manifold.html

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    <h1 id="manifold-top" class="title">Manifold Learning</h1>

    <p>Manifold learning finds a low-dimensional basis for describing
        high-dimensional data. Manifold learning is a popular approach to nonlinear
        dimensionality reduction. Algorithms for this task are based on the idea
        that the dimensionality of many data sets is only artificially high; though
        each data point consists of perhaps thousands of features, it may be
        described as a function of only a few underlying parameters. That is, the
        data points are actually samples from a low-dimensional manifold that is
        embedded in a high-dimensional space. Manifold learning algorithms attempt
        to uncover these parameters in order to find a low-dimensional representation
        of the data.</p>

    <p>Some prominent approaches are locally linear embedding
        (LLE), Hessian LLE, Laplacian eigenmaps, and LTSA. These techniques
        construct a low-dimensional data representation using a cost function
        that retains local properties of the data, and can be viewed as defining
        a graph-based kernel for Kernel PCA. More recently, techniques have been
        proposed that, instead of defining a fixed kernel, try to learn the kernel
        using semidefinite programming. The most prominent example of such a
        technique is maximum variance unfolding (MVU). The central idea of MVU
        is to exactly preserve all pairwise distances between nearest neighbors
        (in the inner product space), while maximizing the distances between points
        that are not nearest neighbors.</p>

    <p>An alternative approach to neighborhood preservation is through the
        minimization of a cost function that measures differences between
        distances in the input and output spaces. Important examples of such
        techniques include classical multidimensional scaling (which is identical
        to PCA), Isomap (which uses geodesic distances in the data space), diffusion
        maps (which uses diffusion distances in the data space), t-SNE (which
        minimizes the divergence between distributions over pairs of points),
        and curvilinear component analysis.</p>

    <h2 id="isomap">Isomap</h2>

    <p>Isometric feature mapping (isomap) is a widely used low-dimensional embedding methods,
        where geodesic distances on a weighted graph are incorporated with the
        classical multidimensional scaling. Isomap is used for computing a
        quasi-isometric, low-dimensional embedding of a set of high-dimensional
        data points. Isomap is highly efficient and generally applicable to a broad
        range of data sources and dimensionalities.</p>

    <p>To be specific, the classical MDS performs low-dimensional embedding based
        on the pairwise distance between data points, which is generally measured
        using straight-line Euclidean distance. Isomap is distinguished by
        its use of the geodesic distance induced by a neighborhood graph
        embedded in the classical scaling. This is done to incorporate manifold
        structure in the resulting embedding. Isomap defines the geodesic distance
        to be the sum of edge weights along the shortest path between two nodes.
        The top <code>n</code> eigenvectors of the geodesic distance matrix, represent the
        coordinates in the new <code>n</code>-dimensional Euclidean space.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_1" data-toggle="tab">Scala</a></li>
        <li><a href="#java_1" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_1" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_1">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def isomap(data: Array[Array[Double]], k: Int, d: Int = 2, CIsomap: Boolean = true): IsoMap
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_1">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    public class IsoMap {
        public static IsoMap of(double[][] data, int k, int d, boolean conformal);
    }
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_1">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    fun isomap(data: Array&lt;DoubleArray&gt;, k: Int, d: Int = 2, CIsomap: Boolean = true): IsoMap
    </code></pre>
            </div>
        </div>
    </div>

    <p>The connectivity of each data point in the neighborhood graph is defined
        as its nearest <code>k</code> Euclidean neighbors in the high-dimensional space. This
        step is vulnerable to "short-circuit errors" if <code>k</code> is too large with
        respect to the manifold structure or if noise in the data moves the
        points slightly off the manifold. Even a single short-circuit error
        can alter many entries in the geodesic distance matrix, which in turn
        can lead to a drastically different (and incorrect) low-dimensional
        embedding. Conversely, if <code>k</code> is too small, the neighborhood graph may
        become too sparse to approximate geodesic paths accurately.</p>

    <p>When the optional parameter <code>CIsomap</code> is true, the method
        performs C-Isomap that involves magnifying the regions
        of high density and shrink the regions of low density of data points
        in the manifold. Edge weights that are maximized in Multi-Dimensional
        Scaling(MDS) are modified, with everything else remaining unaffected.</p>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/swissroll.png" class="enlarge" style="width: 480px;" />
        <div class="caption" style="min-width: 480px;">Swiss Roll</div>
    </div>

    <p>In the below example, we apply Isomap to the famous
        <a href="http://people.cs.uchicago.edu/~dinoj/manifold/swissroll.html">swiss roll data</a> as shown above.
        This data set was created to test out various dimensionality reduction algorithms.
        The idea was to create several points in 2d, and then map them to 3d with some smooth
        function, and then to see what the algorithm would find when it mapped the points back to 2d.
        The data contains 20,000 samples. Because it is computational intensive to calculate
        the shortest path for all samples, the example uses only the first 2,000 samples.</p>

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        <li class="active"><a href="#scala_2" data-toggle="tab">Scala</a></li>
        <li><a href="#java_2" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_2">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    val swissroll = read.csv("data/manifold/swissroll.txt", header=false, delimiter='\t').toArray
    plot(swissroll, '.', BLUE)

    val map = isomap(swissroll.slice(0, 2000), 7)
    val vertices = map.coordinates
    val edges = map.graph.getEdges.asScala.map(edge => Array(edge.v1, edge.v2)).toArray

    wireframe(vertices, edges).window()
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_2">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var swissroll = Read.csv("data/manifold/swissroll.txt", CSVFormat.DEFAULT.withDelimiter('\t')).toArray();
    ScatterPlot.of(swissroll, '.', Color.BLUE).canvas().window();

    var map = IsoMap.of(Arrays.copyOf(swissroll, 2000), 7);
    var vertices = map.coordinates;
    var edges = map.graph.getEdges().stream().map(edge -> new int[]{edge.v1, edge.v2}).toArray(int[][]::new);

    Wireframe.of(vertices, edges).canvas().window();
          </code></pre>
            </div>
        </div>
    </div>

    <p>In this example, we use <code>k = 7</code> for neighborhood graph. In the mapped
        2d space, we also plot the connections between neighbor neighbors.</p>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/isomap.png" class="enlarge" style="width: 480px;" />
        <div class="caption" style="min-width: 480px;">Isomap</div>
    </div>

    <p>Note that Isomap produces strange holes like in a slice of Swiss cheese :).
        This issue can be solved by adding to Isomap a vector quantization step.</p>

    <h2 id="lle">LLE</h2>

    <p>Locally Linear Embedding (LLE) has several advantages over Isomap, including
        faster optimization when implemented to take advantage of sparse matrix
        algorithms, and better results with many problems. LLE also begins by
        finding a set of the nearest neighbors of each point. It then computes
        a set of weights for each point that best describe the point as a linear
        combination of its neighbors. Finally, it uses an eigenvector-based
        optimization technique to find the low-dimensional embedding of points,
        such that each point is still described with the same linear combination
        of its neighbors. LLE tends to handle non-uniform sample densities poorly
        because there is no fixed unit to prevent the weights from drifting as
        various regions differ in sample densities.</p>

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        <li class="active"><a href="#scala_3" data-toggle="tab">Scala</a></li>
        <li><a href="#java_3" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_3" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_3">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def lle(data: Array[Array[Double]], k: Int, d: Int = 2): LLE
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_3">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    public class LLE {
        public static LLE of(double[][] data, int k, int d);
    }
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_3">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    fun lle(data: Array&lt;DoubleArray&gt;, k: Int, d: Int = 2): LLE
    </code></pre>
            </div>
        </div>
    </div>

    <p>The LLE using 8 neighbors on the swiss roll data is as follows.</p>

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        <li class="active"><a href="#scala_4" data-toggle="tab">Scala</a></li>
        <li><a href="#java_4" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_4">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    val map = lle(swissroll.slice(0, 2000), 8)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_4">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var map = LLE.of(Arrays.copyOf(swissroll, 2000), 8);
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/lle.png" class="enlarge" style="width: 480px;" />
        <div class="caption" style="min-width: 480px;">LLE</div>
    </div>

    <h2 id="laplacian">Laplacian Eigenmap</h2>

    <p>Using the notion of the Laplacian of the nearest
        neighbor adjacency graph, Laplacian Eigenmap computes a low dimensional
        representation of the dataset that optimally preserves local neighborhood
        information in a certain sense. The representation map generated by the
        algorithm may be viewed as a discrete approximation to a continuous map
        that naturally arises from the geometry of the manifold.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_5" data-toggle="tab">Scala</a></li>
        <li><a href="#java_5" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_5" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_5">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def laplacian(data: Array[Array[Double]], k: Int, d: Int = 2, t: Double = -1): LaplacianEigenmap
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_5">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    public class LaplacianEigenmap {
        public static LaplacianEigenmap of(double[][] data, int k, int d, double t);
    }
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_5">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    fun laplacian(data: Array&lt;DoubleArray&gt;, k: Int, d: Int = 2, t: Double = -1): LaplacianEigenmap
    </code></pre>
            </div>
        </div>
    </div>

    <p>where <code>t</code> is the smooth/width parameter of heat kernel
        <code>e<sup>-||x-y||<sup>2</sup>/t</sup></code>.
        Non-positive <code>t</code> means discrete weights.</p>

    <p>The locality preserving character of the Laplacian Eigenmap algorithm makes
        it relatively insensitive to outliers and noise. It is also not prone to
        "short circuiting" as only the local distances are used.</p>

    <p>The Laplacian Eigenmap on the swiss roll data is as follows.</p>

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        <li class="active"><a href="#scala_6" data-toggle="tab">Scala</a></li>
        <li><a href="#java_6" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_6">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    val map = laplacian(swissroll.slice(0, 1000), 10, 2, 25.0)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_6">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var map = LaplacianEigenmap.of(Arrays.copyOf(swissroll, 1000), 10, 2, 25.0);
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/laplacian.png" class="enlarge" style="width: 480px;" />
        <div class="caption" style="min-width: 480px;">Laplacian Eigenmap</div>
    </div>

    <h2 id="t-sne">t-SNE</h2>

    <p>The t-distributed stochastic neighbor embedding (t-SNE) is a nonlinear
        dimensionality reduction technique that is particularly well suited
        for embedding high-dimensional data into a space of two or three
        dimensions, which can then be visualized in a scatter plot. Specifically,
        it models each high-dimensional object by a two- or three-dimensional
        point in such a way that similar objects are modeled by nearby points
        and dissimilar objects are modeled by distant points.</p>

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        <li class="active"><a href="#scala_7" data-toggle="tab">Scala</a></li>
        <li><a href="#java_7" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_7" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_7">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def tsne(X: Array[Array[Double]],
             d: Int = 2,
             perplexity: Double = 20.0,
             eta: Double = 200.0,
             iterations: Int = 1000): TSNE
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_7">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-java"><code>
    public class TSNE {
        public TSNE(double[][] X, int d, double perplexity, double eta, int iterations);
    }
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_7">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    fun tsne(X: Array&lt;DoubleArray&gt;,
             d: Int = 2,
             perplexity: Double = 20.0,
             eta: Double = 200.0,
             iterations: Int = 1000): TSNE
    </code></pre>
            </div>
        </div>
    </div>

    <p>where <code>X</code> is input data, <code>perplexity</code> is the perplexity
        of the conditional distribution, <code>eta</code> the learning rate, and <code>iterations</code> is
        the number of iterations. If <code>X</code> is a square matrix, it is assumed
        to be the distance/dissimilarity matrix.</p>

    <p>The t-SNE on the MNIST data is as follows. Note that the input data is preprocessed
        using PCA to reduce the dimensionality to 50 before t-SNE is performed.</p>

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        <li class="active"><a href="#scala_8" data-toggle="tab">Scala</a></li>
        <li><a href="#java_8" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_8">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    val x = read.csv("data/mnist/mnist2500_X.txt", header=false, delimiter=' ').toArray
    val y = read.csv("data/mnist/mnist2500_labels.txt", header=false, delimiter=' ').column(0).toIntArray
    val pc = pca(x)
    pc.setProjection(50)
    val x50 = pc.project(x)
    val sne = tsne(x50, 3, 20, 200, 1000)
    show(plot(sne.coordinates, y, 'o'))
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_8">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var format = CSVFormat.DEFAULT.withDelimiter(' ');
    var mnist = Read.csv(Paths.getTestData("mnist/mnist2500_X.txt"), format).toArray();
    var labels = Read.csv(Paths.getTestData("mnist/mnist2500_labels.txt"), format).column(0).toIntArray();

    var pca = PCA.fit(mnist);
    pca.setProjection(50);
    var X = pca.project(mnist);

    var perplexity = 20;
    var tsne = new TSNE(X, 2, perplexity, 200, 1000);

    var canvas = ScatterPlot.of(tsne.coordinates, labels, '@', Palette).canvas();
    canvas.setTitle("t-SNE of MNIST");
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/tsne.png" class="enlarge" style="width: 480px;" />
        <div class="caption" style="min-width: 480px;">t-SNE</div>
    </div>

    <h2 id="umap">UMAP</h2>

    <p>Uniform Manifold Approximation and Projection can be used for
        visualization similarly to t-SNE, but also for general non-linear
        dimension reduction. The algorithm is founded on three assumptions
        about the data:</p>
 
    <ul>
      <li>The data is uniformly distributed on a Riemannian manifold;</li>
      <li>The Riemannian metric is locally constant (or can be approximated as such);</li>
      <li>The manifold is locally connected.</li>
    </ul>
 
    <p>From these assumptions it is possible to model the manifold with a fuzzy
        topological structure. The embedding is found by searching for a low
        dimensional projection of the data that has the closest possible
        equivalent fuzzy topological structure.</p>

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        <li><a href="#java_9" data-toggle="tab">Java</a></li>
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    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_9">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def umap(data: Array[Array[Double]],
             k: Int = 15,
             d: Int = 2,
             iterations: Int = 0,
             learningRate: Double = 1.0,
             minDist: Double = 0.1,
             spread: Double = 1.0,
             negativeSamples: Int = 5,
             repulsionStrength: Double = 1.0): UMAP
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_9">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-java"><code>
    public class UAMP {
        public of(double[][] data,
                  int k,
                  int d,
                  int iterations,
                  double learningRate,
                  double minDist,
                  double spread,
                  int negativeSamples,
                  double repulsionStrength);
    }
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_9">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    fun umap(data: Array&lt;DoubleArray&gt;,
             k: Int = 15,
             d: Int = 2,
             iterations: Int = 0,
             learningRate: Double = 1.0,
             minDist: Double = 0.1,
             spread: Double = 1.0,
             negativeSamples: Int = 5,
             repulsionStrength: Double = 1.0): UMAP
    </code></pre>
            </div>
        </div>
    </div>

    <p>where <code>data</code> is the input data, <code>k</code> is of
        k-nearest neighbors. Larger values result in more global views
        of the manifold, while smaller values result in more local data
        being preserved. <code>d</code> is the target embedding dimensions.
        <code>iterations</code> is the number of iterations to optimize the
        low-dimensional representation. Larger values result in more
        accurate embedding. Muse be greater than 10. Choose wise value
        based on the size of the input data, e.g, 200 for large
        data (1000+ samples), 500 for small.
        <code>learningRate</code> is the initial learning rate for the
        embedding optimization.
        <code>minDist</code> is the desired separation between close points
        in the embedding space. Smaller values will result in a more
        clustered/clumped embedding where nearby points on the manifold are
        drawn closer together, while larger values will result on a more even
        disperse of points. The value should be set no-greater than
        and relative to the spread value, which determines the scale
        at which embedded points will be spread out.
        <code>spread</code> is the effective scale of embedded points.
        In combination with <code>minDist</code>, this determines how
        clustered/clumped the embedded points are.
        <code>negativeSamples</code> is the number of negative samples to
        select per positive sample in the optimization process. Increasing
        this value will result in greater repulsive force being applied,
        greater optimization cost, but slightly more accuracy.
        <code>repulsionStrength</code>plied to negative samples in low
        dimensional embedding optimization. Values higher than one will result
        in greater weight being given to negative samples.</p>

    <p>UMAP on the MNIST data is as follows.</p>

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        <li class="active"><a href="#scala_10" data-toggle="tab">Scala</a></li>
        <li><a href="#java_10" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_10">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    val map = umap(x)
    val canvas = plot(map.coordinates, map.index.map(i =&gt; y(i)), 'o')
    show(canvas)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_10">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var map = UMAP.of(mnist)
    var y = Arrays.stream(map.index).map(i -&gt; labels[i]).toArray()
    ScatterPlot.of(map.coordinates, y, '@').canvas().window()
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/umap.png" class="enlarge" style="width: 480px;" />
        <div class="caption" style="min-width: 480px;">UMAP</div>
    </div>

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