Manifold learning is a hot research topic in the field of computer science. A crucial issue with current manifold learning methods is that they lack a natural quantitative measure to assess the quality of learned embeddings, which greatly limits their applications to real-world problems. In this paper, a new embedding quality assessment method for manifold learning, named as Normalization Independent Embedding Quality Assessment (NIEQA), is proposed. Compared with current assessment methods which are limited to isometric embeddings, the NIEQA method has a much larger application range due to two features. First, it is based on a new measure which can effectively evaluate how well local neighborhood geometry is preserved under normalization, hence it can be applied to both isometric and normalized embeddings. Second, it can provide both local and global evaluations to output an overall assessment. Therefore, NIEQA can serve as a natural tool in model selection and evaluation tasks for manifold learning. Experimental results on benchmark data sets validate the effectiveness of the proposed method.
Venna and Kaski @cite_42 proposed an assessment method which consists of two measures, one for trustworthiness and one for continuity, based on the change of indices of neighbor samples in @math and @math according to pairwise Euclidean distances, respectively. Aguirre al proposed an alternative approach for quantifying the embedding quality, by evaluating the possible overlaps in the low-dimensional embedding. Their assessment is used for automatic choice of the number of nearest neighbors for LLE @cite_39 and also exploited in @cite_7 to evaluate the embedding quality of LLE with optimal regularization parameter. Akkucuk and Carroll @cite_8 independently developed the Agreement Rate (AR) metric which shares the same form to @math . Based on AR, they suggested another useful assessment method called corrected agreement rate, by randomly reorganize the indices of data in @math . Also with AR, France and Carroll @cite_22 proposed a method using the RAND index to evaluate dimensionality reduction methods.
@cite_22 @cite_7 @cite_8 @cite_42 @cite_39
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In recent years, the effect of the curse of high dimensionality has been studied in great detail on several problems such as clustering, nearest neighbor search, and indexing. In high dimensional space the data becomes sparse, and traditional indexing and algorithmic techniques fail from a efficiency and or effectiveness perspective. Recent research results show that in high dimensional space, the concept of proximity, distance or nearest neighbor may not even be qualitatively meaningful. In this paper, we view the dimensionality curse from the point of view of the distance metrics which are used to measure the similarity between objects. We specifically examine the behavior of the commonly used Lk norm and show that the problem of meaningfulness in high dimensionality is sensitive to the value of k. For example, this means that the Manhattan distance metric (L1 norm) is consistently more preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Using the intuition derived from our analysis, we introduce and examine a natural extension of the Lk norm to fractional distance metrics. We show that the fractional distance metric provides more meaningful results both from the theoretical and empirical perspective. The results show that fractional distance metrics can significantly improve the effectiveness of standard clustering algorithms such as the k-means algorithm. We develop a metric i¾?, based upon the RAND index, for the comparison and evaluation of dimensionality reduction techniques. This metric is designed to test the preservation of neighborhood structure in derived lower dimensional configurations. We use a customer information data set to show how i¾?can be used to compare dimensionality reduction methods, tune method parameters, and choose solutions when methods have a local optimum problem. We show that i¾?is highly negatively correlated with an alienation coefficient K that is designed to test the recovery of relative distances. In general a method with a good value of i¾?also has a good value of K. However the monotonic regression used by Nonmetric MDS produces solutions with good values of i¾?, but poor values of K. Nearest neighbour classification expects the class conditional probabilities to be locally constant, and suffers from bias in high dimensions. We propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality. We use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods. We determine the local decision boundaries from centroid information, and then shrink neighbourhoods in directions orthogonal to these local decision boundaries, and elongate them parallel to the boundaries. Thereafter, any neighbourhood-based classifier can be employed, using the modified neighbourhoods. The posterior probabilities tend to be more homogeneous in the modified neighbourhoods. We also propose a method for global dimension reduction, that combines local dimension information. In a number of examples, the methods demonstrate the potential for substantial improvements over nearest neighbour classification. This paper reviews the classical problem of free-form curve registration and applies it to an efficient RGB-D visual odometry system called Canny-VO, as it efficiently tracks all Canny edge features extracted from the images. Two replacements for the distance transformation commonly used in edge registration are proposed: approximate nearest neighbor fields and oriented nearest neighbor fields. 3-D–2-D edge alignment benefits from these alternative formulations in terms of both efficiency and accuracy. It removes the need for the more computationally demanding paradigms of data-to-model registration, bilinear interpolation, and subgradient computation. To ensure robustness of the system in the presence of outliers and sensor noise, the registration is formulated as a maximum a posteriori problem and the resulting weighted least-squares objective is solved by the iteratively reweighted least-squares method. A variety of robust weight functions are investigated and the optimal choice is made based on the statistics of the residual errors. Efficiency is furthermore boosted by an adaptively sampled definition of the nearest neighbor fields. Extensive evaluations on public SLAM benchmark sequences demonstrate state-of-the-art performance and an advantage over classical Euclidean distance fields.