Probabilistic Sparse Matrix Factorization

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Probabilistic Sparse Matrix Factorization

• Delbert Dueck, Quaid Morris, Brendan
  Frey(Probabilistic Statistical Inference
  Group)
• Tim Hughes(Banting and Best Department of
  Medical Research)

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Objective

• Patterns in gene expression array data can be
  used to help understand gene regulation and
  predict the function of yet-uncharacterized genes
• Objective To develop a method of probabilistic
  sparse matrix factorization (PSMF) and apply it
  to gene expression data to learn the hidden
  structure underlying the data.

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Biological Background

• Genes encode basic information about an organism
• They tend to be highly expressed in tissues
  related to their functional role
• Mouse gene expression data is from Zhang, Morris,
  et al. (2004)
• Gene expression is influenced by the presence of
  transcription factors (TFs)
• Co-expressed genes are likely activated by the
  same TFs
• The activity of each gene can be explained by the
  activities of a small number of transcription
  factors

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Gene Expression Array Dataset
? G22709 genes ?
Entire data set X GT matrix (G22709, T55)
? 100 genes ?
? T55 tissues ?
T55tissues
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Sparse Matrix Factorization

• Gene expression data model
• Each genes expression profile (xg) is
• a linear combination (weighted by ygc, c?sg)
• of a small number (rgltN)
• of C possible transcription factor profiles (zc,
  c?sg)

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Sparse Matrix Factorization
Matrix format (entire dataset)
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Probabilistic Sparse Matrix Factorization

• To express as a distribution, assume
• varying levels of Gaussian noise in the data
• nothing about transcription factor weights
• normally-distributed transcription factor
  profiles
• uniformly-distributed factor assignments
• multinomially-distributed factor counts

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Probabilistic Sparse Matrix Factorization

• To express as a distribution, assume
• varying levels of Gaussian noise in the data
• nothing about transcription factor weights
• normally-distributed transcription factor
  profiles
• uniformly-distributed factor assignments
• multinomially-distributed factor counts
• Multiply together to get joint distribution

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Factorized Variational Inference

• Exact inference is intractable with P()

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Factorized Variational Inference

• Exact inference is intractable with P()
• Approximate it by a simpler distribution, Q(),
  and perform inference on that

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Factorized Variational Inference

• Parameterize Q()
• Accounts for noise in transcription factor
  profiles and uncertainty in transcription factor
  selection

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Factorized Variational Inference

• Parameterize Q()
• Accounts for noise in transcription factor
  profiles and uncertainty in transcription factor
  selection
• Minimize KL-divergence between P(), Q()

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Factorized Variational Inference

• Parameterize Q()
• Accounts for noise in transcription factor
  profiles and uncertainty in transcription factor
  selection
• Minimize KL-divergence between P(), Q()

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Variational EM algorithm

• Use coordinate descent on free energy

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Variational EM Free Energy
iteration
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Visualization
PROBABILISTIC SPARSE MATRIX FACTORIZATION C50
possible factors N3 factors per gene (max)
P(rg).55 .27 .18
Sorted by primary transcription factor (sg1)
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Results p-value histograms

• Genes can be partitioned into primary
  categories (i.e. same sg1 value), secondary
  classes, etc.
• Compare classes with annotated gene ontology
  (GO-BP) categories for statistical significance

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Results mean log10 p-values
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Results count of significant p-values
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Future Directions different Q()
Iterated conditional modes (point estimates)
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Summary

• Introduced probabilistic sparse matrix
  factorization (PSMF), each row is a linear
  combination of a small number of hidden factors
  selected from a larger set.
• Described a variational inference algorithm for
  fitting the PSMF model.
• Evaluated model on a gene functional prediction
  task.