the function performs symmetric t-sne on pairwise similarity matrix p % to c

该函数在配对相似性矩阵P到C上执行对称T-SNE。T-SNE是一种流行的降维技术，用于可视化高维数据。它通过将相似的数据点映射到低维空间中的邻近点来实现。T-SNE可以使用相似性矩阵来计算数据点之间的相似性。对称T-SNE是一种变形，它通过计算两个相似矩阵之间的相似性来建立对称相似矩阵。在此函数中，它将配对相似性矩阵P到C作为输入数据，并使用对称T-SNE算法将其映射到低维空间中，以便对数据进行可视化分析。此外，它还可以计算在低维空间中的数据点之间的相似性，以便更好地理解它们之间的关系。这个函数非常适合在聚类分析和可视化密集高维数据时使用。

相关问题

MATLAB中t-SNE源代码

以下是MATLAB中t-SNE算法的源代码示例：

function mappedX = tsne(X, no_dims, initial_dims, perplexity)
%TSNE t-distributed Stochastic Neighbor Embedding
%
%   mappedX = tsne(X, no_dims, initial_dims, perplexity)
%
% The function runs the t-SNE algorithm on dataset X to reduce its
% dimensionality to no_dims. The initial solution is given by initial_dims
% and the perplexity of the Gaussian kernel is given by perplexity (typically
% a value between 5 and 50). The variable mappedX returns the two-dimensional
% data points in mappedX.
%
% Note: The algorithm is memory intensive; e.g. for N=5000, you will need
% about 2GB of RAM.
%
% (C) Laurens van der Maaten, 2008
% University of California, San Diego

    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 size(X, 2) == 1 && no_dims == 1
        % If X is one-dimensional, we only need to embed it in one dimension
        mappedX = X;
        return
    elseif no_dims > size(X, 2)
        % If the number of input dimensions is smaller than the desired number
        % of output dimensions, simply pad the matrix with zeros.
        warning(['Target dimensionality reduced to ' num2str(size(X, 2)) ' by PCA.']);
        no_dims = size(X, 2);
    end
    if ~exist('Y', 'var') || isempty(Y)
        Y = randn(size(X, 1), no_dims);
    end
    
    % Compute pairwise distances
    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
    mappedX = tsne_p(P, Y, 1000);

这个函数调用了`d2p`函数和`tsne_p`函数。其中`d2p`函数的代码如下：

function P = d2p(D, perplexity, tol)
%D2P Identifies appropriate sigma's to get kk NNs up to some tolerance 
%
%   P = d2p(D, perplexity, tol)
%
% Identifies the appropriate sigma to obtain a Gaussian kernel matrix with a
% certain perplexity (approximately constant conditional entropy) for a
% set of Euclidean input distances D. The desired perplexity is specified
% by perplexity. The function returns the final Gaussian kernel matrix P,
% whose elements P_{i,j} represent the probability of observing
% datapoint j given datapoint i, normalized so that the sum over all i and j
% is 1.
%
% The function iteratively searches for a value of sigma that results in a
% Gaussian distribution over the perplexity-defined number of nearest
% neighbors of each point.
%
% Note: The function is designed for use with the large data sets and
% requires sufficient memory to store the entire NxN distance matrix for
% your NxP data matrix X.
%
% Note: The function may return P=NaN, indicating numerical difficulties.
% In such cases, the 'tol' parameter should be increased and the function
% should be rerun.
%
% The function is based on earlier MATLAB code by Laurens van der Maaten
% (lvdmaaten@gmail.com) and uses ideas from the following paper:
%
%   * D. L. D. Saul and S. T. Roweis. Think globally, fit locally: Unsupervised
%     learning of low dimensional manifolds. Journal of Machine Learning
%     Research 4(2003) 119-155.
%
% (C) Joshua V. Dillon, 2014

    % Initialize some variables
    [n, ~] = size(D);                                    % number of instances
    P = zeros(n, n);                                     % empty probability matrix
    beta = ones(n, 1);                                   % empty precision vector
    logU = log(perplexity);                              % log(perplexity) (H)
    
    % Compute P-values
    disp('Computing P-values...');
    for i=1:n
        
        if mod(i, 500) == 0
            disp(['Computed P-values ' num2str(i) ' of ' num2str(n) ' datapoints...']);
        end
        
        % Compute the Gaussian kernel and entropy for the current precision
        [P(i,:), beta(i)] = gaussiandist(D(i,:), tol, beta(i), logU);
    end    
    
    disp('Mean value of sigma: ');
    disp(mean(sqrt(1 ./ beta)));
    
    % Make sure P-values are symmetric
    P = (P + P') ./ (2 * n);
    
    % Zero any negative values
    P(P < 0) = 0;
end

%-------------------------------------------------------------------------
function [P, beta] = gaussiandist(x, tol, beta, logU)
%GAUSSIANDIST Computes the Gaussian kernel and entropy for a perplexity
%defined by logU.
%
%   [P, beta] = gaussiandist(x, tol, beta, logU)
%
% Returns the Gaussian kernel and entropy for a given perplexity, defined
% by logU, for the NxD matrix X. The function iteratively searches for a
% value of sigma that results in a Gaussian distribution over the
% perplexity-defined number of nearest neighbors of each point.
%
% Note: The function is designed for use with the large data sets and
% requires sufficient memory to store the NxN distance matrix.
%
% Note: The function may return P=NaN, indicating numerical difficulties.
% In such cases, the 'tol' parameter should be increased and the function
% should be rerun.
%
% The function is based on earlier MATLAB code by Laurens van der Maaten
% (lvdmaaten@gmail.com) and uses ideas from the following paper:
%
%   * D. L. D. Saul and S. T. Roweis. Think globally, fit locally: Unsupervised
%     learning of low dimensional manifolds. Journal of Machine Learning
%     Research 4(2003) 119-155.
%
% (C) Joshua V. Dillon, 2014

    % Initialize some variables
    [n, ~] = size(x);                       % number of instances
    P = zeros(1, n);                        % empty probability vector
    sumP = realmin;                          % minimum value to avoid log(0)
    K = 0;                                  % number of nearest neighbors
    
    % Search for good sigma, iterating until we have the perplexity we want
    while abs(sumP - logU) > tol
        
        % Compute Gaussian kernel and entropy for current precision
        P = exp(-beta * x).^2;
        sumP = sum(P);
        H = log(sumP) + beta * sum(x .* P) / sumP;
        
        % Adjust beta according to the perplexity
        if isnan(H)
            beta = beta * 2;
            P = NaN(1, n);
            continue;
        end
        if H > logU
            betaNew = beta * 0.5;
        else
            betaNew = beta * 2;
        end
        
        % Update precision
        beta = betaNew;
    end
    
    % Return final Gaussian kernel row for this point
    P = P / sumP;
end

最后，`tsne_p`函数的代码如下：

function Y = tsne_p(P, labels, no_dims)
%TSNE_P Performs symmetric t-SNE on affinity matrix P
%
%   Y = tsne_p(P, labels, no_dims)
%
% The function performs symmetric t-SNE on pairwise similarity matrix P
% to reduce its dimensionality to no_dims. The matrix P is assumed to be
% symmetric, sum up to 1, and have zeros on its diagonal.
% The labels parameter is an optional vector of labels that can be used to
% color the resulting scatter plot. The function returns the two-dimensional
% data points in Y.
% The perplexity is the only parameter the user normally needs to adjust.
% In most cases, a value between 5 and 50 works well.
%
% Note: This implementation uses the "fast" version of t-SNE. This should
% run faster than the original version but may also have different numerical
% properties.
%
% Note: The function is memory intensive; e.g. for N=5000, you will need
% about 2GB of RAM.
%
% (C) Laurens van der Maaten, 2008
% University of California, San Diego

    if ~exist('labels', 'var')
        labels = [];
    end
    if ~exist('no_dims', 'var') || isempty(no_dims)
        no_dims = 2;
    end
    
    % First check whether we already have an initial solution
    if size(P, 1) ~= size(P, 2)
        error('Affinity matrix P should be square');
    end
    if ~isempty(labels) && length(labels) ~= size(P, 1)
        error('Mismatch in number of labels and size of P');
    end
    
    % Initialize variables
    n = size(P, 1);                                                 % number of instances
    momentum = 0.5;                                                 % initial momentum
    final_momentum = 0.8;                                           % value to which momentum is changed
    mom_switch_iter = 250;                                          % iteration at which momentum is changed
    stop_lying_iter = 100;                                          % iteration at which lying about P-values is stopped
    max_iter = 1000;                                                % maximum number of iterations
    epsilon = 500;                                                  % initial learning rate
    min_gain = .01;                                                 % minimum gain for delta-bar-delta
    
    % Initialize the solution
    Y = randn(n, no_dims);
    dY = zeros(n, no_dims);
    iY = zeros(n, no_dims);
    gains = ones(n, no_dims);
    
    % Compute P-values
    P = P ./ sum(P(:));
    P = max(P, realmin);
    P = P * 4;                                                      % early exaggeration
    P = min(P, 1e-12);
    
    % Lie about the P-vals to find better local minima
    P = P ./ sum(P(:));
    P = max(P, realmin);
    const = sum(P(:) .* log(P(:)));
    for iter = 1:max_iter
        
        % Compute pairwise affinities
        sum_Y = sum(Y .^ 2, 2);
        num = 1 ./ (1 + bsxfun(@plus, sum_Y, bsxfun(@plus, sum_Y', -2 * (Y * Y'))));
        num(1:n+1:end) = 0;
        Q = max(num ./ sum(num(:)), realmin);
        
        % Compute gradient
        PQ = P - Q;
        for i=1:n
            dY(i,:) = sum(bsxfun(@times, PQ(:,i), bsxfun(@minus, Y, Y(i,:))), 1);
        end
        
        % Perform the update
        if iter < stop_lying_iter
            momentum = min_gain * momentum + (1 - min_gain) * dY;
        else
            momentum = final_momentum;
        end
        gains = (gains + .2) .* (sign(dY) ~= sign(iY)) + ...
                (gains * .8) .* (sign(dY) == sign(iY));
        gains(gains < min_gain) = min_gain;
        iY = momentum;
        dY = gains .* momentum;
        Y = Y + dY;
        Y = bsxfun(@minus, Y, mean(Y, 1));
        
        % Compute current value of cost function
        if ~rem(iter, 10)
            C = const - sum(P(:) .* log(Q(:)));
            if ~isempty(labels)
                disp(['Iteration ' num2str(iter) ': error is ' num2str(C) ', norm of gradient is ' num2str(norm(dY))]);
            end
        end
        
        % Stop lying about P-values
        if iter == stop_lying_iter
            P = P ./ 4;
        end
    end
    
    % Return solution
    if iter == max_iter
        disp(['Maximum number of iterations reached (' num2str(max_iter) ')']);
    end
    if ~isempty(labels)
        figure, scatter(Y(:,1), Y(:,2), 9, labels, 'filled');
    end
end

翻译 To ascertain that the improvement of dipIQ is statistically significant, we carry out a two sample T-test (with a 95% confidence) between PLCC values obtained by different models on LIVE [86]. After comparing every possible pairs of OU-BIQA models, the results are summarized in Table V, where a symbol “1” means the row model performs signifi- cantly better than the column model, a symbol “0” means the opposite, and a symbol “-” indicates that the row and column models are statistically indistinguishable. It can be observed that dipIQ is statistically better than dipIQ∗, which is better than all previous OU-BIQA models.

为了确保 dipIQ 的改进在统计上具有显著性，我们在 LIVE 数据集 [86] 上对不同模型得到的 PLCC 值进行了双样本 T 检验（置信度为95%）。在比较了 OU-BIQA 模型的所有可能配对后，结果总结如表 V 所示。其中，“1”表示行模型显著优于列模型，“0”表示相反，而“-”表示行和列模型在统计上无法区分。可以观察到 dipIQ 在统计上优于 dipIQ∗，后者优于所有先前的 OU-BIQA 模型。