Overview of tensor decomposition methods and algebra, followed with a concise overview of how tensors are being used in medical analytics.

1. Colleen M. Farrelly

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

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