当维度大于数据数量时，我可以使用 t-SNE 吗？

Question

我正在将 t-SNE 与来自该网站 (https://lvdmaaten.github.io/tsne/) 的 matlab 代码一起使用。但是，每当我运行这个程序的数据维度大于数据数量时，就会出现错误。下面的代码是我目前使用的代码，这里总是报错

M = M(:,ind(1:initial_dims));

错误是

Index exceeds matrix dimensions.
Error in tsne (line 62)
    M = M(:,ind(1:initial_dims));

我用matlab中的命令调用了这个tsne函数

output = tsne(input, [], 2, 640, 30);

输入大小为(162x640)，维度为640，数据个数为162。下面的程序是上面网站的代码。

function ydata = tsne(X, labels, no_dims, initial_dims, perplexity)
%TSNE Performs symmetric t-SNE on dataset X
%
%   mappedX = tsne(X, labels, no_dims, initial_dims, perplexity)
%   mappedX = tsne(X, labels, initial_solution, perplexity)
%
% The function performs symmetric t-SNE on the NxD dataset X to reduce its 
% dimensionality to no_dims dimensions (default = 2). The data is 
% preprocessed using PCA, reducing the dimensionality to initial_dims 
% dimensions (default = 30). Alternatively, an initial solution     obtained 
% from an other dimensionality reduction technique may be specified in 
% initial_solution. The perplexity of the Gaussian kernel that is     employed 
% can be specified through perplexity (default = 30). The labels of     the
% data are not used by t-SNE itself, however, they are used to color
% intermediate plots. Please provide an empty labels matrix [] if you
% don't want to plot results during the optimization.
% The low-dimensional data representation is returned in mappedX.
%
%
% (C) Laurens van der Maaten, 2010
% University of California, San Diego

if ~exist('labels', 'var')
    labels = [];
end
if ~exist('no_dims', 'var') || isempty(no_dims)
    no_dims = 2;
end
 if ~exist('initial_dims', 'var') || isempty(initial_dims)
    initial_dims = min(50, size(X, 2));
end
if ~exist('perplexity', 'var') || isempty(perplexity)
    perplexity = 30;
end

% First check whether we already have an initial solution
if numel(no_dims) > 1
    initial_solution = true;
    ydata = no_dims;
    no_dims = size(ydata, 2);
    perplexity = initial_dims;
else
    initial_solution = false;
end

% Normalize input data
X = X - min(X(:));
X = X / max(X(:));
X = bsxfun(@minus, X, mean(X, 1));

% Perform preprocessing using PCA
if ~initial_solution
    disp('Preprocessing data using PCA...');
    if size(X, 2) < size(X, 1)
        C = X' * X;
    else
        C = (1 / size(X, 1)) * (X * X');
    end
    [M, lambda] = eig(C);
    [lambda, ind] = sort(diag(lambda), 'descend');
    M = M(:,ind(1:initial_dims));
    lambda = lambda(1:initial_dims);
    if ~(size(X, 2) < size(X, 1))
        M = bsxfun(@times, X' * M, (1 ./ sqrt(size(X, 1) .* lambda))');
    end
    X = bsxfun(@minus, X, mean(X, 1)) * M;
    clear M lambda ind
end

% Compute pairwise distance matrix
sum_X = sum(X .^ 2, 2);
D = bsxfun(@plus, sum_X, bsxfun(@plus, sum_X', -2 * (X * X')));

% Compute joint probabilities
P = d2p(D, perplexity, 1e-5);                                           % compute affinities using fixed perplexity
clear D

% Run t-SNE
if initial_solution
    ydata = tsne_p(P, labels, ydata);
else
    ydata = tsne_p(P, labels, no_dims);
end

我试图理解这段代码，但我无法理解发生错误的部分。

if size(X, 2) < size(X, 1)
    C = X' * X;
else
    C = (1 / size(X, 1)) * (X * X');
end

为什么需要这个条件？由于'X'的大小是(162x640)，所以会执行else语句。我想这就是问题所在。在 else 语句中，'C' 的大小将为 (162x162)。然而，在下一行

M = M(:,ind(1:initial_dims));

使用等于640的'initial_dims'。我是否以错误的方式使用了这段代码？或者它只是不适用于我使用的数据集？

Answer 1

根据文档：使用 PCA 对数据进行预处理，将维度减少到 initial_dims 维（默认值 = 30）。因此，您应该在第一次使用时保持此参数不变。

条件if size(X, 2) < size(X, 1)用于构造经济SVD的矩阵，使得协方差矩阵的大小会更小，从而导致更快的计算。

当维度大于数据数量时，我可以使用 t-SNE 吗？

Can I use t-SNE when the dimension is larger than the number of data?

matlab

dimensionality-reduction