张量分解

@author: slandarer

张量切片及纤维

输入图片说明

单位张量与超对角单位张量

输入图片说明

单位张量生成函数：teye.m

超对角张量生成函数：sdiag.m

使用实例：

% 超对角线为[1,2,3]的超对角张量
A=sdiag([1,2,3],3)

% 3x3x2大小的单位张量
B=teye([3,3,2])

A(:,:,1) =
 1     0     0
 0     0     0
 0     0     0

A(:,:,2) =
 0     0     0
 0     2     0
 0     0     0

A(:,:,3) =
 0     0     0
 0     0     0
 0     0     3


B(:,:,1) =
 1     0     0
 0     1     0
 0     0     1

B(:,:,2) =
 0     0     0
 0     0     0
 0     0     0

模-n展开及模-n积

输入图片说明

张量模-n展开代码：modeN_unfold

使用实例：

X=zeros(3,4,2);
X(1:24)=1:24;

X1=modeN_unfold(X,1)
X2=modeN_unfold(X,2)
X3=modeN_unfold(X,3)

 X1 =   
 1     4     7    10    13    16    19    22
 2     5     8    11    14    17    20    23
 3     6     9    12    15    18    21    24

 X2 =
 1     2     3    13    14    15
 4     5     6    16    17    18
 7     8     9    19    20    21
10    11    12    22    23    24

 X3 =
 1     2     3     4     5     6     7     8     9    10    11    12
13    14    15    16    17    18    19    20    21    22    23    24

输入图片说明 张量模-n展积代码：modeN_prod

使用实例：

X=zeros(3,4,2);
X(1:24)=1:24;

U1=[1,3,5;2,4,6];
U2=[1,0;0,1;1,1];

Y1=modeN_prod(X,U1,1)
Y2=modeN_prod(X,U2,3)

Y1(:,:,1) =
22    49    76   103
28    64   100   136

Y1(:,:,2) =
130   157   184   211
172   208   244   280



Y2(:,:,1) =
1     4     7    10
2     5     8    11
3     6     9    12

Y2(:,:,2) =
13    16    19    22
14    17    20    23
15    18    21    24

Y2(:,:,3) =
14    20    26    32
16    22    28    34
18    24    30    36

所用特殊张量mat文件

超对角线数据抹去张量及正面切片随机旋转张量输入图片说明实验对象为 DIPUM2E 教材中 256×256 大小的原图(Goldhill 及 peppers)，并对其添加不同扰动后拼接而成 256×256×256 大小的张量，其中 Goldhill 图像将图像不同位置增添 50×50 大小的亮度为 0 的矩形区域并拼接而成，而 peppers 则是将图像进行不同程度的旋转后拼接成三维张量的形式。输入图片说明

CP分解

输入图片说明 张量CP分解三维代码：CP3_ALS 使用实例：

% 读取数据
file=load('Peppers.mat');
X=file.XData;

% 截断值60，精确度1e-3CP分解
[A,B,C,lambda]=CP3_ALS(X,60,1e-3);
% 使用模-n积还原拟合张量
% sdiag(lambda)x_1A x_2B x_3C
apprX=modeN_prod(modeN_prod(modeN_prod(sdiag(lambda),A,1),B,2),C,3);

% 展示拟合结果
imshow(uint8(apprX(:,:,1)))
% 展示与原图像差值
figure()
imagesc(abs(X(:,:,1)-apprX(:,:,1)))

colormap(PYCM().inferno())% 使用python colormap渲染，可删去
colorbar

输入图片说明 不同截断值分解误差：

function demo2_moreT_CP
% 读取数据
file=load('Peppers.mat');
X=file.XData;

for i=60:10:500
    % 截断值60，精确度1e-3CP分解
    [A,B,C,lambda]=CP3_ALS(X,i,1e-3);
    % 使用模-n积还原拟合张量
    % sdiag(lambda)x_1A x_2B x_3C
    apprX=modeN_prod(modeN_prod(modeN_prod(sdiag(lambda),A,1),B,2),C,3);
    diffX=abs(X-apprX);

    % 将误差值存入数组
    errRMSE(i)=sqrt(sum(diffX.^2,'all')./numel(diffX));
    errMEAN(i)=mean(diffX,'all');
    errMAX(i)=max(diffX,[],'all');
    disp(['the [',num2str(i),']-term fit is completed'])
end
save demo2_CP_err.mat errRMSE errMEAN errMAX
errSet=load('demo2_CP_err.mat');

t=60:10:500;
colororder([82,124,179;170,64,64]./255)

% 双y轴——左侧图
yyaxis left
hold on
plot(t,errSet.errMEAN(t),'o-','Color',[82,124,179]./255,'MarkerFaceColor',[82,124,179]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)
plot(t,errSet.errRMSE(t),'^-','Color',[62,85,96]./255,'MarkerFaceColor',[62,85,96]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)
ax=gca;
ax.YMinorTick='on';

% 双y轴——右侧图
yyaxis right
plot(t,errSet.errMAX(t),'s-','Color',[170,64,64]./255,'MarkerFaceColor',[170,64,64]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)

% 图例
legend('MEAN error','RMSE error','MAX error')

% 坐标区域修饰
ax=gca;grid on;box off
ax.LineWidth=1.5;
ax.GridLineStyle='-.';
ax.GridColor=[1,1,1].*.2;
ax.Color=[249,250,245]./255;
ax.XMinorTick='on';
ax.YMinorTick='on';
ax.FontName='Cambria';
ax.FontSize=13;
end

输入图片说明

HO-SVD及其截断

输入图片说明 张量HO-SVD N维代码：HO-SVD 使用实例：

% 读取数据
file=load('Peppers.mat');
X=file.XData;

% 截断值60，HO-SVD分解
[G,U]=HO_SVD(X,60);
% 使用模-n积还原拟合张量
% Gx_1U_1 x_2U_2 x_3U_3
apprX=modeN_prod(modeN_prod(modeN_prod(G,U{1},1),U{2},2),U{3},3);

% 展示拟合结果
imshow(uint8(apprX(:,:,1)))
% 展示与原图像差值
figure()
imagesc(abs(X(:,:,1)-apprX(:,:,1)))

colormap(PYCM().inferno())% 使用python colormap渲染，可删去
colorbar

输入图片说明 不同截断值分解误差：

function demo2_moreT_HOSVD
% 读取数据
file=load('Peppers.mat');
X=file.XData;

for i=60:10:250
    % 截断值i，HO-SVD分解
    [G,U]=HO_SVD(X,i);
    % 使用模-n积还原拟合张量
    % Gx_1U_1 x_2U_2 x_3U_3
    apprX=modeN_prod(modeN_prod(modeN_prod(G,U{1},1),U{2},2),U{3},3);
    diffX=abs(X-apprX);

    % 将误差值存入数组
    errRMSE(i)=sqrt(sum(diffX.^2,'all')./numel(diffX));
    errMEAN(i)=mean(diffX,'all');
    errMAX(i)=max(diffX,[],'all');
    disp(['the [',num2str(i),']-term fit is completed'])
end
save demo2_HOSVD_err.mat errRMSE errMEAN errMAX
errSet=load('demo2_HOSVD_err.mat');

t=60:10:250;
colororder([82,124,179;170,64,64]./255)

% 双y轴——左侧图
yyaxis left
hold on
plot(t,errSet.errMEAN(t),'o-','Color',[82,124,179]./255,'MarkerFaceColor',[82,124,179]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)
plot(t,errSet.errRMSE(t),'^-','Color',[62,85,96]./255,'MarkerFaceColor',[62,85,96]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)
ax=gca;
ax.YMinorTick='on';

% 双y轴——右侧图
yyaxis right
plot(t,errSet.errMAX(t),'s-','Color',[170,64,64]./255,'MarkerFaceColor',[170,64,64]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)

% 图例
legend('MEAN error','RMSE error','MAX error')

% 坐标区域修饰
ax=gca;grid on;box off
ax.LineWidth=1.5;
ax.GridLineStyle='-.';
ax.GridColor=[1,1,1].*.2;
ax.Color=[249,250,245]./255;
ax.XMinorTick='on';
ax.YMinorTick='on';
ax.FontName='Cambria';
ax.FontSize=13;
end

输入图片说明

O-SVD及其截断

输入图片说明 张量O-SVD 代码：O-SVD 使用实例：

% 读取数据
file=load('Peppers.mat');
X=file.XData;

% 截断值60，O-SVD分解
[U,S,V,F]=O_SVD(X,60);
% 使用模-n积及切片积还原拟合张量
% U *_3S *_3V x_3F
apprX=modeN_prod(slice_prod(slice_prod(U,S,3),V,3),F,3);

% 展示拟合结果
imshow(uint8(apprX(:,:,1)))
% 展示与原图像差值
figure()
imagesc(abs(X(:,:,1)-apprX(:,:,1)))

colormap(PYCM().inferno())% 使用python colormap渲染，可删去
colorbar

输入图片说明 不同截断值分解误差：

function demo2_moreT_OSVD
% 读取数据
file=load('Peppers.mat');
X=file.XData;

for i=60:10:250
    % 截断值i，O-SVD分解
    [U,S,V,F]=O_SVD(X,i);
    % 使用模-n积及切片积还原拟合张量
    % U *_3S *_3V x_3F
    apprX=modeN_prod(slice_prod(slice_prod(U,S,3),V,3),F,3);
    diffX=abs(X-apprX);

    % 将误差值存入数组
    errRMSE(i)=sqrt(sum(diffX.^2,'all')./numel(diffX));
    errMEAN(i)=mean(diffX,'all');
    errMAX(i)=max(diffX,[],'all');
    disp(['the [',num2str(i),']-term fit is completed'])
end
save demo2_OSVD_err.mat errRMSE errMEAN errMAX
errSet=load('demo2_OSVD_err.mat');

t=60:10:250;
colororder([82,124,179;170,64,64]./255)

% 双y轴——左侧图
yyaxis left
hold on
plot(t,errSet.errMEAN(t),'o-','Color',[82,124,179]./255,'MarkerFaceColor',[82,124,179]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)
plot(t,errSet.errRMSE(t),'^-','Color',[62,85,96]./255,'MarkerFaceColor',[62,85,96]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)
ax=gca;
ax.YMinorTick='on';

% 双y轴——右侧图
yyaxis right
plot(t,errSet.errMAX(t),'s-','Color',[170,64,64]./255,'MarkerFaceColor',[170,64,64]./255,...
    'MarkerEdgeColor',[1,1,1],'LineWidth',2,'MarkerSize',12)

% 图例
legend('MEAN error','RMSE error','MAX error')

% 坐标区域修饰
ax=gca;grid on;box off
ax.LineWidth=1.5;
ax.GridLineStyle='-.';
ax.GridColor=[1,1,1].*.2;
ax.Color=[249,250,245]./255;
ax.XMinorTick='on';
ax.YMinorTick='on';
ax.FontName='Cambria';
ax.FontSize=13;
end

输入图片说明

复杂度

输入图片说明

参考文献

[1]T. G. Kolda, B. W. Bader. Tensor decompositions and applications. SIAM review, 2009, 51(3): 455--500.
[2]Misha E. Kilmer, Karen Braman, Ning Hao, Randy C. Hoover. “Third Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging,”.SIAM J on Matrix Analysis and Applications, 2013, 34(1): 148-172.
[3]TENSOR-TRAIN DECOMPOSITION Society for Industrial and Applied Mathematics . SIAM J. SCI. COMPUT, 2011, 33(5): 2295–2317.
[4]L. R. Tucker. Some mathematical notes on three-mode factor analysis. Psychometrika, 1966, 31: 279–311.
[5]T. G. Kolda. Multilinear operators for higher-order decompositions. Tech. Report SAND. Sandia National Laboratories, 2006: 2006--2081.
[6]J. D. Carroll, J. J. Chang. Analysis of individual differences in multidimensional scaling via an N-way generalization of ‘Eckart-Young’decomposition. Psychometrika, 1970, 35: 283–319.
[7]R. A. Harshman. Foundations of the PARAFAC procedure: models and conditions for an “explanatory” multi-modal factor analysis, UCLA working papers in phonetics, 1970, 16: 1–84.
[8]C. F. Van Loan. The ubiquitous Kronecker product. J. Comput. Appl. Math, 2000, 123: 85–100.
[9]A. Smilde, R. Bro, P. Geladi. Multi-way analysis: applications in the chemical sciences. Wiley, 2004.
[10] L. De Lathauwer, B. De Moor, J. Vandewalle. A multilinear singular value decomposition. SIAM J. Matrix Anal. A, 2000, 21: 1253–1278.
[11] A. Smilde, R. Bro, P. Geladi. Multi-Way Analysis: Applications in the Chemical Sciences. Wiley. West Sussex. England, 2004.
[12] H. A. L. Kiers. Towards a standardized notation and terminology in multiway analysis. J. Chemometrics, 2000, 14: 105–122.
[13] C. Battaglino, G. Ballard, T. G. Kolda. A Practical Randomized CP Tensor Decomposition. SIAM J. Matrix Analysis and Applications，2018, 39(2): 876-901
[14] E. C. Chi, T. G. Kolda. On Tensors, Sparsity, and Nonnegative Factorizations. SIAM J. Matrix Analysis, 2012, 33(4): 1272-1299.
[15] S. Hansen, T. Plantenga, T. G. Kolda. Newton-Based Optimization for Kullback-Leibler Nonnegative Tensor Factorizations. Optimization Methods and Software, 2015.
[16] S. Sherman, T. G. Kolda. Estimating Higher-Order Moments Using Symmetric Tensor Decomposition. SIAM J. Matrix Analysis and Applications, 2020, 41 : 1369-1387.
[17] L. De Lathauwer, B. De Moor, J. Vandewalle. A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl, 2000, 21: 1253–1278.
[18] P. M. Kroonenberg, J. De Leeuw. Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika,1980, 45: 69–97.
[19] L. De Lathauwer, B. De Moor, J. Vandewalle. On the best rank-1 and rank-(R1, R2,...,RN ) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl, 2000, 21: 1324–1342.
[20] C. Zeng and M. K. Ng, Decompositions of third-order tensors: HOSVD, T-SVD, and beyond, Numer. Linear Algebra Appl, 2020, 27. e2290.
[21] R. A. Harshman, PARAFAC2: Mathematical and technical notes, UCLA Working Papers in Phonetics, 1972, 22: 30–47.
[22] Minghui Ding, Pengpeng Xie. A randomized singular value decomposition for third-order oriented tensors. arXiv preprint, 2022, arXiv:2203.02761v1 [math.NA]: 4-5.

slandarer / 张量分解

张量分解

目录

张量切片及纤维

单位张量与超对角单位张量

模-n展开及模-n积

所用特殊张量mat文件

CP分解

HO-SVD及其截断

O-SVD及其截断

复杂度

参考文献

简介

发行版

贡献者

近期动态

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张量分解

目录

张量切片及纤维

单位张量与超对角单位张量

模-n展开及模-n积

所用特殊张量mat文件

CP分解

HO-SVD及其截断

O-SVD及其截断

复杂度

参考文献

简介

发行版

贡献者

近期动态

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slandarer / 张量分解