The document discusses machine learning techniques for search engine relevance and personalized recommendations. It describes using linear regression models to predict relevance scores and regularization methods like Tikhonov regularization to avoid overfitting. It also discusses using empirical Bayesian models like the Poisson gamma model to estimate relevance probabilities. Finally, it mentions using techniques like singular value decomposition and non-negative matrix factorization to find patterns in user behavior data and remove noise.

1. Applied Machine Learning for
Search Engine Relevance
Charles H Martin, PhD

2. Relevance as a Linear Regression
model* query
x=1
r =X†w+e
form X from data
(i.e. group of queries)
x: (tf-idf) bag of words vector
Moore-Penrose Pseudoinverse r: relevance score (i.e. 1/-1)
w: weight vector
w = X†r/X†X
*Actually will model and predict pairwise
relations and not exact rank. ..stay tuned.
Solve as a numerical minimization
(i.e. iterative methods like SOR, CG, etc )
 X†w-r  2
min  w  2 : 2-norm of w

3. Relevance as a Linear Regression:
Tikhonov Regularization
Problem: inverse may be not exist (numerical instabilities, poles)
w = (X†X)-1 X†r
Solution: add constant a to diagonal of (X†X)-1
a: single, adjustable
aI)-1 X†r
w = (X X + †
smoothing parameter
Equivalent minimization problem
X†w-r 2 + a w2
min
More generally : form (something like) X†X + G †G + aI,
which is a self-adjoint , bounded operator =>
min  X†w-r 2 + a Gw 2 i.e. G chosen to avoid over-fitting

4. The Representer Theorem Revisited:
Kernels and Greens Functions
Problem: to estimate a function f(x) from training data (xi)
f(x) = S aiR(x, xi) R := Kernel
Solution: solve a general minimization problem
min Loss[(f(xi), yi)] + aT Ka Kij = R(xi, xj)
Equivalent to: given a Linear regularization operator ( G: H->L2(x) )
f(x) = S aiR(x, xi) + S buu (xi) ; u span null space of G
min Loss[(f(xi), yi)] + a Gx2
K is an integral operator: (Kf)(y) = R(x,y)f(x)dx
where
so K is the Green’s Function for (GG) †, or G = (K1/2)†
R(x,y) = <y|(GG) †|x>
in Dirac Notation:
Machine Learning Methods for Estimating Operator Equations (Steinke & Scholkpf 2006)

5. Personalized Relevance Algorithms:
eSelf Personality Subspace
p q personality traits
pages ads
Rock-n-roll
Cars: 0.4
Sports cars
Learned Traits:
0.0 =>0.3
User Hard rock
(Likes cars 0.4)
Reading
(Sports cars 0.3)
Present to user
(music site)
(used sports car ad)
Compute personality traits during user visit to web site
q values = stored learned “personality traits”
Provide relevance rankings (for pages or ads) which include personality traits

6. Personalized Relevance Algorithms:
eSelf Personality Subspace
p: output nodes q: hidden nodes
(observables) (not observed)
Individualized
Personality
Traits
u: user segmentation
h: history (observed outputs)
Web pages, Classified Ads, …
L [p,q] = [h, u]
model:
where L is a square matrix

7. Personalized Search:
Effective Regression Problem
[p, q](t) = (Leff[q(t-1)]) -1 • [h, u](t) on each time step (t)
PLP PLQ p = h PLP p + PLQ q = h
=>
QLP QLQ q u QLP p +QLQ q = 0
Leff p = h Formal solution:
p = (Leff [q,u])-1 h Leff = (PLP + PLQ (QLQ)-1 QLP)
Adapts on each visit, finding relevant pages p(t) based on the links L, and the
learned personality traits (q(t-1))
Regularization of PLP achieved with “Green’s Function / Resolvent Operator”
i.e. G †G ~= PLQ (QLQ)-1 QLP
Equivalent to Gaussian Process on a Graph, and/or Bayesian Linear Regression

8. Related Dimensional Noise Reductions:
Rank (k) Approximations of a Matrix
Latent Semantic Analysis (LSA)
PDP PDQ
(Truncated) Singular Value Decomposition (SVD):
Diagonalize the Density operator D = A†A
QDP QDQ
Retain a subset of (k) eigenvalues/vectors
Equivalent relations for SVD
(D-X)  2 2
X s.t. min
Optimal rank(k) apprx.
Decomposition: A = U∑ V† A†A = V (∑† ∑) V†
Can generalize to various noise models: i.e. VLSA*, PLSA**
]
min E [ qT (D-X)  2 2
VLSA* provides a rank (k) apprx. to any q query:
min DKL [P -P(data)]
PLSA* provides a rank (k) apprx. of classes (z)
P = U∑ V†
P(d,w) = P(d|z) P(z)P(w|z) D KL = Kullback–Leibler divergence
*Variable Latent Semantic Indexing (Yahoo! Research Labs) **Probabilistic Latent Semantic Indexing (Recommind, Inc)
http://www.cs.cornell.edu/people/adg/html/papers/vlsi.pdf http://www.cs.brown.edu/~th/papers/Hofmann-UAI99.pdf

9. Personalized Relevance:
Mobile Services and Advertising
France Telecom: Last Inch Relevance Engine
time
location
suggest
play game send msg play song …

10. KA for Comm Services
Comm
p Events q Personal Context
Services (Sun. mornings)
Call [who]
Mom (5)
SMS [who]
Bob (3)
MMS [who]
Phone company (1)
Events
Learned Traits:
Map to a contextual Suggestions for user
On Sunday morning,
comm service (Call, SMS, MMS, E-mail) most likely to call Mom
• Based on Empirical Bayesian score and Suggestion mapping table,
a decision is made to one or more possible Comm services
• Based on Business Intelligence (BI) data mining and/or Pattern
Recognition algorithms (i.e. supervised or unsupervised learning)
, we compute statistical scores indicating who are the most likely
people to Call, send an SMS, MMS, or E-Mail.

11. Comm/Call Patterns
POD Day of Week
LOC
calls to different #'s
p( |POD ) > p( |SUN); p( |POD) < p( |SUN)
p( , P|POD) > 0; p( , )=0

12. Bayesian Score Estimation
To estimate p(call|POD)
frequency:
p(call|POD) = # of times user called
someone at that POD
Bayesian:
p(call|POD) = p(POD|call)p(call)
Sq p(POD|q)p(q)
where q = call, sms, mms, or email

13. i.e. Bayesian Choice Estimator
• We seek to know the probability of a quot;callquot; (choice) at a given POD.
• We quot;borrow informationquot; from other PODs, assuming this is less
biased, to improve our statistical estimate
f( | 1) = 2/5 frequency estimator
1
2
p( | 1) = (2/5)(3/15) .
3
(2/5)(3/15)+(2/5)(3/15)+(1/5)(11/15)
5 days Bayesian choice estimator
3 PODs = 6/23 ~ 1/4
Note: the Bayesian estimate is
3 choices
significantly lower because we now
expect we might see a at POD 1

14. Incorporating Feedback
• It is not enough to simple recognize call patterns in the Event Facts—it
is also necessary to incorporate feedback into our suggestions scores
• p( c | user, pod, loc, facts, feedback ) = ?
Event Facts A: Simply Factorize:
p( c | user, pod, facts, feedback ) =
p( c | user, pod, facts ) p ( c | user, pod, feedback)
Suggestions
Evaluate probabilities independently,
irrelevant perhaps using different Bayesian models
poor
good
random

15. Personalized Relevance:
Empirical Bayesian Models
Closed form models:
Correct a sample estimate (mean m, variance ) with a
i.e:
weighted average of sample + complete data set
B shrinkage
m = Bm + (1-B) m factor
user
individual
segment
sample
Can rank order mobile services
based on estimated likelihood (m , )
play game send msg play song
1
3 1 2

16. Personalized Relevance:
Empirical Bayesian Models
What is Empirical Bayes modeling?
specify Likelihood L(y|) and Prior () distributions
estimate the posterior () = L(y|) () L(y|) ()
 L(y|) ()d (marginal)
Combines Bayesianism and frequentism:
Approximates marginal using (or posterior) using point estimate(MLE), Monte Carlo, etc.
Estimates marginal using empirical data
Uses empirical data to infer prior, plug into likelihood to make predictions
Note: Special case of Effective Operator Regression:
P space ~ Q space ; PLQ = I ; u  0
Q-space defines prior information

17. Empirical Bayesian Methods:
Poisson Gamma Model
Likelihood L(y| ) = Poisson distribution ( y e- )/y!
Conjugate Prior ( a,b) = Gamma distribution ( a-1 b a e-b a )/ G(a) ;  > 0
posterior (k) L(y|) () = (( y e- )/y!) (( a-1 b a e-b a )/ G(a) )
y+a-1 e-(1+(1/b))
also a Gamma distribution(a’,b’)
a’ = y + a ; b’ = (1+1/b)-1
Take MLE estimate of Marginal = mean (m) of the posterior (ab)
Obtain a,b from the mean (m = ab) and variance (ab2) of complete data
Final Point estimate E(y)= a’b’ for a sample is a weighted average of
sample mean y=my and prior mean m
E(y) = ( my + a ) (1+1/b)-1
E(y) = (b/1+b) my + (1/1+b) m

18. Linear Personality Matrix
suggestions
actio
n
events
Linear (or non-linear)
Matrix transformation: M s = a
Over time, we can estimate the Ma,s = prob( a | s )
and can then solve for the prob( s ) using a computational linear solver: s = M-1a
Notice: the personality matrix may or may not mix suggestions across events, can include semantic information
i.e. calls
i.e. for a given time and location…
s1 = call count how many times we suggested a call
but the user chose an email instead
s2 = sms
s3 = mms Obviously we would like M to be diagonal…or as good as possible !
s4 = email Can we devise an algorithm that will learn to give quot;optimalquot; suggestions?

19. Matrices for Pattern Recognition
(Statistical Factor Analysis)
We can use apply Computational Linear Algebra to remove noise and find patterns in data.
Called Factor Analysis by Statisticians, Singular Value Decomposition (SVD) by Engineers.
Implemented in Oracle Data Mining (ODM) as Non-Negative Matrix Factorization
Week 1 2 3 4 5 …
Call on Mon @ pod 1
4. Weekly patterns are
Call on Mon @ pod 2
collapsed into the density
Call on Mon @ pod 3
Matrix AtA
…
…
They can be detected
using spectral analysis
Sms on Tue @ pod 1
(i.e. principal eigenvalues)
…
2. Count # of times a choice 3. Form weekly choice
1. Enumerate all choices is made each week
density Matrix AtA
All weekly patterns Pure Noise
Similar to Latent (Multinomial) Dirichlet Algorithm (LDA), but much simpler to implement.
Suitable when the number (#) of choices is not too large, and patterns are weekly.

20. Search Engine Relevance : Listing on
Which 5 items to list at bottom of page ?

21. Statistical Machine Learning:
Support Vector Machines (SVM)
From Regression to Classification: Maximum Margin Solutions
2
w2 Classification := Find the line
that separates the points with
the maximum margin
w
min ½ w2 2 subject to constraints
all “above” line
all “below” line
perhaps within some slack (i.e. min ½ w2 2 + C S I )
constraint specifications:
“above” : w.xi –b >= 1 + I
“below” : w.xi –b <= 1 + i
Simple minimization (regression) becomes a convex optimization (classification)

22. SVM Light: Multivariate Rank Constraints
wTx
x y
Multivariate Classification: x1 - 0.1 -1
x2 +1.2 +1
let Ψ(x,y’) = S y’x be a linear fcn sgn
… … …
xn -0.7 -1
maps docs to relevance scores (+1/-1)
learn weights (w) s.t. max wT Ψ(x,y’)
is correct for training set
wT Ψ(x,y’)
max
(within a single slack constraint )
½ w2 2 +C s.t.
min
for all y’: wT Ψ(x,y) - wT Ψ(x,y’) >= D(y,y’) - 
Ψ(x,y’) is a linear discriminant function: (i.e. sum of ordered pairs S S yij (xi -xj) )
ij
D(y,y’) is a multivariate loss function: (i.e. 1- Average Precision(y,y’) )

23. SvmLight Ranking SVMs
SVMrank : Ordinal Regression
Stnd Classification on pairwise differences
S I,j,k
min ½ w2 2 + C s.t
for all queries qk (later, may not be query specific in SVMstruct)
wT Ψ(qk,di) - wT Ψ(qk,dj) >= 1- I,j,k
doc pairs di, dj
SVMperf : ROC Area, F1 Score, Precision/Recall
DROCArea = 1- # swapped pairs
SVMmap : Mean Average Precision ( warning: buggy ! )
Enforces a directed ordering
1 2 3 4 5 6 7 8 MAP ROC Area
1 0 0 0 0 1 1 0
8 7 6 5 4 3 2 1 0.56 0.47
1 2 3 4 5 6 7 8 0.51 0.53
A Support Vector Method for Optimizing Average Precision (Joachims et. Al. 2007)

24. Large Scale, Linear SVMs
• Solving the Primal
– Conjugate Gradient
– Joachims: cutting plane algorithm
– Nyogi
• Handling Large Numbers of Constraints
• Cutting Plane Algorithm
• Open Source Implementations:
– LibSVM
– SVMLight

25. Search Engine Relevance : Listing on
A ranking SVM consistently improves Shopping.com <click rank> by %12

26. Various Sparse Matrix Problems:
Google Page Rank algorithm
Ma = a rank a series of web pages by simulating user
browsing patterns (a) based on probabilistic
model (M) of page links
Pattern Recognition, Inference
Lp=h estimate unknown probabilities (p) based on
historical observations (h) and probability
model (L) of links between hidden nodes
Quantum Chemistry
H =E compute color of dyes, pigments given empirical
information on realted molecules and/or solving
massive eigenvalue problems

27. Quantum Chemistry:
the electronic structure eigenproblem
Solve a massive eigenvalue problem (109-1012)
H  (, , …) =   (, , …)
H nergy Matrix
 quantum state eigenvector E
,  , … electrons
Methods can have general applicability:
Davidson method for dominant eigenvalue / eigenvectors
Motivation for Personalization Technology
From solution of understanding the conceptual foundations
of semi-empirical models (noiseless dimensional reduction)

28. Relations between Quantum Mechanics and
Probabilistic Language Models
• Quantum States  resemble the states (strings, words, phrases)
in probabilistic language models (HMMs, SCFGs), except:
 is a sum* of strings of electrons:
 (, ) = 0. |  1  2  1  2 | + 0.2 |  2  3  1  2 | + …
*Not just a single string!
• Energy Matrix H is known exactly, but large. Models of H can be
inferred from empirical data to simplify computations.
• Energies ~= Log [Probabilities], un-normalized

29. Dimensional Reduction in Quantum Chemistry:
where do semi-empirical Hamiltonians come from?
Ab initio (from first principles):
Solve entire H  (, ) =   (, ) …approximately
OR
Semi-empirical:
Assume (, ) electrons statistically independent:
 (, ) = p() q()
Treat  -electrons explicitly, ignore  (hidden):
PHP p() =  p() much smaller problem
Parameterize PHP matrix => Heff with empirical data using a small set
of molecules, then apply to others (dyes, pigments)

30. Effective Hamiltonians:
Semi-Empirical Pi-Electron Methods
Heff [] p() =  p()
implicit / hidden
PHP PHQ p= E p PHP p + PHQ q = E p
=>
QHP QHQ q q QHP p + QHQ q = E q
Heff [] = (PHP – PHQ (E-QHQ)-1 QHP)
Final Heff can be solved iteratively (as with eSelf Leff),
or perturbatively in various forms
Solution is formally exact =>
Dimensional Reduction / “Renormalization”

31. Graphical Methods
Vij = + + +…
Decompose Heff into effective interactions between  electrons
(Expand (E-QHQ)-1 in an infinite series, remove E dependence)
Represent diagrammatically, ~300 diagrams to evaluate
Precompile using symbolic manipulation:
~35 MG executable; 8-10 hours to compile
run time: 3-4 hours/parameter

32. Effective Hamiltonians:
Numerical Calculations
-only effective empirical
VCC 16 11.5 11-12 (eV)
example
Compute ab initio empirical parameters :
Can test all basic assumptions of semi-empirical theory ,
“from first principles”
Also provides highly accurate eigenvalue spectra
Augment commercial packages (i.e. Fujitsu MOPAC) to model
spectroscopy of photoactive proteins