Contents

## Combined Central in addition to Subspace Clustering as long as Computer Vision Applications Le Lu

Parker, Ray, Mesa Schools Reporter has reference to this Academic Journal, PHwiki organized this Journal Combined Central in addition to Subspace Clustering as long as Computer Vision Applications Le Lu René Vidal Computer Science Department Center as long as Imaging Science Johns Hopkins University Johns Hopkins University Motivation: Central in addition to subspace clustering methods are at the core of many segmentation problems in computer vision. However, both methods fail to give the correct segmentation in many practical scenarios, e.g. when data points are close to the intersection of two subspaces or when two cluster centers in different subspaces are spatially close. A natural metric of considering both types of constraints Contributions: We address the problem of clustering a set of points lying in a union of subspaces in addition to distributed around multiple cluster centers inside each subspace. We propose a generalization of Kmeans in addition to Ksubspaces that clusters the data by minimizing a cost function that combines both central in addition to subspace distances.

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Two Toy Examples (1): A set of points in R3 drawn from 4 clusters labeled as A1, A2, B1, B2. Clusters B1 in addition to B2 lie in the X-Y plane in addition to clusters A1 in addition to A2 lie in the Y-Z plane. Note that some points in A2 in addition to B2 are drawn from the intersection of the two planes (Y-axis). Two Toy Examples (1): Left: Subspace clustering by GPCA assigns all the points in the Y-axis to the Y-Z plane, thus it misclassifies some points in B2. Right: Subspace clustering using GPCA followed by central clustering inside each plane using Kmeans misclassifies some points in B2. Two Toy Examples (2): A set of points in R3 distributed around 4 clusters labeled as A1, A2 B1, B2. Clusters B1 in addition to B2 lie in the X-Y plane in addition to clusters A1 in addition to A2 lie in the Y-Z plane. Note that cluster B2 (in blue) is spatially close to cluster A2 (in red).

Two Toy Examples (2): Left: Central clustering by Kmeans assigns some points in A2 to B2. Right: Subspace clustering using GPCA followed by central clustering inside each subspace using Kmeans gives the correct clustering into four groups. Problem Statement: Combined Central-Subspace Clustering Objective Functions: Central Clustering (K-Means): Subspace Clustering (K-Subspace):

Objective Functions: Joint Central + Subspace Clustering: Objective Functions: By using Lagrange multipliers: Algorithm: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships.

Computing the memberships: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Computing the cluster centers: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Computing the normal vectors: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships.

Remark 1: Extension from hyperplanes to subspaces: Bj = Null Space of Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Remark 2: Maximum Likelihood Solution: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Remark 2: Covariances (variances inside in addition to orthogonal to the hyperplanes): Variances: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships.

Clustering Per as long as mance (simulated): Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Clustering Per as long as mance (simulated): Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Top: Clustering error as a function of noise in the data. Bottom: Error in the estimation of the normal vectors (degrees) as a function of the level of noise in the data. Illumination-Invariant Face Clustering: YALE Face Database B (64 images per subject with fixed pose in addition to changing illumination): Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships.

Illumination-Invariant Face Clustering: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. GPCA (1): Illumination-Invariant Face Clustering: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Mixture of PPCA (2): Illumination-Invariant Face Clustering: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. GPCA-KMeans + Joint Optimization (3):

Video Shot Segmentation: http://www.open-video.org Mountain Sequence Drama Sequence Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Video Shot Segmentation: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. GPCA (1) as long as Mountain Sequences: Video Shot Segmentation: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Kmeans (2) as long as Mountain Sequences :

Video Shot Segmentation: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. GPCA-KMeans + Joint Optimization (3) as long as Mountain Sequences : Video Shot Segmentation: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. GPCA (1) as long as Drama Sequences: Video Shot Segmentation: Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. Initialization: Obtain an initial estimate of the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n; k=1 mj using GPCA followed by Kmeans in each subspace. Computing the memberships: Given the normal vectors {bj}j=1 n in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the memberships {wijk}. Computing the cluster centers: Given the memberships {wijk} in addition to the normal vectors {bj}j=1 n, compute the cluster centers {µjk}j=1 n;k=1 mj. Computing the normal vectors: Given the memberships {wijk} in addition to the cluster centers {µjk}j=1 n;k=1 mj, compute the normal vectors {bj}j=1 n. Iterate: Repeat steps 2,3,4 until convergence of the memberships. GPCA-KMeans + Joint Optimization (2) as long as Drama Sequences :

Conclusion in addition to Discussion: A simple, geometrically intuitive clustering method of combining central in addition to subspace constraints to solve computer vision problems. Insights to solve the intrinsic subspace clustering ambiguity of subspace intersections. Model Selection: Duality to mixture of factor analysers in addition to variational bayesian approach [Ghahramani, Beal, 2000] Interesting Related Work: Hyper-Graph Clustering [Agarwal, Belongie, et al. 2005, 2006]

## Parker, Ray Mesa Schools Reporter

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