Learning the parts of object by NMF

本文为Letters to nature上文章Learning the parts of objects by non-negativematrix factorization的读书笔记，针对如何基于NMF在神经网络中学习一个object的各层part做出理论上的分析，并在人脸part学习和text语义特征学习上做了相应实验。本文不含如何去解NMF，只给出非负约束下矩阵分解的结果。

Learning the parts of object by NMF

Rachel Zhang

1. Theoretical basis and Motivation

1. Part-based representation

There is psychological and physiological evidence for parts-based representations in the brain, and certaincomputational theories of object recognition rely on such representations.

2. What is NMF? Why use NFM in NeuralNetwork?

NMF: non-negative matrix factorization

Difference: PCA, VQ learn holistic, not part-based representations, NMFdifferent from them via non-negative constrains.

Virtue in Neural Network:

1. Firing rates(number of spikes in a window) of neurons are nevernegative.

2. Synaptic strengths do not changesign.

2. Applied Experiments

3. Applied result of PCA, VQ and NMF

Figure 1 shows the 3 methods learn to represent a face as a linearcombination of basis images.

VQ: discovers a basisconsisting of whole-face prototypes.

PCA: discovers a basis of ‘eigenfaces’,some of which resemble distorted versions of wholefaces.

NMF: discovers a basis consisting localized features that correspond better withintuitive notions of the parts of faces.

Figure 1.Basis of NMF, VQ and PCA

对于图中的encoding，红色表示负数，灰黑表示正数，颜色程度表示大小。

4. Matrix Factorization framework

为什么NMF会得出与PCA和VQ迥然不同的基呢？我们这里将这三种方法展示在Matrix factorizationframework里，首先看一下这个框架。图像数据库用n*m的矩阵V表示，每列包括m张人脸图中一张的n个非负像素值。这三种方法构造近似矩阵分解：V≈WH，or

V_iu≈(WH)_iu≈Σ_aW_iaH_au

W∈R^n*r中的r列是basis images. H∈R^r*m中的每一列是编码，对应V中的一张脸。秩r的选择遵循(n+m)r<nm。下面说说VQ，PCA和NMF的区别。

5. Difference in MF framework

In VQ，each column in H is constrained to be a unary vector(一个非0元),也就是说每张脸只由一个基进行估计。PCA 在VQ的基础上做了松弛, it constrains the column of W to be orthonormal and the rows of H to beorthogonal to each other. 这样的松弛就允许了每张脸可以由一些基图像（即eigenface）通过线性组合生成了。尽管eigenfaces在统计上有最大variance的解释，但由于PCA对矩阵W和H的赋值是任意sign的，所以从视觉角度去想还不是很说得通。

与一元向量约束的VQ不同，非负矩阵分解约束允许多个基图像组成一张人脸，但由于约束了W和H都非负，所以只允许增加基（不许减少）。所以基于NMF的方法可以构造出一个parts-based representation，这刚好符合了将不同parts组成一个整体的直觉想法。从图1中就可以看出，NMF的基和编码包含了一个大片0系数，或者说basis和encoding都是稀疏的。Basis的稀疏性源于基是局部信息（嘴巴、鼻子和其他人脸部分）在不同位置、形态的版本；由于不同人脸对这些versionsof faces’ parts 进行了组合、重用，所以encoding也是稀疏的，这和unaryencoding的VQ与fully distributed PCA都有很大不同。

6. NMF的优化函数

NMF优化目标函数：

该函数中，V_iu表示一个pixel，它由加入泊松噪声的（WH）_iu而得。因此目标函数实际上表示由基W和编码H产生图像V的概率。通过update算法，F将达到一个局部极大值。关于update的具体说明请见[2,3].

Figure 3. Iterativeupdate algorithm for NMF

图3中为updata算法，其单调收敛可以通过类似EM算法的收敛证明。更新算法保证了W和H的非负性及正交性（和W的标准正则性）。

7. image pixels和编码变量的依赖关系：

Figure 4. Probabilistichidden variables model underlying NMF

图4这个网络描述了底层可见变量Vi如何由生成高层隐变量生成。根据这个模型，V_i由Σ_aW_iaH_a生成。或者说H_a对V_i的影响可以用权值W_ia表示。在人脸图像应用中（右图），Vi就是V的第i列像素（底层的一个单元），隐变量（顶层）H_a(H的第a列)为第a个part-based基图像（中层，如某特定形态的眼睛）的编码. 给定a，W_1a…W_na就构成了一个特定basis image(中层)，和其他basis images一起构成了一个完整的人脸图像。

PS：[1]illustrate the versatility of NMF by applying NMF to the semantic analysis oftext documents. Refer to [1] ,P3, left part.

3. Method Analysis

8. 用NMF处理复杂问题<重点分析>

尽管NMF在人脸识别和语义topic分析中比较成功，但这不表示它可以用于任何数据，比如采集到非固定点拍摄的图像，或者高清晰物体就不适合用NMF做。对于这种复杂问题part学习的处理，就需要一个多层隐变量的结构模型(类似DL)，而不像NMF中只用一层表示隐变量。另外，尽管非负这个约束可以进行part-based representation的学习，它们在编码的相关性方面也是有不足的，NMF只约束了W和H的非负性（这是唯一先验，只要求满足这个），而没有考虑到对于该先验，H内部元素间的相关性。比如，第一列第5幅，跟第4列最后一幅，那两列是有重合的地方：眼睛。它们线性组合表达人脸的时候，h对应这两位置的码应该相互间“有所协调”，否则就“重复”了。但是NMF不考虑的那么复杂，简单起见，假设他们不相关了。考虑的话是更复杂，更完善的模型，已有不少工作就是在这样方向上的改进，包括sparse的|X|1，|X|0。

需要注意的是，这样无相关性仅仅是先验的无相关性，在v=Wh的或min |v-Wh|约束下h内部还是相互影响，相互制约的。有人叫你用W表达一幅人脸，但他先不给你看到那幅人脸，人脸V=Wh，表达人脸的码只事先要求是非负的，没有其它要求。所以你知道了客户没有关心h内部元素相关性，在先验里头它们是相互独立的。然后，他把人脸给你了，要满足你写的约束，这里h内部要相互协调线性组合出V来，所以约束下（而非先验下）还是有相关性的。

9. ICA

ICA是PCA的变种，假设隐变量相互独立且非高斯分布。将ICA用于人脸图像学出来的基是整体的(类似PCA)。ICA的独立假设不适于part-basedrepresentation学习，因为很多部分可能会同时出现，假设h内部元素独立就不能表示比较复杂的相关性。

10．神经科学上的应用总结

1). The consequence of non-negative constraints is that synapses are either excitatory or inhibitory,but do not change sign.

2). In neural science, the non-negativity of the hidden and visible variables corresponds to the physiological fact that firing rates of neurons cannot be negative.

3). One-sided constraints on neural activity and synaptic strengths in the brain may be important for developing sparsely distributed, parts-based representations for perception.

这里解释下第一条，在神经元产生的时候，excitatory为正，inhibitory为0（而不为负）。因为如果刺激成功，产生spike，否则正常波动firing=0。所以在非负约束下H要么>0要么=0，不改变sign，使得NMF适用于神经信号firing rate矩阵分解。

Appendix - Code: implemented in MATLAB

1. Gradient Descent. 这个代码是台湾大学林智仁老师写的。

function [W,H] = nmf(V,Winit,Hinit,tol,timelimit,maxiter)% NMF by alternative non-negative least squares using projected gradients
% Author: Chih-Jen Lin, National Taiwan University% W,H: output solution
% Winit,Hinit: initial solution
% tol: tolerance for a relative stopping condition
% timelimit, maxiter: limit of time and iterationsW = Winit; H = Hinit; initt = cputime;gradW = W*(H*H') - V*H'; gradH = (W'*W)*H - W'*V;
initgrad = norm([gradW; gradH'],'fro');
fprintf('Init gradient norm %f\n', initgrad);
tolW = max(0.001,tol)*initgrad; tolH = tolW;for iter=1:maxiter,% stopping conditionprojnorm = norm([gradW(gradW<0 | W>0); gradH(gradH<0 | H>0)]);if projnorm < tol*initgrad | cputime-initt > timelimit,break;end[W,gradW,iterW] = nlssubprob(V',H',W',tolW,1000); W = W'; gradW = gradW';if iterW==1,tolW = 0.1 * tolW;end[H,gradH,iterH] = nlssubprob(V,W,H,tolH,1000);if iterH==1,tolH = 0.1 * tolH; endif rem(iter,10)==0, fprintf('.'); end
end
fprintf('\nIter = %d Final proj-grad norm %f\n', iter, projnorm);function [H,grad,iter] = nlssubprob(V,W,Hinit,tol,maxiter)% H, grad: output solution and gradient
% iter: #iterations used
% V, W: constant matrices
% Hinit: initial solution
% tol: stopping tolerance
% maxiter: limit of iterationsH = Hinit; WtV = W'*V; WtW = W'*W; alpha = 1; beta = 0.1;
for iter=1:maxiter,  grad = WtW*H - WtV;projgrad = norm(grad(grad < 0 | H >0));if projgrad < tol,breakend% search step size for inner_iter=1:20,Hn = max(H - alpha*grad, 0); d = Hn-H;gradd=sum(sum(grad.*d)); dQd = sum(sum((WtW*d).*d));suff_decr = 0.99*gradd + 0.5*dQd < 0;if inner_iter==1,decr_alpha = ~suff_decr; Hp = H;endif decr_alpha, if suff_decr,H = Hn; break;elsealpha = alpha * beta;endelseif ~suff_decr | Hp == Hn,H = Hp; break;elsealpha = alpha/beta; Hp = Hn;endendend
endif iter==maxiter,fprintf('Max iter in nlssubprob\n');
end

2. Multiplier Update

multiplier update对应上面本文提到的更新方程。

function [A, S] = nmf(X, K, type)
NMF  Nonnegative Matrix Factorization.[A, S] = NMF(X, K, TYPE) performs nonnegative matrix factorization on the
nonnegative M-by-N matrix X using K components. The M-by-K matrix A and
the K-by-N matrix S are computed such that a divergence between X and A*S
is minimized while preserving element-wise nonnegativity of both
matrices.[A, S] = NMF(X, K, 'euc') uses the Euclidean distance.
[A, S] = NMF(X, K, 'kl') uses the Kullback-Leibler divergence.
[A, S] = NMF(X, K, 'is') uses the Itakura-Saito divergence.Author: Steve Tjoa
Institution: University of Maryland (Signals and Information Group)
Created: July 1, 2009
Last modified: July 2, 2009This code was written during the workshop on Music Information Retrieval
at the Center for Computer Research in Music and Acoustics (CCRMA) at
Stanford University.Initialize parameters.
maxiter = 100;
[M, N] = size(X);
O = ones(M, N);
c = 1;  % safety parameter% Initialize outputs.
A = rand(M, K);
S = rand(K, N);if strcmp(type, 'euc')for iter=1:maxiter% Euclidean distanceA = A.*(X*S' + c)./(A*(S*S') + c);S = S.*(A'*X + c)./((A'*A)*S + c);[A, S] = rescaledict(A, S);endelseif strcmp(type, 'kl')for iter=1:maxiter% KL DivergenceA = A.*((X./(A*S))*S' + c)./(O*S' + c);S = S.*(A'*(X./(A*S)) + c)./(A'*O + c);[A, S] = rescaledict(A, S);endelseif strcmp(type, 'is')for iter=1:maxiter% IS DivergenceA = A.*((X./(A*S).^2)*S' + c)./((1./(A*S))*S' + c);S = S.*(A'*(X./(A*S).^2) + c)./(A'*(1./(A*S)) + c);[A, S] = rescaledict(A, S);end
end

function [A, S] = rescaledict(A,S)
% RESCALEDICT  Rescale dictionary.
%
% Author: Steve Tjoa
% Institution: University of Maryland (Signals and Information Group)
% Created: July 1, 2009
% Last modified: July 2, 2009
%
% This code was written during the workshop on Music Information Retrieval
% at the Center for Computer Research in Music and Acoustics (CCRMA) at
% Stanford University.if nargin==2K = size(A,2);for k=1:Kg = norm(A(:,k));A(:,k) = A(:,k)./g;S(k,:) = S(k,:).*g;end
end
end

11. Reference

[1]. Daniel D.Lee* & H. Sebastian Seung* Learningthe parts of objects by Non-negative Matrix Factorization

[2]. Daniel D.Lee* & H. Sebastian Seung* Algorithms for Non-negative Matrix Factorization

[3]. Chih-Jen Lin On the Convergence of Multiplicative Update Algorithms for Non-negative Matrix Factorization

[4]. Phoyer U. NMF with sparse constraints. 论文中matlab代码实现和这只牛的主页。

本文尚不成熟，希望大家提出宝贵意见。

关于Machine Learning更多的学习资料与相关讨论将继续更新，敬请关注本博客和新浪微博Rachel____Zhang.

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