CIS 5590: Large-Scale Matrix Decomposition Tensors and Applications

Slides:

Advertisements

Similar presentations

2.3 Modeling Real World Data with Matrices

Advertisements

Section 13-4: Matrix Multiplication

Maths for Computer Graphics

Matrices. Special Matrices Matrix Addition and Subtraction Example.

Subspaces, Basis, Dimension, Rank

CE 311 K - Introduction to Computer Methods Daene C. McKinney

Little Linear Algebra Contents: Linear vector spaces Matrices Special Matrices Matrix & vector Norms.

Introduction to tensor, tensor factorization and its applications

Row 1 Row 2 Row 3 Row m Column 1Column 2Column 3 Column 4.

Unit 3: Matrices.

Matrices. Definitions  A matrix is an m x n array of scalars, arranged conceptually as m rows and n columns.  m is referred to as the row dimension.

AIM: How do we perform basic matrix operations? DO NOW:  Describe the steps for solving a system of Inequalities  How do you know which region is shaded?

If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding elements of A and B. add these.

4.3 Matrix Multiplication 1.Multiplying a Matrix by a Scalar 2.Multiplying Matrices.

Slide Copyright © 2009 Pearson Education, Inc. 7.3 Matrices.

3.6 Solving Systems Using Matrices You can use a matrix to represent and solve a system of equations without writing the variables. A matrix is a rectangular.

Matrix Operations.

1.3 Solutions of Linear Systems

Algebra Matrix Operations. Definition Matrix-A rectangular arrangement of numbers in rows and columns Dimensions- number of rows then columns Entries-

Warm Up Perform the indicated operations. If the matrix does not exist, write impossible

1.7 Linear Independence. in R n is said to be linearly independent if has only the trivial solution. in R n is said to be linearly dependent if there.

What is Matrix Multiplication? Matrix multiplication is the process of multiplying two matrices together to get another matrix. It differs from scalar.

4.8 Rank Rank enables one to relate matrices to vectors, and vice versa. Definition Let A be an m  n matrix. The rows of A may be viewed as row vectors.

3.6 Multiplying Matrices Homework 3-17odd and odd.

Linear System of Simultaneous Equations Warm UP First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct:

PRESENT BY BING-HSIOU SUNG A Multilinear Singular Value Decomposition.

Notes Over 4.2 Finding the Product of Two Matrices Find the product. If it is not defined, state the reason. To multiply matrices, the number of columns.

Unit 3: Matrices. Matrix: A rectangular arrangement of data into rows and columns, identified by capital letters. Matrix Dimensions: Number of rows, m,

4-3 Matrix Multiplication Objective: To multiply a matrix by a scalar multiple.

Matrices. Variety of engineering problems lead to the need to solve systems of linear equations matrixcolumn vectors.

Introduction to Vectors and Matrices

Umm Al-Qura University

Lesson 43: Working with Matrices: Multiplication

12-1 Organizing Data Using Matrices

Multiplying Matrices.

Linear Algebra review (optional)

Matrix Operations.

Unit 1: Matrices Day 1 Aug. 7th, 2012.

Warm up (5, 7 4 ) Write the solution of the following system:

Warm-Up - 8/30/2010 Simplify. 1.) 2.) 3.) 4.) 5.)

Linear Transformations

Linear Transformations

Matrix Operations SpringSemester 2017.

Warm Up Use scalar multiplication to evaluate the following:

LINEAR MODELS AND MATRIX ALGEBRA

Multiplying Matrices.

WarmUp 2-3 on your calculator or on paper..

Section 3.1 – Operations with Matrices

Warmup Solve each system of equations. 4x – 2y + 5z = 36 2x + 5y – z = –8 –3x + y + 6z = 13 A. (4, –5, 2) B. (3, –2, 4) C. (3, –1, 9) D. no solution.

Linear Equations in Linear Algebra

CSCI N207 Data Analysis Using Spreadsheet

MATRICES MATRIX OPERATIONS.

1.3 Vector Equations.

2.2 Introduction to Matrices

Multiplying Matrices.

Introduction to Vectors and Matrices

Multi-Dimensional Arrays

3.6 Multiply Matrices.

Linear Algebra review (optional)

Matrix Operations SpringSemester 2017.

Matrix A matrix is a rectangular arrangement of numbers in rows and columns Each number in a matrix is called an Element. The dimensions of a matrix are.

Multiplying Matrices.

Multiplying Matrices.

Introduction to Matrices

Multiplying Matrices.

Presentation transcript:

CIS 5590: Large-Scale Matrix Decomposition Tensors and Applications Instructor: Kai Zhang CIS @ Temple University, Spring 2018

What is a Tensor? A tensor is a multi-way extension of a matrix A multi-dimensional array A multi-linear map Examples Scalar, Vector, Matrices color images, videos data matrices, recorded at different times t ECG: electrode x ECG beat x time action recognition

Epilespy Sedure Detection

Fibers and Slices Third First Second

Tensor Matrication Definition: unfolds an N-way tensor into a matrix Mode-n matricization arranges the mode-n fibers as columns of a matrix Denoted X(n) As many rows as is the dimensionality of the nth mode As many columns as is the product of the dimensions of other modes

Tensor Multiplication Definition

Pictorial view of Tensor Multiplication

Mode-n Product Example

Kronecker Product

Khatri–Rao Matrix Product

Khatri–Rao Matrix Product

Tensor Decomposition CP decomposition

Tensor Decomposition Using matricization, we can re-write the CP decomposition ALS Algorithm

Tucker Decomposition

ALS Solution for Turker

Tucker Decomposition