The Geometry of Learning

The Geometry of Learning
November 17th, 2009, Utrecht, The Netherlands
Fridolin WildKMi, The Open University

(createdwith http://www.wordle.net)

Outline
Context & Framing Theories
Latent Semantic Analysis (LSA)
Social Network Analysis (SNA)
Meaningful Interaction Analysis (MIA)
Conclusion & Outlook

Context & Theories

Information
(96dpi)
Information could be the quality of a certain signal.
Information could be a logical abstractor, the release mechanism.
Information & Knowledge
Knowledge could be the delta at the receiver (a paper, a human, a library).

Learning is change
Learning is about competence development
Competence becomes visible in performance
Professional competence is mainly about (re-)constructing and processing information and knowledge from cues
Professional competence development is much about learning concepts from language
Professional performance is much about demonstrating conceptual knowledge with language
Language!
What is learning about?

Tying shoelaces
Douglas Adams’ ‘meaning of liff’:
Epping: The futile movements of forefingers and eyebrows used when failing to attract the attention of waiters and barmen.
Shoeburyness: The vague uncomfortable feeling you get when sitting on a seat which is still warm from somebody else's bottom
I have been convincingly Sapir-Whorfed by this book.
Non-textual concepts things we can’t (easily) learn from language

Latent Semantic Analysis

Word Choice
Educated adult understands ~100,000 word forms
An average sentence contains 20 tokens.
Thus 100,00020 possible combinationsof words in a sentence
 maximum of log2 100,00020= 332 bits in word choice alone.
20! = 2.4 x 1018 possible orders of 20 words = maximum of 61 bits from order of the words.
332/(61+ 332) = 84%word choice
(Landauer, 2007)

Latent Semantic Analysis
“Humans learn word meanings and how to combine them into passage meaning through experience with ~paragraph unitized verbal environments.”
“They don’t remember all the separate words of a passage; they remember its overall gist or meaning.”
“LSA learns by ‘reading’ ~paragraph unitized texts that represent the environment.”
“It doesn’t remember all the separate words of a text it; it remembers its overall gist or meaning.”
(Landauer, 2007)

Latent Semantics
latent-semantic space
In other words:Assumption: language utterances have a semantic structure
Problem: structure is obscured by word usage(noise, synonymy, polysemy, …)
Solution: map doc-term matrix using conceptual indices derived statistically (truncated SVD) and make similarity comparisons using angles

Input (e.g., documents)
term = feature
vocabulary = ordered set of features
Only the red terms appear in more than one document, so strip the rest.
TEXTMATRIX
{ M } =
Deerwester, Dumais, Furnas, Landauer, and Harshman (1990): Indexing by Latent Semantic Analysis, In: Journal of the American Society for Information Science, 41(6):391-407

Singular Value Decomposition
=

Truncated SVD
… we will get a different matrix (different values, but still of the same format as M).

(Landauer, 2007)

Reconstructed, Reduced Matrix
m4: Graphminors: A survey

Similarity in a Latent-Semantic Space
Query
Y dimension
Target 1
Angle 1
Angle 2
Target 2
X dimension
(Landauer, 2007)

doc2doc - similarities
Unreduced = pure vector space model
- Based on M = TSD’
- Pearson Correlation over document vectors
reduced
- based on M2 = TS2D’
- Pearson Correlation over document vectors

Typical, simple workflow
tm = textmatrix(‘dir/‘)
tm = lw_logtf(tm) * gw_idf(tm)
space = lsa(tm, dims=dimcalc_share())
tm3 = fold_in(tm, space)
as.textmatrix(tm)

Processing Pipeline (with Options)
4 x 12 x 7 x 2 x 3 = 2016 Combinations

b) SVD is computationally expensive
From seconds (lower hundreds of documents, optimised linear algebra libraries, truncated SVD)
To minutes (hundreds to thousands of documents)
To hours (tens and hundreds of thousands)
a) SVD factor stability
SVD calculates factorsover a given text base; different texts – different factors
Problem: avoid unwanted factor changes
Solution: folding-in of instead of recalculating
Projecting by Folding-In

2
1
vT
Folding-In in Detail
(cf. Berry et al., 1995)
Mk
(2) convert
„Dk“-format
vector to
„Mk“-format
Tk
Sk
Dk
(1) convert
Original
Vector to
„Dk“-format

The Value of Singular Values
Pearson(jahr, wien)
Pearson(eu, österreich)

Simple LSA application

Summary Writing: Working Principle
(Landauer, 2007)

Summary Writing
Gold Standard 1
Gold
Standard 2
Y dimension
Gold Standard 3
Essay 1
Essay 2
X dimension

‘Dumb’ Summary Writing (Code)
library( "lsa“ )# load package
# load training texts
trm = textmatrix( "trainingtexts/“ )
trm = lw_bintf( trm ) * gw_idf( trm )# weighting
space = lsa( trm )# create an LSA space
# fold-in summaries to be tested (including gold standard text)
tem = textmatrix( "testessays/", vocabulary=rownames(trm) )
tem_red = fold_in( tem, space )
# score a summary by comparing with
# gold standard text (very simple method!)
cor( tem_red[,"goldstandard.txt"], tem_red[,"E1.txt"] )
=> 0.7

Evaluating Effectiveness
Compare Machine Scores with Human Scores
Human-to-Human Correlation
Usually around .6
Increased by familiarity between assessors, tighter assessment schemes, …
Scores vary even stronger with decreasing subject familiarity (.8 at high familiarity, worst test -.07)
Test Collection: 43 German Essays, scored from 0 to 5 points (ratio scaled), average length: 56.4 words
Training Collection: 3 ‘golden essays’, plus 302 documents from a marketing glossary, average length: 56.1 words

(Positive) Evaluation Results
LSA machinescores:
Spearman's rank correlationrho
data: humanscores[names(machinescores), ] and machinescores
S = 914.5772, p-value = 0.0001049
alternative hypothesis: truerhoisnotequal to 0
sampleestimates:
rho
0.687324
Pure vectorspacemodel:
Spearman's rank correlationrho
data: humanscores[names(machinescores), ] and machinescores
S = 1616.007, p-value = 0.02188
alternative hypothesis: truerhoisnotequal to 0
sampleestimates:
rho
0.4475188

Social Network Analysis
Existing for a long time (term coined 1954)
Basic idea:
Actors and Relationships between them (e.g. Interactions)
Actors can be people (groups, media, tags, …)
Actors and Ties form a Graph (edges and nodes)
Within that graph, certain structures can be investigated
Betweenness, Degree of Centrality, Density, Cohesion
Structural Patterns can be identified (e.g. the Troll)

Forum Messages

Incidence Matrix
msg_id = incident, authorsappear in incidents

DeriveAdjacency Matrix
= t(im) %*% im

Visualization: Sociogramme

Measuring Techniques (Sample)
Closenesshow close to all others
Degree Centralitynumber of (in/out) connections to others
Betweennesshow often intermediary
Componentse.g. kmeans cluster (k=3)

SNA applications

Co-Authorship Network WI (2005)

Paper Collaboration Prolearn
e.g. co-authorships of ~30 deliverables of three work packages (ProLearn NoE)
Roles: reviewer (red), editor (green), contributor
Size: Prestige()
But: type of interaction? Content of interaction? => not possible!

TEL Project Cooperation (2004-2007)

iCamp Collaboration (Y1)
Shades of yellow: WP leadership
Red: coordinator

Meaningful Interaction Analysis (MIA)
Fusion: Combining LSA with SNA
Terms and Documents (or anything else represented with column vectors or row vectors) are mapped into same space by LSA
Semantic proximity can be measured between them: how close is a term to a document?
(S)NA allows to analyse these resulting graph structures
By e.g. cluster or component analysis
By e.g. identifying central descriptors for these

The mathemagics behind
Meaning Interaction Analysis

Truncated SVD
… we will get a different matrix (different values, but still of the same format as M).

Knowledge Proxy: LSA Part
Tk= left-hand sided matrix = ‚term loadings‘ on the singular value
Dk= right-hand sided matrix = ‚document loadings‘ on the singular value
Multiply them into same space
VT = TkSk
VD = DkTSk
Cosine Distance Matrix over ... = a graph
Extension: add author vectors VAthrough cluster centroids or vector addition of their publication vectors
Ofcourse:useexistingspaceandfold inthewholesetsofvectors

Knowledge Proxy: SNA Part:Filter the Network
Every vectorhas a cosinedistancetoeveryother (maybe negative)!
So: filter forthedesiredsimilaritystrength

ConSpectmonitoring conceptual development

TopicProxy (30 people, 2005)

Spot unwanted fragmentation
e.g. two authors work on the same topic, but with different collaborator groups and with different literature
Intervention Instrument: automatically recommend to hold a flashmeeting
Bringing together what belongs together
Wild, Ochoa, Heinze, Crespo, Quick (2009, to appear)

//eof.

The Geometry of Learning

by The Open University on Nov 19, 2009

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