This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
KDD Cup 2021で開催された時系列異常検知コンペ
Multi-dataset Time Series Anomaly Detection (https://compete.hexagon-ml.com/practice/competition/39/) に参加して
5位入賞した解法の紹介と上位解法の整理のための資料です.
9/24のKDD2021参加報告&論文読み会 (https://connpass.com/event/223966/) の発表資料です.
KDD Cup 2021で開催された時系列異常検知コンペ
Multi-dataset Time Series Anomaly Detection (https://compete.hexagon-ml.com/practice/competition/39/) に参加して
5位入賞した解法の紹介と上位解法の整理のための資料です.
9/24のKDD2021参加報告&論文読み会 (https://connpass.com/event/223966/) の発表資料です.
SchNet: A continuous-filter convolutional neural network for modeling quantum...Kazuki Fujikawa
The document summarizes a paper about modeling quantum interactions using a continuous-filter convolutional neural network called SchNet. Some key points:
1) SchNet performs convolution using distances between nodes in 3D space rather than graph connectivity, allowing it to model interactions between arbitrarily positioned nodes.
2) This is useful for cases where graphs have different configurations that impact properties, or where graph and physical distances differ.
3) The paper proposes a continuous-filter convolutional layer and interaction block to incorporate distance information into graph convolutions performed by the SchNet model.
This document discusses self-supervised representation learning (SRL) for reinforcement learning tasks. SRL learns state representations by using prediction tasks as an auxiliary objective. The key ideas are: (1) SRL learns an encoder that maps observations to states using a prediction task like modeling future states or actions; (2) The learned state representations improve generalization and exploration in reinforcement learning algorithms; (3) Several SRL methods are discussed, including world models, inverse models, and causal infoGANs.
SchNet: A continuous-filter convolutional neural network for modeling quantum...Kazuki Fujikawa
The document summarizes a paper about modeling quantum interactions using a continuous-filter convolutional neural network called SchNet. Some key points:
1) SchNet performs convolution using distances between nodes in 3D space rather than graph connectivity, allowing it to model interactions between arbitrarily positioned nodes.
2) This is useful for cases where graphs have different configurations that impact properties, or where graph and physical distances differ.
3) The paper proposes a continuous-filter convolutional layer and interaction block to incorporate distance information into graph convolutions performed by the SchNet model.
This document discusses self-supervised representation learning (SRL) for reinforcement learning tasks. SRL learns state representations by using prediction tasks as an auxiliary objective. The key ideas are: (1) SRL learns an encoder that maps observations to states using a prediction task like modeling future states or actions; (2) The learned state representations improve generalization and exploration in reinforcement learning algorithms; (3) Several SRL methods are discussed, including world models, inverse models, and causal infoGANs.
A guide to Tensor and its applications in Machine Learning.pdfVanessa Bridge
Analysis of applications of tensor based algorithms to improve performance of many machine learning algorithms for training, detection and classification.
A Novel Methodology for Designing Linear Phase IIR FiltersIDES Editor
This paper presents a novel technique for
designing an Infinite Impulse Response (IIR) Filter with
Linear Phase Response. The design of IIR filter is always a
challenging task due to the reason that a Linear Phase
Response is not realizable in this kind. The conventional
techniques involve large number of samples and higher
order filter for better approximation resulting in complex
hardware for implementing the same. In addition, an
extensive computational resource for obtaining the inverse
of huge matrices is required. However, we propose a
technique, which uses the frequency domain sampling along
with the linear programming concept to achieve a filter
design, which gives a best approximation for the linear
phase response. The proposed method can give the closest
response with less number of samples (only 10) and is
computationally simple. We have presented the filter design
along with its formulation and solving methodology.
Numerical results are used to substantiate the efficiency of
the proposed method.
A probabilistic model for recursive factorized image features pptirisshicat
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This document is a past exam paper for an EEC-404 Signals and Systems course. It contains 3 sections with multiple choice and numerical problems related to signals and systems concepts. Section A contains 10 short answer questions testing definitions, properties and operations related to signals, systems, Laplace transforms, Z-transforms, Fourier transforms, convolution, and block diagrams. Section B contains 5 longer answer questions on properties, Fourier transforms, convolution, correlation and filter responses. Section C contains 5 multi-part problems involving plots, proofs, transforms, convolution, correlation and filter characteristics. The document provides an exam for students to test their understanding of key concepts across time, frequency and transform domains for continuous and discrete time signals and systems.
This document describes a new algorithm for dual tree kernel conditional density estimation (KCDE) that provides fast and accurate density predictions. The algorithm extends previous work on univariate KCDE to allow for multivariate labels (Y) and conditioning variables (X). It applies Gray's dual tree approach separately to the numerator and denominator of the KCDE formula, and uses error bounds to ensure the quotient estimates have bounded relative error. This new algorithm provides the fastest known method for kernel conditional density estimation for prediction tasks.
The paper examines the problem of systems redesign within the context of passive electrical networks and through analogies provides also the means of addressing issues of re-design of mechanical networks. The problem addressed here are special cases of the more general network redesign problem. Redesigning autonomous passive electric networks involves changing the network natural dynamics by modification of the types of elements, possibly their values, interconnection topology and possibly addition, or elimination of parts of the network. We investigate the modelling of systems, whose structure is not fixed but evolves during the system lifecycle. As such, this is a problem that differs considerably from a standard control problem, since it involves changing the system itself without control and aims to achieve the desirable system properties, as these may be expressed by the natural frequencies by system re-engineering. In fact, this problem involves the selection of alternative values for dynamic elements and non-dynamic elements within a fixed interconnection topology and/or alteration of the network interconnection topology and possible evolution of the cardinality of physical elements (increase of elements, branches). The aim of the paper is to define an appropriate representation framework that allows the deployment of control theoretic tools for the re-engineering of properties of a given network. We use impedance and admittance modelling for passive electrical networks and develop a systems framework that is capable of addressing “life-cycle design issues” of networks where the problems of alteration of existing topology and values of the elements, as well as issues of growth, or death of parts of the network are addressed.
We use the Natural Impedance/ Admittance (NI-A) models and we establish a representation of the different types of transformations on such models. This representation provides the means for an appropriate formulation of natural frequencies assignment using the Determinantal Assignment Problem framework defined on appropriate structured transformations. The developed natural representation of transformations are expressed as additive structured transformations. For the simpler case of RL or RC networks it is shown that the single parameter variation problem (dynamic or non-dynamic) is equivalent to Root Locus problems.
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Existence of Hopf-Bifurcations on the Nonlinear FKN ModelIJMER
This document discusses Hopf bifurcations, which occur when the stability of an equilibrium point in a nonlinear dynamical system changes as a parameter is varied, resulting in the emergence of periodic solutions. It first provides background on limit cycles and the Hopf bifurcation theorem. It then determines the indicator k, whose sign indicates whether a Hopf bifurcation is supercritical (k<0) or subcritical (k>0). The analysis is extended to three-dimensional systems by reducing them to a two-dimensional system near the equilibrium point. Finally, the document applies this analysis to the Field-Körös-Noyen (FKN) chemical reaction model to determine its supercritical and subcritical Hopf bifurcations.
The document summarizes key concepts in social network analysis including metrics like degree distribution, path lengths, transitivity, and clustering coefficients. It also discusses models of network growth and structure like random graphs, small-world networks, and preferential attachment. Computational aspects of analyzing large networks like calculating shortest paths and the diameter are also covered.
X-ray diffraction is a technique used to analyze the crystal structure of materials. It works by firing X-rays at a crystalline sample and observing the scattered rays, which are determined by the positions of atoms in the crystal structure. Bragg's law describes the conditions under which constructive interference of X-rays scattered from crystal planes occurs, producing a diffraction pattern. Analysis of diffraction patterns can provide information about a material's lattice structure, symmetry, and chemical composition. Miller indices are used to describe crystallographic planes and directions within a crystal lattice.
In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales.
To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation.
This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.
This document discusses various topics related to sphere packings, lattices, spherical codes, and energy minimization on the sphere. It defines sphere packings, lattices, and spherical codes. It describes problems like finding the densest sphere packing in each dimension, determining optimal spherical codes, and minimizing potential energy on the sphere. Linear programming bounds are introduced as a technique for proving optimality of codes. Properties of positive definite kernels and Gegenbauer polynomials are also summarized.
Data Complexity in EL Family of Description LogicsAdila Krisnadhi
The document summarizes data complexity results for reasoning in extensions of the EL family of description logics. It shows that instance checking is coNP-hard, and thus data intractable, for several extensions including EL∀r.⊥, EL∀r.C, EL∃¬r.C, ELC∪D, EL∃r+.C, and EL(≥kr) for k ≥ 2. The reductions are from the NP-complete 2+2SAT problem and use partitioning or covering concepts in the TBox along with a polynomial-sized ABox to encode truth assignments. Instance checking remains tractable for the data-tractable logics ELIf and extensions of DL
The document summarizes the k-means clustering algorithm. It describes how k-means aims to group data into k clusters by minimizing the distance between data points and their assigned cluster centroid. The algorithm works by iteratively assigning points to the closest centroid and moving each centroid to the mean of its assigned points until convergence. While k-means converges, finding the global minimum is not guaranteed as it can get stuck in local optima, so it is best to run it multiple times.
Measures of risk on variability with application in stochastic activity networksAlexander Decker
This document discusses measures of risk and variability that can be applied to stochastic activity networks. It proposes a simple measure, Δ(F), to rank commonly used probability distributions based on their variability. Δ(F) is defined as the difference between the mean squared and variance. Distributions with a larger Δ(F) value have lower variability. The document outlines several probability distributions commonly used in project management, including the beta, uniform, triangular, exponential and Erlang distributions. It proves that under certain conditions of symmetry, the uniform distribution has the highest variability while the beta-PERT distribution has the lowest, based on their respective Δ(F) values. The measure can help compare risks when different distributions are used to model activity
This document provides an overview of Linear Discriminant Analysis (LDA) for dimensionality reduction. LDA seeks to perform dimensionality reduction while preserving class discriminatory information as much as possible, unlike PCA which does not consider class labels. LDA finds a linear combination of features that separates classes best by maximizing the between-class variance while minimizing the within-class variance. This is achieved by solving the generalized eigenvalue problem involving the within-class and between-class scatter matrices. The document provides mathematical details and an example to illustrate LDA for a two-class problem.
System 1 and System 2 were basic early systems for image matching that used color and texture matching. Descriptor-based approaches like SIFT provided more invariance but not perfect invariance. Patch descriptors like SIFT were improved by making them more invariant to lighting changes like color and illumination shifts. The best performance came from combining descriptors with color invariance. Representing images as histograms of visual word occurrences captured patterns in local image patches and allowed measuring similarity between images. Large vocabularies of visual words provided more discriminative power but were costly to compute and store.
This document summarizes a study on degeneracy in random Boolean networks. It introduces random Boolean networks and discusses experiments measuring the effects of degeneracy, redundancy, and other factors on network properties like number of attractors, states in attractors, and sensitivity to initial conditions. The discussion suggests that degeneracy and redundancy can increase robustness and evolvability by allowing variations in node functions or structures without significantly changing dynamics.
The k-means clustering algorithm aims to group data points into k clusters based on their distances from initial cluster centroid points. It works by alternating between assigning each point to its nearest centroid, and updating the centroid locations to be the mean of their assigned points. This process monotonically decreases the distortion score measuring distances from points to centroids, and is guaranteed to converge, though possibly to local optima rather than the global minimum. Running it multiple times can help avoid bad initial results.
This document discusses beam deflections and summarizes a method for calculating beam deflection using multiple integration. It provides an example of using this method to calculate the deflection of a beam under three-point bending. The maximum deflection occurs at the beam's midpoint and is given by the equation P L3/48EI. It also discusses analyzing statically indeterminate beams by writing slope and deflection equations with unknown reaction forces and solving for the forces using boundary conditions. An example is provided of calculating the deflection of a beam supported at three points.
Similar to Tensor Decomposition and its Applications (20)
Reading Seminar (140515) Spectral Learning of L-PCFGsKeisuke OTAKI
1. The document presents a spectral learning method for latent-variable PCFGs (L-PCFGs) that uses tensor factorization.
2. It defines observable representations based on features of tree structures that can be computed from training data alone, without hidden variables.
3. The tensor parameter C of the L-PCFG can be recovered from the observable representations, allowing for spectral learning of the L-PCFG from a treebank via tensor methods.
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This document discusses various machine learning topics including supervised learning techniques like support vector machines, decision trees, and neural networks. It also discusses unsupervised learning techniques like clustering algorithms. It provides short code examples for algorithms like quicksort in Haskell and OCaml. Finally, it introduces other concepts like probably approximately correct learning and boosting.
The document discusses wavelet transforms and related concepts like mother wavelets, scaling functions, and two-scale relationships. It covers definitions of wavelet transforms and wavelets, properties of wavelets like orthogonality, and applications of wavelet transforms such as signal analysis and image compression. Sections 2.1 through 2.11 each explore an aspect of wavelet transforms and wavelets.
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The document discusses greedy algorithms. It defines greedy algorithms as choosing the locally optimal choice at each step in the hope of finding a global optimum. The document outlines the steps of greedy algorithms as having optimal substructure, making greedy choices at each step, and being iterative or recursive. It provides examples of activity selection problems and the 0-1 knapsack problem to illustrate greedy algorithms.
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?Speck&Tech
ABSTRACT: A prima vista, un mattoncino Lego e la backdoor XZ potrebbero avere in comune il fatto di essere entrambi blocchi di costruzione, o dipendenze di progetti creativi e software. La realtà è che un mattoncino Lego e il caso della backdoor XZ hanno molto di più di tutto ciò in comune.
Partecipate alla presentazione per immergervi in una storia di interoperabilità, standard e formati aperti, per poi discutere del ruolo importante che i contributori hanno in una comunità open source sostenibile.
BIO: Sostenitrice del software libero e dei formati standard e aperti. È stata un membro attivo dei progetti Fedora e openSUSE e ha co-fondato l'Associazione LibreItalia dove è stata coinvolta in diversi eventi, migrazioni e formazione relativi a LibreOffice. In precedenza ha lavorato a migrazioni e corsi di formazione su LibreOffice per diverse amministrazioni pubbliche e privati. Da gennaio 2020 lavora in SUSE come Software Release Engineer per Uyuni e SUSE Manager e quando non segue la sua passione per i computer e per Geeko coltiva la sua curiosità per l'astronomia (da cui deriva il suo nickname deneb_alpha).
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
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In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
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Mental Health: Maintaining balance and not feeling pressured by user demands.
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GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
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Tensor Decomposition and its Applications
1. Applications of tensor
(multiway array)
factorizations and
decompositions in data mining
機械学習班輪講
11/10/25
@taki__taki__
2. Paper
Mørup, M. (2011), Applications of
tensor (multiway array) factorizations
and decompositions in data mining.
Wiley Interdisciplinary Reviews: Data
Mining and Knowledge Discovery, 1: 24–
40. doi: 10.1002/widm.1
こちらの論文からいくつか図を引用します.
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A, B =
i, j,k
j,kbi, j,k
ai, j,kbi, j,k i, j,k
(3)
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dels that ten-well
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ructure.
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the Frobenius = of a of a tensor is given
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andecomp/ limitdecompo- basic√ten-
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eration maps = A A,= . A, a .The nth mode matricizing and
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rview will ap- basic the basic ten- The nthFmode matricizing and unmatricizing op- unmatricizing op-
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mit extract
ell itself models such as the
position
troductory
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limit inherent
such asTucker asathe as well as As such, the Frobenius norm of a tensor is given by
tensor, respectively, i.e.,
ition models suchmodel, Candecomp/ their ap- √ eration maps a tensor into a matrix and a matrix into
the Candecomp/
and Tucker model, their ap- ten- ap- A F =
eration maps a into into a matrix(n)番目を中心に開く
eration maps a tensorA . a matrix and a matrix intointo
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eirdataitselfOther great introductory N arespectively, i.e.,n ×Ii.e., ···In−1 ·In+1 ···Iunmatricizing op-
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→
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els tensoras
or such the decomposition X and their applica- X ×I2 ×...×IN → (4)
Ref 24
position and their applica- matricizingX I1 (n) n ×I1 X2I···Imatrix and(n) matrix into
→ → ·I n n−1 2 n+1 ···IN X a
···I
In
ensor decomposition and their applica- eration1 ×I2 ×...×Ia tensorIinto a×I1 ·I·I n−1 ·In+1 ···IN×I1 ·I2 ···In−1 ·In+1 ···IN
I maps N (4)
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r found asthe recent of Ref 24 X 2 ×...×I
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lultiway analysis introductory 28 a tensor,matricizing i.e.,
sciences for thefor the chemical sciences28 respectively,
ecent review of Ref 24 the
g. Other greatsciences×Isciences ·In+1 ···IN
way analysis chemical 1 ·I2 ···In−1
In 28 → ···I n+1
for the chemical
the book
book on applied X applica- analysis(n)×I ·I2 ···INX In ···I ·I2→ → I1 IN1×IX·II21···I2→ ·In+1 ···IN
analysis on applied (n) analysis Iof 1 ·I2 ···IIn−1 ·I1n+1···In−1 ·I(n)×I1N → n−1 ·IX ···I×I2 ×...×I×...×IN(5) X I1 ×I2 ×...×IN (5)
of multiway In
mposition and their multiway n ×I X 1 ofun-matricizing X I ×I12 ×...×IN (5) (5)
n+1 ×I N
plied multiway analysis of nonnega- X ×I2 ×...×IN
X (n) un-matricizing
X (n) un-matricizing
n n−1
(4)
orecent review ofintroduction
nonnega-
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e introduction to nonnega- the to nonnega-
thermore, a good Ref 24 un-matricizing
matricizing
d and decompositions can be foundbe found in
s their the chemical sciences28 in The matricizing matricizing operation for a
The operation for a third-order tensor is
be for theircan model estimation The matricizing operation a third-order tensorisisof
is found in be found in can re- illustrated in Figure for a third-order tensor third-order tensor is
mpositions
decompositions
present paper, The matricizing operation for 1. The n-mode multiplication
is
the present paper,analysis of →
model estimation Iis re- 2 ···In−1 ·In+1 ···IN I1 ×I2 ×...×IN 1. The I1 ×I2 ×...×IN (5)
multiway illustrated and Figure·I1. Nillustrated in Figure a matrix M J ×Inmultiplication of
appliedis estimation is re-
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r, model re-
nimum considering only the simple in
n ×I1
X anin Figuretensor X n-mode multiplication of
illustrated order The n-mode multiplication of
1. The with X n-mode
18. The n mode matricizing and unmatricizing op-
s the Candecomp/
eration maps a tensor into a matrix and a matrix into
行列化のイメージ
as well as their ap-
great introductory
a tensor, respectively, i.e.,
n and their applica- → In ×I ·I ···I ·I ···I
Overview X I1 ×I2 ×...×IN X (n) 1 2 n−1 n+1 N (4) wires.wiley.com/widm
view of Ref 24 the matricizing
chemical sciences28
In ×I ·I ···I ·I ···I →
ultiway analysis of X (n) 1 2 n−1 n+1 N X I1 ×I2 ×...×IN (5)
un-matricizing
uction to nonnega-
(3)
ons can be found in The matricizing operation for a third-order tensor is
el estimation is re- illustrated in Figure 1. The n-mode multiplication of
only the simple and (2) an order N tensor X I1 ×I2 ×...×IN with a matrix M J ×In
res (ALS) approach. is given by
sor model estima-
the reader consult X ×n M = X •n M = Z I1 ×...×In−1 ×J ×In+1 ×...×IN , (6)
s therein. In
ollows: In ‘Tensor
zi1 ,...,in−1 , j,in+1 ,...,i N = xi1 ,...,in−1 ,in ,in+1 ,...,i N m j,in . (7)
andard tensor no-
i n =1
ucker and Cande-
ibe(1) two most
the Using the matricizing operation, this operation cor-
approaches namely responds to Z(n) = MX (n) . As a result, the matrix
ns as well as some products underlying the singular value decomposi-
torization for Data tion (SVD) can be written as U SV = S ×1 U ×2 V =
applications 1 | The matricizing S ×2 V ×third-order tensor oforder 4of 4. multiplication does
F I G U R E of ten- operation on a
1 U as the size 4 × × the
on in data mining. not matter. The outer product of the three vectors a,
whereas the Khatri–Rao product is defined as a to O(max{I J 2 , K J 2 , J 3 , L3 })(本文中より引用)
and O(I K J 2 ) to
m ofcolumn-wise Kroneckerb, and c is given by
this article is product 2 2 3
O(max{I K J, I J , K J , J }), respectively. For addi-
21. Penrose inverse (i.e., A = ( A A) A ) of Kronecker
and Khatri–Rao products are THE TUCKER AND
CANDECOMP/PARAFAC MODELS
( P ⊗ Q)† = ( P † ⊗ Q† ) (11)
The two most widely used tensor decomposition
methods are the Tucker model29 and Canonical De-
(A B)† = [( A A)∗ (B B)]−1 ( A B) (12)
composition (CANDECOMP)30 also known as Parallel
where ∗ denotes elementwise multiplication. Factor Analysis (PARAFAC)31 jointly abbreviated CP.
This reduces the complexity from O(J 3 L3 ) In the following section, we describe the models for
T A B L E 1 Summary of the Utilized Variables and Operations. X , X, x, and x are Used to Denote
Tensors, Matrices, Vectors, and Scalars Respectively.
Operator Name Operation
A, B Inner product A, B = i , j ,k a i , j ,k bi , j ,k
√
A F Frobenius norm A, A
I n × I · I ··· I · I ··· I
X(n ) Matricizing X I 1 × I 2 ×...× I N → X (n ) 1 2 n −1 n +1 N
×n or •n n-mode product X ×n M = Z where Z(n ) = MX (n )
◦ outer product a ◦ b = Z where z i , j = a i b j
⊗ Kronecker product A ⊗ B = Z where z k + K (i −1),l + L ( j −1) = a i j bkl
or | ⊗ | Khatri–Rao product A B = Z, where z k + K (i −1), j = a i j bk j .
kA k-rank Maximal number of columns of A guaranteed to be linearly independent.
26 c 2011 John Wiley & Sons, Inc. Volume 1, January/February 2011
(本文中より引用)
23. TuckerモデルとCPモデルの紹介
• TuckerモデルとCPモデルは広く利用されている
テンソル分解手法.論文では3階のテンソルの
場合について説明する.
Tuckerモデル
WIREs Data Mining and Knowledge Discovery
CPモデル
Applications of tensor (multiway array) factorizations and decompositions in data mining
WIREs Data Mining and Knowledge Discovery Applications o
multiplication by orthogonal/orthonormal matrices
Q, R, and S. Using the n-mode matricizing and an
Kronecker product operation, the Tucker model can ces
be written as cal
X (1) ≈ AG(1) (C ⊗ B) be
po
X (2) ≈ B G(2) (C ⊗ A)
tor
X ≈ CG (B ⊗ A) . rep
F I G U R E 2 | Illustration of the Tucker model of a third-order tensor F I G U R E 3 |(3) (3)
Illustration of the CANDECOMP/PARAFAC (CP) model of a
X . The model decomposes the tensor into loading matrices with a
The third-order tensor X . The model decomposes a tensor into a sum of
above decomposition for a third-order tensor is
mode specific number of components as well as a core array also rank one components and the model is very appealing due to its
denoted a Tucker3 model, the Tucker2 model ap
accounting for all multilinear interactions between the components of and uniquenessmodels are given by
Tucker1 properties. cu
each mode. The Tucker model is particularly useful for compressing Tucker2: X ≈ G × 1 A ×2 B ×3 I , are
tensors into a reduced representation given by the smaller core array G . sol
R D×D, and S D×D, we find×2 I ×3 I ,
Tucker1: X ≈ G ×1 A ma
a third-order tensor but they trivially generalize to
where X ≈ the ×1 Q ×2 R ×3 S) ×1 (本文中より引用)
I is (D identity matrix. Thus, (the Tucker1(B R−1 )
A Q−1 ) ×2 to
general Nth order arrays by introducing additional
model is equivalent −1 regular matrix decomposition
to nu
mode-specific loadings. × (CS ) = D × A × B × C.
24. T A B L E 2 Overview of the Most Common Tensor Decomposition Models, Details of the Models as well
as References to Their Literature can be Found in Refs 24, 28, and 44
Model name Decomposition Unique
CP x i , j ,k ≈ d a i ,d b j ,d c k ,d Yes
The minimal D for which approximation is exact is called the rank of a tensor, model in general unique.
Tucker x i , j ,k ≈ l ,m ,n gl ,m ,n a i ,l b j ,m c k ,n No
The minimal L , M , N for which approximation is exact is called the multilinear rank of a tensor.
Tucker2 x i , j ,k ≈ l m gl ,m ,k a i ,l b j ,m No
Tucker model with identity loading matrix along one of the modes.
Tucker1 x i , j ,k ≈ l ,m ,n gl , j ,k a i ,l No
Tucker model with identity loading matrices along two of the modes.
The model is equivalent to regular matrix decomposition.
PARAFAC2 x i , j ,k ≈ d a i ,d b (jk ) c k ,d , s.t.
D
,d
(k ) (k )
j b j ,d b j ,d = ψd ,d Yes
Imposes consistency in the covariance structure of one of the modes. The model is well suited to account for shape changes;
furthermore, the second mode can potentially vary in dimensionality.
INDSCAL x i , j ,k ≈ d a i ,d a j ,d c k ,d Yes
Imposing symmetry on two modes of the CP model.
Symmetric CP x i , j ,k ≈ d a i ,d a j ,d a k ,d Yes
Imposing symmetry on all modes in the CP model useful in the analysis of higher order statistics.
CANDELINC ˆ ˆ
x i , j ,k ≈ l mn ( d al ,d bm ,d c n ,d )a i ,l b j ,m c k ,n
ˆ No
CP with linear constraints can be considered a Tucker decomposition where the Tucker core has CP structure.
DEDICOM x i , j ,k ≈ d ,d a i ,d bk ,d r d ,d bk ,d a j ,d Yes
Can capture asymmetric relationships between two modes that index the same type of object.
PARATUCK2 x i , j ,k ≈ d ,e a i ,d bk ,d r d ,e sk ,e t j ,e Yes55
A generalization of DEDICOM that can consider interactions between two possible different sets of objects.
Block Term Decomp. x i , j ,k ≈ r l mn gl(r ) a i(,n b (jr,)m c kr,)n
mn
r) (
Yes56
A sum over R Tucker models of varying sizes where the CP and Tucker models are natural special cases.
ShiftCP x i , j ,k ≈ d a i ,d b j −τi ,d ,d c k ,d Yes6
Can model latency changes across one of the modes.
T
ConvCP x i , j ,k ≈ τ d a i ,d ,τ b j −τ,d c k ,d Yes
Can model shape and latency changes across one of the modes. When T = J the model can be reduced to regular matrix
factorization; therefore, uniqueness is dependent on T. (本文中より引用)
25. Tuckerモデル (1)
• Tuckerモデルは3階のテンソル
(core-array)
を核配列
と3つのmodeに分ける.
n-mode積による定義
WIREs Data Mining and Knowledge Discovery Applications of tensor (multiway array) factorizations and decomp
multiplication by orthogonal/orthon
Q, R, and S. Using the n-mode m
Kronecker product operation, the Tu
be written as
X (1) ≈ AG(1) (C ⊗ B)
X (2) ≈ B G(2) (C ⊗ A)
X (3) ≈ CG(3) (B ⊗ A)
F I G U R E 2 | Illustration of the Tucker model of a third-order tensor
X . The model decomposes the tensor into loading matrices with a
The above decomposition for a third
mode specific number of components as well as a core array also denoted a Tucker3 model, the
and Tucker1 models are given by
27. on the basis the solution of each is denoted solved Xfitting (C ⊗ B) )
A ← (1) (G(1)
entsminimization, of updating the elements be ALS. Bymode the model using ALS, the
of each mode. To indicate how mode can of each
Tuckerモデル (3)
ainin turn modality, it is customary estimation reducesX (2) (G(2) (C ⊗ A) of regular matrix
to each that for the least squares objective B ← to a sequence )
by pseudoinverses, i.e., commonly †
ai,l b j,mc Tucker(L, M, N) model. factorization problems. As a result, for least squares
,nmodel a k,n ,
e is denoted ALS. By fitting the model using ← X the (B ⊗ A) )†C ALS,(3) (G(3)
A ← X (1) (G(1) (C ⊗ B) )†
tensor product reduces the model minimization, the matrix †of each mode can be solved
to a sequence of regular solution × B † × C† .
29,32
estimation ×n ,
• Tuckerモデルの推定は各モード(mode)の成分を
re factorization problems. (C ⊗ A) )† for least← X ×1 A 2
array G L×M×N with (2) (G(2) As a result,
B ← X elements G
by pseudoinverses, i.e.,
squares
possibleI×L ×2 B J ← X 3 C K×N. ⊗ A) The analysis←simplifies (C ⊗ B) orthogonality is
×N
×1 A linear C ×M ×(3) (G(3) (B
interactions be-
minimization, the solution of each mode can be X (1) (G(1) when )†
順番に更新していく.最小二乗法の目的関数を )† A solved
3
s of each mode. To indicate how † imposed24 such that the estimation of the core can be
array pseudoinverses,×1 A† ×by ×omitted. Orthogonality can be⊗ A) )† by estimating
by ismodality, it is X spanned 2 B 3 C† .
持つ場合,ALSと呼ばれる
o each approximately
i.e.,
G ← customary B ← X (2) (G(2) (C imposed
odelThe analysis M,suchmodel. (C ⊗orthogonalityof is X mode(B ⊗ A) )† SVD forming
tricesafor that mode N) that the
A simplifies when the loadings ←each (G(3) through the
Tucker(L, ← X (1) (G(1) B) )† C
odality interact with the vectors of
24 ,29,32 the model
(3)
the Higher-order Orthogonal Iteration (HOOI),10,24
sor imposed ×such that the estimation of the core can be
product n
dalities with strengths given (G the ⊗i.e., )†
B ← X (2) by (2) (C A)by estimating X ×1 A† ×2 B † ×3 C† .
G←
omitted. Orthogonality can be imposed
also Figure 2.
• Tuckerモデルに直交性を課す条件がある.この
the
⊗ A) SVD forming
the loadings×M each K×N through the analysis AS(1) V (1) = X (1) (C ⊗ orthogonality is
of mode
×1 AI×LHigher-order 3Orthogonal(B The )HOOI),10,24
×2 B J C ← such, multi- Iteration (
As X
model is not unique.× C (3) (G(3) .
†
24
simplifies when B),
e matrices QL×L, R M×M , and S N×N imposed †such S(2) Vthe estimation of the core can be
条件は解析を簡素化させる
that (2)
i.e.,
ayrepresentation, G ← X ×1 by ×2omitted.C .
t is approximately spanned A
i.e.,
† † B
B ×3 Orthogonality canX (2) imposed by estimating
=
be
(C ⊗ A),
es for that mode −1(1) V (1) =−1 (1) (C ⊗ B),
ASsuch that the X CS(3) V mode through the
the loadings of each (3) = X (3) (B ⊗ A). SVD forming
2 R ×3 S) analysis ) simplifies
lity interact with the×2 (B R of when Higher-order Orthogonal Iteration (HOOI),10,24
The ×1 ( A Q vectors )) orthogonality is
24 the thatA,B,Cの初期値として,SVDのM, and
A, B, and C are
ies= G ×1 A ×2such3that by the (C ⊗suchof the core can be found as the first L,
1 imposed
with strengths× C. the X (2) i.e.,
B Sgiven = estimation
(2) (2)
)) B V A),
o Figure 2. Orthogonality can(B ⊗ A).
omitted. (3) (3) be imposed by estimating by solving the right hand
N left singular vectors given
左特異ベクトル列を使う
CS V
ctors of the unconstrained = X (3)Tucker side by SVD.AS(1) V (1) array is estimated upon con-
The core =
el the loadings As such, multi- through the SVD forming †X (1) (C ⊗ B),
• 「解析」はより良い分解を探索するというイ
is not unique. of each mode
trained orthogonal and C areN×N as the first L, by G ← X ×1 A ×2 B † ×3 C† . The above
such that ,A, M×M , orthonormal
L×L B, or and S found vergence M, and 10,24
atrices Higher-order Orthogonal Iteration (B S(2) V (2) = X not ⊗ A),
or the left singularwithout given by solving the rightare unfortunately (C guaranteed to con-
Q R
compression) vectors hamper-
Nメージで,簡素化したモデルはHOSVDと呼ぶ. procedures hand ),
HOOI
(2)
presentation, i.e.,
i.e.,
tionside by However, core arrayor- estimated to the global optimum.
error. SVD. The imposing is verge upon con- (3)
normalty does not ×2(1) ×1the ×2 B ×3 C . The above
−1 resolve (1) † lack †
−1 CS(3) V
† A special case of the X (3) (B ⊗ A). is given by
= Tucker model
×3 S) ×1 ( A Q GAS X V A = X (C ⊗ B), 29,32
vergence by ) ← (B R ))
the procedures are unfortunately to (1) the that A, B, and C are found as the first L, M, and
solution is still ambiguous not such HOSVDto con-
guaranteed where the loadings of each mode is
= G ×1 A ×2 B ×3 C.(2)
B S V (2) =
verge to the global optimum. X N left singular vectors given by solving the right hand
(C ⊗ A),
29. t is that the nonrotatability char- written as
CPモデル (2)
d even when the number of factors
r than every dimension of the three-
X I×J ×K ≈ d
J
aI ◦ bd ◦ cK ,
d such that
d
• CPモデルの推定は次のようになる b j,d ck,d .
been generalized to order N arrays xi, j,k ≈ ai,d
d
スケーリングのあいまい性を排除するために
ess property of the optimal CP so-
対角成分を全て1としたモデル Khatri–Rao product, this
the most appealing aspect of the
Using the matricizing and
equivalent to
ness of matrix decomposition has
ng challenge that has spurred a X (1) ≈ A(C B) ,
arch early on in the psychomet-
X (2) ≈ B(C A) ,
re rotational approaches such as
Overview
oposed (see, also Refs 1, 2, 31, X (3) ≈ C(B A) .
• 最小二乗法のためには For the least squares objective we, thus, find rank is that t
determine the
bruary 2011 c 2 0 1 A J←n X (1) (C
1 oh Wiley & B)(Cc . C ∗ B B)−1
Sons, In 2
a tensor is de
B ← X (2) (C A)(C C ∗ A A)−1 CP models for
C ← X (3) (B A)(B B ∗ A A)−1 representation
multilinear ra
However, some calculations are redundant between 2 rank, and m
the alternating steps. Thus, the following approach respectively10
based on premultiplying the largest mode(s) with the
27
31. テンソル分解の応用例
• 論文中の応用例としては心理学,化学,神経科
学,信号処理,バイオインフォマティクス,コ
ンピュータビジョン,Webマイニングの7つ
wires.wiley.com/widm
and handling of sparse
is provided by the
nal software, see also
TION FOR DATA
or decomposition was
y in the 1970s when
ed to alleviate the ro-
analysis, whereas the
ss higher order inter-
Davidson3 pioneered
hemistry for the anal- F I G U R E 4 | Example of a Tucker(2, 3, 2) analysis of the chopin
reas Mocks47 demon-
¨ data X 24 Preludes×20 Scales×38 Subjects described in Ref 49. The overall
odel was useful in the mean of the data has been subtracted prior to analysis. Black and
white boxes indicate negative and positive variables, whereas the size
32. WIREs Data Mining and Knowledge Discovery Applications of tensor (multiway array) factorizations and decompositions in data mining
F I G U R E 7 | Left panel: Tutorial dataset two of ERPWAVELAB50 given by X 64 Channels×61 Frequency bins×72 Time points×11Subjects×2Conditions . Right
panel a three component nonnegativity constrained three-way CP decomposition of Channel × Time − Frequency × Subject − Condition and a
three component nonnegative matrix factorization of Channel × Time − Frequency − Subject − Condition. The two models account for 60% and
76% of the variation in the data, respectively. The matrix factorization assume spatial consistency but individual time-frequency patterns of
activation across the subjects and conditions, whereas the three-way CP analysis impose consistency in the time-frequency patterns across the
subjects and conditions. As such, these most consistent patterns of activations are identified by the model.
down-weighted in the extracted estimates of the con- such that S is statistically independent and E residual
sistent event-related activations. noise can be solved through the CP decomposition of
D some higher-order cumulants due to the important
9,52