Tensor-Based Signal Processing with Applications in Biomedical Signal Analysis Technische Universität Ilmenau Fachgebiete Nachrichtentechnik & Biosignalverarbeitung

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

 Why tensors? dn = ? dn = ?? Well, why even matrices? Example: DFT Matrix equations are usually more compact  insights, manipulations Example: DFT Not a different data model but a more compact representation More than two dimensions: tensors even more compact  new insights dn = ? dn = ?? 

“Classical” Communications Research Description of the Mobile Radio Channel resolve, characterize individual propagation paths of the mobile radio channel

Biomedical engineering For example: EEG data diagnostics (neurology, ophthalmology) human-machine interface

Automotive engineering Wind tunnel analysis Audio Source Localization find sources of disturbance to optimize aerodynamic behavior

Motivation More applications Signal Processing (sensor arrays, blind multi-user detection, source separation, CDMA, SONAR and seismo-acoustic signal processing) Computer vision (Face and facial expression recognition, handwritten text recognition) Data mining (weblink analysis, personalized web search, cross-language information retrieval) Neuroscience (Multisubject fMRI anlaysis, concurrent EEG/fMRI) Chemical engineering (food industry, NIR spectroscopy) Geophysics (moment tensor inversion) Data compression (image coding, video coding)

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

? The term “tensor” Mathematics: 1846: W. Voigt  very abstract definition Physics: 1915: M. Grossmann and A. Einstein  describe physical quantities Here: Intuitive definition: ”A tensor of order p is a collection of elements referenced by p incides“  multi-way array … Scalars Vectors Matrices Order-3-tensors Order-4-tensors ?

Notation Symbols Matrix operations

Matrix unfoldings n-mode matrix unfoldings vary the n-th along rows, the others along columns e.g., R = 3: n-rank of . In general, 1-, 2-, and 3-rank can differ. M1 M2 M3 “1-mode vectors” “2-mode vectors” “3-mode vectors”

n-mode products n-mode product between a tensor and a matrix i.e., all the n-mode vectors multiplied from the left-hand-side by 1 2 outer product between two tensors: all pair-wise products between elements

The tensor rank Definition of the tensor rank A tensor is rank one, iff A tensor is rank r iff it can be decomposed into a sum of r and not less than r rank one tensors (Only) connection to the n-ranks: The rank of a tensor can exceed its size (which is a good thing and a bad thing) 2 x (maximum rank, cf. [Kolda08])

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

The Higher-Order SVD (HOSVD) Singular Value Decomposition Higher-Order SVD (Tucker3) “Full SVD” “Full HOSVD” [Tucker: 1966] [de Lathauwer: 2000]

Computing the HOSVD Computing the HOSVD The core tensor 4 5 3 not necessarily any zero elements (only if n-rank-deficient) all-orthogonality condition: 4 3 5 also three sets of singular values: n-mode singular values

The Higher-Order SVD (HOSVD) Singular Value Decomposition Higher-Order SVD (Tucker3) “Full SVD” “Full HOSVD” “Economy size SVD” “Economy size HOSVD” Low-rank approximation Low-rank approximation (truncated HOSVD)

Summary HOSVD The HOSVD … is an extension of the SVD to tensors. generalizes the concept of row-space and column-space to r-spaces. the r-mode singular vectors are an orthonormal basis for the r-space of the tensor. is very easy to compute (Matrix-SVD of the unfoldings). the remaining core tensor is not diagonal, it may be full of non-zero elements (same size as original data) not a decomposition into rank-one components

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

= = PARAFAC: Motivation + + + + Another way to look at the SVD decomposition into a sum of rank one matrices also referred to as principal components (PCA) Tensor case: = + + [Carroll, Chang 1970] [Harshman 1970] Canonical Decomposition (CANDECOMP) Parallel Factor Analysis (PARAFAC) = + +

PARAFAC expressions Many equations to express the same model: HOSVD:

HOSVD vs. PARAFAC Example: HOSVD PARAFAC Core tensor not necessarily diagonal, can be full. Direct, easy computation via matrix-SVDs “Factors” always tall or square Reveals the n-ranks Often used for “compressing” data 1 2 PARAFAC “Core tensor” diagonal Not easy to find the factors Factors may be flat (underdetermined) Reveals the tensor rank Often used for analyzing data 1 2

Uniqueness When is the PARAFAC decomposition of X into A,B,C unique? Let be the Kruskal-rank of A. Then, given that … … the PARAFAC decomposition is unique up to scaling and permutation. [Kruskal, 1966] scaling and permutation can be removed if additional constraints are imposed or prior knowledge is used scaling: permutation:

Finding the parallel factors Since we only have the noisy tensor, we restate the goal: “Plain vanilla” approach: ALS Many years of research to improve convergence speed smart initializations smart updates: Enhanced line search … “PARAFAC”, “COMFAC” “Least Squares Fit” works, but very slow convergence requires good initial solution

Finding the parallel factors Closed-form solutions? GRAM (generalized rank annihilation method): An exact closed-form solution if either of M1, M2 or M3 = 2 (only two slices). Can be used as initialization for other methods. DTLD (direct trilinear decomposition): A suboptimal approximation, mostly as initialization to PARAFAC. Based on Tucker3 (HOSVD) and GRAM. Very fast though. Ours: Reduced the problem onto joint diagonalization of matrices (which by itself is a very well studied problem).

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

PARAFAC via Joint Diagonalization First, consider the case where The transform matrices diagonalize the core tensor: We can estimate the transform matrices via joint diagonalization the fundamental link between the HOSVD and PARAFAC

PARAFAC via Joint Diagonalization One slide on the six diagonalization problems

One slide on the R-D extension

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

Time-Frequency- Analysis Processing Chain Biomedical process Time-Frequency- Analysis Component Analysis Measurement Time-Frequency-Analysis Wavelet-basiert Wigner-basiert Channel Frequency Frequency Channel Channel Time Time Time

≈ + + Component Analysis Given a three-way tensor (time, frequency, channel), we decompose it into a predefined number of components for each component: time-, frequency-, and spatial characteristics Zeit Frequenz Raum ≈ + +

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

Outline Motivation: Applications of multi-linear signal processing Introduction to multi-linear algebra Tensor decompositions Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP Other decompositions PARAFAC via Joint Diagonalization 3-way PARAFAC for EEG data Methodology and current status Open issues and questions Discussion Status of the project proposals

Geplante Folgeprojekte (1) BMBF - Innovationswettbewerb Medizintechnik Modul I „Innovationswettbewerb - BASIS“ Schlüsselexperiment zum Nachweis der Machbarkeit: „Tensor-basierte Analyse von polygraphischen Biosignalen zur Anfallsvorhersage bei Epilepsie“ Projektpartner TU Ilmenau: FG BSV & FG NT GJB Datentechnik GmbH, Langewiesen Zentralklinik Bad Berka GmbH, Klinik für Neurologie Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer/Herbst 2008 Modul II „Innovationswettbewerb – Transfer“ F&E-Vorhaben „Neue Methoden der Tensor-basierten Analyse von polygraphischen Biosignalen“ Psychotherapeutische und Neurologische Praxen Projektskizze vorzubereiten im Herbst 2008

Geplante Folgeprojekte (2) BMBF - Innovationswettbewerb Medizintechnik „Frühdiagnostik und Intervention von Essanfällen mittels Polygraphie bei Patienten mit Bulimia Nervosa“ (BuPoly) Projektpartner TU Ilmenau: FG BSV & FG NT NeuroConn GmbH, Ilmenau Praxis Dr. Braun, Gotha Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer/Herbst 2008 „Zeitvariable Niederfeldmagnetstimulation in der Therapie depressiver Erkrankungen und deren Wirkung auf die Herzratenvariabilität“ (DeNFMagS) Neurologische Praxis Henkel/Müller, Ilmenau

Geplante Folgeprojekte (3) BMBF - Ernährungsforschung „Polygraphiebasierte methodische und experimentelle Untersuchung der Reizreaktion auf lebensmittelbezogene visuelle Stimulationen bei Personen mit und ohne psychogene Essstörungen“ Projektpartner TU Ilmenau: FG BSV & FG NT Praxis Dr. Braun, Gotha Neurologische Praxis Henkel/Müller, Ilmenau Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer 2008

Geplante Folgeprojekte (4) LUBOM – Thüringen „Zeitvariable Niederfeldmagnetstimulation in der Therapie depressiver Erkrankungen und deren Wirkung auf die Herzratenvariabilität“ Projektpartner in der ersten Phase TU Ilmenau: FG BSV & FG NT in der nächsten Phase zusätzlich neurologische und psychotherapeutische Praxen Status Projektbeginn bei Bewilligung Anfang 2009 „Die Wirkungsweise von Eye Movement Desensitization and Reprocessing analysiert anhand polygraphischer Untersuchungen multimodaler Biosignale“

Geplante Folgeprojekte (5) LUBOM – Thüringen „EKG-Analyse zur Bestimmung der anaeroben Schwelle anhand der Absenkung des ST-Komplexes“ Projektpartner TU Ilmenau: FG BSV & FG NT Status Projektbeginn bei Bewilligung Herbst 2008 TAB – Thüringen „Erkennung von psychischen Verarbeitungsprozessen angstgestörter Patienten zur Unterstützung der lnterventionstherapie mittels mobiler onIinefähiger Biofeedbackgeräte“ GJB Datentechnik GmbH, Langewiesen Psychotherapie Dr. Wilms, Erfurt

Geplante Folgeprojekte (6) European Research Council: Advanced Investigators Grants im FP 7 „Mikrosensoren zur Erfassung wichtiger Lebensfunktionen“ Projektpartner TU Ilmenau: FG BSV & FG NT & IMN & FG NIKR IDMT Fraunhofer, Ilmenau Status Projektskizze in Vorbereitung einzureichen im Winter 2008

Geplante Folgeprojekte (7) DFG Forschungsprojekt zur dynamischen tensorbasierten Analyse von nichtlinearen zeitvariablen Prozessen Projektpartner TU Ilmenau: FG BSV & FG NT Zentralklinik Bad Berka GmbH, Klinik für Neurologie GJB Datentechnik GmbH, Langewiesen Psychotherapeutische und Neurologische Praxen Status Projektskizze vorzubereiten im Herbst 2008