Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

FRAMEWORK SYNTHESIS FOR SYMBOLIC EXECUTION OF EVENT-DRIVEN FRAMEWORKS

Symbolic execution is a powerful program analysis technique, but it is very challenging to apply to programs built using event-driven frameworks, such as Android. The main reason is that the framework code itself is too complex to symbolically execute. The standard solution is to manually create a framework model that is simpler and more amenable to symbolic execution. However, developing and maintaining such a model by hand is difficult and error-prone. We claim that we can leverage program synthesis to introduce a high-degree of automation to the process of framework modeling. To support this thesis, we present three pieces of work. First, we introduced SymDroid, a symbolic executor for Android. While Android apps are written in Java, they are compiled to Dalvik bytecode format. Instead of analyzing an app’s Java source, which may not be available, or decompiling from Dalvik back to Java, which requires significant engineering effort and introduces yet another source of potential bugs in an analysis, SymDroid works directly on Dalvik bytecode. Second, we introduced Pasket, a new system that takes a first step toward automatically generating Java framework models to support symbolic execution. Pasket takes as input the framework API and tutorial programs that exercise the framework. From these artifacts and Pasket's internal knowledge of design patterns, Pasket synthesizes an executable framework model by instantiating design patterns, such that the behavior of a synthesized model on the tutorial programs matches that of the original framework. Lastly, in order to scale program synthesis to framework models, we devised adaptive concretization, a novel program synthesis algorithm that combines the best of the two major synthesis strategies: symbolic search, i.e., using SAT or SMT solvers, and explicit search, e.g., stochastic enumeration of possible solutions. Adaptive concretization parallelizes multiple sub-synthesis problems by partially concretizing highly influential unknowns in the original synthesis problem. Thanks to adaptive concretization, Pasket can generate a large-scale model, e.g., thousands lines of code. In addition, we have used an Android model synthesized by Pasket and found that the model is sufficient to allow SymDroid to execute a range of apps.