Humans Learn Using Manifolds, Reluctantly
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Humans Learn Using Manifolds, Reluctantly
Bryan Gibson, Xiaojin Zhu, Timothy Rogers, Charles Kalish, and Joseph
Harrison
University of Wisconsin-Madison
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A Familiar Task
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if we choose supervised learning semi-supervised learning
Question: which one would humans do?
Contribution: an empirical study of human manifold learning behaviors
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Manifolds are Common in Life
(University of Wisconsin)
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Manifolds are Common in Life
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y=“Bryan Gibson”
y=?
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The Stimuli of Our Behavioral Experiments Not faces: avoid existing knowledge
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x1=0
x2=0.10 1
1
0
x1=1
x2=0.9
x1=0.39
x2=0.41
x1=0.61
x2=0.59
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Human Behavioral Experiments
Batch learning (instead of sequential)Humans Learn Using Manifolds (University of
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labeled
unlabeled
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Aggregated Results: Majority Vote
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n subjects classified this stimuluso: majority vote blue+: majority vote red*: equal blue, red
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Aggregated Results: Majority Vote
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condition: L=28 subjects
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Individual Subject Fit
If best fitting model m has accuracy >75% then subject potentially uses m, Otherwise subject uses “other”
Percentage of subjects potentially using each model:
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graph[Belkin et al.’06]
1NN l2[Nosofsky’86]
1NN l1 vertical horizontal
graph 1NN l2
1NN l1
v h other
L=2 0 0.12 0 0.5 0.25 0.12
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New Experiment: 4 Labeled Points Intention: remove “vertical” or “horizontal” L=4:
These 4 labeled points chosen to maximize differences in model predictions:
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1NN l11NN l2graph multi-v multi-h v h
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The Interface for L=4
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Results for L=4 Majority vote:
Individual fit:
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L=4, 24 subjects L=2, 8 subjects
graph 1NN l2
1NN l1 multi-v
multi-h v h other
L=4 0.25 0.25 0.12 0.12 0 0.04 0.08 0.38L=2 0 0.12 0 - - 0.5 0.25 0.12
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NN graph
2 connected components with pure labels
New Experiment: Give People the Graph!
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=0.07
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Giving People the Graph via Highlighting Clicking on any item highlights all its neighbors
Instructions to subjects: highlighting not necessarily mean “same class”
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Results for (L=4, Hi) Majority vote:
Individual fit:
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L=4, 24 subjects L=2, 8 subjects
graph 1NN l2
1NN l1 multi-v
multi-h v h other
L=4 Hi
0.39 0.09 0.09 0.04 0.04 0 0.13 0.22
L=4 0.25 0.25 0.12 0.12 0 0.04 0.08 0.38L=2 0 0.12 0 - - 0.5 0.25 0.12
(L=4, Hi) 23 sub
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Humans simply follow highlighting?!
No. Three pieces of evidence
This is Awfully Suspicious
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o
++
++++
+o
ooo
oo
oo
+
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Evidence 1: Leap-of-Faith Against Highlight A leap-of-faith (lof) move goes against labeled
neighbors, or has no labeled neighbors
If people simply follow highlighting, #lof=0 In (L=4, Hi), average #lof=17 (20% of their
moves)Humans Learn Using Manifolds (University of
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highlightedlabeledneighbor
leap-of-faith
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Evidence 2: New Experiment (L=2, Hi) Same highlighting, but 2 labeled points Majority vote:
Individual fit:
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L=4, 24 subjects
L=2, 8 subjects
(L=4, Hi) 23 sub
(L=2, Hi) 8 sub
graph 1NN l2
1NN l1 multi-v
multi-h v h other
L=2 Hi
0.12 0 0 - - 0.38 0.25 0.25
L=4 Hi
0.39 0.09 0.09 0.04 0.04 0 0.13 0.22
L=4 0.25 0.25 0.12 0.12 0 0.04 0.08 0.38L=2 0 0.12 0 - - 0.5 0.25 0.12
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Evidence 3: New Experiment (Isomorphic Graph) Randomly permute the unlabeled nodes,
“bring the edges with them”
If people simply follow highlighting, then everything should be the same: 1 is easy blue, 2 is harder but still blue, …
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12
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Evidence 3: New Experiment (Isomorphic Graph) (L=4, Iso) isomorphic to (L=4, Hi):
There are really two “pure” connected components!
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Evidence 3: New Experiment (Isomorphic Graph)
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The interface for (L=4, Iso)
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Evidence 3: New Experiment (Isomorphic Graph)
Highlighting ignored if it contradicts with similarity
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L=4, Hi (23 subjects) L=4, Iso (30 subjects)
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Why not Always Manifold Learning?
Possible explanation: Bayesian model selection
data-dependent prior:
likelihood:
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highlighting:
none
manifold
isomorphic random
L=2 L=4 (neither
graphs)
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Summary
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weaken linear hypotheses
(L=4)
provide graph
(highlighting)
graph agrees w. similarity
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Thank you
(University of Wisconsin)
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(L=4, Iso)
(University of Wisconsin)
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Majority Vote (30 subjects):
Individual fit:grap
h1NN
l21NN l1 multi-
vmulti-h v h other
L=4 Iso
0.13 0.03 0.07 0 0 0.07 0.03 0.67
L=2 Hi
0.12 0 0 - - 0.38 0.25 0.25
L=4 Hi
0.39 0.09 0.09 0.04 0.04 0 0.13 0.22
L=4 0.25 0.25 0.12 0.12 0 0.04 0.08 0.38L=2 0 0.12 0 - - 0.5 0.25 0.12