Within statistics, Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA). MPCA is employed in the analysis of n-way arrays, i.e. a cube or hyper-cube of numbers, also informally referred to as a "data tensor". N-way arrays may be decomposed, analyzed, or modeled by * linear tensor models such as CANDECOMP/Parafac, or * multilinear tensor models, such as multilinear principal component analysis (MPCA), or multilinear independent component analysis (MICA), etc. The origin of MPCA can be traced back to the Tucker decomposition and Peter Kroonenberg's "M-mode PCA/3-mode PCA" work. In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear and concise numerical computational terms in their SIAM paper entitled "", (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R1, R2, ..., RN ) Approximation of Higher-order Tensors". Circa 2001, Vasilescu reframed the data analysis, recognition and synthesis problems as multilinear tensor problems based on the insight that most observed data are the compositional consequence of several causal factors of data formation, and are well suited for multi-modal data tensor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in terms of their causal factors of data formation in the following works: Human Motion Signatures(CVPR 2001, ICPR 2002), face recognition โ ,(ECCV 2002, CVPR 2003, etc.) and computer graphics โ (Siggraph 2004). Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg in 1980. In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work that computed 2nd order statistics associated with each data tensor mode(axis), and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics associated with each tensor mode/axis. Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized, or whose observations are treated as matrix and concatenated into a data tensor. MPCA computes a set of orthonormal matrices associated with each mode of the data tensor which are analogous to the orthonormal row and column space of a matrix computed by the matrix SVD. This transformation aims to capture as high a variance as possible, accounting for as much of the variability in the data associated with each data tensor mode(axis). (Wikipedia).
Ideas for a Complex World - Anna Seigal
Science is full of smart tools for explaining the world around us. But those tools can sometimes feel far removed from the way the rest of us understand that world. How can we reconcile the two approaches? Oxford Mathematician Anna Seigal provides some pertinent answers in this Oxford Math
From playlist Oxford Mathematics Public Lectures
QED Prerequisites Geometric Algebra 13 Tensors
In this lesson we make contact with the standard concept of tensors using spacetime algebra. Please consider supporting this channel on Patreon: https://www.patreon.com/XYLYXYLYX The software I usually use to produce the lectures is: https://apps.apple.com/us/app/vittle-pro-video-whit
From playlist QED- Prerequisite Topics
Solving a multi step equation using distributive property
๐ Learn how to solve multi-step equations with parenthesis and variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To
From playlist How to Solve Multi Step Equations with Parenthesis on Both Sides
Solving a multi-step equation with fractions and variable on both sides
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Gregoire Montavon - Towards Higher-Order and Disentangled Explainable AI - IPAM at UCLA
Recorded 11 January 2023. Gregoire Montavon of Freie Universitรคt Berlin presents "Towards Higher-Order and Disentangled Explainable AI" at IPAM's Explainable AI for the Sciences: Towards Novel Insights Workshop. Abstract: The field of Explainable AI has produced methods that can robustly i
From playlist 2023 Explainable AI for the Sciences: Towards Novel Insights
Solving a multi step equation with variables on both sides 5+3r=5rโ19
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Ming Yuan: "Low Rank Tensor Methods in High Dimensional Data Analysis (Part 1/2)"
Watch part 2/2 here: https://youtu.be/5IA4z9On3Mg Tensor Methods and Emerging Applications to the Physical and Data Sciences Tutorials 2021 "Low Rank Tensor Methods in High Dimensional Data Analysis (Part 1/2)" Ming Yuan - Columbia University, Statistics Abstract: Large amount of multid
From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021
How to solve multi step equations with fractional coefficients
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Solving an equation with distributive property on both sides
๐ Learn how to solve multi-step equations with parenthesis and variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To
From playlist Solve Multi-Step Equations......Help!
Quantitative Inverse Theorem for Gowers Uniformity Norms ๐ด5 and ๐ด6 in ๐ฝn2 - Luka Milicevic
Workshop on Additive Combinatorics and Algebraic Connections Topic: Quantitative Inverse Theorem for Gowers Uniformity Norms ๐ด5 and ๐ด6 in ๐ฝn2 Speaker: Luka Milicevic Affiliation: Serbian Academy of Sciences and Arts Date: October 27, 2022ย In this talk, I will discuss a proof of a quant
From playlist Mathematics
Solving an equation with fraction where your variable is on both sides
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Lek-Heng Lim: "What is a tensor? (Part 2/2)"
Watch part 1/2 here: https://youtu.be/MkYEh0UJKcE Tensor Methods and Emerging Applications to the Physical and Data Sciences Tutorials 2021 "What is a tensor? (Part 2/2)" Lek-Heng Lim - University of Chicago, Statistics Abstract: We discuss the three best-known definitions of a tensor:
From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021
Hรผlya Argรผz - Gromov-Witten Theory of Complete Intersections 1/3
I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. This uses a monodromy analysis, as well as new degeneration and splitting formulas for nodal Gromov--Witten invariants
From playlist Workshop on Quantum Geometry
Solving a multi-step equation by multiplying by the denominator
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Seminar In the Analysis and Methods of PDE (SIAM PDE): Andrea R. Nahmod
Title: Gibbs measures and propagation of randomness under the flow of nonlinear dispersive PDE Date: Thursday, May 5, 2022, 11:30 am EDT Speaker: Andrea R. Nahmod, University of Massachusetts Amherst The COVID-19 pandemic and consequent social distancing call for online venues of research
From playlist Seminar In the Analysis and Methods of PDE (SIAM PDE)
Solving a multi step equation using distributive property one solution
๐ Learn how to solve multi-step equations with parenthesis and variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To
From playlist How to Solve Multi Step Equations with Parenthesis on Both Sides
How to solve a multi step equation with rational terms - (b-4)/6 = b/2
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Statistical Programming with R by Connor Harris
Statisticians, scientists, and data analysts often use R, an free, open-source interpreted language with matchless support for statistical analysis and data visualization. This seminar will provide an introduction to Rโs syntax and capabilities, with special attention to common pitfalls an
From playlist CS50 Seminars 2015
With a variable on both sides solving a multi step equation two ways
๐ Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Polynomial Ergodic Theorems for Strongly Mixing Commuting Transformations - Rigoberto Zelada
Special Year Research Seminar 2:00pm|Simonyi 101 and Remote Access Topic: Polynomial Ergodic Theorems for Strongly Mixing Commuting Transformations Speaker: Rigoberto Zelada Affiliation: University of Maryland Date: April 04, 2023ย The goal of this talk is to present new results dealing
From playlist Mathematics