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Colleen M. Farrelly
Linear Algebra in Analytics
 Data Analysis
 Matrix Singular Value Decomposition (SVD)
 Factor analysis/latent modeling
 Cluster of individuals
 Single time, longitudinal
 Signal processing
 Recommenders and Collaborative Filtering
 User-item matrices
 Adjacency matrix and PageRank
 Mathematical Statistics
 Rich theoretical history
 Moment statistics
 Optimization
 Physics and physical chemistry modeling
Left, right, and
diagonal of
matrix’s
singular values
SVD
Extensions of Linear Algebra
 Vectors and matrices are
common in statistics and
machine learning
 1-D and 2-D representations
of data relationships
 Many theoretical results
leveraged in algorithms
 What about 3-D or 4-D or 100-
D representations of data
relationships?
 MRI slice sequence
 Multimodal signals
 These objects are tensors.
Vector
Matrix
Tensor
Tensor Algebra
 Rich history
 Physics
 Gravity
 Field/string theory
 Fluid mechanics
 Multilinear algebra
 Grassman algebra
 Differential forms
 Extends familiar linear algebra tools and
constructs to higher-dimensional spaces
 Determinants/traces
 Linear mapping (space and basis
transformations of topological space)
 Inner products
 Building complex topological objects
Tensors and the SVD
 The SVD has multilinear
extensions.
 Rank of tensors an open problem
considered to be NP-hard
 Rank approximation algorithms
exist for tensor decomposition
 Many nice theoretical results
(bounds, statistical properties…)
 Can be used exploited for analytics
 User/item/time tensor construction
for recommendation
 Latent transition analysis
 Multimodal signal integrated analysis
Factor loading matrices
Reduced
tensor
Full tensor
Tensor
Decomposition
Tensor Decomposition Methods Graph-Based Composite
Likelihoods
 Hidden node conditional
likelihood to estimate
parameters
 Latent Tree Graph Models
 Latent Tree Graph Model
formulation followed by
iterative, hierarchical
decomposition collapsing
into matrix SVD
 TripleRank
 PARAFAC followed by HITS
authority score on resulting
graph (PageRank variant)
 Latent Schatten Norms
 Group LASSO/Tucker
hybrid
 General Decomposition
 Alternating least squares
 Gradient-based
 Eigenvalue decomposition
Level 1
Level 2
 High Order Singular Value
Decomposition
 Full dimensionality control using
extended SVD method
 Canonical Decomposition/
PARAFAC (Tucker)
 Linear combination with no
orthogonality constraints (least
squares algorithms like HOSVD)
 Regularization and truncation
 Weaker requirements for
uniqueness
 Tensor Unfolding
 Unfold tensor along mode and
perform SVD on unfolded tensor
matrix
 Non-Negative Tensor Factorization
 Independent polynomial
formulation
 Follows non-negative matrix
factorization algorithm (least
squares)
Tensor Software
 R
 Tensor, rTensor, tensorA,
dti, PTAk, Debian
packages
 Python
 Scikit, TensorToolbox,
PyTensor
 Matlab
 Tensor Toolbox,
Tensorlab, htucker
Image Integration/Analysis
 MRI data with many components of images per patient
 Tensor decomposition to reduce dimensionality
 Noise filtration
 Less computationally-intensive data mining/predictive modeling
 Control over dimensionality to obtain standard size
components across individuals (integrate with prediction)
 Integration/analysis of many types of image data (ex. MRI
+ PET + fMRI)
 Tensor decomposition to identify key elements within each
patient’s images
 Data mining
 Highlighting potentially useful information for clinicians
 Factor analysis extension for identifying similar components
across images
 Partition images corresponding to anatomy or function
 Data mine factors to identify functional areas
Extensions to Signal Data
 These principles of
integrating image data
extends to other types of
signals:
 EEG
 EKG
 Pulse Oxygenation
 Other biometric data
collected over time from
patients
 Set up problem as high-
dimensional tensor and
apply algorithms as before

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Tensor decompositions for medical analytics

  • 2.
  • 3. Linear Algebra in Analytics  Data Analysis  Matrix Singular Value Decomposition (SVD)  Factor analysis/latent modeling  Cluster of individuals  Single time, longitudinal  Signal processing  Recommenders and Collaborative Filtering  User-item matrices  Adjacency matrix and PageRank  Mathematical Statistics  Rich theoretical history  Moment statistics  Optimization  Physics and physical chemistry modeling Left, right, and diagonal of matrix’s singular values SVD
  • 4. Extensions of Linear Algebra  Vectors and matrices are common in statistics and machine learning  1-D and 2-D representations of data relationships  Many theoretical results leveraged in algorithms  What about 3-D or 4-D or 100- D representations of data relationships?  MRI slice sequence  Multimodal signals  These objects are tensors. Vector Matrix Tensor
  • 5. Tensor Algebra  Rich history  Physics  Gravity  Field/string theory  Fluid mechanics  Multilinear algebra  Grassman algebra  Differential forms  Extends familiar linear algebra tools and constructs to higher-dimensional spaces  Determinants/traces  Linear mapping (space and basis transformations of topological space)  Inner products  Building complex topological objects
  • 6. Tensors and the SVD  The SVD has multilinear extensions.  Rank of tensors an open problem considered to be NP-hard  Rank approximation algorithms exist for tensor decomposition  Many nice theoretical results (bounds, statistical properties…)  Can be used exploited for analytics  User/item/time tensor construction for recommendation  Latent transition analysis  Multimodal signal integrated analysis Factor loading matrices Reduced tensor Full tensor Tensor Decomposition
  • 7. Tensor Decomposition Methods Graph-Based Composite Likelihoods  Hidden node conditional likelihood to estimate parameters  Latent Tree Graph Models  Latent Tree Graph Model formulation followed by iterative, hierarchical decomposition collapsing into matrix SVD  TripleRank  PARAFAC followed by HITS authority score on resulting graph (PageRank variant)  Latent Schatten Norms  Group LASSO/Tucker hybrid  General Decomposition  Alternating least squares  Gradient-based  Eigenvalue decomposition Level 1 Level 2  High Order Singular Value Decomposition  Full dimensionality control using extended SVD method  Canonical Decomposition/ PARAFAC (Tucker)  Linear combination with no orthogonality constraints (least squares algorithms like HOSVD)  Regularization and truncation  Weaker requirements for uniqueness  Tensor Unfolding  Unfold tensor along mode and perform SVD on unfolded tensor matrix  Non-Negative Tensor Factorization  Independent polynomial formulation  Follows non-negative matrix factorization algorithm (least squares)
  • 8. Tensor Software  R  Tensor, rTensor, tensorA, dti, PTAk, Debian packages  Python  Scikit, TensorToolbox, PyTensor  Matlab  Tensor Toolbox, Tensorlab, htucker
  • 9.
  • 10. Image Integration/Analysis  MRI data with many components of images per patient  Tensor decomposition to reduce dimensionality  Noise filtration  Less computationally-intensive data mining/predictive modeling  Control over dimensionality to obtain standard size components across individuals (integrate with prediction)  Integration/analysis of many types of image data (ex. MRI + PET + fMRI)  Tensor decomposition to identify key elements within each patient’s images  Data mining  Highlighting potentially useful information for clinicians  Factor analysis extension for identifying similar components across images  Partition images corresponding to anatomy or function  Data mine factors to identify functional areas
  • 11. Extensions to Signal Data  These principles of integrating image data extends to other types of signals:  EEG  EKG  Pulse Oxygenation  Other biometric data collected over time from patients  Set up problem as high- dimensional tensor and apply algorithms as before