less useful in explaining the dataset for other featuresthe values might vary considerably--these features are worth exploring in greater detail since they will be better at helping the model we design separate the data in pcathe algorithm finds low-dimensional representation of the data while retaining as much of the variation as possible the number of dimensions we are left with is considerably smaller than the number of dimensions of the full dataset ( the number of total featureswe lose some of the variance by moving to this lowdimensional spacebut the underlying structure of the data is easier to identifyallowing us to perform tasks like clustering more efficiently there are several variants of pcawhich we will explore later in the book these include mini-batch variants such as incremental pcanonlinear variants such as kernel pcaand sparse variants such as sparse pca singular value decomposition (svdanother approach to learning the underlying structure of the data is to reduce the rank of the original matrix of features to smaller rank such that the original matrix can be recreated using linear combination of some of the vectors in the smaller rank matrix this is known as svd to generate the smaller rank matrixsvd keeps the vectors of the original matrix that have the most information ( the highest singular valuethe smaller rank matrix captures the most important elements of the original feature space random projection similar dimensionality reduction algorithm involves projecting points from high-dimensional space to space of much lower dimensions in such way that the scale of distances between the points is preserved we can use either random gaussian matrix or random sparse matrix to accomplish this manifold learning both pca and random projection rely on projecting the data linearly from highdimensional space to low-dimensional space instead of linear projectionit may be better to perform nonlinear transformation of the data--this is known as manifold learning or nonlinear dimensionality reduction isomap isomap is one type of manifold learning approach this algorithm learns the intrinsic geometry of the data manifold by estimating the geodesic or curved distance between each point and its neighbors rather than the euclidean distance isomap uses this to then embed the original high-dimensional space to low-dimensional one -distributed stochastic neighbor embedding ( -sneanother nonlinear dimensionality reduction--known as -sne--embeds high-dimensional data into space of just two or three dimensionsallowing the transformed data to be visualized in this twoor unsupervised learning in the machine learning ecosystem