An incremental multilinear system for human face learning and recognition

This dissertation establishes a novel system for human face learning and recognition based on incremental multilinear Principal Component Analysis (PCA). Most of the existing face recognition systems need training data during the learning process. The system as proposed in this dissertation utilizes an unsupervised or weakly supervised learning approach, in which the learning phase requires a minimal amount of training data. It also overcomes the inability of traditional systems to adapt to the testing phase as the decision process for the newly acquired images continues to rely on that same old training data set. Consequently when a new training set is to be used, the traditional approach will require that the entire eigensystem will have to be generated again. However, as a means to speed up this computational process, the proposed method uses the eigensystem generated from the old training set together with the new images to generate more effectively the new eigensystem in a so-called incremental learning process. In the empirical evaluation phase, there are two key factors that are essential in evaluating the performance of the proposed method: (1) recognition accuracy and (2) computational complexity. In order to establish the most suitable algorithm for this research, a comparative analysis of the best performing methods has been carried out first. The results of the comparative analysis advocated for the initial utilization of the multilinear PCA in our research. As for the consideration of the issue of computational complexity for the subspace update procedure, a novel incremental algorithm, which combines the traditional sequential Karhunen-Loeve (SKL) algorithm with the newly developed incremental modified fast PCA algorithm, was established. In order to utilize the multilinear PCA in the incremental process, a new unfolding method was developed to affix the newly added data at the end of the previous data. The results of the incremental process based on these two methods were obtained to bear out these new theoretical improvements. Some object tracking results using video images are also provided as another challenging task to prove the soundness of this incremental multilinear learning method.

Algebraic theory of minimal nondeterministic finite automata with applications

Since the 1950s, the theory of deterministic and nondeterministic finite automata (DFAs and NFAs, respectively) has been a cornerstone of theoretical computer science. In this dissertation, our main object of study is minimal NFAs. In contrast with minimal DFAs, minimal NFAs are computationally challenging: first, there can be more than one minimal NFA recognizing a given language; second, the problem of converting an NFA to a minimal equivalent NFA is NP-hard, even for NFAs over a unary alphabet. Our study is based on the development of two main theories, inductive bases and partials, which in combination form the foundation for an incremental algorithm, ibas, to find minimal NFAs. An inductive basis is a collection of languages with the property that it can generate (through union) each of the left quotients of its elements. We prove a fundamental characterization theorem which says that a language can be recognized by an n-state NFA if and only if it can be generated by an n-element inductive basis. A partial is an incompletely-specified language. We say that an NFA recognizes a partial if its language extends the partial, meaning that the NFA’s behavior is unconstrained on unspecified strings; it follows that a minimal NFA for a partial is also minimal for its language. We therefore direct our attention to minimal NFAs recognizing a given partial. Combining inductive bases and partials, we generalize our characterization theorem, showing that a partial can be recognized by an n-state NFA if and only if it can be generated by an n-element partial inductive basis. We apply our theory to develop and implement ibas, an incremental algorithm that finds minimal partial inductive bases generating a given partial. In the case of unary languages, ibas can often find minimal NFAs of up to 10 states in about an hour of computing time; with brute-force search this would require many trillions of years.

Algebraic Theory of Minimal Nondeterministic Finite Automata with Applications

Since the 1950s, the theory of deterministic and nondeterministic finite automata (DFAs and NFAs, respectively) has been a cornerstone of theoretical computer science. In this dissertation, our main object of study is minimal NFAs. In contrast with minimal DFAs, minimal NFAs are computationally challenging: first, there can be more than one minimal NFA recognizing a given language; second, the problem of converting an NFA to a minimal equivalent NFA is NP-hard, even for NFAs over a unary alphabet. Our study is based on the development of two main theories, inductive bases and partials, which in combination form the foundation for an incremental algorithm, ibas, to find minimal NFAs. An inductive basis is a collection of languages with the property that it can generate (through union) each of the left quotients of its elements. We prove a fundamental characterization theorem which says that a language can be recognized by an n-state NFA if and only if it can be generated by an n-element inductive basis. A partial is an incompletely-specified language. We say that an NFA recognizes a partial if its language extends the partial, meaning that the NFA's behavior is unconstrained on unspecified strings; it follows that a minimal NFA for a partial is also minimal for its language. We therefore direct our attention to minimal NFAs recognizing a given partial. Combining inductive bases and partials, we generalize our characterization theorem, showing that a partial can be recognized by an n-state NFA if and only if it can be generated by an n-element partial inductive basis. We apply our theory to develop and implement ibas, an incremental algorithm that finds minimal partial inductive bases generating a given partial. In the case of unary languages, ibas can often find minimal NFAs of up to 10 states in about an hour of computing time; with brute-force search this would require many trillions of years.