Slide 1Dimensionality ReductionExample…Types of structure in multivariate data..Concept of ManifoldsManifolds of Perception..Human Visual SystemLinear methods..MultiDimensional Scaling..Example of MDS…How to get dot product matrix from pairwise distance matrix?MDS..MDS is more general..Nonlinear Manifolds..Slide 14Two methodsIsomapIsomap - AlgorithmSlide 19Slide 20Slide 21Slide 22Residual VarianceSlide 24Locally Linear EmbeddingFit Locally…Important property...Think Globally…Slide 29Slide 30Slide 31Slide 32Summary..Short Circuit Problem??????Conformal & Isometric EmbeddingSlide 37C-IsomapSlide 39Slide 40NonLinear Dimensionality Reductionor Unfolding ManifoldsTennenbaum|Silva|Langford [Isomap]Roweis|Saul [Locally Linear Embedding] Presented by Vikas C. Raykar | University of Maryland, CollegeParkDimensionality ReductionNeed to analyze large amounts multivariate data.Human Faces.Speech Waveforms.Global Climate patterns.Gene Distributions.Difficult to visualize data in dimensions just greater than three.Discover compact representations of high dimensional data.Visualization.Compression.Better Recognition.Probably meaningful dimensions.Example…Types of structure in multivariate data..•Clusters.–Principal Component Analysis–Density Estimation Techniques.•On or around low Dimensional Manifolds–Linear–NonLinearConcept of Manifolds•“A manifold is a topological space which is locally Euclidean.”•In general, any object which is nearly "flat" on small scales is a manifold. •Euclidean space is a simplest example of a manifold.•Concept of submanifold.•Manifolds arise naturally whenever there is a smooth variation of parameters [like pose of the face in previous example]•The dimension of a manifold is the minimum integer number of co-ordinates necessary to identify each point in that manifold.Embed data in a higher dimensional space to a lower dimensional manifoldConcept of Dimensionality Reduction:Manifolds of Perception..Human Visual SystemYou never see the same face twice.Preceive constancy whenraw sensory inputs are in flux..Linear methods..•Principal Component Analysis (PCA)One DimensionalManifoldMultiDimensional Scaling..•Here we are given pairwise distances instead of the actual data points.–First convert the pairwise distance matrix into the dot product matrix –After that same as PCA.If we preserve the pairwise distances do we preserve the structure??Example of MDS…How to get dot product matrix from pairwise distance matrix?kijdkjdkidjiMDS..•MDS—origin as one of the points and orientation arbitrary.Centroid as originMDS is more general..•Instead of pairwise distances we can use paiwise “dissimilarities”.•When the distances are Euclidean MDS is equivalent to PCA.•Eg. Face recognition, wine tasting•Can get the significant cognitive dimensions.Nonlinear Manifolds..AUnroll the manifoldPCA and MDS see the EuclideandistanceWhat is important is the geodesic distanceTo preserve structure preserve the geodesic distance and not the euclidean distance.Two methods•Tenenbaum et.al’s Isomap Algorithm–Global approach.–On a low dimensional embedding•Nearby points should be nearby.•Farway points should be faraway.•Roweis and Saul’s Locally Linear Embedding Algorithm–Local approach•Nearby points nearbyIsomap•Estimate the geodesic distance between faraway points.•For neighboring points Euclidean distance is a good approximation to the geodesic distance.•For farway points estimate the distance by a series of short hops between neighboring points.–Find shortest paths in a graph with edges connecting neighboring data pointsOnce we have all pairwise geodesic distances use classical metric MDSIsomap - Algorithm•Determine the neighbors.–All points in a fixed radius.–K nearest neighbors• Construct a neighborhood graph.–Each point is connected to the other if it is a K nearest neighbor.–Edge Length equals the Euclidean distance•Compute the shortest paths between two nodes–Floyd’s Algorithm–Djkastra’s ALgorithm•Construct a lower dimensional embedding.–Classical MDSIsomapResidual VarianceFace ImagesSwisRollHand Images 2Locally Linear Embedding manifold is a topological space which is locally Euclidean.”Fit Locally , Think GloballyWe expect each data point and its neighbours to lie on or close to a locally linear patch of themanifold.Each point can be written as a linear combination of its neighbors.The weights choosen tominimize the reconstructionError.Derivation on boardFit Locally…Important property...•The weights that minimize the reconstruction errors are invariant to rotation, rescaling and translation of the data points.–Invariance to translation is enforced by adding the constraint that the weights sum to one.•The same weights that reconstruct the datapoints in D dimensions should reconstruct it in the manifold in d dimensions.–The weights characterize the intrinsic geometric properties of each neighborhood.Derivation on boardThink Globally…Grolliers EncyclopediaSummary..ISOMAP LLEDo MDS on the geodesic distance matrix.Model local neighborhoods as linear a patches and then embed in a lower dimensional manifold.Global approach Local aproachDynamic programming approachesComputationally efficient..sparse matricesConvergence limited by the manifold curvature and number of points.Good representational capacityShort Circuit Problem??? Unstable?Only free parameter isHow many neighbours?–How to choose neighborhoods.•Susceptible to short-circuit errors if neighborhood is larger than the folds in the manifold.•If small we get isolated patches.???•Does Isomap work on closed manifold, manifolds with holes?•LLE may be better..•Isomap Convergence Proof?•How smooth should the manifold be?•Noisy Data?•How to choose K?•Sparse Data?Conformal & Isometric EmbeddingC-Isomap•Isometric mapping–Intrinsically flat manifold–Invariants??•Geodesic distances are reserved.•Metric space under geodesic distance.•Conformal Embedding–Locally isometric upo a scale factor s(y)–Estimate s(y) and rescale.–C-Isomap–Original data should be uniformly denseThank You ! | Questions