz0NYDl3s4517mzvETzqSyrG9jYXFpeWW1sFZc39jc2jZ3du9lnApCHRLzWDR8LClnEXUUU5w2EkFx6HNa93vXmV9/oEKyOLpTw4S6Ie5ELGAEKy155n7fsweefdL3KgOvopt1aRU9s2SVrQnQPLFzUoIcNc/8arVjkoY0UoRjKZu2lSh3hIVihNNxsZVKmmDSwx3a1DTCIZXuaHL9GB1ppY2CWOiKFJqovzdGOJRyGPp6MsSqK2e9TPzPa6YqOHdHLEpSRSMyfShIOVIxyqJAbSYoUXyoCSaC6VsR6WKBidKBZSHYs1+eJ06lfFG2b09L1as8jQIcwCEcgw1nUIUbqIEDBB7hGV7hzXgyXox342M6umDkO3vwB8bnDxURkzs=</latexit><latexit sha1_base64="YSO/gynGP/X0WLp/5vtFOjEa/F0=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpgrqRohuXFYwttCFMppN26OTBzMS2luKvuHGh4tYPceffOGmz0NYDl3s4517mzvETzqSyrG9jYXFpeWW1sFZc39jc2jZ3du9lnApCHRLzWDR8LClnEXUUU5w2EkFx6HNa93vXmV9/oEKyOLpTw4S6Ie5ELGAEKy155n7fsweefdL3KgOvopt1aRU9s2SVrQnQPLFzUoIcNc/8arVjkoY0UoRjKZu2lSh3hIVihNNxsZVKmmDSwx3a1DTCIZXuaHL9GB1ppY2CWOiKFJqovzdGOJRyGPp6MsSqK2e9TPzPa6YqOHdHLEpSRSMyfShIOVIxyqJAbSYoUXyoCSaC6VsR6WKBidKBZSHYs1+eJ06lfFG2b09L1as8jQIcwCEcgw1nUIUbqIEDBB7hGV7hzXgyXox342M6umDkO3vwB8bnDxURkzs=</latexit><latexit sha1_base64="YSO/gynGP/X0WLp/5vtFOjEa/F0=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpgrqRohuXFYwttCFMppN26OTBzMS2luKvuHGh4tYPceffOGmz0NYDl3s4517mzvETzqSyrG9jYXFpeWW1sFZc39jc2jZ3du9lnApCHRLzWDR8LClnEXUUU5w2EkFx6HNa93vXmV9/oEKyOLpTw4S6Ie5ELGAEKy155n7fsweefdL3KgOvopt1aRU9s2SVrQnQPLFzUoIcNc/8arVjkoY0UoRjKZu2lSh3hIVihNNxsZVKmmDSwx3a1DTCIZXuaHL9GB1ppY2CWOiKFJqovzdGOJRyGPp6MsSqK2e9TPzPa6YqOHdHLEpSRSMyfShIOVIxyqJAbSYoUXyoCSaC6VsR6WKBidKBZSHYs1+eJ06lfFG2b09L1as8jQIcwCEcgw1nUIUbqIEDBB7hGV7hzXgyXox342M6umDkO3vwB8bnDxURkzs=</latexit><latexit sha1_base64="YSO/gynGP/X0WLp/5vtFOjEa/F0=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpgrqRohuXFYwttCFMppN26OTBzMS2luKvuHGh4tYPceffOGmz0NYDl3s4517mzvETzqSyrG9jYXFpeWW1sFZc39jc2jZ3du9lnApCHRLzWDR8LClnEXUUU5w2EkFx6HNa93vXmV9/oEKyOLpTw4S6Ie5ELGAEKy155n7fsweefdL3KgOvopt1aRU9s2SVrQnQPLFzUoIcNc/8arVjkoY0UoRjKZu2lSh3hIVihNNxsZVKmmDSwx3a1DTCIZXuaHL9GB1ppY2CWOiKFJqovzdGOJRyGPp6MsSqK2e9TPzPa6YqOHdHLEpSRSMyfShIOVIxyqJAbSYoUXyoCSaC6VsR6WKBidKBZSHYs1+eJ06lfFG2b09L1as8jQIcwCEcgw1nUIUbqIEDBB7hGV7hzXgyXox342M6umDkO3vwB8bnDxURkzs=</latexit>
w1x1 + w2x2 + w0 < 0<latexit sha1_base64="8Kx3hYRWdh+HI8oqkpTUnhITmPs=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpggouim5cVjC20IYwmU7aoZMHMxPbWoq/4saFils/xJ1/46TNQlsPXO7hnHuZO8dPOJPKsr6NhcWl5ZXVwlpxfWNza9vc2b2XcSoIdUjMY9HwsaScRdRRTHHaSATFoc9p3e9dZ379gQrJ4uhODRPqhrgTsYARrLTkmft9zx549knfqwy8im7WpVX0zJJVtiZA88TOSQly1Dzzq9WOSRrSSBGOpWzaVqLcERaKEU7HxVYqaYJJD3doU9MIh1S6o8n1Y3SklTYKYqErUmii/t4Y4VDKYejryRCrrpz1MvE/r5mq4NwdsShJFY3I9KEg5UjFKIsCtZmgRPGhJpgIpm9FpIsFJkoHloVgz355njiV8kXZvj0tVa/yNApwAIdwDDacQRVuoAYOEHiEZ3iFN+PJeDHejY/p6IKR7+zBHxifPxIHkzk=</latexit><latexit sha1_base64="8Kx3hYRWdh+HI8oqkpTUnhITmPs=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpggouim5cVjC20IYwmU7aoZMHMxPbWoq/4saFils/xJ1/46TNQlsPXO7hnHuZO8dPOJPKsr6NhcWl5ZXVwlpxfWNza9vc2b2XcSoIdUjMY9HwsaScRdRRTHHaSATFoc9p3e9dZ379gQrJ4uhODRPqhrgTsYARrLTkmft9zx549knfqwy8im7WpVX0zJJVtiZA88TOSQly1Dzzq9WOSRrSSBGOpWzaVqLcERaKEU7HxVYqaYJJD3doU9MIh1S6o8n1Y3SklTYKYqErUmii/t4Y4VDKYejryRCrrpz1MvE/r5mq4NwdsShJFY3I9KEg5UjFKIsCtZmgRPGhJpgIpm9FpIsFJkoHloVgz355njiV8kXZvj0tVa/yNApwAIdwDDacQRVuoAYOEHiEZ3iFN+PJeDHejY/p6IKR7+zBHxifPxIHkzk=</latexit><latexit sha1_base64="8Kx3hYRWdh+HI8oqkpTUnhITmPs=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpggouim5cVjC20IYwmU7aoZMHMxPbWoq/4saFils/xJ1/46TNQlsPXO7hnHuZO8dPOJPKsr6NhcWl5ZXVwlpxfWNza9vc2b2XcSoIdUjMY9HwsaScRdRRTHHaSATFoc9p3e9dZ379gQrJ4uhODRPqhrgTsYARrLTkmft9zx549knfqwy8im7WpVX0zJJVtiZA88TOSQly1Dzzq9WOSRrSSBGOpWzaVqLcERaKEU7HxVYqaYJJD3doU9MIh1S6o8n1Y3SklTYKYqErUmii/t4Y4VDKYejryRCrrpz1MvE/r5mq4NwdsShJFY3I9KEg5UjFKIsCtZmgRPGhJpgIpm9FpIsFJkoHloVgz355njiV8kXZvj0tVa/yNApwAIdwDDacQRVuoAYOEHiEZ3iFN+PJeDHejY/p6IKR7+zBHxifPxIHkzk=</latexit><latexit sha1_base64="8Kx3hYRWdh+HI8oqkpTUnhITmPs=">AAAB/HicbVDLSsNAFL3xWesrPnZuBosgCCUpggouim5cVjC20IYwmU7aoZMHMxPbWoq/4saFils/xJ1/46TNQlsPXO7hnHuZO8dPOJPKsr6NhcWl5ZXVwlpxfWNza9vc2b2XcSoIdUjMY9HwsaScRdRRTHHaSATFoc9p3e9dZ379gQrJ4uhODRPqhrgTsYARrLTkmft9zx549knfqwy8im7WpVX0zJJVtiZA88TOSQly1Dzzq9WOSRrSSBGOpWzaVqLcERaKEU7HxVYqaYJJD3doU9MIh1S6o8n1Y3SklTYKYqErUmii/t4Y4VDKYejryRCrrpz1MvE/r5mq4NwdsShJFY3I9KEg5UjFKIsCtZmgRPGhJpgIpm9FpIsFJkoHloVgz355njiV8kXZvj0tVa/yNApwAIdwDDacQRVuoAYOEHiEZ3iFN+PJeDHejY/p6IKR7+zBHxifPxIHkzk=</latexit>
Set:
y=1 if f(x)>0 .5
y=0 if f(x)<0 .5
Therefore:
y=1 if z> 0
y=0 if z<0
Optimise for w
C =
1
n
X
x
y ln( (z))+
(1 y) ln(1 (z))<latexit sha1_base64="cLaIz7Of9aZ2LEqmMCk2O7roAiw=">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</latexit><latexit sha1_base64="cLaIz7Of9aZ2LEqmMCk2O7roAiw=">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</latexit><latexit sha1_base64="cLaIz7Of9aZ2LEqmMCk2O7roAiw=">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</latexit><latexit sha1_base64="cLaIz7Of9aZ2LEqmMCk2O7roAiw=">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</latexit>
1
1 + ez
<latexit sha1_base64="EHt0D5r5E17mwuH7HaOlapKR6bA=">AAAB+HicbVBNS8NAEN34WeNHox69LBZBEEoignorevFYwdhCG8tmO2mXbjZhdyPUkF/iSVDEq3/Fk//GbZuDtj4YeLw3w8y8MOVMadf9tpaWV1bX1isb9ubW9k7V2d27V0kmKfg04Ylsh0QBZwJ8zTSHdiqBxCGHVji6nvitR5CKJeJOj1MIYjIQLGKUaCP1nGo3koTmXpF7J/DwVPScmlt3p8CLxCtJDZVo9pyvbj+hWQxCU06U6nhuqoOcSM0oh8LuZgpSQkdkAB1DBYlBBfn08AIfGaWPo0SaEhpP1d8TOYmVGseh6YyJHqp5byL+53UyHV0EORNppkHQ2aIo41gneJIC7jMJVPOxIYRKZm7FdEhMEtpkZdsmBW/+50Xin9Yv697tWa1xVcZRQQfoEB0jD52jBrpBTeQjijL0jF7Rm5VbL9a79TFrXbLKmX30B9bnD2irktc=</latexit><latexit sha1_base64="EHt0D5r5E17mwuH7HaOlapKR6bA=">AAAB+HicbVBNS8NAEN34WeNHox69LBZBEEoignorevFYwdhCG8tmO2mXbjZhdyPUkF/iSVDEq3/Fk//GbZuDtj4YeLw3w8y8MOVMadf9tpaWV1bX1isb9ubW9k7V2d27V0kmKfg04Ylsh0QBZwJ8zTSHdiqBxCGHVji6nvitR5CKJeJOj1MIYjIQLGKUaCP1nGo3koTmXpF7J/DwVPScmlt3p8CLxCtJDZVo9pyvbj+hWQxCU06U6nhuqoOcSM0oh8LuZgpSQkdkAB1DBYlBBfn08AIfGaWPo0SaEhpP1d8TOYmVGseh6YyJHqp5byL+53UyHV0EORNppkHQ2aIo41gneJIC7jMJVPOxIYRKZm7FdEhMEtpkZdsmBW/+50Xin9Yv697tWa1xVcZRQQfoEB0jD52jBrpBTeQjijL0jF7Rm5VbL9a79TFrXbLKmX30B9bnD2irktc=</latexit><latexit sha1_base64="EHt0D5r5E17mwuH7HaOlapKR6bA=">AAAB+HicbVBNS8NAEN34WeNHox69LBZBEEoignorevFYwdhCG8tmO2mXbjZhdyPUkF/iSVDEq3/Fk//GbZuDtj4YeLw3w8y8MOVMadf9tpaWV1bX1isb9ubW9k7V2d27V0kmKfg04Ylsh0QBZwJ8zTSHdiqBxCGHVji6nvitR5CKJeJOj1MIYjIQLGKUaCP1nGo3koTmXpF7J/DwVPScmlt3p8CLxCtJDZVo9pyvbj+hWQxCU06U6nhuqoOcSM0oh8LuZgpSQkdkAB1DBYlBBfn08AIfGaWPo0SaEhpP1d8TOYmVGseh6YyJHqp5byL+53UyHV0EORNppkHQ2aIo41gneJIC7jMJVPOxIYRKZm7FdEhMEtpkZdsmBW/+50Xin9Yv697tWa1xVcZRQQfoEB0jD52jBrpBTeQjijL0jF7Rm5VbL9a79TFrXbLKmX30B9bnD2irktc=</latexit><latexit sha1_base64="EHt0D5r5E17mwuH7HaOlapKR6bA=">AAAB+HicbVBNS8NAEN34WeNHox69LBZBEEoignorevFYwdhCG8tmO2mXbjZhdyPUkF/iSVDEq3/Fk//GbZuDtj4YeLw3w8y8MOVMadf9tpaWV1bX1isb9ubW9k7V2d27V0kmKfg04Ylsh0QBZwJ8zTSHdiqBxCGHVji6nvitR5CKJeJOj1MIYjIQLGKUaCP1nGo3koTmXpF7J/DwVPScmlt3p8CLxCtJDZVo9pyvbj+hWQxCU06U6nhuqoOcSM0oh8LuZgpSQkdkAB1DBYlBBfn08AIfGaWPo0SaEhpP1d8TOYmVGseh6YyJHqp5byL+53UyHV0EORNppkHQ2aIo41gneJIC7jMJVPOxIYRKZm7FdEhMEtpkZdsmBW/+50Xin9Yv697tWa1xVcZRQQfoEB0jD52jBrpBTeQjijL0jF7Rm5VbL9a79TFrXbLKmX30B9bnD2irktc=</latexit>

8.
Artificial Neural Networks
!7
Neural networks with hidden layers can approximate complex non-linear
functions
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Layer 1
3 neurons
Layer 2
2 neurons
Multi Layer Perceptron (MLP)

9.
Artificial Neural Networks
!8
https://medium.com/@vivek.yadav/how-
neural-networks-learn-nonlinear-
functions-and-classify-linearly-non-
separable-data-22328e7e5be1
• 3 hidden nodes
• learn 3 linear decision
boundaries
• Combined by the output
function
Neural nets as universal
approximators: http://
neuralnetworksanddeeplearning.com/
chap4.html
Neural network playground
(tensorflow):
https://goo.gl/qAaR1i

10.
Image classification
!9
e.g.
• Object recognition
• Localisation
• Semantic segmentation
Artificial Neural Networks

11.
Image classification
!9
e.g.
• Object recognition
• Localisation
• Semantic segmentation
Artificial Neural Networks
Training of fully connected networks on images would require
prohibitive numbers of parameters

12.
Convolutional neural networks reflect
the brain’s visual processing system
!10
Van Essen, David C., and Jack L. Gallant. "Neural mechanisms of form and motion processing in the primate visual
system." Neuron 13.1 (1994): 1-10.

13.
Convolutional neural networks reflect
the brain’s visual processing system
!11
Yamins, Daniel LK, and James J. DiCarlo. "Using goal-driven deep learning models to understand sensory cortex." Nature
neuroscience 19.3 (2016): 356.

14.
Convolutional network design
!12
Convolutional Layer

15.
Convolutional network design
!12
Convolutional Layer
• Learns filters (kernel weights w, size [h,w,d])
d=input layer depth i.e. 3 for RGB
• Via convolutions strided across image
• Output y= activation map
Each output ‘neuron’ ‘sees’ only
one location in space

16.
Convolutional network design
!12
Convolutional Layer
• Learns filters (kernel weights w, size [h,w,d])
d=input layer depth i.e. 3 for RGB
• Via convolutions strided across image
• Output y= activation map
Each output ‘neuron’ ‘sees’ only
one location in space
• Parameters are shared between
neurons
Learns consistent set of filters
Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. "Imagenet classification with deep convolutional neural
networks." Advances in neural information processing systems. 2012.

17.
Convolutional network design
!13
Layers (classification network)
• INPUT
• CONV
• POOL/STRIDED CONVOLUTION
• MLP (fully connected)

18.
Convolutional network design
!13
Layers (classification network)
• INPUT
• CONV
• POOL/STRIDED CONVOLUTION
• MLP (fully connected)
Multiple blocks

19.
Convolutional network design
!13
Layers (classification network)
• INPUT
• CONV
• POOL/STRIDED CONVOLUTION
• MLP (fully connected)
Multiple blocks

20.
Convolutional networks mimics
spatial smoothness observed for fMRI
!14

21.
Convolutional networks mimics
spatial smoothness observed for fMRI
!14
• Functional MRI is low
spatial and temporal
resolution
Brain activation maps are coarse and smooth

22.
Convolutional networks mimics
spatial smoothness observed for fMRI
!14
• Functional MRI is low
spatial and temporal
resolution
• But still can decode
visual representations
Brain activation maps are coarse and smooth
Nishimoto, Shinji, et al. "Reconstructing visual experiences from brain activity evoked by natural movies." Current Biology21.19 (2011): 1641-1646.

23.
Convolutional networks mimics
spatial smoothness observed for fMRI
!14
• Functional MRI is low
spatial and temporal
resolution
• But still can decode
visual representations
• Thus neural code must
be coarse &
functionally smooth
(at some level)
Brain activation maps are coarse and smooth
Nishimoto, Shinji, et al. "Reconstructing visual experiences from brain activity evoked by natural movies." Current Biology21.19 (2011): 1641-1646.

24.
Convolutional networks mimics
spatial smoothness observed for fMRI
!14
• Functional MRI is low
spatial and temporal
resolution
• But still can decode
visual representations
• Thus neural code must
be coarse &
functionally smooth
(at some level)
• This also is true of
convolutional
networks
Guest, Olivia, and Bradley C. Love. "What the success of brain imaging implies about the neural code." Elife 6 (2017).
Brain activation maps are coarse and smooth
Network representations of stimulus similarity
fall away with addition of distortion as expected

25.
Enhancing AI through even more sophisticated
models of neurological processes
!15
Models how biological vision
• Factorised representation of shape
and appearance
• Has lateral connections
• Feedforward & feedback
connections
Within loopy belief propagation
inference model
Recursive Cortical Network
George, Dileep, et al. "A generative vision model that trains with high data efficiency and breaks text-based
CAPTCHAs." Science 358.6368 (2017): eaag2612.

26.
Enhancing AI through even more sophisticated
models of neurological processes
!15
Models how biological vision
• Factorised representation of shape
and appearance
• Has lateral connections
• Feedforward & feedback
connections
Within loopy belief propagation
inference model
Recursive Cortical Network
George, Dileep, et al. "A generative vision model that trains with high data efficiency and breaks text-based
CAPTCHAs." Science 358.6368 (2017): eaag2612.
Generalises better to complex object recognition tasks e.g. captcha recognition

27.
!16
A. Attentional mechanisms
(glimpses
B. Artificial hippocampi for
Deep RL algorithms
C. LSTMs build from
working memory
systems
D. Brain plasticity
modelled by elastic
weight consolidation
(EWC) algorithms
Enhancing AI through even more sophisticated
models of neurological processes
Hassabis, Demis, et al. "Neuroscience-inspired artificial intelligence."
Neuron 95.2 (2017): 245-258.

28.
The limits of current AI relative to
the capabilities of the human brain
!17
The human brain is a vastly complex system
• Multiple forward and feedback systems
• Parallel processing of multiple different interacting
systems
Roelfsema, Pieter R., and Anthony Holtmaat. "Control of synaptic plasticity in deep cortical networks." Nature
Reviews Neuroscience 19.3 (2018): 166.

29.
The limits of understanding of the
human brain
!18
• 100 billion nerve cells; 100 trillion connections
• Vast scales involved make comprehensive making of even a
single human brain impossible

30.
Studying the brain over multiple
scales
!19
2. Coarse scale models of the whole brain:
• Derived from MRI imaging data
Full brain coverage
In vivo
Non invasive
Multiple properties:
Morphology
Function
Microstructure

31.
Studying the brain over multiple
scales
!19
2. Coarse scale models of the whole brain:
• Derived from MRI imaging data
Full brain coverage
In vivo
Non invasive
Multiple properties:
Morphology
Function
Microstructure

32.
1. Data Limitations
The Quest for a Macroscale Model of
Human Brain Networks
!20
• Low spatial resolution
• Voxel sizes 1mm^3 or 2mm^3
• Low Temporal resolution
• fMRI measures ‘haemodynamic
response’ function (HRF)
• Flow of blood to supply nutrients
to neutrons for firing
• Indirect and approximate
HRF

33.
Macroscale network models
!21
Brain network model:
• Functionally specialised
regions
• Connected through
axon bundles
• Complex brain
functions result from
coordinated actions of
multiple regions

34.
!22
i. Reconstruction
ii. Denoising/artifact correction
iii. Image registration
iv. Modelling of the signal
2. Processing
The Quest for a Macroscale Model of
Human Brain Networks

35.
!22
i. Reconstruction
ii. Denoising/artifact correction
iii. Image registration
iv. Modelling of the signal
2. Processing
The Quest for a Macroscale Model of
Human Brain Networks

36.
ML models for fMRI denoising
!23
The fMRI signal

37.
ML models for fMRI denoising
!24
Sources of noise
• Scanner Artifacts
• Head Motion
• Physiological (breathing/cardiac pulsation)

38.
ML models for fMRI denoising
!24
Sources of noise
• Scanner Artifacts
• Head Motion
• Physiological (breathing/cardiac pulsation)
Corrected separately

39.
ML models for fMRI denoising
!25
• Use Independent Component Analysis (ICA)
• Decompose data (X) into a reduced set of spatial maps and
temporal components
• Train classifier to distinguish true signal from noise
• Regress noise components from X

40.
ML models for fMRI denoising
!26
• ICA assumes linear model of the brain
• Optimises to find matrix A-1 that un-mixes independent
sources S (Spatial ICA)
• Assumption is valid data is spatially sparse
X(t, v) =
LX
l
Al(t)Sl(v)
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S = A 1
X = WX<latexit sha1_base64="hdBfqgKQHcwT9cShfRXFwaIL0xM=">AAACEHicbZDNSsNAFIUn/tb6F3XpZrAI3VgSEdRFoerGZUVjA20sk+mkHTqZhJmJUEJewY2v4saFiluX7nwbJ20EbT0w8HHuvcy9x48Zlcqyvoy5+YXFpeXSSnl1bX1j09zavpVRIjBxcMQi4fpIEkY5cRRVjLixICj0GWn5w4u83ronQtKI36hRTLwQ9TkNKEZKW12z2gmRGvgBvK4XlJ7dpQd25mZ1+OO03KzcNStWzRoLzoJdQAUUanbNz04vwklIuMIMSdm2rVh5KRKKYkaycieRJEZ4iPqkrZGjkEgvHV+UwX3t9GAQCf24gmP390SKQilHoa878x3ldC03/6u1ExWceCnlcaIIx5OPgoRBFcE8HtijgmDFRhoQFlTvCvEACYSVDjEPwZ4+eRacw9ppzb46qjTOizRKYBfsgSqwwTFogEvQBA7A4AE8gRfwajwaz8ab8T5pnTOKmR3wR8bHN6tInHw=</latexit><latexit sha1_base64="hdBfqgKQHcwT9cShfRXFwaIL0xM=">AAACEHicbZDNSsNAFIUn/tb6F3XpZrAI3VgSEdRFoerGZUVjA20sk+mkHTqZhJmJUEJewY2v4saFiluX7nwbJ20EbT0w8HHuvcy9x48Zlcqyvoy5+YXFpeXSSnl1bX1j09zavpVRIjBxcMQi4fpIEkY5cRRVjLixICj0GWn5w4u83ronQtKI36hRTLwQ9TkNKEZKW12z2gmRGvgBvK4XlJ7dpQd25mZ1+OO03KzcNStWzRoLzoJdQAUUanbNz04vwklIuMIMSdm2rVh5KRKKYkaycieRJEZ4iPqkrZGjkEgvHV+UwX3t9GAQCf24gmP390SKQilHoa878x3ldC03/6u1ExWceCnlcaIIx5OPgoRBFcE8HtijgmDFRhoQFlTvCvEACYSVDjEPwZ4+eRacw9ppzb46qjTOizRKYBfsgSqwwTFogEvQBA7A4AE8gRfwajwaz8ab8T5pnTOKmR3wR8bHN6tInHw=</latexit><latexit sha1_base64="hdBfqgKQHcwT9cShfRXFwaIL0xM=">AAACEHicbZDNSsNAFIUn/tb6F3XpZrAI3VgSEdRFoerGZUVjA20sk+mkHTqZhJmJUEJewY2v4saFiluX7nwbJ20EbT0w8HHuvcy9x48Zlcqyvoy5+YXFpeXSSnl1bX1j09zavpVRIjBxcMQi4fpIEkY5cRRVjLixICj0GWn5w4u83ronQtKI36hRTLwQ9TkNKEZKW12z2gmRGvgBvK4XlJ7dpQd25mZ1+OO03KzcNStWzRoLzoJdQAUUanbNz04vwklIuMIMSdm2rVh5KRKKYkaycieRJEZ4iPqkrZGjkEgvHV+UwX3t9GAQCf24gmP390SKQilHoa878x3ldC03/6u1ExWceCnlcaIIx5OPgoRBFcE8HtijgmDFRhoQFlTvCvEACYSVDjEPwZ4+eRacw9ppzb46qjTOizRKYBfsgSqwwTFogEvQBA7A4AE8gRfwajwaz8ab8T5pnTOKmR3wR8bHN6tInHw=</latexit><latexit sha1_base64="hdBfqgKQHcwT9cShfRXFwaIL0xM=">AAACEHicbZDNSsNAFIUn/tb6F3XpZrAI3VgSEdRFoerGZUVjA20sk+mkHTqZhJmJUEJewY2v4saFiluX7nwbJ20EbT0w8HHuvcy9x48Zlcqyvoy5+YXFpeXSSnl1bX1j09zavpVRIjBxcMQi4fpIEkY5cRRVjLixICj0GWn5w4u83ronQtKI36hRTLwQ9TkNKEZKW12z2gmRGvgBvK4XlJ7dpQd25mZ1+OO03KzcNStWzRoLzoJdQAUUanbNz04vwklIuMIMSdm2rVh5KRKKYkaycieRJEZ4iPqkrZGjkEgvHV+UwX3t9GAQCf24gmP390SKQilHoa878x3ldC03/6u1ExWceCnlcaIIx5OPgoRBFcE8HtijgmDFRhoQFlTvCvEACYSVDjEPwZ4+eRacw9ppzb46qjTOizRKYBfsgSqwwTFogEvQBA7A4AE8gRfwajwaz8ab8T5pnTOKmR3wR8bHN6tInHw=</latexit>

41.
ML models for fMRI denoising
!27
• Achieved by optimising for non Gaussianty
e.g. FastICA Hyvärinen 1999
Two stages:
Pre-whitening
Iterative component extraction

42.
ML models for fMRI denoising
!28
• Centre (demean)
• Whiten
• Transform X to s.t
• Performed using eigenvalue
decomposition of covariance matrix (PCA)
Pre-whitening
xij xij
1
M
X
j0
xij
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Y = WWX<latexit sha1_base64="0LhMhzBZ47SDxhG+nUZ0f7ZYIz8=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoB6EohePFYyttCFstpt26WYTdjeFEvpPvHhQ8epP8ea/cdPmoK0DC8PMe7zZCVPOlHacb6uysrq2vlHdrG1t7+zu2fsHjyrJJKEeSXgiOyFWlDNBPc00p51UUhyHnLbD0W3ht8dUKpaIBz1JqR/jgWARI1gbKbDtXoz1MIzyp+t20O5Ma4FddxrODGiZuCWpQ4lWYH/1+gnJYio04Vipruuk2s+x1IxwOq31MkVTTEZ4QLuGChxT5eez5FN0YpQ+ihJpntBopv7eyHGs1CQOzWSRUy16hfif1810dOnnTKSZpoLMD0UZRzpBRQ2ozyQlmk8MwUQykxWRIZaYaFNWUYK7+OVl4p01rhru/Xm9eVO2UYUjOIZTcOECmnAHLfCAwBie4RXerNx6sd6tj/loxSp3DuEPrM8fMkCS6g==</latexit><latexit sha1_base64="0LhMhzBZ47SDxhG+nUZ0f7ZYIz8=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoB6EohePFYyttCFstpt26WYTdjeFEvpPvHhQ8epP8ea/cdPmoK0DC8PMe7zZCVPOlHacb6uysrq2vlHdrG1t7+zu2fsHjyrJJKEeSXgiOyFWlDNBPc00p51UUhyHnLbD0W3ht8dUKpaIBz1JqR/jgWARI1gbKbDtXoz1MIzyp+t20O5Ma4FddxrODGiZuCWpQ4lWYH/1+gnJYio04Vipruuk2s+x1IxwOq31MkVTTEZ4QLuGChxT5eez5FN0YpQ+ihJpntBopv7eyHGs1CQOzWSRUy16hfif1810dOnnTKSZpoLMD0UZRzpBRQ2ozyQlmk8MwUQykxWRIZaYaFNWUYK7+OVl4p01rhru/Xm9eVO2UYUjOIZTcOECmnAHLfCAwBie4RXerNx6sd6tj/loxSp3DuEPrM8fMkCS6g==</latexit><latexit sha1_base64="0LhMhzBZ47SDxhG+nUZ0f7ZYIz8=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoB6EohePFYyttCFstpt26WYTdjeFEvpPvHhQ8epP8ea/cdPmoK0DC8PMe7zZCVPOlHacb6uysrq2vlHdrG1t7+zu2fsHjyrJJKEeSXgiOyFWlDNBPc00p51UUhyHnLbD0W3ht8dUKpaIBz1JqR/jgWARI1gbKbDtXoz1MIzyp+t20O5Ma4FddxrODGiZuCWpQ4lWYH/1+gnJYio04Vipruuk2s+x1IxwOq31MkVTTEZ4QLuGChxT5eez5FN0YpQ+ihJpntBopv7eyHGs1CQOzWSRUy16hfif1810dOnnTKSZpoLMD0UZRzpBRQ2ozyQlmk8MwUQykxWRIZaYaFNWUYK7+OVl4p01rhru/Xm9eVO2UYUjOIZTcOECmnAHLfCAwBie4RXerNx6sd6tj/loxSp3DuEPrM8fMkCS6g==</latexit><latexit sha1_base64="0LhMhzBZ47SDxhG+nUZ0f7ZYIz8=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIoB6EohePFYyttCFstpt26WYTdjeFEvpPvHhQ8epP8ea/cdPmoK0DC8PMe7zZCVPOlHacb6uysrq2vlHdrG1t7+zu2fsHjyrJJKEeSXgiOyFWlDNBPc00p51UUhyHnLbD0W3ht8dUKpaIBz1JqR/jgWARI1gbKbDtXoz1MIzyp+t20O5Ma4FddxrODGiZuCWpQ4lWYH/1+gnJYio04Vipruuk2s+x1IxwOq31MkVTTEZ4QLuGChxT5eez5FN0YpQ+ihJpntBopv7eyHGs1CQOzWSRUy16hfif1810dOnnTKSZpoLMD0UZRzpBRQ2ozyQlmk8MwUQykxWRIZaYaFNWUYK7+OVl4p01rhru/Xm9eVO2UYUjOIZTcOECmnAHLfCAwBie4RXerNx6sd6tj/loxSp3DuEPrM8fMkCS6g==</latexit>
Cov(Y) = I<latexit sha1_base64="nF8UTxFTGSjaXUfDM+2Psmjxn1E=">AAACA3icbZDNSsNAFIUn9a/Wv6jLbgaLUDclEUFdCMVudFfB2EobymQ6aYdOJmFmUighCze+ihsXKm59CXe+jZM2grYeGPg4917m3uNFjEplWV9GYWl5ZXWtuF7a2Nza3jF39+5kGAtMHByyULQ9JAmjnDiKKkbakSAo8BhpeaNGVm+NiZA05LdqEhE3QANOfYqR0lbPLDfCcbUbIDX0/OQ+Pbr44eu01DMrVs2aCi6CnUMF5Gr2zM9uP8RxQLjCDEnZsa1IuQkSimJG0lI3liRCeIQGpKORo4BIN5kekcJD7fShHwr9uIJT9/dEggIpJ4GnO7MV5XwtM/+rdWLln7kJ5VGsCMezj/yYQRXCLBHYp4JgxSYaEBZU7wrxEAmElc4tC8GeP3kRnOPaec2+OanUL/M0iqAMDkAV2OAU1MEVaAIHYPAAnsALeDUejWfjzXiftRaMfGYf/JHx8Q1gUpd7</latexit><latexit sha1_base64="nF8UTxFTGSjaXUfDM+2Psmjxn1E=">AAACA3icbZDNSsNAFIUn9a/Wv6jLbgaLUDclEUFdCMVudFfB2EobymQ6aYdOJmFmUighCze+ihsXKm59CXe+jZM2grYeGPg4917m3uNFjEplWV9GYWl5ZXWtuF7a2Nza3jF39+5kGAtMHByyULQ9JAmjnDiKKkbakSAo8BhpeaNGVm+NiZA05LdqEhE3QANOfYqR0lbPLDfCcbUbIDX0/OQ+Pbr44eu01DMrVs2aCi6CnUMF5Gr2zM9uP8RxQLjCDEnZsa1IuQkSimJG0lI3liRCeIQGpKORo4BIN5kekcJD7fShHwr9uIJT9/dEggIpJ4GnO7MV5XwtM/+rdWLln7kJ5VGsCMezj/yYQRXCLBHYp4JgxSYaEBZU7wrxEAmElc4tC8GeP3kRnOPaec2+OanUL/M0iqAMDkAV2OAU1MEVaAIHYPAAnsALeDUejWfjzXiftRaMfGYf/JHx8Q1gUpd7</latexit><latexit sha1_base64="nF8UTxFTGSjaXUfDM+2Psmjxn1E=">AAACA3icbZDNSsNAFIUn9a/Wv6jLbgaLUDclEUFdCMVudFfB2EobymQ6aYdOJmFmUighCze+ihsXKm59CXe+jZM2grYeGPg4917m3uNFjEplWV9GYWl5ZXWtuF7a2Nza3jF39+5kGAtMHByyULQ9JAmjnDiKKkbakSAo8BhpeaNGVm+NiZA05LdqEhE3QANOfYqR0lbPLDfCcbUbIDX0/OQ+Pbr44eu01DMrVs2aCi6CnUMF5Gr2zM9uP8RxQLjCDEnZsa1IuQkSimJG0lI3liRCeIQGpKORo4BIN5kekcJD7fShHwr9uIJT9/dEggIpJ4GnO7MV5XwtM/+rdWLln7kJ5VGsCMezj/yYQRXCLBHYp4JgxSYaEBZU7wrxEAmElc4tC8GeP3kRnOPaec2+OanUL/M0iqAMDkAV2OAU1MEVaAIHYPAAnsALeDUejWfjzXiftRaMfGYf/JHx8Q1gUpd7</latexit><latexit sha1_base64="nF8UTxFTGSjaXUfDM+2Psmjxn1E=">AAACA3icbZDNSsNAFIUn9a/Wv6jLbgaLUDclEUFdCMVudFfB2EobymQ6aYdOJmFmUighCze+ihsXKm59CXe+jZM2grYeGPg4917m3uNFjEplWV9GYWl5ZXWtuF7a2Nza3jF39+5kGAtMHByyULQ9JAmjnDiKKkbakSAo8BhpeaNGVm+NiZA05LdqEhE3QANOfYqR0lbPLDfCcbUbIDX0/OQ+Pbr44eu01DMrVs2aCi6CnUMF5Gr2zM9uP8RxQLjCDEnZsa1IuQkSimJG0lI3liRCeIQGpKORo4BIN5kekcJD7fShHwr9uIJT9/dEggIpJ4GnO7MV5XwtM/+rdWLln7kJ5VGsCMezj/yYQRXCLBHYp4JgxSYaEBZU7wrxEAmElc4tC8GeP3kRnOPaec2+OanUL/M0iqAMDkAV2OAU1MEVaAIHYPAAnsALeDUejWfjzXiftRaMfGYf/JHx8Q1gUpd7</latexit>
WW = E 1
D 1/2
ET
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⌃ = Cov(X) = EDE 1
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43.
ML models for fMRI denoising
!29
• Optimise for negative entropy
• a gaussian variable has the largest entropy among all
random variables of equal variance (Cover and Thomas, 1991);
• Computationally difficult to estimate; therefore
approximate as
Component Estimation
J(y) = H(ygauss) H(y)<latexit sha1_base64="nGBPMfEXSg7/uQojXGhQ56c9htk=">AAACAXicbVDLSsNAFJ34rPUVdSVuBovQLiyJCOpCKLoprioYW2hDmEwn7dDJJMxMhBCKG3/FjQsVt/6FO//GSZuFth64cOace5l7jx8zKpVlfRsLi0vLK6ultfL6xubWtrmzey+jRGDi4IhFouMjSRjlxFFUMdKJBUGhz0jbH13nfvuBCEkjfqfSmLghGnAaUIyUljxz/6aa1i6b1dTLBiiRclyDx1A/a2XPrFh1awI4T+yCVECBlmd+9foRTkLCFWZIyq5txcrNkFAUMzIu9xJJYoRHaEC6mnIUEulmkxPG8EgrfRhEQhdXcKL+nshQKGUa+rozRGooZ71c/M/rJio4dzPK40QRjqcfBQmDKoJ5HrBPBcGKpZogLKjeFeIhEggrnVoegj178jxxTuoXdfv2tNK4KtIogQNwCKrABmegAZqgBRyAwSN4Bq/gzXgyXox342PaumAUM3vgD4zPHx5TlN8=</latexit><latexit sha1_base64="nGBPMfEXSg7/uQojXGhQ56c9htk=">AAACAXicbVDLSsNAFJ34rPUVdSVuBovQLiyJCOpCKLoprioYW2hDmEwn7dDJJMxMhBCKG3/FjQsVt/6FO//GSZuFth64cOace5l7jx8zKpVlfRsLi0vLK6ultfL6xubWtrmzey+jRGDi4IhFouMjSRjlxFFUMdKJBUGhz0jbH13nfvuBCEkjfqfSmLghGnAaUIyUljxz/6aa1i6b1dTLBiiRclyDx1A/a2XPrFh1awI4T+yCVECBlmd+9foRTkLCFWZIyq5txcrNkFAUMzIu9xJJYoRHaEC6mnIUEulmkxPG8EgrfRhEQhdXcKL+nshQKGUa+rozRGooZ71c/M/rJio4dzPK40QRjqcfBQmDKoJ5HrBPBcGKpZogLKjeFeIhEggrnVoegj178jxxTuoXdfv2tNK4KtIogQNwCKrABmegAZqgBRyAwSN4Bq/gzXgyXox342PaumAUM3vgD4zPHx5TlN8=</latexit><latexit sha1_base64="nGBPMfEXSg7/uQojXGhQ56c9htk=">AAACAXicbVDLSsNAFJ34rPUVdSVuBovQLiyJCOpCKLoprioYW2hDmEwn7dDJJMxMhBCKG3/FjQsVt/6FO//GSZuFth64cOace5l7jx8zKpVlfRsLi0vLK6ultfL6xubWtrmzey+jRGDi4IhFouMjSRjlxFFUMdKJBUGhz0jbH13nfvuBCEkjfqfSmLghGnAaUIyUljxz/6aa1i6b1dTLBiiRclyDx1A/a2XPrFh1awI4T+yCVECBlmd+9foRTkLCFWZIyq5txcrNkFAUMzIu9xJJYoRHaEC6mnIUEulmkxPG8EgrfRhEQhdXcKL+nshQKGUa+rozRGooZ71c/M/rJio4dzPK40QRjqcfBQmDKoJ5HrBPBcGKpZogLKjeFeIhEggrnVoegj178jxxTuoXdfv2tNK4KtIogQNwCKrABmegAZqgBRyAwSN4Bq/gzXgyXox342PaumAUM3vgD4zPHx5TlN8=</latexit><latexit sha1_base64="nGBPMfEXSg7/uQojXGhQ56c9htk=">AAACAXicbVDLSsNAFJ34rPUVdSVuBovQLiyJCOpCKLoprioYW2hDmEwn7dDJJMxMhBCKG3/FjQsVt/6FO//GSZuFth64cOace5l7jx8zKpVlfRsLi0vLK6ultfL6xubWtrmzey+jRGDi4IhFouMjSRjlxFFUMdKJBUGhz0jbH13nfvuBCEkjfqfSmLghGnAaUIyUljxz/6aa1i6b1dTLBiiRclyDx1A/a2XPrFh1awI4T+yCVECBlmd+9foRTkLCFWZIyq5txcrNkFAUMzIu9xJJYoRHaEC6mnIUEulmkxPG8EgrfRhEQhdXcKL+nshQKGUa+rozRGooZ71c/M/rJio4dzPK40QRjqcfBQmDKoJ5HrBPBcGKpZogLKjeFeIhEggrnVoegj178jxxTuoXdfv2tNK4KtIogQNwCKrABmegAZqgBRyAwSN4Bq/gzXgyXox342PaumAUM3vgD4zPHx5TlN8=</latexit>
Hyvärinen, 1998b New approximations of differential entropy for independent
component analysis and projection pursuit. NIPS (10) pages 273–279.MIT Press
J(y) / [E{G(y)} E{G(v)}]2
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G(u) = e u2
/2
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44.
ML models for fMRI denoising
!30
FastICA is then optimised as:
1. Choose an initial (e.g. random) weight vector w
2. Let
3. Let
4. If not converged, go back to 2.
Component Estimation
! = !+
/||!+
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Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications". Neural
Networks. 13 (4–5): 411–430.
!+
= E{xg(!T
x)} E{g0
(!T
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45.
ML models for fMRI denoising
!30
FastICA is then optimised as:
1. Choose an initial (e.g. random) weight vector w
2. Let
3. Let
4. If not converged, go back to 2.
Component Estimation
! = !+
/||!+
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Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications". Neural
Networks. 13 (4–5): 411–430.
!+
= E{xg(!T
x)} E{g0
(!T
x)}!<latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit>
* is first derivative of G(.)
* is second derivative
g(!T
x)<latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit><latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit><latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit><latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit>
g0
(!T
x)<latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit><latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit><latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit><latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit>

46.
ML models for fMRI denoising
!30
FastICA is then optimised as:
1. Choose an initial (e.g. random) weight vector w
2. Let
3. Let
4. If not converged, go back to 2.
Component Estimation
! = !+
/||!+
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Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications". Neural
Networks. 13 (4–5): 411–430.
!+
= E{xg(!T
x)} E{g0
(!T
x)}!<latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit>
* is first derivative of G(.)
* is second derivative
g(!T
x)<latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit><latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit><latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit><latexit sha1_base64="24tfcyyTNRPz2Byp2jgq5V27EPY=">AAACA3icbVC7TsMwFHV4lvIKMHaJqJDKUiUICdgqWBiL1NBKTagcx02tOnZkO4gqysDCr7AwAGLlJ9j4G5w2A7QcyfLROffq3nuChBKpbPvbWFpeWV1br2xUN7e2d3bNvf1byVOBsIs45aIXQIkpYdhVRFHcSwSGcUBxNxhfFX73HgtJOOuoSYL9GEaMDAmCSksDsxY1vIDTUE5i/WUej3EE77JO/pAfD8y63bSnsBaJU5I6KNEemF9eyFEaY6YQhVL2HTtRfgaFIojivOqlEicQjWGE+5oyGGPpZ9MjcutIK6E15EI/pqyp+rsjg7EsttSVMVQjOe8V4n9eP1XDcz8jLEkVZmg2aJhSS3GrSMQKicBI0YkmEAmid7XQCAqIlM6tqkNw5k9eJO5J86Lp3JzWW5dlGhVQA4egARxwBlrgGrSBCxB4BM/gFbwZT8aL8W58zEqXjLLnAPyB8fkDtQeYVw==</latexit>
g0
(!T
x)<latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit><latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit><latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit><latexit sha1_base64="7rI5Yp4ZMpNDKnPKm3I0mlBBirI=">AAACBHicbVC7TsMwFHXKq5RXgBGGiApRlipBSMBWwcJYpIZWakLlOE5q1Ykj20FUURYWfoWFARArH8HG3+C0GaDlSJaPzrlX997jJZQIaZrfWmVhcWl5pbpaW1vf2NzSt3duBUs5wjZilPGeBwWmJMa2JJLiXsIxjDyKu97oqvC795gLwuKOHCfYjWAYk4AgKJU00PfDo4bjMeqLcaS+zGERDuFd1skf8uOBXjeb5gTGPLFKUgcl2gP9y/EZSiMcS0ShEH3LTKSbQS4JojivOanACUQjGOK+ojGMsHCzyRW5cagU3wgYVy+WxkT93ZHBSBRbqsoIyqGY9QrxP6+fyuDczUicpBLHaDooSKkhmVFEYviEYyTpWBGIOFG7GmgIOURSBVdTIVizJ88T+6R50bRuTuutyzKNKtgDB6ABLHAGWuAatIENEHgEz+AVvGlP2ov2rn1MSyta2bML/kD7/AEZnpiI</latexit>
Maxima of the approximation of the negentropy of wTx are obtained at certain
optima of E{G(wTx)}, which under constraints:
E{xg(!T
x)} ! = 0<latexit sha1_base64="pEuPR6pigWO5ZsW3dVHDaA3LpUM=">AAACM3icbVDLSgMxFM3UV62vqks3wSLUhWVGBHUhFEUQ3FRobaFTSya9bUMzkyHJSMswH+XGD3EjggsVt/6D6WNRWw+EnJx7Lrn3eCFnStv2m5VaWFxaXkmvZtbWNza3sts790pEkkKFCi5kzSMKOAugopnmUAslEN/jUPV6V8N69RGkYiIo60EIDZ90AtZmlGgjNbO3127seoK31MA3F+7jTn7qHbvChw55KON+cugm+Ai7HmiCpyxjB77AdjObswv2CHieOBOSQxOUmtkXtyVo5EOgKSdK1R071I2YSM0ohyTjRgpCQnukA3VDA+KDasSjpRN8YJQWbgtpTqDxSJ3uiImvhhMap090V83WhuJ/tXqk22eNmAVhpCGg44/aEcda4GGCuMUkUM0HhhAqmZkV0y6RhGqTc8aE4MyuPE8qx4XzgnN3kiteTtJIoz20j/LIQaeoiG5QCVUQRU/oFX2gT+vZere+rO+xNWVNenbRH1g/v038qsk=</latexit><latexit sha1_base64="pEuPR6pigWO5ZsW3dVHDaA3LpUM=">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</latexit><latexit sha1_base64="pEuPR6pigWO5ZsW3dVHDaA3LpUM=">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</latexit><latexit sha1_base64="pEuPR6pigWO5ZsW3dVHDaA3LpUM=">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</latexit>
E{(!T
x)2
} = ||!||2
= 1<latexit sha1_base64="Ld2L1TfjtL2FPO4WiWBFKcHTMgo=">AAACJnicbVDLSsNAFJ34rPUVdelmsAh1U5IiqItCUQSXFRpbaNoymU7boZNMmJmIJc3XuPJTXAkWEXd+ipM2C209MMzhnHu59x4vZFQqy/oyVlbX1jc2c1v57Z3dvX3z4PBB8khg4mDOuGh6SBJGA+IoqhhphoIg32Ok4Y1uUr/xSISkPKircUjaPhoEtE8xUlrqmpVbNy66Hmc9Ofb1F7vcJwPUqcOn5KxTdpMKnEyW/WQy6ZQrdtcsWCVrBrhM7IwUQIZa15y6PY4jnwQKMyRly7ZC1Y6RUBQzkuTdSJIQ4REakJamAfKJbMezMxN4qpUe7HOhX6DgTP3dESNfplvqSh+poVz0UvE/rxWp/mU7pkEYKRLg+aB+xKDiMM0M9qggWLGxJggLqneFeIgEwkonm8/rFOzFm5eJUy5dlez780L1OosjB47BCSgCG1yAKrgDNeAADJ7BK3gHU+PFeDM+jM956YqR9RyBPzC+fwBdaaY8</latexit><latexit sha1_base64="Ld2L1TfjtL2FPO4WiWBFKcHTMgo=">AAACJnicbVDLSsNAFJ34rPUVdelmsAh1U5IiqItCUQSXFRpbaNoymU7boZNMmJmIJc3XuPJTXAkWEXd+ipM2C209MMzhnHu59x4vZFQqy/oyVlbX1jc2c1v57Z3dvX3z4PBB8khg4mDOuGh6SBJGA+IoqhhphoIg32Ok4Y1uUr/xSISkPKircUjaPhoEtE8xUlrqmpVbNy66Hmc9Ofb1F7vcJwPUqcOn5KxTdpMKnEyW/WQy6ZQrdtcsWCVrBrhM7IwUQIZa15y6PY4jnwQKMyRly7ZC1Y6RUBQzkuTdSJIQ4REakJamAfKJbMezMxN4qpUe7HOhX6DgTP3dESNfplvqSh+poVz0UvE/rxWp/mU7pkEYKRLg+aB+xKDiMM0M9qggWLGxJggLqneFeIgEwkonm8/rFOzFm5eJUy5dlez780L1OosjB47BCSgCG1yAKrgDNeAADJ7BK3gHU+PFeDM+jM956YqR9RyBPzC+fwBdaaY8</latexit><latexit sha1_base64="Ld2L1TfjtL2FPO4WiWBFKcHTMgo=">AAACJnicbVDLSsNAFJ34rPUVdelmsAh1U5IiqItCUQSXFRpbaNoymU7boZNMmJmIJc3XuPJTXAkWEXd+ipM2C209MMzhnHu59x4vZFQqy/oyVlbX1jc2c1v57Z3dvX3z4PBB8khg4mDOuGh6SBJGA+IoqhhphoIg32Ok4Y1uUr/xSISkPKircUjaPhoEtE8xUlrqmpVbNy66Hmc9Ofb1F7vcJwPUqcOn5KxTdpMKnEyW/WQy6ZQrdtcsWCVrBrhM7IwUQIZa15y6PY4jnwQKMyRly7ZC1Y6RUBQzkuTdSJIQ4REakJamAfKJbMezMxN4qpUe7HOhX6DgTP3dESNfplvqSh+poVz0UvE/rxWp/mU7pkEYKRLg+aB+xKDiMM0M9qggWLGxJggLqneFeIgEwkonm8/rFOzFm5eJUy5dlez780L1OosjB47BCSgCG1yAKrgDNeAADJ7BK3gHU+PFeDM+jM956YqR9RyBPzC+fwBdaaY8</latexit><latexit sha1_base64="Ld2L1TfjtL2FPO4WiWBFKcHTMgo=">AAACJnicbVDLSsNAFJ34rPUVdelmsAh1U5IiqItCUQSXFRpbaNoymU7boZNMmJmIJc3XuPJTXAkWEXd+ipM2C209MMzhnHu59x4vZFQqy/oyVlbX1jc2c1v57Z3dvX3z4PBB8khg4mDOuGh6SBJGA+IoqhhphoIg32Ok4Y1uUr/xSISkPKircUjaPhoEtE8xUlrqmpVbNy66Hmc9Ofb1F7vcJwPUqcOn5KxTdpMKnEyW/WQy6ZQrdtcsWCVrBrhM7IwUQIZa15y6PY4jnwQKMyRly7ZC1Y6RUBQzkuTdSJIQ4REakJamAfKJbMezMxN4qpUe7HOhX6DgTP3dESNfplvqSh+poVz0UvE/rxWp/mU7pkEYKRLg+aB+xKDiMM0M9qggWLGxJggLqneFeIgEwkonm8/rFOzFm5eJUy5dlez780L1OosjB47BCSgCG1yAKrgDNeAADJ7BK3gHU+PFeDM+jM956YqR9RyBPzC+fwBdaaY8</latexit>

47.
ML models for fMRI denoising
!31
FastICA is then optimised as:
1. Choose an initial (e.g. random) weight vector w
2. Let
3. Let
4. If not converged, go back to 2.
Component Estimation
! = !+
/||!+
||<latexit sha1_base64="99vFKgnJTWxwf9CqRWTgwDu4IQg=">AAACK3icbVDLSsNAFJ34rPUVdelmsAiCUBMR1IVQ7MZlBWMLTSyTyaQdOsmEmYlQ0n6QG39FEBdW3PofTtostO2BYQ7n3Mu99/gJo1JZ1thYWl5ZXVsvbZQ3t7Z3ds29/UfJU4GJgznjouUjSRiNiaOoYqSVCIIin5Gm36/nfvOZCEl5/KAGCfEi1I1pSDFSWuqYddfnLJCDSH/Q5RHpohs4rz2dng2HC3U4HHbMilW1JoDzxC5IBRRodMx3N+A4jUisMENStm0rUV6GhKKYkVHZTSVJEO6jLmlrGqOISC+bHDuCx1oJYMiFfrGCE/VvR4Yime+oKyOkenLWy8VFXjtV4ZWX0ThJFYnxdFCYMqg4zJODARUEKzbQBGFB9a4Q95BAWOl8yzoEe/bkeeKcV6+r9v1FpXZbpFECh+AInAAbXIIauAMN4AAMXsAb+ARj49X4ML6M72npklH0HIB/MH5+AeYhqDM=</latexit><latexit sha1_base64="99vFKgnJTWxwf9CqRWTgwDu4IQg=">AAACK3icbVDLSsNAFJ34rPUVdelmsAiCUBMR1IVQ7MZlBWMLTSyTyaQdOsmEmYlQ0n6QG39FEBdW3PofTtostO2BYQ7n3Mu99/gJo1JZ1thYWl5ZXVsvbZQ3t7Z3ds29/UfJU4GJgznjouUjSRiNiaOoYqSVCIIin5Gm36/nfvOZCEl5/KAGCfEi1I1pSDFSWuqYddfnLJCDSH/Q5RHpohs4rz2dng2HC3U4HHbMilW1JoDzxC5IBRRodMx3N+A4jUisMENStm0rUV6GhKKYkVHZTSVJEO6jLmlrGqOISC+bHDuCx1oJYMiFfrGCE/VvR4Yime+oKyOkenLWy8VFXjtV4ZWX0ThJFYnxdFCYMqg4zJODARUEKzbQBGFB9a4Q95BAWOl8yzoEe/bkeeKcV6+r9v1FpXZbpFECh+AInAAbXIIauAMN4AAMXsAb+ARj49X4ML6M72npklH0HIB/MH5+AeYhqDM=</latexit><latexit sha1_base64="99vFKgnJTWxwf9CqRWTgwDu4IQg=">AAACK3icbVDLSsNAFJ34rPUVdelmsAiCUBMR1IVQ7MZlBWMLTSyTyaQdOsmEmYlQ0n6QG39FEBdW3PofTtostO2BYQ7n3Mu99/gJo1JZ1thYWl5ZXVsvbZQ3t7Z3ds29/UfJU4GJgznjouUjSRiNiaOoYqSVCIIin5Gm36/nfvOZCEl5/KAGCfEi1I1pSDFSWuqYddfnLJCDSH/Q5RHpohs4rz2dng2HC3U4HHbMilW1JoDzxC5IBRRodMx3N+A4jUisMENStm0rUV6GhKKYkVHZTSVJEO6jLmlrGqOISC+bHDuCx1oJYMiFfrGCE/VvR4Yime+oKyOkenLWy8VFXjtV4ZWX0ThJFYnxdFCYMqg4zJODARUEKzbQBGFB9a4Q95BAWOl8yzoEe/bkeeKcV6+r9v1FpXZbpFECh+AInAAbXIIauAMN4AAMXsAb+ARj49X4ML6M72npklH0HIB/MH5+AeYhqDM=</latexit><latexit sha1_base64="99vFKgnJTWxwf9CqRWTgwDu4IQg=">AAACK3icbVDLSsNAFJ34rPUVdelmsAiCUBMR1IVQ7MZlBWMLTSyTyaQdOsmEmYlQ0n6QG39FEBdW3PofTtostO2BYQ7n3Mu99/gJo1JZ1thYWl5ZXVsvbZQ3t7Z3ds29/UfJU4GJgznjouUjSRiNiaOoYqSVCIIin5Gm36/nfvOZCEl5/KAGCfEi1I1pSDFSWuqYddfnLJCDSH/Q5RHpohs4rz2dng2HC3U4HHbMilW1JoDzxC5IBRRodMx3N+A4jUisMENStm0rUV6GhKKYkVHZTSVJEO6jLmlrGqOISC+bHDuCx1oJYMiFfrGCE/VvR4Yime+oKyOkenLWy8VFXjtV4ZWX0ThJFYnxdFCYMqg4zJODARUEKzbQBGFB9a4Q95BAWOl8yzoEe/bkeeKcV6+r9v1FpXZbpFECh+AInAAbXIIauAMN4AAMXsAb+ARj49X4ML6M72npklH0HIB/MH5+AeYhqDM=</latexit>
Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications". Neural
Networks. 13 (4–5): 411–430.
!+
= E{xg(!T
x)} E{g0
(!T
x)}!<latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">AAACYHicfVHPS8MwGE3r1G3+2KY3vRSHOBFHK4J6EIYieFSwOljnSLO0C0ubkqSyUfJPevPixX/EdOtBN/GDkMd77yPf9+InlAhp2x+GuVJaXVsvV6obm1vbtXpj51mwlCPsIkYZ7/pQYEpi7EoiKe4mHMPIp/jFH9/m+ssb5oKw+ElOE9yPYBiTgCAoNTWoTzyf0aGYRvrKPBbhEL5mJ0pd33mZF0E58oNsosLWX7YnNVHHnjrV1vDoP8eypKqDetNu27OyloFTgCYo6mFQf/eGDKURjiWiUIieYyeyn0EuCaJYVb1U4ASiMQxxT8MYRlj0s1lCyjrUzNAKGNcnltaM/dmRwUjkA2pnvrRY1HLyL62XyuCyn5E4SSWO0fyhIKWWZFYetzUkHCNJpxpAxIme1UIjyCGS+lPyEJzFlZeBe9a+ajuP583OTZFGGeyDA9ACDrgAHXAPHoALEPg0SsaWsW18mRWzZjbmVtMoenbBrzL3vgHiiLnu</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit><latexit sha1_base64="hIR5qzUv5xPlOEj1aeTWIlXqbcE=">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</latexit>

48.
ML models for fMRI denoising
!31
FastICA is then optimised as:
1. Choose an initial (e.g. random) weight vector w
2. Let
3. Let
4. If not converged, go back to 2.
Component Estimation
! = !+
/||!+
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Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications". Neural
Networks. 13 (4–5): 411–430.
For one component
!+
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