Inferring Data InterRelationships

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Inferring Data Inter-Relationships Via Fast
Hierarchical Models
Lawrence Carin Duke University www.ece.duke.edu/l
carin
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Sensor Deployed Previously Across Globe
Previous deployments
New deployment
Deploy to New Location. Can Algorithm Infer Which
Data from Past is Most Relevant for New Sensing
Task?
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Semi-Supervised Active Learning

• Enormous quantity of unlabeled data - exploit
  context via semi-supervised learning
• Focus the analyst on most-informative data -
  active learning

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Technology Employed Motivation

• Appropriately exploit related data from previous
  experience over sensor lifetime
• - Transfer learning
• Place learning with labeled data in the context
  of unlabeled data, thereby
• exploiting manifold information
• - Semi-supervised learning
• Reduce load on analyst only request labeled
  data on subset of data for which
• label acquisition would be most informative
• - Active learning

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Bayesian Hierarchical Models Dirichlet Processes

• Principled setting for transfer learning
• Avoids problems with model selection
• Number of mixture components
• Number of HMM states

iGMM Rasmussan, 00, iHMM Teh et al., 04,06,
Escobar West, 95
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Data Sharing Stick-Breaking View of DP 1/2

• The Dirichlet process (DP) is a prior on a
  density function, i.e., G(T) DPa,Go(T)
• One draw of G(T) from DPa,Go(T)

Sethuraman, 94
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Data Sharing Stick-Breaking View of DP 2/2
1

• As a ? 0, the more likely that Beta(1, a) yields
  large ?k , implying more sharing
• a few larger sticks, with corresponding
  likely parameters
• As a ? 8, sticks very small and roughly the same
  size, so reduces to Go

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Non-Parametric Mixture Models

• Data sample di drawn from a Gaussian/HMM with
  associated parameters
• - Posterior on model parameters indicates which
  parameters are shared, yielding a
• Gaussian/HMM mixture model no model selection
  on number of mixture components

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Dirichlet Process as a Shared Prior

• Cumulative set of data Dd1, d2, ,dn, with
  associated parameters
• When parameters are shared then the associated
  data are also shared data sharing implies
• learning from previous/other experiences ?
  Life-long learning
• Posterior reflects a balance between the
  DP-based desire for sharing, constituted by the
• prior ,
  against the likelihood function
• that rewards parameters that match the data well

DP Desire for Sharing Parameters
Likelihoods Desire to Fit Data
Posterior Balances these Objectives
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Hierarchical Dirichlet Process 1/2

• A DP prior on the parameters of a Gaussian model
  yields a GMM in which the number
• of mixture components need not be set a priori
  (non-parametric)
• Assume we wish to build N GMMs, each designed
  using a DP prior
• We link the N GMMs via an overarching DP hyper
  prior

Teh et al., 06
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Hierarchical Dirichlet Process 2/2

• Coefficients an,k represent the probability of
  transitioning from state n to state k
• Naturally yields the structure of an HMM number
  of large amplitude coefficients an,k
• implicitly determines the most-probable
  number of states

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Computational Challenges in Performing Inference

• We have the general challenge of estimating the
  posterior

• The denominator is typically of high dimension
  (number of parameters in model), and
• cannot be computed exactly in reasonable time
• Approximations required

MCMC
Variational Bayes (VB)
Accuracy
Laplace
Computational Complexity
Blei Jordan, 05
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Graphical Model of the nDP-iHMM
Ni, Dunson, Carin ICML 07
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How Do You Convince Navy Data Search
Works? Validation Not as Simple as Text
Search Consider Special Kind of Acoustic Data
Music
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Multi-Task HMM Learning

• Assume we have N sequential data sets
• Wish to learn HMM for each of the data sets
• Believe that data can be shared between the
  learning tasks not independent task
• All N HMMs learned jointly, with appropriate
  data sharing
• Use of iHMM avoids the problem of selecting
  number of states in HMM
• Validation on large music database VB yields
  fast inference

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Demonstration Music Database 525 Jazz
975 Classical 997
Rock
Rock
Jazz
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Classical
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Inter-Task Similarity Matrix
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Typical Recommendations from Three Genres
Classical Jazz Rock
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Applications of Interest to Navy

• Music search provides a fairly good objective
  demonstration of the technology
• Other than use of acoustic/speech features
  (MFCCs), nothing in previous
• analysis specifically tied to music simply
  data search
• Use similar technology for underwater acoustic
  sensing (MCM) - generative
• Use related technology for synthetic aperture
  radar and EO/IR detection
• and classification discriminative
• Technology delivered to NSWC Panama City, and
  demonstrated independently
• on mission-relevant MCM data

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Underwater Mine Counter Measures (MCM)
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Generative Model - iHMM
Ni Carin, 07
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Full Posterior on Number of HMM States
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Anti-Submarine Warfare (ASW)
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Design HMM for all Targets of Interest Over
Sensor Lifetime
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State Sharing Between ASW Targets
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Semi-Supervised Multi-Task Learning
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Semi-Supervised Discriminative Multi-Task
Learning

• Semi-supervised learning implemented via
  graphical techniques
• Multi-task learning implemented via DP
• Exploits all available data-driven context
• - Data available from previous collections,
  labeled unlabeled
• - Labeled and unlabeled data from current
  data set

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Graph representation of partially labeled data
manifolds (1/2)

• Construct the graph G(X,W), with the affinity
  matrix W, where the (i, j)-th element of W is
  defined by a Gaussian kernel
• Define a Markov random walk on the graph by the
  transition matrix A,
• where the (i, j)-th element
• which gives the probability of walking
  from xi to xj by a single step
• Markov random walk.
• The one-step Markov random walk provides a local
  similarity measure between data points.

Lu, Liao, Carin 07 Szummer Jaakkola, 02
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Graph representation (2/2)

• To account for global similarity between data
  points, we consider a t-step random walk, where
  the transition matrix is given by A raised to the
  power of t
• It was demonstrated1 that the t-step Markov
  random walk would result in a volume of paths
  connecting the data points in stead of the
  shortest path that are susceptible to noise
  thus it permits us to incorporate global manifold
  structure in the training data set.
• The t-step neighborhood of xi is defined as the
  set of data points xj with
• and denoted as

1 Tishby and Slonim, Data clustering by
Markovian relaxation and the information
bottleneck Method. NIPS 13, 2000
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Semi-Supervised Learning Algorithm (1/2)

• Neighborhood-based classifier Define the
  probability of label yi given the t-step
  neighborhood of xi as
• where is probability
  of labeling yi given a single data point xj and
  is represented by a standard probabilistic
  classifier parameterized by
• The label yi implicitly propagates over the
  neighborhood. Thus it is possible to learn a
  classifier with only a few labels present.

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The Algorithm (2/2)

• For binary classification problems, we choose the
  form of as logistic regression
  classifier
• To enforce sparseness, we impose a normal prior
  with zero mean and diagonal precision matrix
  on , and each
  hyper-parameter has an independent Gamma prior.
• Important for transfer learning The
  semi-supervised algorithm is inductive and
  parametric
• Place a DP prior on parameters, shared among all
  tasks

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Toy Data for Tasks 1-6
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Sharing Data
Pooling tasks 1-3
Pooling tasks 1-6
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Task similarity for MTL tasks 1-6
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Navy-Relevant Data Synthetic Aperture Radar
(SAR) Data Collected At 19 Different Locations
Across USA
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Real Radar Sensor Data

• Data from 19 tasks or geographical regions
• 10 of these regions are relatively highly
  foliated
• 9 regions bare earth, or desert
• Algorithm adaptively and autonomously clusters
  the task-dependent classifier
• weights into two basic pools, which agree with
  truth
• Active learning used to define labels of
  interest for the site under test
• Other sites used as auxiliary data, in a
  life-long-learning setting

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Supervised MTL JMLR 07
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Previous deployments
New deployment

• Classifier at new site placed appropriately
  within context of all available previous data
• Both labeled and unlabeled data employed
• Found that the algorithm relatively insensitive
  to particular labeled data selected
• Validation with relatively large music database

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Reconstruction of Random-Bars with hybrid CS.
Example (a) is from 3, and (b-c) are the
modified images from (a) by us to represent
similar tasks for simultaneous CS inversion. The
intensities of all the rectangles are randomly
permuted, and the positions of all the rectangles
are shifted by distances randomly sampled from a
uniform distribution of -10,10.
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