This paper proposes a novel metric for evaluating the quality of images of the same category generated by a synthetic image generator. The proposed method assumes that the points representing synthetic images in a feature space lie on a manifold, and it aims to determine the curvatures of the manifold at those points. The lower the angle, the higher the quality of the generated images is considered to be. Let $\mathcal{F}$ be the set of data points in a D-dimensional feature space where each point represents one synthetically generated image. Let $\mathbb{M}$ be the corresponding manifold to match $\mathcal{F}$. For each data point $y\in\mathbb{R}^{D}$ on $\mathbb{M}$, a number of neighboring points are determined. A local subspace $\mathbb{S}$ is matched for y and a local subspace is also matched for each of its neighboring points. Then, a set of weighted angles between $\mathbb{S}$ and each neighboring local subspace are computed. The average of those weighted angles is used as a measure of curvature of $\mathbb{M}$ at y. Experimental results demonstrate the effectiveness of the proposed metric in evaluating the quality of synthetically generated images.

This paper introduces a novel method for creating a metric to measure curvature of a discretized manifold. For each data point xion a manifold, a subspace ${{\mathcal{S}}_i}$ is matched using a number of neighboring points of xi. A local subspace is also matched to each neighboring point of xi. Then, a set of weighted angles between ${{\mathcal{S}}_i}$ and each neighboring local subspace are computed and the minimum of those weighted angles is used as a measure of curvature of the manifold at xi. The average curvature for all data points on the manifold is used as metric for the manifold’s curvature estimation. This research also uses the proposed metric to show that each layer of a neural network maps an input manifold to a flatter manifold during the training process. It is observed that each successive block in a neural network generates a manifold with less curvature than that of the previous layer. Another observation is that convolutional layers always generate flatter manifolds. Furthermore, it is shown that this metric can be used as a robustness measure for a neural network. The method has been tested successfully using two datasets MNIST and Extended YaleB datasets.

This paper introduces a clustering algorithm, called principal coordinate clustering. It takes in a similarity matrix SWof a data matrix W and computes the singular value decomposition of SWto determine the principal coordinates to convert the clustering problem to a simpler domain. It is a relative of spectral clustering, however, principal coordinate clustering is easier to interpret, and gives a clear understanding of why it performs well. In a fashion, this gives intuition behind why spectral clustering works from a more simple, linear algebra perspective, beyond the typical explanations via graph cuts, or other techniques. Moreover, it was demonstrated through experimentation on real and synthetic data that the proposed method performs equally well on average as spectral clustering, and that the method has the ability to scale quite easily to truly large data.

This paper presents a novel technique for the segmentation of data W = [w1· · · wn] ⊂ RDdrawn from a union u = ∪Mi=1of subspaces {Si}Mi=1. First, an existing subspace segmentation algorithm is used to perform an initial data clustering {Ci}Mi=1, where Ci= {wi1· · ·wik} ⊂ W is the set of data from the ithcluster. Then, a local subspace LSiis matched for each Ciand the distance dijbetween LSiand each point wij∊ Ciis computed. A data-driven threshold η is computed and the data points (in Ci) whose distances to LSiare larger than η are eliminated since they are considered as outliers or erroneously clustered data points in Ci. The remaining data points Ci⊂ Ciare considered to be coming from the same subspace with high confidence. Then, {Ci}Mi=1are used in unsupervised way to train a convolution neural network to obtain a deep learning model, which is in turn used to re-cluster W. The system has been successfully implemented using the MNIST dataset and it improved the segmentation accuracy of a particular algorithm (EnSC-ORGEN) from 93.79% to 96.52%.

This paper provides a comprehensive analysis of skeleton decomposition used for segmentation of data W = [w1···WN] ⊂ ℝddrawn from a union U = ∪i=1MSiof linearly independent subspaces {Si}i=1Mof dimensions of {di}i=1M. Our previous work developed a generalized theoretical framework for computing similarity matrices by matrix factorization. Skeleton decomposition is a special case of this general theory. First, a square sub-matrix A ϵ ℝr×rof W with the same rank r as W is found. Then, the corresponding row restriction R of W is constructed. This leads to P = A-1ℝ and corresponding similarity matrix SW= (pTp)dmax, where dmaxis the maximum subspace dimension. Since most of the data matrices are low-rank in many subspace segmentation problems, this is computationally efficient compared to the other constructions of similarity matrices. It is also shown (with some limitations) that center-of-mass based sorting of data columns in SWcan be used to quickly assess clustering performance while algorithm development in both noisy or noise-free cases.