Nate Strawn

Posterior consistency in linear models under shrinkage priors

We investigate the asymptotic behaviour of posterior distributions of regression coefficients in high-dimensional linear models as the number of dimensions grows with the number of observations. We show that the posterior distribution concentrates in neighbourhoods of the true parameter under simple sufficient conditions. These conditions hold under popular shrinkage priors given some sparsity assumptions. Copyright 2013, Oxford University Press.

Constructing all self-adjoint matrices with prescribed spectrum and diagonal

The Schur–Horn Theorem states that there exists a self-adjoint matrix with a given spectrum and diagonal if and only if the spectrum majorizes the diagonal. Though the original proof of this result was nonconstructive, several constructive proofs have subsequently been found. Most of these constructive proofs rely on Givens rotations, and none have been shown to be able to produce every example of such a matrix. We introduce a new construction method that is able to do so. This method is based on recent advances in finite frame theory which show how to construct frames whose frame operator has a given prescribed spectrum and whose vectors have given prescribed lengths. This frame construction requires one to find a sequence of eigensteps, that is, a sequence of interlacing spectra that satisfy certain trace considerations. In this paper, we show how to explicitly construct every such sequence of eigensteps. Here, the key idea is to visualize eigenstep construction as iteratively building a staircase. This visualization leads to an algorithm, dubbed Top Kill, which produces a valid sequence of eigensteps whenever it is possible to do so. We then build on Top Kill to explicitly parametrize the set of all valid eigensteps. This yields an explicit method for constructing all self-adjoint matrices with a given spectrum and diagonal, and moreover all frames whose frame operator has a given spectrum and whose elements have given lengths.

Topological and statistical behavior classifiers for tracking applications

This paper introduces a method to integrate target behavior into the multiple hypothesis tracker (MHT) likelihood ratio. In particular, a periodic track appraisal based on behavior is introduced. The track appraisal uses elementary topological data analysis coupled with basic machine-learning techniques, and it adjusts the traditional kinematic data association likelihood (i.e., track score) using an established formulation for feature-aided data association. The proposed method is tested and demonstrated on synthetic vehicular data representing an urban traffic scene generated by the Simulation of Urban Mobility package. The vehicles in the scene exhibit different driving behaviors. The proposed method distinguishes those behaviors and shows improved data association decisions relative to a conventional, kinematic MHT.

Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

We affirm the conjectures in [K. Dykema and N. Strawn, Int. J. Pure Appl. Math., 28 (2006), pp. 217--256] by demonstrating the connectivity of spaces of finite unit norm tight frames (FUNTFs). Our central technique involves explicit continuous lifts of paths from the polytope of eigensteps (see [J. Cahill et al., Appl. Comput. Harmon. Anal., 35 (2013), pp. 52--73]) to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this to show that spaces of FUNTFs are irreducible in the algebro-geometric sense. This last result allows us to show that generic FUNTFs are full spark, and hence the full spark FUNTFs are dense in the space of FUNTFs. This resolves an important theoretical question regarding the application of FUNTFs in the field of compressed sensing.

Feature-aided multiple hypothesis tracking using topological and statistical behavior classifiers

This paper introduces a method to integrate target behavior into the multiple hypothesis tracker (MHT) likelihood ratio. In particular, a periodic track appraisal based on behavior is introduced that uses elementary topological data analysis coupled with basic machine learning techniques. The track appraisal adjusts the traditional kinematic data association likelihood (i.e., track score) using an established formulation for classification-aided data association. The proposed method is tested and demonstrated on synthetic vehicular data representing an urban traffic scene generated by the Simulation of Urban Mobility package. The vehicles in the scene exhibit different driving behaviors. The proposed method distinguishes those behaviors and shows improved data association decisions relative to a conventional, kinematic MHT.

Algebraic Geometry and Finite Frames

Interesting families of finite frames often admit characterizations in terms of algebraic constraints, and thus it is not entirely surprising that powerful results in finite frame theory can be obtained by utilizing tools from algebraic geometry. In this chapter, our goal is to demonstrate the power of these techniques. First, we demonstrate that algebro-geometric ideas can be used to explicitly construct local coordinate systems that reflect intuitive degrees of freedom within spaces of finite unit norm tight frames (and more general spaces), and that optimal frames can be characterized by useful algebraic conditions. In particular, we construct locally well-defined real-analytic coordinate systems on spaces of finite unit norm tight frames, and we demonstrate that many types of optimal Parseval frames are dense and that further optimality can be discovered through embeddings that naturally arise in algebraic geometry.

Connectivity of spaces of finite unit-norm tight frames

We show that the spaces of finite unit norm tight frames are connected, which verifies a conjecture first appearing in Dykema and Strawn (2006). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al., 2012), or Gelfand-Tsetlin patterns, to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense.

Geometric multi-resolution analysis for dictionary learning

We present an efficient algorithm and theory for Geometric Multi-Resolution Analysis (GMRA), a procedure for dictionary learning. Sparse dictionary learning provides the necessary complexity reduction for the critical applications of compression, regression, and classification in high-dimensional data analysis. As such, it is a critical technique in data science and it is important to have techniques that admit both efficient implementation and strong theory for large classes of theoretical models. By construction, GMRA is computationally efficient and in this paper we describe how the GMRA correctly approximates a large class of plausible models (namely, the noisy manifolds).

Geometric multi-resolution analysis and data-driven convolutions

We introduce a procedure for learning discrete convolutional operators for generic datasets which recovers the standard block convolutional operators when applied to sets of natural images. They key observation is that the standard block convolutional operators on images are intuitive because humans naturally understand the grid structure of the self-evident functions over images spaces (pixels). This procedure first constructs a Geometric Multi-Resolution Analysis (GMRA) on the set of variables giving rise to a dataset, and then leverages the details of this data structure to identify subsets of variables upon which convolutional operators are supported, as well as a space of functions that can be shared coherently amongst these supports.

Geometric optimization on spaces of finite frames

A finite (μ; S)-frame variety consists of the real or complex matrices F = [f1...fN] with frame operator FF* = S, and satisfying IIfiII = μi for all i = 1,...,N. Here, S is a fixed Hermitian positive definite matrix and μ = [μ1,..., μN] is a fixed list of lengths. These spaces generalize the well-known spaces of finite unit norm tight frames. We explore the local geometry of these spaces and develop geometric optimization algorithms based on the resulting insights.