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Clustering Multimedia Data
Published in Charu C. Aggarwal, Chandan K. Reddy, Data Clustering, 2018
Shen-Fu Tsai, Guo-Jun Qi, Shiyu Chang, Min-Hsuan Tsai, Thomas S. Huang
To reduce the users’ burden on large scale face annotation, Suh and Bederson [28] introduced a bulk annotation framework that incorporates hierarchical clustering techniques to cluster faces according to timestamps and torso information. With the clustered faces, the users can do annotation at the cluster level instead of single-image annotation. However, the clustering based on event and torso information has certain limitations and may produce inconsistent results, which would thus increase the workload of labeling the face images. To solve this problem, Tian et al. [29] proposed a partial clustering algorithm with the goal of providing small sized and high accuracy (evident) clusters to reduce the labeling effort instead of improving the overall clustering performance. They substitute the classic k-means clustering with their novel partial clustering algorithm after the spectral embedding onto the unit hypersphere. Their partial clustering algorithm focuses only on finding evident clusters and excludes all noisy samples into a litter-bin cluster by adding a uniform background noise distribution in their Gaussian Mixture Model (GMM) based clustering.
Preliminaries
Published in Stephen Marsland, Machine Learning, 2014
The curse of dimensionality is a very strong name, so you can probably guess that it is a bit of a problem. The essence of the curse is the realisation that as the number of dimensions increases, the volume of the unit hypersphere does not increase with it. The unit hypersphere is the region we get if we start at the origin (the centre of our coordinate system) and draw all the points that are distance 1 away from the origin. In 2 dimensions we get a circle of radius 1 around (0, 0) (drawn in Figure 2.2), and in 3D we get a sphere around (0, 0, 0) (Figure 2.3). In higher dimensions, the sphere becomes a hypersphere. The following table shows the size of the unit hypersphere for the first few dimensions, and the graph in Figure 2.4 shows the same thing, but also shows clearly that as the number of dimensions tends to infinity, so the volume of the hypersphere tends to zero.
A fast optimal Latin hypercube design method using an improved translational propagation algorithm
Published in Engineering Optimization, 2020
Yibo Sun, Xiuyun Meng, Teng Long, Yufei Wu
Using a sampling problem as an example, Figures 6 and 7 show the differences between the resizing process of TPA and iTPA (note that for two-dimensional problems, the translating process of TPA and iTPA are same). Using the Euclidean norm, as shown in Figure 6(a), TPA remains points in a circle (sphere or hypersphere in high dimensions) field inside the square (cube or hypercube in high dimensions) field of the design space. The points in pink-donut shape will be removed. The point distances change in the white background region while keeping the same in the shadow. As shown in Figure 6(b), after the resizing process, the minimal distances among points change to three levels in the vertical dimension in the white background region, while the minimal distances are still five levels in the shadow region. The space-filling quality of the resized LHD is weakened after the resizing process of TPA. On the contrary, shown in Figure 7(a), iTPA remains points in a square field. As shown in Figure 7(b), the square field is the same as the square field of the design space. The distance changes are even among points in the design space. Compared with original TPA, the negative effects on space-filling quality from the resizing process of iTPA is much smaller.
Design and analysis of confirmation experiments
Published in Journal of Quality Technology, 2019
Nathaniel T. Stevens, Christine M. Anderson-Cook
Since the available budget for confirmation runs will be dependent on the study constraints, a general strategic approach is needed for how to construct a confirmation design. There are two ideas for where runs should be placed once an optimal location in the input space has been identified through estimation of the response surface: (a) one or more runs at the ideal location, and (b) points uniformly spread on the surface of a circle (two dimensions), sphere (three dimensions), or hypersphere (four or more dimensions) in the input space. Designs that satisfy criterion (b) are known as equiradial designs. In choosing a final design, a number of aspects need to be specified: (i) the total number of runs in the experiment, (ii) the partitioning of runs allocated to the equiradial design versus the center runs (CR), (iii) the radius of the hypersphere centered at the estimated optimum, and (iv) the particular locations in the input space based on choices made in (ii) and (iii).
Convergence acceleration for the multilevel Hartree–Fock model
Published in Molecular Physics, 2020
The ARH approach is in LSDALTON implemented within a framework of trust-region optimisation [46]. In trust-region optmization, steps are restricted to be inside or on the boundary of a trust-region. The point of the trust-region is to only take steps where the quadratic model of the energy (Equation (19)) is a good approximation to the energy (also when using the ARH approximation to the linear transformation [40]). The trust-region is defined by a hypersphere of radius h around the current expansion point. To constrain the step to be within or to the boundary of this region, a constraint on the step length is added. The constraint may be on e.g. the Frobenius norm, , or the max norm, . The max norm is a size intensive measure which is used in the global region, whereas the size extensive Frobenius norm is used in the local region. Exemplifying by using the Frobenius norm constraint, we obtain the Lagrangian, The minimum of Equation (26) is defined by the level-shifted Newton equations where and are in the ARH approximation given by The dynamic determination of a level-shift, μ, that gives a step to the minimum of Λ on the boundary of the trust-region is discussed by Salek et al. [47] and will not be re-iterated here. A measure of the quality of the quadratic model at the expansion points can be evaluated after each step, by the ratio of the actual energy change versus the energy change predicted by the quadratic model. A discussion of this ratio and the update of the trust-region is given by Høst et al. [40].