We developed a subspace classifier for measurement classification to provide an alternative to current deep learning approaches. Many modern neural networks cannot provide an understandable explanation for their classification. The subspace classifier provides a decomposition of the classification problem making it computationally simpler. We first use a Bayesian method in which all the class conditional probabilities for the entire measurement space can be stored in memory.Then we made experiments with simulated class conditional distributions and defined a subspace classifier that only stores the class conditional probabilities for the subspaces. This can use much larger distributions than the previous model as it uses much less memory so we expanded to cases where the measurement space is generated sequentially and everything does not have to be in the memory at the same time.For cases with distributions that fit in the memory we also compared a Bayesian approach with the subspace approach. The Bayesian subspace classifiers consistently outperformed the subspace classifiers without Bayes rule by a large margin. We also compare the subspace classifier with 3 Python Machine Learning Models, namely a Ridge Classifier, a Multi-Layer Perceptron (MLP) classifier (neural network), and a Support Vector Machine (SVM) on a set of tuples and class conditional probability distributions with 4 classes. The subspace classifier had an average probability of correct identification of 0.25172, the SVM model had an average accuracy of 0.20987, the neural network MLP classifier had an average accuracy of 0.2140 and the Ridge Classifier had an average accuracy of 0.2798 over 10,000 trials.

We present a new measure of dependence suitable for time series forecasting: Partial Monotone Correlation (PMC) that generalizes Monotone Correlation. Unlike the Monotone Correlation, the new measure of dependence uses piecewise strictly monotone transformations that increase the value of the correlation coefficient. We explore its properties, its relationship with Monotone and Maximal Correlation, and present an algorithm that calculates it based on the Simultaneous Perturbation Stochastic Approximation method. We also demonstrate how to apply Partial Monotone Correlation for time series analysis and forecasting introducing Partial Monotone Autoregressive model (PMAR) of order 1. Its performance is then evaluated against the baseline of linear and nonlinear autoregressive models (AR, LSTAR) on 150 time series produced from 3 datasets: Yellow Taxi pickups, Citi Bike pickups, and Cellular Network hits. Overall, PMAR model outperforms the baseline with the average sMAPE of about 1.7-4% lower.

In this paper, we discourse an analysis of classical first-order predicate logic as a constraint satisfaction problem, CSP. First, we will offer our general framework for CSPs, and then apply it to first-order logic. We claim it would function as a new semantics, constraint semantics, for logic. Then, we prove the soundness and completeness theorems with respect to the constraint semantics. The latter theorem, which will be proven by a proof-search method, implies the cut-elimination theorem. Furthermore, using the constraint semantics, we make a new proof of the Craig interpolation theorem. Also, we will provide feasible algorithms to generate interpolants for some decidable fragments of first-order logic: the propositional logic and the monadic fragments. The algorithms, reflecting a ‘projection’ of an indexed relation, will show how to transform given formulas syntactically to obtain interpolants.

This article is written in recognition of W. Bledsoe, who with Browning, introduced the N-tuple subspace classifier in 1959. This 1959 article was the first article to introduce subspace classifiers and the sum rule to combine the outputs of the classifiers. A mathematical notation is given to easily express in a precise and unambiguous way everything going on in the N-tuple subspace classifier. Extensions of the N-tuple method are discussed using a generalized product expression and we relate the generalization to graphical models. We discuss the sum rule, the product rule, and the plurality voting rule for combining the scores of the subspace classifiers. We selected a representative sample of papers that the 1959 N-tuple subspace classifier inspired. Some of the papers introduced specialized improvements. Many of the papers showed the value of the N-tuple subspace classifier in all kinds of applications and compared the results of one or more varieties of the N-tuple subspace classifier with other state-of-the-art classifiers. Their experiments showed that the N-tuple subspace classifier was competitive with the state-of-the-art classifiers and often had a higher accuracy. Finally, we highlight some papers that describe experiments of the N-tuple subspace classifier executing in a quantum computer.

The first level of document structural decomposition consists of zone delineation and zone classification. The purpose of zone delineation is to define the regions which cover the page such that each region, excluding its holes, {the regions which it contains), is entirely a text region or a non-text region and such that in each text region no text line is fragmented. After zone delineation, each text zone is then further decomposed into text lines and words. The purpose of zone classification is to identify each delineated zone into one of the classes: text or non-text. Non-text regions may be further classified into line drawing, graph; half-tone, etc. Zones which have been classified as text regions can then be given to an OCR system to produce computer readable text

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis on polycyclic and metabelian groups that are of interest to non-commutative cryptography. As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. The very high accuracy of these classifiers suggests an underlying mathematical relationship with respect to conjugacy in the tested groups.

We discuss a more powerful probabilistic graphical model for discovering semantic patterns from sequential text data, such as sentences. It is developed based on the idea that each word (or each symbol) in a sentence itself might carry lexical, semantic, or syntactic information, which can be used to replace conditional dependences in existing methods. Hence, our method has fewer conditional independence assumptions in contrast to these existing probabilistic graphical methods, such as CRFs, HMMs, MEMMs and Naive Bayes. Moreover, our method does not need to employ dynamic programming and therefore the on-line time complexity and memory complexity are reduced. We test the method on discovering noun phrases, the meaning of an ambiguous word, and semantic arguments of a verb in a sentence. We find that the misclassification rate is smaller compared to previously published results on the same data sets. For example, the method achieves an average f-measure of 98.25% for recognizing noun phrases on WSJ data from Penn Treebank; an average accuracy of 81.12% for recognizing the six sense word line; an average f-measure of 93.61% for classifying semantic argument boundaries of a verb in a sentence on WSJ data from Penn Treebank and PropBank. (C) 2018 Elsevier B.V. All rights reserved.

The speed and storage advantages of performing vision tasks at downsampled levels of multiresolution data structures can incur the cost of distortion caused by the sampling process. We analyze the constraints on sampling and on image objects, in order to speed up morphological operations without sacrificing accurate shape analysis. The following results are shown to be true under reasonable morphological sampling conditions. Sets which are 'Bampled can be reconstructed in two ways, by either a closing or dilation. In both reconstructions, the sampled reconstructed sets are equal to the sampled sets . A set contains its reconstruction by closing and is contained in its reconstruction by dilation; indeed, these are extremal bounding sets. That is, the largest set which downsamples to a given set is its reconstruction by dilation; the smallest is its reconstruction by closing. Furthermore, the distance from the maximal reconstruction to the minimal reconstruction is no more than the diameter of the reconstruction structurililg element. Morphological sampling thus provides reconstructions positioned only to within some spatial tolerance, in contrast to sampling in signal processing which reconstructs only those frequencies bandlimited to Nyquist frequency. All sampling results in the binary cise generalize to the greyscale case. For greyscale, the boundedness constraint on the reconstruction translates to a boundedness both spatially and in the greyscale image. This work provides a basis for multi-resolution shape analysis in shapebased hierarchical structures, supplementing the advantages of multi -resolution techniques with those of morphological operators on cellular arrays.

In this paper we discuss several heuristic strategies which allow one to solve the Whitehead’s minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice leads to several interesting mathematical conjectures. In particular, we conjecture that the length of most non-minimal elements in a free group can be reduced by a Nielsen automorphism which can be identified by inspecting the structure of the corresponding Whitehead Graph.

This chapter presents some useful properties of B-coded restricted domains that will be used in later proofs. Line drawings have been commonly used to represent the boundaries of two-dimensional objects. Restricted domains can be represented in terms of the intersections of discrete half-planes. The half-planes that are not redundant and form the sides of the polygon will be called primary. If the input restricted domains are in their B-code representations or if the output restricted domains are needed in their B-code representation, the results can be used for the interconversion between representations. The algorithms for opening and closing can be easily obtained by applying the dilation and erosion algorithms in the appropriate order. The algorithm for opening consists of two stages—an erosion stage followed by a dilation stage. Since erosion and dilation algorithms are constant in time, the algorithm for opening is also constant in time.

Given a data set taken over a population, the question of how can we construct possible explanatory models for the interactions and dependencies in the population is a discovery question. Projection and Relation Join is a way of addressing this question in a non-deterministic context with mathematical relations. In this paper, we apply projection and relation join to music harmonic sequences to generate new sequences in a given composer or genre style. Instead of first learning the patterns, and then making replications as early music generation work did, we introduce a completely new data driven methodology to generate music. Then we discuss exploring the difference between the original music and synthetic music sequences using information theory based techniques.

Relation Join has been applied to generate music harmonic sequences in a given composer or genre style. Compared to other music generation method, Relation Join does not require expert knowledge or estimation of any probabilities while generating a massive number of sequences. However, whether the generated compositions are distinguishable from the original ones is still a question that has not been explored. The evaluation of synthetic music sequences is considered subjective and hard to quantify. In this paper, we formed the evaluation problem as a classification task where the original and synthetic music sequences are assigned with two different labels. The evaluation of synthetic music sequences can then be quantified using the accuracy of classification experiment. In this paper, besides the traditional state-of-art classifiers, we propose a Seq2Tree network based deep learning model to classify synthetic and original music sequences. Our model extends the Long Short Term Memory network (LSTM) to take advantage of the characteristics of music data. The results show that our methods successfully classify 98.75% of the corpus that contain both original and synthetic Bach sequences, while the best performance of other machine learning classifiers is 81.63%.

This paper discusses different kinds of dependency. For numerically valued variables our discussion centers on the maximal correlation coefficient and its cousin the monotone correlation coefficient. We show how to calculate the maximal correlation coefficient in the case the random variables take on a finite set of values. For non-numerically valued variables our discussion centers on information theoretic measures related to mutual information and we describe some that are also metrics. We visually illustrate the difference between these two kinds of measures with a texture example that computes the joint probability image: an image in which the gray level of each pixel is the joint probability of the gray levels of the pixels in its neighborhood. Neighborhoods can be regular such as 5 x 5 or they can be irregular. Finally, we discuss manifold methods for classification: the N-tuple method, the subspace classifiers, the subspace ensemble classifiers, including the graphical model for representing the class conditional probability distribution. We describe a procedure to convert an N-tuple classifier to a graphical model classifier. We also conjecture that there is new form of a universal approximation theorem by which not too complex classification functions from measurement space to the set of classes can be approximately represented in the form of a subspace classifier using multiple subspaces such as the N-tuple method. (C) 2017 Published by Elsevier B.V.

In this paper, we present an Automatic Target Detection system that operates on a simulated E3D(Exploitation of 3D Data) image dataset. Simulated E3D images are range images where each value represents the height above the ground. In our work, we treat the 3D data as if the height values were pixel intensity values(2D) and a set of mathematical morphology operations are applied to each image to generate features and then a decision tree classification algorithm is used to distinguish between the target and nontarget pixels in test images. The experimental results show over target detection performance.

We present a regularization technique based on the minimum description length (MDL) principle for the linear manifold clustering. We suggest an inexact minimum description length method based on describing the data structure as linear manifold clusters. We examine the behavior of the proposed method and compare it performance against simulated clustering results of various dimensionality and structure. Finally, we empirically evaluate the proposed technique on a climate data.

Information retrieval methods represent query results in a ranked, one dimensional list without revealing connections among documents and document groups. We propose a new model of document representation and extend the notion of similarity to consider document length and word synonyms to organize documents into topically relevant groups. Matches to a user query are presented in an intuitive, interactive map that facilitates browsing information and finding the most relevant matches within the overall landscape of results. Current research is aimed at finding the best metric to group documents, and in defining the visual model. Our system, named GIST, is built using the PREFUSE toolkit for graphical display [1].

In this study, we revisit quadratic discriminant analysis (QDA). For this purpose, we present a majorize-minimize (MM) optimization algorithm to estimate parameters for generative classifiers, of which conditional distributions are from the exponential family. Furthermore, we propose a block-coordinate descent algorithm to sequentially update parameters of QDA in each iteration of the MM algorithm; for each update, we apply a trust region method, of which each iteration has a simple closed form solution. Numerical experiments show that: when compared with conjugate gradient method, the new proposed method is faster in 9 of 10 benchmark data sets; when compared with other widely used quadratic classifiers in the literature, QDA trained with the proposed method is either the best or not statistically significantly different from the best ones in 8 of 10 benchmark data sets.

We present a new multiscale surface representation for 3D shape matching that is based on scale-space theory. The representation, Curvature Scale-Space 3D (CS3), is well-suited for measuring dissimilarity between (partial) surfaces having unknown position, orientation, and scale. The CS3 representation is obtained by evolving the surface curvatures according to the heat equation. This evolution process yields a stack of increasingly smoothed surface curvatures that is useful for keypoint extraction and descriptor computations. We augment this information with an associated scale parameter at each stack level to define our multiscale CS3 surface representation. The scale parameter is necessary for automatic scale selection, which has proven to be successful in 2D scale-invariant shape matching applications. We show that our keypoint and descriptor computation approach outperforms many of the leading methods. The main advantages of our representation are its computational efficiency, lower memory requirements, and ease of implementation.