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Why Spectral Retrieval Works

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### Why Spectral Retrieval Works

SIGIR 2005 in Salvador, Brazil, August 15 – 19

Holger Bast

Max-Planck-Institut für Informatik (MPII)

Saarbrücken, Germany

joint work with Debapriyo Majumdar

What we mean by spectral retrieval

- Ranked retrieval in the term space

1.00

1.00

0.00

0.50

0.00

"true" similarities to query

qTd2

———|q||d2|

qTd1

———|q||d1|

cosine similarities

0.82

0.00

0.00

0.38

0.00

What we mean by spectral retrieval

- Ranked retrieval in the term space

1.00

1.00

0.00

0.50

0.00

"true" similarities to query

cosine similarities

0.82

0.00

0.00

0.38

0.00

- Spectral retrieval = linear projection to an eigensubspace

L q

projection matrix L

cosine similarities in the subspace

(Lq)T(Ld1)——————|Lq| |Ld1|

0.98

0.98

-0.25

0.73

0.01

…

Why and when does this work?

- Previous work: if the term-document matrix is a slight perturbation of a rank-k matrix then projection to ak-dimensional subspace works
- Papadimitriou, Tamaki, Raghavan, Vempala PODS'98
- Ding SIGIR'99
- Ando and Lee SIGIR'01
- Azar, Fiat, Karlin, McSherry, Saia STOC'01

- Our explanation: spectral retrieval works through its ability to identify pairs of terms with similar co-occurrence patterns
- no single subspace is appropriate for all term pairs
- we fix that problem

Spectral retrieval — alternative view

- Ranked retrieval in the term space

- Spectral retrieval = linear projection to an eigensubspace

L q

projection matrix L

(Lq)T(Ld1)——————|Lq||Ld1|

cosine similarities in the subspace

…

=

qT(LTLd1)——————|Lq||LTLd1|

Spectral retrieval — alternative view

- Ranked retrieval in the term space

expansion matrix LTL

- Spectral retrieval = linear projection to an eigensubspace

L q

projection matrix L

cosine similarities in the subspace

…

qT(LTLd1)——————|Lq||LTLd1|

Spectral retrieval — alternative view

- Ranked retrieval in the term space

expansion matrix LTL

qT(LTLd1)——————|q||LTLd1|

…

similarities after document expansion

- Spectral retrieval = linear projection to an eigensubspace

L q

projection matrix L

qT(LTLd1)——————|Lq||LTLd1|

cosine similarities in the subspace

…

Spectral retrieval = document expansion (not query expansion)

Why document "expansion"

add "internet" if "web" is present

internet

surfing

beach

web

=

·

0-1 expansion matrix

Why document "expansion"

- Ideal expansion matrix has
- high scores for intuitively related terms
- low scores for intuitively unrelated terms

add "internet" if "web" is present

internet

surfing

beach

web

=

·

matrix L projectingto 2 dimensions

expansion matrix LTL

expansion matrixdepends heavily on the subspace dimension!

Why document "expansion"

- Ideal expansion matrix has
- high scores for intuitively related terms
- low scores for intuitively unrelated terms

add "internet" if "web" is present

internet

surfing

beach

web

=

·

matrix L projectingto 3 dimensions

expansion matrix LTL

expansion matrixdepends heavily on the subspace dimension!

logics

node /

vertex

logic /

vertex

0

200

400

600

0

200

400

600

0

200

400

600

subspace dimension

subspace dimension

subspace dimension

Our Key Observation- We studied how the entries in the expansion matrix depend on the dimension of the subspace to which documents are projected

expansion matrix entry

0

no single dimension is appropriate for all term pairs

logics

node /

vertex

logic /

vertex

0

200

400

600

0

200

400

600

0

200

400

600

subspace dimension

subspace dimension

subspace dimension

Our Key Observation- We studied how the entries in the expansion matrix depend on the dimension of the subspace to which documents are projected

expansion matrix entry

0

no single dimension is appropriate for all term pairs

but the shape of the curve is a good indicator for relatedness!

200

400

600

0

0

200

200

400

400

600

600

subspace dimension

subspace dimension

subspace dimension

Curves for related terms- We call two terms perfectly related if they have an identical co-occurrence pattern

term 1

term 2

proven shape for perfectly related terms

provably small change after slight perturbation

half way to a real matrix

expansion matrix entry

0

point of fall-off is different for every term pair!

up-and-then-down shape remains

0

0

200

200

200

400

400

400

600

600

600

subspace dimension

subspace dimension

subspace dimension

Curves for unrelated terms- Co-occurrence graph:
- vertices = terms
- edge = two terms co-occur

- We call two terms perfectly unrelated if no path connects them in the graph

provably small changeafter slight perturbation

proven shape forperfectly unrelated terms

half way to a real matrix

expansion matrix entry

0

curves for unrelated terms are random oscillations around zero

Telling the shapes apart — TN

- Normalize term-document matrix so that theoretical point of fall-off is equal for all term pairs
- For each term pair: if curve is never negative before this point, set entry in expansion matrix to 1, otherwise to 0

expansion matrix entry

0

set entry to 1

set entry to 1

set entry to 0

0

200

400

600

0

200

400

600

0

200

400

600

subspace dimension

subspace dimension

subspace dimension

a simple 0-1 classification, no fractional entries!

An alternative algorithm — TM

- Again, normalize term-document matrix so that theoretical point of fall-off is equal for all term pairs
- For each term pair compute the monotonicity of its initial curve (= 1 if perfectly monotone, 0 as number of turns increase)
- If monotonicity is above some threshold, set entry in expansion matrix to 1, otherwise to 0

0.07

0.07

0.69

0.69

0.82

0.82

expansion matrix entry

0

set entry to 1

set entry to 1

set entry to 0

0

200

400

600

0

200

400

600

0

200

400

600

subspace dimension

subspace dimension

subspace dimension

again: a simple 0-1 classification!

Experimental results

(average precision)

425 docs3882 terms

Baseline: cosine similarity in term space

Latent Semantic Indexing Dumais et al. 1990

Term-normalized LSI Ding et al. 2001

Correlation-based LSI Dupret et al. 2001

Iterative Residual Rescaling Ando & Lee 2001

our non-negativity test

our monotonicity test

* the numbers for LSI, LSI-RN, CORR, IRR are for the best subspace dimension!

Experimental results

(average precision)

425 docs3882 terms

21578 docs5701 terms

233445 docs99117 terms

* the numbers for LSI, LSI-RN, CORR, IRR are for the best subspace dimension!

Conclusions

- Main message: spectral retrieval works through its ability to identify pairs of termswith similar co-occurrence patterns
- a simple 0-1 classification that considers a sequence of subspaces is at least as good as schemes that commit to a fixed subspace

- Some useful corollaries …
- new insights into the effect of term-weighting and other normalizations for spectral retrieval
- straightforward integration of known word relationships
- consequences for spectral link analysis?

Conclusions

- Main message: spectral retrieval works through its ability to identify pairs of terms with similar co-occurrence patterns
- a simple 0-1 classification that considers a sequence of subspaces is at least as good as schemes that commit to a fixed subspace

- Some useful corollaries …
- new insights into the effect of term-weighting and other normalizations for spectral retrieval
- straightforward integration of known word relationships
- consequences for spectral link analysis?

Obrigado!

Why document "expansion"

- Ideal expansion matrix has
- high scores for related terms
- low scores for unrelated terms

- Expansion matrix LTL depends on the subspace dimension

add "internet" if "web" is present

internet

surfing

beach

web

=

·

matrix L projectingto 4 dimensions

expansion matrix LTL

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