A Graph-Based Semi-Supervised k Nearest-Neighbor Method for Nonlinear Manifold Distributed Data Classification

$k$ Nearest Neighbors ($k$NN) is one of the most widely used supervised learning algorithms to classify Gaussian distributed data, but it does not achieve good results when it is applied to nonlinear manifold distributed data, especially when a very limited amount of labeled samples are available. In this paper, we propose a new graph-based $k$NN algorithm which can effectively handle both Gaussian distributed data and nonlinear manifold distributed data. To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by constructing an $R$-level nearest-neighbor strengthened tree over the graph, and then compute a TRW matrix for similarity measurement purposes. After this, the nearest neighbors are identified according to the TRW matrix and the class label of a query point is determined by the sum of all the TRW weights of its nearest neighbors. To deal with online situations, we also propose a new algorithm to handle sequential samples based a local neighborhood reconstruction. Comparison experiments are conducted on both synthetic data sets and real-world data sets to demonstrate the validity of the proposed new $k$NN algorithm and its improvements to other version of $k$NN algorithms. Given the widespread appearance of manifold structures in real-world problems and the popularity of the traditional $k$NN algorithm, the proposed manifold version $k$NN shows promising potential for classifying manifold-distributed data.Comment: 32 pages, 12 figures, 7 table

Adaptive Nonparametric Image Parsing

In this paper, we present an adaptive nonparametric solution to the image parsing task, namely annotating each image pixel with its corresponding category label. For a given test image, first, a locality-aware retrieval set is extracted from the training data based on super-pixel matching similarities, which are augmented with feature extraction for better differentiation of local super-pixels. Then, the category of each super-pixel is initialized by the majority vote of the $k$-nearest-neighbor super-pixels in the retrieval set. Instead of fixing $k$ as in traditional non-parametric approaches, here we propose a novel adaptive nonparametric approach which determines the sample-specific k for each test image. In particular, $k$ is adaptively set to be the number of the fewest nearest super-pixels which the images in the retrieval set can use to get the best category prediction. Finally, the initial super-pixel labels are further refined by contextual smoothing. Extensive experiments on challenging datasets demonstrate the superiority of the new solution over other state-of-the-art nonparametric solutions.Comment: 11 page

Declutter and Resample: Towards parameter free denoising

In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth $K$ in a metric space, but it got corrupted with noise so that some of the data points lie far away from $K$ creating outliers also termed as {\em ambient noise}. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of $K$. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice

Local feature weighting in nearest prototype classification

The distance metric is the corner stone of nearest neighbor (NN)-based methods, and therefore, of nearest prototype (NP) algorithms. That is because they classify depending on the similarity of the data. When the data is characterized by a set of features which may contribute to the classification task in different levels, feature weighting or selection is required, sometimes in a local sense. However, local weighting is typically restricted to NN approaches. In this paper, we introduce local feature weighting (LFW) in NP classification. LFW provides each prototype its own weight vector, opposite to typical global weighting methods found in the NP literature, where all the prototypes share the same one. Providing each prototype its own weight vector has a novel effect in the borders of the Voronoi regions generated: They become nonlinear. We have integrated LFW with a previously developed evolutionary nearest prototype classifier (ENPC). The experiments performed both in artificial and real data sets demonstrate that the resulting algorithm that we call LFW in nearest prototype classification (LFW-NPC) avoids overfitting on training data in domains where the features may have different contribution to the classification task in different areas of the feature space. This generalization capability is also reflected in automatically obtaining an accurate and reduced set of prototypes.Publicad