# Research

**Research Interests:**computational neuroscience, compressed sensing, signal processing, bayesian statistics, mathematical biology

**Advisors:**Dr. Rachel Ward, Department of Mathematics, UT Austin and Dr. Jonathan Pillow, Departments of Psychology and Neurobiology, UT Austin

More specifically and most recently, I have been working on problems in one-bit compressed sensing, where one seeks to reconstruct a sparse signal given linear measurements that are subsequently quantized to one bit. My other current projects involve applications of compressed sensing to neural spike sorting, and estimating information theoretic quantities like entropy and mutual information rate from neural spike trains using Bayesian techniques. If you are interested, please get in touch.

**Publications**

- Knudson, K., Saab, R., and Ward, R. (2014). "One-bit compressive sensing with norm estimation." arXiv preprint, arXiv:1404.6853.

**Abstract:**Compressive sensing typically involves the recovery of a sparse signal \( \mathbf{x} \) from linear measurements \( \langle \mathbf{a}_i, \mathbf{x} \rangle \), but recent research has demonstrated that recovery is also possible when the observed measurements are quantized to \( \mbox{sign}(\langle \mathbf{a}_i, \mathbf{x} \rangle) \in \{\pm1\} \). Since such measurements give no information on the norm of \( \mathbf{x} \), recovery methods geared towards such one-bit compressive sensing measurements typically assume that \( \| \mathbf{x} \|_2=1 \). We show that if one allows more generally for quantized*affine*measurements of the form \( \mbox{sign}(\langle \mathbf{a}_i, \mathbf{x} \rangle + b_i) \), and if such measurements are random, an appropriate choice of the affine shifts \( b_i \) allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Alternatively, if one is interested in norm estimation alone, we show that the fraction of measurements quantized to +1 (versus -1) can be used to estimate \( \| \mathbf{x} \|_2 \) through a single evaluation of the inverse Gaussian error function, providing a computationally efficient method for norm estimation from 1-bit compressive measurements. Finally, all of our recovery guarantees are*universal*over sparse vectors, in the sense that with high probability, one set of measurements will successfully estimate all sparse vectors simultaneously.

**Code (MATLAB):** - Knudson, K. and Pillow, J.W. (2013). "Spike train entropy-rate estimation using hierarchical Dirichlet
process priors." Advances in Neural Information Processing Systems 26 pdf

**Abstract:**Entropy rate quantifies the amount of disorder in a stochastic process. For spiking neurons, the entropy rate places an upper bound on the rate at which the spike train can convey stimulus information, and a large literature has focused on the problem of estimating entropy rate from spike train data. Here we present Bayes least squares and empirical Bayesian entropy rate estimators for binary spike trains using hierarchical Dirichlet process (HDP) priors. Our estimator leverages the fact that the entropy rate of an ergodic Markov Chain with known transition probabilities can be calculated analytically, and many stochastic processes that are non-Markovian can still be well approximated by Markov processes of sufficient depth. Choosing an appropriate depth of Markov model presents challenges due to possibly long time dependencies and short data sequences: a deeper model can better account for long time dependencies, but is more difficult to infer from limited data. Our approach mitigates this difficulty by using a hierarchical prior to share statistical power across Markov chains of different depths. We present both a fully Bayesian and empirical Bayes entropy rate estimator based on this model, and demonstrate their performance on simulated and real neural spike train data. - Adams, C. and Knudson, K. (2013). "Unknotting tunnels, bracelets and the elder sibling property for hyperbolic 3-manifolds." Journal of the Australian Mathematical Society: 1-19.
**Abstract:**An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic α in a hyperbolic 3-manifold M, we find sufficient conditions for it to be an unknotting tunnel. In particular, if α corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic α that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at ∞ is connected to a larger horoball by a lift of α. Such an α with length less than ln(2) is then shown to be an unknotting tunnel.