實驗要求可以參考deeplearning的tutorial，Exercise:Sparse Autoencoder。稀疏自動編碼的原理可以參照之前的博文，神經網絡， 稀疏自動編碼 。
1. 神經網絡結構：
實驗是實現三層的稀疏自動編碼神經網絡，神經網絡結構包括輸入層64個neuron，隱含層25個neuron（都不包括bias結點），輸出層和輸入層相同的neuron的個數。
2. 訓練數據：
實驗中的原始數據是10副512×512大小的灰度圖像，存儲在“IMAGES.mat”文件中。下圖是其中的一張圖像。
我們隨機中10副圖像中挑選一個，然後再從該圖像中隨機的選一個8*8的圖像patch，得到的這個patch作爲一個訓練數據，重複這個隨機過程10000，最後得到10000數據組成的訓練集，存儲在64×10000 的矩陣，每一列代表一個數據。
下圖是總10000個數據中隨機挑選出100個進行顯示。
3. 損失函數（loss function）
稀疏自動編碼的損失函數由三部分組成，公式如下：
4. 偏導數（partial derivatives）
偏導數的計算是通過BP算法，需要注意是，由於在實驗中使用的是batch的優化方法，所以偏導數計算的時候對所有樣本的偏導數做了一個求和操作。
5.梯度檢驗（Gradient checking）
有網友說他們在自己的電腦上跑Gradient checking這步的時候用了1個多小時，但不知道是不是我的電腦配置比較高，我才跑了不到4分鐘。
6. 用L-BFGS算法進行訓練
因爲我們已經有了損失函數和相應參數的偏導數，所以和直接用成熟的L-BFGS算法的包，最後算法到底最大迭代次數400的時候，停止迭代。
7. 實驗結果：
下圖顯示的是第一層到第二層的參數，由於輸入層爲64個neuron，隱含層爲25個neuron，所以下面共有25個patch，每一個patch大小爲8*8.
sampleIMAGES.m
function patches = sampleIMAGES()
% sampleIMAGES
% Returns 10000 patches for training
load IMAGES; % load images from disk
patchsize = 8; % we'll use 8x8 patches
numpatches = 10000;
% Initialize patches with zeros. Your code will fill in this matrix--one
% column per patch, 10000 columns.
patches = zeros(patchsize*patchsize, numpatches);
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Fill in the variable called "patches" using data
% from IMAGES.
%
% IMAGES is a 3D array containing 10 images
% For instance, IMAGES(:,:,6) is a 512x512 array containing the 6th image,
% and you can type "imagesc(IMAGES(:,:,6)), colormap gray;" to visualize
% it. (The contrast on these images look a bit off because they have
% been preprocessed using using "whitening." See the lecture notes for
% more details.) As a second example, IMAGES(21:30,21:30,1) is an image
% patch corresponding to the pixels in the block (21,21) to (30,30) of
% Image 1
for i = 1:numpatches
x = ceil(rand(1)*505);
y = ceil(rand(1)*505);
z = ceil(rand(1)*10);
patch = IMAGES(x : x+patchsize-1, y : y+patchsize-1, z);
patches(:,i) = patch(:);
end
%% ---------------------------------------------------------------
% For the autoencoder to work well we need to normalize the data
% Specifically, since the output of the network is bounded between [0,1]
% (due to the sigmoid activation function), we have to make sure
% the range of pixel values is also bounded between [0,1]
patches = normalizeData(patches);
end
%% ---------------------------------------------------------------
function patches = normalizeData(patches)
% Squash data to [0.1, 0.9] since we use sigmoid as the activation
% function in the output layer
% Remove DC (mean of images).
patches = bsxfun(@minus, patches, mean(patches));
% Truncate to +/-3 standard deviations and scale to -1 to 1
pstd = 3 * std(patches(:));
patches = max(min(patches, pstd), -pstd) / pstd;
% Rescale from [-1,1] to [0.1,0.9]
patches = (patches + 1) * 0.4 + 0.1;
end
sparseAutoencoderCost.m
function [cost,grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ...
lambda, sparsityParam, beta, data)
% visibleSize: the number of input units (probably 64)
% hiddenSize: the number of hidden units (probably 25)
% lambda: weight decay parameter
% sparsityParam: The desired average activation for the hidden units (denoted in the lecture
% notes by the greek alphabet rho, which looks like a lower-case "p").
% beta: weight of sparsity penalty term
% data: Our 64x10000 matrix containing the training data. So, data(:,i) is the i-th training example.
% The input theta is a vector (because minFunc expects the parameters to be a vector).
% We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this
% follows the notation convention of the lecture notes.
W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);
b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end);
% Cost and gradient variables (your code needs to compute these values).
% Here, we initialize them to zeros.
cost = 0;
W1grad = zeros(size(W1));
W2grad = zeros(size(W2));
b1grad = zeros(size(b1));
b2grad = zeros(size(b2));
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Compute the cost/optimization objective J_sparse(W,b) for the Sparse Autoencoder,
% and the corresponding gradients W1grad, W2grad, b1grad, b2grad.
%
% W1grad, W2grad, b1grad and b2grad should be computed using backpropagation.
% Note that W1grad has the same dimensions as W1, b1grad has the same dimensions
% as b1, etc. Your code should set W1grad to be the partial derivative of J_sparse(W,b) with
% respect to W1. I.e., W1grad(i,j) should be the partial derivative of J_sparse(W,b)
% with respect to the input parameter W1(i,j). Thus, W1grad should be equal to the term
% [(1/m) \Delta W^{(1)} + \lambda W^{(1)}] in the last block of pseudo-code in Section 2.2
% of the lecture notes (and similarly for W2grad, b1grad, b2grad).
%
% Stated differently, if we were using batch gradient descent to optimize the parameters,
% the gradient descent update to W1 would be W1 := W1 - alpha * W1grad, and similarly for W2, b1, b2.
%
% data = [ones(1,size(data,2)); data];
[n m] = size(data);
z2 = W1* data + repmat(b1,1,m);
a2 = sigmoid(z2);
z3 = W2 * a2 + repmat(b2,1,m);
a3 = sigmoid(z3);
phat = mean(a2,2);
p = repmat(sparsityParam, size(phat));
sparse = p .* log(p ./ phat) + (1-p) .* log((1-p) ./ (1-phat));
%J = trace((data - a3)' * (data - a3)) / (size(data,2)*2);
J = sum(sum((a3-data).^2)) / (m*2);
regu = (W1(:)'*W1(:) + W2(:)'*W2(:))/2;
cost = J + lambda*regu + beta * sum(sparse);
delta3 = -1* (data-a3).*a3.*(1-a3);
delta2 = (W2'*delta3+beta*repmat(-p./phat+(1-p)./(1-phat),1,size(data,2))).*a2.*(1-a2);
W2grad = delta3*a2'/m;
b2grad = mean(delta3,2);
W1grad = delta2*data'/m;
b1grad = mean(delta2,2);
W2grad = W2grad + lambda*W2;
W1grad = W1grad + lambda*W1;
%-------------------------------------------------------------------
% After computing the cost and gradient, we will convert the gradients back
% to a vector format (suitable for minFunc). Specifically, we will unroll
% your gradient matrices into a vector.
grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];
end
%-------------------------------------------------------------------
% Here's an implementation of the sigmoid function, which you may find useful
% in your computation of the costs and the gradients. This inputs a (row or
% column) vector (say (z1, z2, z3)) and returns (f(z1), f(z2), f(z3)).
function sigm = sigmoid(x)
sigm = 1 ./ (1 + exp(-x));
end
