A geometric analysis of convex demixing

Demixing is the task of identifying multiple signals given only their sum and prior information about their structures. Examples of demixing problems include (i) separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis; (ii) decomposing an observed matrix into low-rank and sparse components; and (iii) identifying a binary codeword with impulsive corruptions. This thesis describes and analyzes a convex optimization framework for solving an array of demixing problems.

Our framework includes a random orientation model for the constituent signals that ensures the structures are incoherent. This work introduces a summary parameter, the statistical dimension, that reflects the intrinsic complexity of a signal. The main result indicates that the difficulty of demixing under this random model depends only on the total complexity of the constituent signals involved: demixing succeeds with high probability when the sum of the complexities is less than the ambient dimension; otherwise, it fails with high probability.

The fact that a phase transition between success and failure occurs in demixing is a consequence of a new inequality in conic integral geometry. Roughly speaking, this inequality asserts that a convex cone behaves like a subspace whose dimension is equal to the statistical dimension of the cone. When combined with a geometric optimality condition for demixing, this inequality provides precise quantitative information about the phase transition, including the location and width of the transition region.

An examination of strategic opportunities provided by the conference committee procedure in the U.S. Congress

Deference to committees in Congress has been a much studied phenomena for close to 100 years. This deference can be characterized as the unwillingness of a potentially winning coalition on the House floor to impose its will on a small minority, a standing committee. The congressional scholar is then faced with two problems: observing such deference to committees, and explaining it. Shepsle and Weingast have proposed the existence of an ex-post veto for standing committees as an explanation of committee deference. They claim that as conference reports in the House and Senate are considered under a rule that does not allow amendments, the conferees enjoy agenda-setting power. In this paper I describe a test of such a hypothesis (along with competing hypotheses regarding the effects of the conference procedure). A random-utility model is utilized to estimate legislators' ideal points on appropriations bills from 1973 through 1980. I prove two things: 1) that committee deference can not be said to be a result of the conference procedure; and moreover 2) that committee deference does not appear to exist at all.

Neural and computational representations of decision variables

These studies explore how, where, and when representations of variables critical to decision-making are represented in the brain. In order to produce a decision, humans must first determine the relevant stimuli, actions, and possible outcomes before applying an algorithm that will select an action from those available. When choosing amongst alternative stimuli, the framework of value-based decision-making proposes that values are assigned to the stimuli and that these values are then compared in an abstract “value space” in order to produce a decision. Despite much progress, in particular regarding the pinpointing of ventromedial prefrontal cortex (vmPFC) as a region that encodes the value, many basic questions remain. In Chapter 2, I show that distributed BOLD signaling in vmPFC represents the value of stimuli under consideration in a manner that is independent of the type of stimulus it is. Thus the open question of whether value is represented in abstraction, a key tenet of value-based decision-making, is confirmed. However, I also show that stimulus-dependent value representations are also present in the brain during decision-making and suggest a potential neural pathway for stimulus-to-value transformations that integrates these two results.

More broadly speaking, there is both neural and behavioral evidence that two distinct control systems are at work during action selection. These two systems compose the “goal-directed system”, which selects actions based on an internal model of the environment, and the “habitual” system, which generates responses based on antecedent stimuli only. Computational characterizations of these two systems imply that they have different informational requirements in terms of input stimuli, actions, and possible outcomes. Associative learning theory predicts that the habitual system should utilize stimulus and action information only, while goal-directed behavior requires that outcomes as well as stimuli and actions be processed. In Chapter 3, I test whether areas of the brain hypothesized to be involved in habitual versus goal-directed control represent the corresponding theorized variables.

The question of whether one or both of these neural systems drives Pavlovian conditioning is less well-studied. Chapter 4 describes an experiment in which subjects were scanned while engaged in a Pavlovian task with a simple non-trivial structure. After comparing a variety of model-based and model-free learning algorithms (thought to underpin goal-directed and habitual decision-making, respectively), it was found that subjects’ reaction times were better explained by a model-based system. In addition, neural signaling of precision, a variable based on a representation of a world model, was found in the amygdala. These data indicate that the influence of model-based representations of the environment can extend even to the most basic learning processes.

Knowledge of the state of hidden variables in an environment is required for optimal inference regarding the abstract decision structure of a given environment and therefore can be crucial to decision-making in a wide range of situations. Inferring the state of an abstract variable requires the generation and manipulation of an internal representation of beliefs over the values of the hidden variable. In Chapter 5, I describe behavioral and neural results regarding the learning strategies employed by human subjects in a hierarchical state-estimation task. In particular, a comprehensive model fit and comparison process pointed to the use of "belief thresholding". This implies that subjects tended to eliminate low-probability hypotheses regarding the state of the environment from their internal model and ceased to update the corresponding variables. Thus, in concert with incremental Bayesian learning, humans explicitly manipulate their internal model of the generative process during hierarchical inference consistent with a serial hypothesis testing strategy.

Delivery of Targeted Nanoparticles Across the Blood-Brain Barrier Using a Detachable Targeting Ligand

Chronic diseases of the central nervous system are poorly treated due to the inability of most therapeutics to cross the blood-brain barrier. The blood-brain barrier is an anatomical and physiological barrier that severely restricts solute influx, including most drugs, from the blood to the brain. One promising method to overcome this obstacle is to use endogenous solute influx systems at the blood-brain barrier to transport drugs. Therapeutics designed to enter the brain through transcytosis by binding the transferrin receptor, however, are restricted within endothelial cells. The focus of this work was to develop a method to increase uptake of transferrin-containing nanoparticles into the brain by overcoming these restrictive processes.

To accomplish this goal, nanoparticles were prepared with surface transferrin molecules bound through various liable chemical bonds. These nanoparticles were designed to shed the targeting molecule during transcytosis to allow increased accumulation of nanoparticles within the brain.

Transferrin was added to the surface of nanoparticles through either redox or pH sensitive chemistry. First, nanoparticles with transferrin bound through disulfide bonds were prepared. These nanoparticles showed decreased avidity for the transferrin receptor after exposure to reducing agents and increased ability to enter the brain in vivo compared to those lacking the disulfide link.

Next, transferrin was attached through a chemical bond that cleaves at mildly acidic pH. Nanoparticles containing a cleavable link between transferrin and gold nanoparticle cores were found to both cross an in vitro model of the blood-brain barrier and accumulate within the brain in significantly higher numbers than similar nanoparticles lacking the cleavable bond. Also, this increased accumulation was not seen when using this same strategy with an antibody to transferrin receptor, indicating that behavior of nanoparticles at the blood-brain barrier varies depending on what type of targeting ligand is used.

Finally, polymeric nanoparticles loaded with dopamine and utilizing a superior acid-cleavable targeting chemistry were investigated as a potential treatment for Parkinson’s disease. These nanoparticles were capable of increasing dopamine quantities in the brains of healthy mice, highlighting the therapeutic potential of this design. Overall, this work describes a novel method to increase targeted nanoparticle accumulation in the brain.