Tensor Methods for

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**M. Alex O. Vasilescu**

MIT

maov@mit.edu**II. Factorizations and Statistical
Modeling/Inference: **

(2 hours, Amnon Shashua)

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UROPS

MIT

maov@mit.edu

**Amnon Shashua**

Hebrew University of Jerusalem

shashua@cs.huji.ac.il

**Description:**

Tensor factorizations of higher order tensors have been successfully applied
in numerous machine learning, vision, graphics and signal processing tasks
in recent years and are drawing a lot of attention. There are two main types of
higher order tensor decompositions which generalize different concepts of
the matrix SVD, the rank-R decomposition (open problem) and Rank-(R_{1},R_{2},...,R_{M}) decomposition,
plus various tensor factorizations under convex constraints relevant to classical
inference and clustering tasks.

In the first part of the tutorial, we will define the linear tensor rank, rank-R
and the multilinear tensor rank, rank-(R_{1},R_{2},...,R_{M}). The linear tensor rank,
rank-R, generalizes the matrix concept of rank, while the multilinear rank,
rank-(R_{1},R_{2},...,R_{M}), generalizes the matrix concepts of orthonormal row/column
subspaces. We will discuss several multilinear representations, Multilinear PCA,
Multilinear ICA, etc. and introduce the multilinear projection operator, tensor
pseudo-inverse and the identity tensor which are important in performing
recognition in a tensor framework. Furthermore, we will discuss why images have
been traditionally vectorized in statistical learning, and discuss the
advantages and disadvantages of treating images as vectors, matrices and higher
order objects in the context of a tensor framework. In the second part this
tutorial, we will discuss factorizations relevant to statistical inference and
clustering where orthogonality are replaced by convex constraints. We will
address general low-rank tensors (arise form latent class models),
super-symmetric (arise from clustering over hypergraphs) and semi-symmetric
(arise from latent clustering models) and introduce factorization algorithms.

The tutorial will cover the application of these techniques to compression, face recognition, multi-object detection in supervised and unsupervised settings, gait recognition, and computer graphics.

**Tutorial Outline:**

**I. Multilinear Factorizations:** - (2 hours, M. Alex O. Vasilescu)

- Definitions:
- a. Linear Tensor Rank, Rank-R
- b. Multilinear Tensor Rank, Rank-(R
_{1},R_{2},...,R_{M}) - Multilinear factorizations:
- a. Multilinear SVD/PCA (M-mode SVD/PCA), Rank-(R
_{1},R_{2},...,R_{M}) - b. Multilinear ICA (M-mode ICA)
- Multilinear Projection Operator:
- a. Pseudo-Inverse Tensor (Mode-M Pseudo-Inverse)
- b. Identity Tensor (Mode-M Identity Tensor)
- c. Response Tensor, Contribution Tensor
- Image as a vector, image as matrix and higher order data tensors in statistical learning

(2 hours, Amnon Shashua)

- Tensor factorization under simplex constraints: Latent Class Models
- a. Traditional 2D models: LSA, NMF
- b. Choice of error model: L2 versus Relative-entropy
- c. Algorithms
- Super-Symmetric Tensor Factorization: Clustering over Hypergraphs.
- a. Completely Positive matrices and clustering: relationship to K- means, N-cuts, Ration-cuts, Min-cut.
- b. The role of double-stochastic normalization
- c. Super-symmetric Tensors: beyond pairwise clustering
- d. Algorithms
- Symmetric Embeddings and Multi-object detection
- a. Clustering with a latent variable = lifting 2D to 3D = NTF
- b. Multi-object detection = symmetric embedding factorization
- c. Feature sharing = clustering with latent variable

**Intended audience:**
All people in machine learning will benefit from a deep understanding of basic techniques such as SVD, LDA, CP model, Multilinear PCA, Multilinear ICA, Multilinear Projections, concepts of rank and multilinear rank for tensors, etc.

**Bios:**

**M. Alex O. Vasilescu** (www.media.mit.edu/~maov) was educated at MIT and the University of Toronto. She has done research at the MIT Artificial Intelligence Lab, NYU’s Courant Institute and at research centers of IBM, Intel, Compaq, and Schlumberger corporations. She is currently a research scientist at MIT Media Lab. She has published research papers in computer vision and computer graphics, particularly in the areas of face recognition, human motion analysis/synthesis, image-based rendering, and physics-based modeling (deformable models). She has given several invited talks about her work and has four patents pending. Her face recognition research, known as TensorFaces, was funded by the TSWG, the Department of Defense's Combating Terrorism Support Program. She has been named by MIT's Technology Review Magazine to their 2003 TR100 List of Top Young Innovators.

**Amnon Shashua** was the head of the School of Engineering and Computer Science at the
Hebrew University of Jerusalem during the term 2003--2005. He has been a Professor since 2003, Associate Professor since 1999, and joined the Hebrew University as a Senior Lecturer in the fall of 1996. He received his Ph.D. degree in Computational Neuroscience, working at the Artificial Intelligence Laboratory, from the Massachusetts Institute of Technology (MIT), in 1993; his M.Sc. degree in Mathematics and Computer Science from the Weizmann Institute of Science, Rehovot, Israel, in 1989; and his B.Sc. degree in Mathematics and Computer Science from Tel-Aviv University, Tel-Aviv, Israel, in 1986. During 2001/2 he spent a Sabbatical at the Computer Science department of Stanford University.
His research interests are in Computer Vision and Machine Learning. His work includes early visual processing of Saliency and Grouping mechanisms, Visual Recognition and Learning, Image Synthesis for Animation and Graphics, theory of Computer Vision in the areas of multiple-view geometry and multi-view tensors, and multilinear algebraic systems in Vision and Learning. His work on multiple-view geometry received the "best paper award" at ECCV'2000 and the honorable mention to the MARR prize in ICCV'2001.
Amnon Shashua received the first prize of the 2004 Kaye Innovation award, and the 2005 Landau award for Science and Research in the area of exact sciences - Robotics.

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