Three problems in high-dimensional statistical parameter estimation [electronic resource]
Keywords: Electrical Engineering.

Statistical parameter estimation is the process of estimating parameters of a system from direct or indirect observations. As noted by Bradly Efron in his recent book [29], at its inception parameter estimation was about simple questions using large datasets, e.g., estimating mean and variance of the age of a population using survey data. Later it evolved into problems involving still simple questions but now small amount of data. Examples included hypothesis testing and estimating parameters of a communication channel using short training sequences. This setting resulted in a fascinating literature and promotion of ideas like statistical eciency and power. In recent years, a new class of parameter estimation has been emerging which is best exemplified by the Netflix challenge. These problems are characterized by vast amount of input data but at the same time complex and large set of parameters to be estimated. In the case of the Netflix challenge, the data consist of about 10^8 known movie ratings and the challenge was to estimate about 10^10 unknown ratings. In this work, we consider three problems of this type, namely, learning and controlling high- dimensional dynamical systems, and statistical signal processing for time of flight mass spectrometry. We present both theoretical and experimental results. First, we consider the problem of learning high-dimensional dynamical systems. A continuous time autonomous dynamical system is described by a stochastic differential equation while a discrete time system is described by a state evolution equation. We consider the problem of learning the drift coefficient of a p-dimensional stochastic differential equation from a sample path of length T. We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem in the limit in which both p and T tend to infinity. In particular, we prove a general lower iv bound on the sample-complexity T by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a p-̳>̳ p matrix which describes which degrees of freedom interact under the dynamics. In this case, we analyze an `1-regularized least squares estimator and prove an upper bound on T that nearly matches the lower bound on specific classes of sparse matrices. Linear dynamical systems are important modeling tools for many applications, ranging from robotics and control to biology and chemistry. However, in some applications the ultimate goal is to efficiently control an unknown or poorly known system, i.e., the reinforcement learning of the underlying dynamical system. A linear quadratic dynamical system is describe by a state evolution equation and a cost function. The dynamics is similar to that of an autonomous linear dynamical system with the addition of a control term. The cost at each time is a quadratic function of the state and the control exerted at that step. In the second part of this thesis we study the problem of adaptive control of a high dimensional linear quadratic system. Previous work established the asymptotic convergence to an optimal controller for various adaptive control schemes. More recently, for the average cost LQ problem, a regret bound of O(\sqrt{T}) was shown, apart form logarithmic factors. However, this bound scales exponentially with p, the dimension of the state space. In this work we consider the case where the matrices describing the dynamic of the LQ system are sparse and their dimensions are large. We present an adaptive control scheme that achieves a regret bound of O(p\sqrt{T}), apart from logarithmic factors. In particular, our algorithm has an average cost of (1+\epsilon) times the optimum cost after T = polylog(p)O(1/\epsilon^2). This is in comparison to previous work on the dense dynamics where the algorithm requires time that scales exponentially with dimension in order to achieve regret of e̳ times the optimal cost. We describe the applications of this result in the emerging area of computational advertising, in particular targeted online advertising and advertising in social networks. In the third part of this thesis we tend to the problem of statistical signal processing for time of flight mass spectrometry. Time of flight mass spectrometry is a technique for measuring the mass of ionized chemical species. Traditionally, because of the high data rate and the requirement for the online processing of the signal by the instrument, time of fight mass spectrometers employed only rudimentary signal processing. We employ a simple modification to the conventional time of flight mass spectrometry where a variable and (pseudo)-random pulsing rate is used which allows for traces from different pulses to overlap. This modification requires little alteration to the currently employed hardware. However, it requires a reconstruction method to recover the spectrum from highly aliased traces. We propose and demonstrate an efficient algorithm that can process massive time of flight mass spectrometry data using computational resources that can be considered modest with today's standards. We demonstrate how our method can be employed to increase the throughput of an instrument by an order of magnitude. We expect this to extend the applicability of TOFMS to new domains.