Uniqueness of the polar factorisation and projection of a vector-valued mapping

G.R. BURTON and R.J. DOUGLAS, Uniqueness of the polar factorisation and projection of a vector-valued mapping. Ann. I.H. Poincare ? A.N. 20 (2003), 405-418.

Problems of maximal mean resistance on the plane

Plakhov, A.Y.; Gouveia, P.D.F., (2007) 'Problems of maximal mean resistance on the plane', Nonlinearity 20(9) pp.2271-2287 RAE2008

Exact Geosedics and Shortest Paths on Polyhedral Surface

We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.

A Neural Network Model of Speech Acquisition and Motor Equivalent Speech Production

This article describes a neural network model that addresses the acquisition of speaking skills by infants and subsequent motor equivalent production of speech sounds. The model learns two mappings during a babbling phase. A phonetic-to-orosensory mapping specifies a vocal tract target for each speech sound; these targets take the form of convex regions in orosensory coordinates defining the shape of the vocal tract. The babbling process wherein these convex region targets are formed explains how an infant can learn phoneme-specific and language-specific limits on acceptable variability of articulator movements. The model also learns an orosensory-to-articulatory mapping wherein cells coding desired movement directions in orosensory space learn articulator movements that achieve these orosensory movement directions. The resulting mapping provides a natural explanation for the formation of coordinative structures. This mapping also makes efficient use of redundancy in the articulator system, thereby providing the model with motor equivalent capabilities. Simulations verify the model's ability to compensate for constraints or perturbations applied to the articulators automatically and without new learning and to explain contextual variability seen in human speech production.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

Reasoning with imprecise trade-offs in decision making under certainty and uncertainty

In many real world situations, we make decisions in the presence of multiple, often conflicting and non-commensurate objectives. The process of optimizing systematically and simultaneously over a set of objective functions is known as multi-objective optimization. In multi-objective optimization, we have a (possibly exponentially large) set of decisions and each decision has a set of alternatives. Each alternative depends on the state of the world, and is evaluated with respect to a number of criteria. In this thesis, we consider the decision making problems in two scenarios. In the first scenario, the current state of the world, under which the decisions are to be made, is known in advance. In the second scenario, the current state of the world is unknown at the time of making decisions. For decision making under certainty, we consider the framework of multiobjective constraint optimization and focus on extending the algorithms to solve these models to the case where there are additional trade-offs. We focus especially on branch-and-bound algorithms that use a mini-buckets algorithm for generating the upper bound at each node of the search tree (in the context of maximizing values of objectives). Since the size of the guiding upper bound sets can become very large during the search, we introduce efficient methods for reducing these sets, yet still maintaining the upper bound property. We define a formalism for imprecise trade-offs, which allows the decision maker during the elicitation stage, to specify a preference for one multi-objective utility vector over another, and use such preferences to infer other preferences. The induced preference relation then is used to eliminate the dominated utility vectors during the computation. For testing the dominance between multi-objective utility vectors, we present three different approaches. The first is based on a linear programming approach, the second is by use of distance-based algorithm (which uses a measure of the distance between a point and a convex cone); the third approach makes use of a matrix multiplication, which results in much faster dominance checks with respect to the preference relation induced by the trade-offs. Furthermore, we show that our trade-offs approach, which is based on a preference inference technique, can also be given an alternative semantics based on the well known Multi-Attribute Utility Theory. Our comprehensive experimental results on common multi-objective constraint optimization benchmarks demonstrate that the proposed enhancements allow the algorithms to scale up to much larger problems than before. For decision making problems under uncertainty, we describe multi-objective influence diagrams, based on a set of p objectives, where utility values are vectors in Rp, and are typically only partially ordered. These can be solved by a variable elimination algorithm, leading to a set of maximal values of expected utility. If the Pareto ordering is used this set can often be prohibitively large. We consider approximate representations of the Pareto set based on ϵ-coverings, allowing much larger problems to be solved. In addition, we define a method for incorporating user trade-offs, which also greatly improves the efficiency.

Sorting under partial information (without the ellipsoid algorithm)

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical. We develop efficient algorithms for sorting under partial information. Like Kahn and Kim, our approach relies on graph entropy. However, our algorithms differ in essential ways from theirs. Rather than resorting to convex programming for computing the entropy, we approximate the entropy, or make sure it is computed only once in a restricted class of graphs, permitting the use of a simpler algorithm. Specifically, we present: an O(n2) algorithm performing O(log n·log e(P)) comparisons; an O(n2.5) algorithm performing at most (1+ε) log e(P) + Oε(n) comparisons; an O(n2.5) algorithm performing O(log e(P)) comparisons. All our algorithms are simple to implement. © 2010 ACM.

Sequential anomaly detection in the presence of noise and limited feedback

This paper describes a methodology for detecting anomalies from sequentially observed and potentially noisy data. The proposed approach consists of two main elements: 1) filtering, or assigning a belief or likelihood to each successive measurement based upon our ability to predict it from previous noisy observations and 2) hedging, or flagging potential anomalies by comparing the current belief against a time-varying and data-adaptive threshold. The threshold is adjusted based on the available feedback from an end user. Our algorithms, which combine universal prediction with recent work on online convex programming, do not require computing posterior distributions given all current observations and involve simple primal-dual parameter updates. At the heart of the proposed approach lie exponential-family models which can be used in a wide variety of contexts and applications, and which yield methods that achieve sublinear per-round regret against both static and slowly varying product distributions with marginals drawn from the same exponential family. Moreover, the regret against static distributions coincides with the minimax value of the corresponding online strongly convex game. We also prove bounds on the number of mistakes made during the hedging step relative to the best offline choice of the threshold with access to all estimated beliefs and feedback signals. We validate the theory on synthetic data drawn from a time-varying distribution over binary vectors of high dimensionality, as well as on the Enron email dataset. © 1963-2012 IEEE.

Identification of fluorescent beads using a coded aperture snapshot spectral imager.

We apply a coded aperture snapshot spectral imager (CASSI) to fluorescence microscopy. CASSI records a two-dimensional (2D) spectrally filtered projection of a three-dimensional (3D) spectral data cube. We minimize a convex quadratic function with total variation (TV) constraints for data cube estimation from the 2D snapshot. We adapt the TV minimization algorithm for direct fluorescent bead identification from CASSI measurements by combining a priori knowledge of the spectra associated with each bead type. Our proposed method creates a 2D bead identity image. Simulated fluorescence CASSI measurements are used to evaluate the behavior of the algorithm. We also record real CASSI measurements of a ten bead type fluorescence scene and create a 2D bead identity map. A baseline image from filtered-array imaging system verifies CASSI's 2D bead identity map.

Millimeter-wave compressive holography.

We describe an active millimeter-wave holographic imaging system that uses compressive measurements for three-dimensional (3D) tomographic object estimation. Our system records a two-dimensional (2D) digitized Gabor hologram by translating a single pixel incoherent receiver. Two approaches for compressive measurement are undertaken: nonlinear inversion of a 2D Gabor hologram for 3D object estimation and nonlinear inversion of a randomly subsampled Gabor hologram for 3D object estimation. The object estimation algorithm minimizes a convex quadratic problem using total variation (TV) regularization for 3D object estimation. We compare object reconstructions using linear backpropagation and TV minimization, and we present simulated and experimental reconstructions from both compressive measurement strategies. In contrast with backpropagation, which estimates the 3D electromagnetic field, TV minimization estimates the 3D object that produces the field. Despite undersampling, range resolution is consistent with the extent of the 3D object band volume.

A soft robust model for optimization under ambiguity

In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate theme thodology on an asset allocation example consisting of historical market data over a 25-year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk-return trade-off: each case results in a higher out-of-sample expected return for a relatively minor degradation of out-of-sample downside performance. © 2010 INFORMS.

Characterization of Image Quality for 3D Scatter Corrected Breast CT Images.

The goal of this study was to characterize the image quality of our dedicated, quasi-monochromatic spectrum, cone beam breast imaging system under scatter corrected and non-scatter corrected conditions for a variety of breast compositions. CT projections were acquired of a breast phantom containing two concentric sets of acrylic spheres that varied in size (1-8mm) based on their polar position. The breast phantom was filled with 3 different concentrations of methanol and water, simulating a range of breast densities (0.79-1.0g/cc); acrylic yarn was sometimes included to simulate connective tissue of a breast. For each phantom condition, 2D scatter was measured for all projection angles. Scatter-corrected and uncorrected projections were then reconstructed with an iterative ordered subsets convex algorithm. Reconstructed image quality was characterized using SNR and contrast analysis, and followed by a human observer detection task for the spheres in the different concentric rings. Results show that scatter correction effectively reduces the cupping artifact and improves image contrast and SNR. Results from the observer study indicate that there was no statistical difference in the number or sizes of lesions observed in the scatter versus non-scatter corrected images for all densities. Nonetheless, applying scatter correction for differing breast conditions improves overall image quality.

POCS-based reconstruction of multiplexed sensitivity encoded MRI (POCSMUSE): A general algorithm for reducing motion-related artifacts.

PURPOSE: A projection onto convex sets reconstruction of multiplexed sensitivity encoded MRI (POCSMUSE) is developed to reduce motion-related artifacts, including respiration artifacts in abdominal imaging and aliasing artifacts in interleaved diffusion-weighted imaging. THEORY: Images with reduced artifacts are reconstructed with an iterative projection onto convex sets (POCS) procedure that uses the coil sensitivity profile as a constraint. This method can be applied to data obtained with different pulse sequences and k-space trajectories. In addition, various constraints can be incorporated to stabilize the reconstruction of ill-conditioned matrices. METHODS: The POCSMUSE technique was applied to abdominal fast spin-echo imaging data, and its effectiveness in respiratory-triggered scans was evaluated. The POCSMUSE method was also applied to reduce aliasing artifacts due to shot-to-shot phase variations in interleaved diffusion-weighted imaging data corresponding to different k-space trajectories and matrix condition numbers. RESULTS: Experimental results show that the POCSMUSE technique can effectively reduce motion-related artifacts in data obtained with different pulse sequences, k-space trajectories and contrasts. CONCLUSION: POCSMUSE is a general post-processing algorithm for reduction of motion-related artifacts. It is compatible with different pulse sequences, and can also be used to further reduce residual artifacts in data produced by existing motion artifact reduction methods.

Design of Scheduling Algorithms Using Game Theoretic Ideas

Scheduling a set of jobs over a collection of machines to optimize a certain quality-of-service measure is one of the most important research topics in both computer science theory and practice. In this thesis, we design algorithms that optimize {\em flow-time} (or delay) of jobs for scheduling problems that arise in a wide range of applications. We consider the classical model of unrelated machine scheduling and resolve several long standing open problems; we introduce new models that capture the novel algorithmic challenges in scheduling jobs in data centers or large clusters; we study the effect of selfish behavior in distributed and decentralized environments; we design algorithms that strive to balance the energy consumption and performance.

The technically interesting aspect of our work is the surprising connections we establish between approximation and online algorithms, economics, game theory, and queuing theory. It is the interplay of ideas from these different areas that lies at the heart of most of the algorithms presented in this thesis.

The main contributions of the thesis can be placed in one of the following categories.

1. Classical Unrelated Machine Scheduling: We give the first polygorithmic approximation algorithms for minimizing the average flow-time and minimizing the maximum flow-time in the offline setting. In the online and non-clairvoyant setting, we design the first non-clairvoyant algorithm for minimizing the weighted flow-time in the resource augmentation model. Our work introduces iterated rounding technique for the offline flow-time optimization, and gives the first framework to analyze non-clairvoyant algorithms for unrelated machines.

2. Polytope Scheduling Problem: To capture the multidimensional nature of the scheduling problems that arise in practice, we introduce Polytope Scheduling Problem (\psp). The \psp problem generalizes almost all classical scheduling models, and also captures hitherto unstudied scheduling problems such as routing multi-commodity flows, routing multicast (video-on-demand) trees, and multi-dimensional resource allocation. We design several competitive algorithms for the \psp problem and its variants for the objectives of minimizing the flow-time and completion time. Our work establishes many interesting connections between scheduling and market equilibrium concepts, fairness and non-clairvoyant scheduling, and queuing theoretic notion of stability and resource augmentation analysis.

3. Energy Efficient Scheduling: We give the first non-clairvoyant algorithm for minimizing the total flow-time + energy in the online and resource augmentation model for the most general setting of unrelated machines.

4. Selfish Scheduling: We study the effect of selfish behavior in scheduling and routing problems. We define a fairness index for scheduling policies called {\em bounded stretch}, and show that for the objective of minimizing the average (weighted) completion time, policies with small stretch lead to equilibrium outcomes with small price of anarchy. Our work gives the first linear/ convex programming duality based framework to bound the price of anarchy for general equilibrium concepts such as coarse correlated equilibrium.

Statistical Techniques In Stereology

Stereology typically concerns estimation of properties of a geometric structure from plane section information. This paperprovides a brief review of some statistical aspects of this rapidly developing field, with some reference to applications in the earth sciences. After an introductory discussion of the scope of stereology, section 2 briefly mentions results applicable when no assumptions can be made about the stochastic nature of the sampled matrix, statistical considerations then arising solelyfrom the ‘randomness’ of the plane section. The next two sections postulate embedded particles of specific shapes, the particular case of spheres being discussed in some detail. References are made to results for ‘thin slices’ and other prob-ing mechanisms. Randomly located convex particles, of otherwise arbitrary shape, are discussed in section 5 and the review concludes with a specific application of stereological ideas to some data on neolithic mining.