December 23, 2015 8:51 WSPC/WS-IJWMIP ODEAI Semi-Supervised Learning Using Multiple One-Dimensional Embedding Based Adaptive Interpolation Jianzhong Wang Department of Mathematics and Statistics, Sam Houston State University Huntsville, TX 77341, USA jzwang@shsu.edu. Received 12 November 2014 Revised 15 March 2015 Accepted 20 March 2015 Abstract We propose a novel semi-supervised learning scheme using adaptive interpola- tion on multiple one-dimensional (1-D) embedded data. For a give high dimensional data set, we smoothly map it onto several different one-dimensional (1-D) sequences, so that the labeled subset is converted to a 1-D subset for each of these sequences. Applying the cubic interpolation of the labeled subset, we obtain a subset of unlabeled points, which are assigned to the same label in all interpolations. Selecting a proportion of these points at random and adding them to the current labeled subset, we build a larger labeled sub- set for the next interpolation. Repeating the embedding and interpolation, we enlarge the labeled subset gradually, and finally reach a labeled set with a reasonable large size, based on which the final classifier is constructed. We explore the use of the proposed scheme in the classification of handwritten digits and show promising results. Keywords: multiple 1-D embedding; 1-D multi-embedding; interpolation; semi- supervised learning. AMS Subject Classification: 22E46, 53C35, 57S20 1. Introduction In recent years, analysis of high dimensional data is a major challenge for the statis- tics and machine learning communities. Many traditional statistical methods can- not directly applied to high dimensional data analysis. The usual machine learning framework often does not directly exploit the rich geometric structure of high di- mensional data. To capitalize on the geometric structure, new learning approaches to high dimensional data have been developed. Their main theme is embedding high dimensional data into a lower dimensional manifold so that the analysis can be carried out on the reorganized data. Smooth ordering of image patches intro- duced in Ref. 16, 17, 18 originally was developed for image processing. The key idea of smooth-ordering is to deal with high-dimensional data in a one-dimensional framework. We find that this idea can also be applied to semi-supervised learning (SSL). The purpose of this paper is to introduce a novel SSL algorithm based on data 1-D representation. Let X = {x i } n i=1 ⊂ R m be a given data set in m-space, 37