使用MATLAB中的自动编码器进行函数逼近

Question

我有一个简单的非线性函数y=x^2，其中x和y是n维向量，平方是分量平方。我想在Matlab中使用自动编码器用低维向量近似y。问题是，即使低维空间设置为n-1，我也会得到失真的重建y。我的训练数据看起来像这和这里是从低维空间重建的典型结果。下面给出了我的Matlab代码。

%% Training data
inputSize=100;
hiddenSize1 = 80;
epo=1000;
dataNum=6000;
rng(123);
y=rand(2,dataNum);
xTrain=zeros(inputSize,dataNum);
for i=1:dataNum
    xTrain(:,i)=linspace(y(1,i),y(2,i),inputSize).^2;
end
%scaling the data to [-1,1]
for i=1:inputSize
    meanX=0.5; %mean(xTrain(i,:));
    sd=max(xTrain(i,:))-min(xTrain(i,:));
    xTrain(i,:) = (xTrain(i,:)- meanX)./sd;
end
%% Training the first Autoencoder
% Create the network. 
autoenc1 = feedforwardnet(hiddenSize1);
autoenc1.trainFcn = 'trainscg';
autoenc1.trainParam.epochs = epo;
% Do not use process functions at the input or output
autoenc1.inputs{1}.processFcns = {};
autoenc1.outputs{2}.processFcns = {};
% Set the transfer function for both layers to the logistic sigmoid
autoenc1.layers{1}.transferFcn = 'tansig';
autoenc1.layers{2}.transferFcn = 'tansig';
% Use all of the data for training
autoenc1.divideFcn = 'dividetrain';
autoenc1.performFcn = 'mae';
%% Train the autoencoder
autoenc1 = train(autoenc1,xTrain,xTrain);
%%
% Create an empty network
autoEncoder = network;
% Set the number of inputs and layers
autoEncoder.numInputs = 1;
autoEncoder.numlayers = 1;
% Connect the 1st (and only) layer to the 1st input, and also connect the
% 1st layer to the output
autoEncoder.inputConnect(1,1) = 1;
autoEncoder.outputConnect = 1;
% Add a connection for a bias term to the first layer
autoEncoder.biasConnect = 1;
% Set the size of the input and the 1st layer
autoEncoder.inputs{1}.size = inputSize;
autoEncoder.layers{1}.size = hiddenSize1;
% Use the logistic sigmoid transfer function for the first layer
autoEncoder.layers{1}.transferFcn = 'tansig';
% Copy the weights and biases from the first layer of the trained
% autoencoder to this network
autoEncoder.IW{1,1} = autoenc1.IW{1,1};
autoEncoder.b{1,1} = autoenc1.b{1,1};

%%
% generate the features
feat1 = autoEncoder(xTrain);
%%
% Create an empty network
autoDecoder = network;
% Set the number of inputs and layers
autoDecoder.numInputs = 1;
autoDecoder.numlayers = 1;
% Connect the 1st (and only) layer to the 1st input, and also connect the
% 1st layer to the output
autoDecoder.inputConnect(1,1) = 1;
autoDecoder.outputConnect(1) = 1;
% Add a connection for a bias term to the first layer
autoDecoder.biasConnect(1) = 1;
% Set the size of the input and the 1st layer
autoDecoder.inputs{1}.size = hiddenSize1;
autoDecoder.layers{1}.size = inputSize;
% Use the logistic sigmoid transfer function for the first layer
autoDecoder.layers{1}.transferFcn = 'tansig';
% Copy the weights and biases from the first layer of the trained
% autoencoder to this network
autoDecoder.IW{1,1} = autoenc1.LW{2,1};
autoDecoder.b{1,1} = autoenc1.b{2,1};
%% Reconstruction
desired=xTrain(:,50);
input=feat1(:,50);
output = autoDecoder(input);
figure
plot(output)
hold on
plot(desired,'r')

Answer 1

我不是Matlab用户，但你的代码让我觉得你有一个标准的浅层自动编码器。使用单个自动编码器无法真正近似非线性，因为它不会比纯线性PCA重建更优化（如果你需要，我可以提供更详细的数学推理，尽管这不是math.stackexchange）。你需要构建一个深度网络，通过几层线性变换来近似非线性。然后，自动编码器是一个不好选择的模型（现在几乎没有人在实践中使用它们），当你有去噪自动编码器时，它们往往会通过尝试从噪声版本重建先验来学习更重要的表示。尝试构建一个深度去噪自动编码器。本视频介绍了去噪自动编码器的概念。该课程还有一个关于深度去噪自动编码器的视频。