As machine learning has more closely interacted with optimization, the concept of convexity has loomed large. Two properties beyond simple convexity have received particularly close attention: strong smoothness and strong convexity. These properties (and their relatives) underlie machine learning analyses from convergence rates to generalization bounds --- they are central and fundamental. This thesis takes as its focus properties from operator theory that, in specific instances, relate to broadened conceptions of convexity, strong smoothness, and strong convexity. Some of the properties we consider coincide with strong smoothness and strong convexity in some settings, but represent broadenings of these concepts in other situations of interest. Our intention throughout is to take an approach that balances theoretical generality with ease of use and subsequent extension. Through this approach we establish a framework, novel in its scope of application, in which a single analysis serves to recover standard convergence rates (typically established via a variety of separate arguments) for convex optimization methods prominent in machine learning. The framework is based on a perspective in which the iterative update for each convex optimization method is regarded as the application of some operator. We establish a collection of correspondences, novel in its comprehensiveness, that exist between "contractivity- type'' properties of the iterative update operator and "monotonicity-type'' properties of the associated displacement operator. We call particular attention to the comparison between the broader range of properties that we discuss and the more restricted range considered in the contemporary literature, demonstrating as well the relationship between the broader and narrower range. In support of our discussion of these property correspondences and the optimization method analyses based on them, we relate operator theory concepts that may be unfamiliar to a machine learning audience to more familiar concepts from convex analysis. In addition to grounding our discussion of operator theory, this turns out to provide a fresh perspective on many touchstone concepts from convex analysis

In many problem domains data may come from multiple sources (or views), such as video and audio from a camera or text on and links to a web page. These multiple views of the data are often not directly comparable to one another, and thus a principled method for their integration is warranted. In this paper we develop a new algorithm to leverage information from multiple views for unsupervised clustering by constructing a custom kernel. We generate a multipartite graph (with the number of parts given by the number of views) that induces a kernel we then use for spectral clustering. Our algorithm can be seen as a generalization of co-clustering and spectral clustering and a relative of Kernel Canonical Correlation Analysis. We demonstrate the algorithm on four data sets: an illustrative artificial data set, synthetic fMRI data, voxels from an fMRI study, and a collection of web pages. Finally, we compare its performance to common alternatives.