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.
In @cite_15 , posited that pattern recognition techniques are an appropriate methodology for solving problems in combinatorial group theory. To demonstrate, they constructed a machine learning system for discovering effective heuristics for the Whitehead automorphism problem, a search problem in free groups that uses the successive application of the namesake automorphisms to reduce a word to its minimal length. As mentioned in @cite_15 , every machine learning system must contend with the following tasks: data generation, feature extraction, model selection, and evaluation. Once the system is constructed, analysis of the system's performance can yield insights into the nature of the problem at hand, and potentially be used to improve upon it. In the following sections we will delve into each of these aforementioned tasks, showing in the process how these techniques can be extended from free groups to finitely presented groups, and ultimately be adapted to solving the conjugacy decision problem The primary difference in the construction of machine learning systems for free and not-free groups is in feature extraction, which is the focus of the next section.
@cite_15
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We review some basic methodologies from pattern recognition that can be applied to helping solve combinatorial problems in free group theory. We illustrate how this works with recognizing Whitehead minimal words in free groups of rank 2. The methodologies reviewed include how to form feature vectors, principal components, distance classifers, linear classifiers, regression classifiers, Fisher linear discriminant, support vector machines, quantizing, classification trees, and clustering techniques. We aim to apply sparse autoencoder-based unsupervised feature learning to emotional semantic analysis for textile images. To tackle the problem of limited training data, we present a cross-domain feature learning scheme for emotional textile image classification using convolutional autoencoders. We further propose a correlation-analysis-based feature selection method for the weights learned by sparse autoencoders to reduce the number of features extracted from large size images. First, we randomly collect image patches on an unlabeled image dataset in the source domain and learn local features with a sparse autoencoder. We then conduct feature selection according to the correlation between different weight vectors corresponding to the autoencoder’s hidden units. We finally adopt a convolutional neural network including a pooling layer to obtain global feature activations of textile images in the target domain and send these global feature vectors into logistic regression models for emotional image classification. The cross-domain unsupervised feature learning method achieves 65 to 78 average accuracy in the cross-validation experiments corresponding to eight emotional categories and performs better than conventional methods. Feature selection can reduce the computational cost of global feature extraction by about 50 while improving classification performance. The main contribution of this paper is a systematic analysis of a minimally supervised machine learning method for relation extraction grammars. The method is based on a bootstrapping approach in which the bootstrapping is triggered by semantic seeds. The starting point of our analysis is the pattern-learning graph which is a subgraph of the bipartite graph representing all connections between linguistic patterns and relation instances exhibited by the data. It is shown that the performance of such general learning framework for actual tasks is dependent on certain properties of the data and on the selection of seeds. Several experiments have been conducted to gain explanatory insights into the interaction of these two factors. From the investigation of more effective seeds and benevolent data we understand how to improve the learning in less fortunate configurations. A relation extraction method only based on positive examples cannot avoid all false positives, especially when the data properties yield a high recall. Therefore, negative seeds are employed to learn negative patterns, which boost precision. Machine learning research has, to a great extent, ignored an important aspect of many real world applications: time. Existing concept learners predominantly operate on a static set of attributes; for example, classifying flowers described by leaf size, pet al colour and pet al count. However, many real datasets are not static; they cannot sensibly be represented as a fixed set of attributes. Rather, the examples are expressed as features that vary temporally, and it is the temporal variation itself that is used for classification. Consider a simple gesture recognition domain, in which the temporal features are the position of the hands, finger bends, and so on. Looking at the position of the hand at one point in time is not likely to lead to a successful classification; it is only by analysing changes in position that recognition is possible. This thesis presents a new technique for temporal classification. By extracting sub-events from the training instances and parameterising them to allow feature construction for a subsequent learning process, it is able to employ background knowledge and express learnt concepts in terms of the background knowledge. The novel results of the thesis are: a temporal learner capable of producing comprehensible and accurate classifiers for multivariate time series that can learn from a small number of instances and can integrate non-temporal features; a feature construction technique that parameterises sub-events of the training set and clusters them to construct features for a propositional learner; and a technique for post-processing classification rules produced by the learner to give a comprehensible description expressed in the same form as the original background knowledge. The thesis discusses the implementation of TClass, a temporal learner. Results show rules that are comprehensible in many cases and accuracy results close to or better than existing technique—over 98 per cent for sign language and 72 per cent for ECGs (equivalent to the accuracy of a human cardiologist). One further surprising result is that a small set of very primitive sub-events proves to be functional, avoiding the need for labour-intensive background knowledge if it is not available.