Shifting : one-inclusion mistake bounds and sample compression

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d = VC(F) bound on the graph density of a subgraph of the hypercube—oneinclusion graph. The first main result of this paper is a density bound of n [n−1 <=d-1]/[n <=d] < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d contractible simplicial complexes, extending the well-known characterization that d = 1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth—the second part to a conjectured proof of correctness for Peeling—that every class has one-inclusion minimum degree at most its VCdimension. Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(logn) and is shown to be optimal up to an O(logk) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout.

Learning to act in uncertain environments : technical perspective

The problem of decision making in an uncertain environment arises in many diverse contexts: deciding whether to keep a hard drive spinning in a net-book; choosing which advertisement to post to a Web site visitor; choosing how many newspapers to order so as to maximize profits; or choosing a route to recommend to a driver given limited and possibly out-of-date information about traffic conditions. All are sequential decision problems, since earlier decisions affect subsequent performance; all require adaptive approaches, since they involve significant uncertainty. The key issue in effectively solving problems like these is known as the exploration/exploitation trade-off: If I am at a cross-roads, when should I go in the most advantageous direction among those that I have already explored, and when should I strike out in a new direction, in the hopes I will discover something better?

Shifting : one-inclusion mistake bounds and sample compression

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d=VC(F) bound on the graph density of a subgraph of the hypercube—one-inclusion graph. The first main result of this report is a density bound of n∙choose(n-1,≤d-1)/choose(n,≤d) < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d-contractible simplicial complexes, extending the well-known characterization that d=1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth—the second part to a conjectured proof of correctness for Peeling—that every class has one-inclusion minimum degree at most its VC-dimension. Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(log n) and is shown to be optimal up to a O(log k) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout

Multi-objective optimisation of bijective s-boxes

In this paper we investigate the heuristic construction of bijective s-boxes that satisfy a wide range of cryptographic criteria including algebraic complexity, high nonlinearity, low autocorrelation and have none of the known weaknesses including linear structures, fixed points or linear redundancy. We demonstrate that the power mappings can be evolved (by iterated mutation operators alone) to generate bijective s-boxes with the best known tradeoffs among the considered criteria. The s-boxes found are suitable for use directly in modern encryption algorithms.

Some novel techniques of parameter estimation for the dynamical models in biological systems

Inverse problems based on using experimental data to estimate unknown parameters of a system often arise in biological and chaotic systems. In this paper, we consider parameter estimation in systems biology involving linear and non-linear complex dynamical models, including the Michaelis–Menten enzyme kinetic system, a dynamical model of competence induction in Bacillus subtilis bacteria and a model of feedback bypass in B. subtilis bacteria. We propose some novel techniques for inverse problems. Firstly, we establish an approximation of a non-linear differential algebraic equation that corresponds to the given biological systems. Secondly, we use the Picard contraction mapping, collage methods and numerical integration techniques to convert the parameter estimation into a minimization problem of the parameters. We propose two optimization techniques: a grid approximation method and a modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO) for non-linear parameter estimation. The two techniques are used for parameter estimation in a model of competence induction in B. subtilis bacteria with noisy data. The MH-NMSS-PSO scheme is applied to a dynamical model of competence induction in B. subtilis bacteria based on experimental data and the model for feedback bypass. Numerical results demonstrate the effectiveness of our approach.

Adaptive cache aware multiprocessor scheduling framework

Computer resource allocation represents a significant challenge particularly for multiprocessor systems, which consist of shared computing resources to be allocated among co-runner processes and threads. While an efficient resource allocation would result in a highly efficient and stable overall multiprocessor system and individual thread performance, ineffective poor resource allocation causes significant performance bottlenecks even for the system with high computing resources. This thesis proposes a cache aware adaptive closed loop scheduling framework as an efficient resource allocation strategy for the highly dynamic resource management problem, which requires instant estimation of highly uncertain and unpredictable resource patterns. Many different approaches to this highly dynamic resource allocation problem have been developed but neither the dynamic nature nor the time-varying and uncertain characteristics of the resource allocation problem is well considered. These approaches facilitate either static and dynamic optimization methods or advanced scheduling algorithms such as the Proportional Fair (PFair) scheduling algorithm. Some of these approaches, which consider the dynamic nature of multiprocessor systems, apply only a basic closed loop system; hence, they fail to take the time-varying and uncertainty of the system into account. Therefore, further research into the multiprocessor resource allocation is required. Our closed loop cache aware adaptive scheduling framework takes the resource availability and the resource usage patterns into account by measuring time-varying factors such as cache miss counts, stalls and instruction counts. More specifically, the cache usage pattern of the thread is identified using QR recursive least square algorithm (RLS) and cache miss count time series statistics. For the identified cache resource dynamics, our closed loop cache aware adaptive scheduling framework enforces instruction fairness for the threads. Fairness in the context of our research project is defined as a resource allocation equity, which reduces corunner thread dependence in a shared resource environment. In this way, instruction count degradation due to shared cache resource conflicts is overcome. In this respect, our closed loop cache aware adaptive scheduling framework contributes to the research field in two major and three minor aspects. The two major contributions lead to the cache aware scheduling system. The first major contribution is the development of the execution fairness algorithm, which degrades the co-runner cache impact on the thread performance. The second contribution is the development of relevant mathematical models, such as thread execution pattern and cache access pattern models, which in fact formulate the execution fairness algorithm in terms of mathematical quantities. Following the development of the cache aware scheduling system, our adaptive self-tuning control framework is constructed to add an adaptive closed loop aspect to the cache aware scheduling system. This control framework in fact consists of two main components: the parameter estimator, and the controller design module. The first minor contribution is the development of the parameter estimators; the QR Recursive Least Square(RLS) algorithm is applied into our closed loop cache aware adaptive scheduling framework to estimate highly uncertain and time-varying cache resource patterns of threads. The second minor contribution is the designing of a controller design module; the algebraic controller design algorithm, Pole Placement, is utilized to design the relevant controller, which is able to provide desired timevarying control action. The adaptive self-tuning control framework and cache aware scheduling system in fact constitute our final framework, closed loop cache aware adaptive scheduling framework. The third minor contribution is to validate this cache aware adaptive closed loop scheduling framework efficiency in overwhelming the co-runner cache dependency. The timeseries statistical counters are developed for M-Sim Multi-Core Simulator; and the theoretical findings and mathematical formulations are applied as MATLAB m-file software codes. In this way, the overall framework is tested and experiment outcomes are analyzed. According to our experiment outcomes, it is concluded that our closed loop cache aware adaptive scheduling framework successfully drives co-runner cache dependent thread instruction count to co-runner independent instruction count with an error margin up to 25% in case cache is highly utilized. In addition, thread cache access pattern is also estimated with 75% accuracy.

Reliability of partial ambiguity fixing with multiple GNSS constellations

Reliable ambiguity resolution (AR) is essential to Real-Time Kinematic (RTK) positioning and its applications, since incorrect ambiguity fixing can lead to largely biased positioning solutions. A partial ambiguity fixing technique is developed to improve the reliability of AR, involving partial ambiguity decorrelation (PAD) and partial ambiguity resolution (PAR). Decorrelation transformation could substantially amplify the biases in the phase measurements. The purpose of PAD is to find the optimum trade-off between decorrelation and worst-case bias amplification. The concept of PAR refers to the case where only a subset of the ambiguities can be fixed correctly to their integers in the integer least-squares (ILS) estimation system at high success rates. As a result, RTK solutions can be derived from these integer-fixed phase measurements. This is meaningful provided that the number of reliably resolved phase measurements is sufficiently large for least-square estimation of RTK solutions as well. Considering the GPS constellation alone, partially fixed measurements are often insufficient for positioning. The AR reliability is usually characterised by the AR success rate. In this contribution an AR validation decision matrix is firstly introduced to understand the impact of success rate. Moreover the AR risk probability is included into a more complete evaluation of the AR reliability. We use 16 ambiguity variance-covariance matrices with different levels of success rate to analyse the relation between success rate and AR risk probability. Next, the paper examines during the PAD process, how a bias in one measurement is propagated and amplified onto many others, leading to more than one wrong integer and to affect the success probability. Furthermore, the paper proposes a partial ambiguity fixing procedure with a predefined success rate criterion and ratio-test in the ambiguity validation process. In this paper, the Galileo constellation data is tested with simulated observations. Numerical results from our experiment clearly demonstrate that only when the computed success rate is very high, the AR validation can provide decisions about the correctness of AR which are close to real world, with both low AR risk and false alarm probabilities. The results also indicate that the PAR procedure can automatically chose adequate number of ambiguities to fix at given high-success rate from the multiple constellations instead of fixing all the ambiguities. This is a benefit that multiple GNSS constellations can offer.

Evaluation of the "reconceptualising early mathematics learning" project

The Pattern and Structure Mathematics Awareness Project (PASMAP) has investigated the development of patterning and early algebraic reasoning among 4 to 8 year olds over a series of related studies. We assert that an awareness of mathematical pattern and structure enables mathematical thinking and simple forms of generalisation from an early age. The project aims to promote a strong foundation for mathematical development by focusing on critical, underlying features of mathematics learning. This paper provides an overview of key aspects of the assessment and intervention, and analyses of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in the evaluation throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ assessment and a Pattern and Structure Assessment (PASA) interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students; those categorised as low ability developed improved structural responses over a relatively short period of time.

Exponential asymptotics of free surface flow due to a line source

The steady problem of free surface flow due to a submerged line source is revisited for the case in which the fluid depth is finite and there is a stagnation point on the free surface directly above the source. Both the strength of the source and the fluid speed in the far field are measured by a dimensionless parameter, the Froude number. By applying techniques in exponential asymptotics, it is shown that there is a train of periodic waves on the surface of the fluid with an amplitude which is exponentially small in the limit that the Froude number vanishes. This study clarifies that periodic waves do form for flows due to a source, contrary to a suggestion by Chapman & Vanden-Broeck (2006, J. Fluid Mech., 567, 299--326). The exponentially small nature of the waves means they appear beyond all orders of the original power series expansion; this result explains why attempts at describing these flows using a finite number of terms in an algebraic power series incorrectly predict a flat free surface in the far field.

Fixed failure rate ambiguity validation methods for GPS and compass

Ambiguity resolution plays a crucial role in real time kinematic GNSS positioning which gives centimetre precision positioning results if all the ambiguities in each epoch are correctly fixed to integers. However, the incorrectly fixed ambiguities can result in large positioning offset up to several meters without notice. Hence, ambiguity validation is essential to control the ambiguity resolution quality. Currently, the most popular ambiguity validation is ratio test. The criterion of ratio test is often empirically determined. Empirically determined criterion can be dangerous, because a fixed criterion cannot fit all scenarios and does not directly control the ambiguity resolution risk. In practice, depending on the underlying model strength, the ratio test criterion can be too conservative for some model and becomes too risky for others. A more rational test method is to determine the criterion according to the underlying model and user requirement. Miss-detected incorrect integers will lead to a hazardous result, which should be strictly controlled. In ambiguity resolution miss-detected rate is often known as failure rate. In this paper, a fixed failure rate ratio test method is presented and applied in analysis of GPS and Compass positioning scenarios. A fixed failure rate approach is derived from the integer aperture estimation theory, which is theoretically rigorous. The criteria table for ratio test is computed based on extensive data simulations in the approach. The real-time users can determine the ratio test criterion by looking up the criteria table. This method has been applied in medium distance GPS ambiguity resolution but multi-constellation and high dimensional scenarios haven't been discussed so far. In this paper, a general ambiguity validation model is derived based on hypothesis test theory, and fixed failure rate approach is introduced, especially the relationship between ratio test threshold and failure rate is examined. In the last, Factors that influence fixed failure rate approach ratio test threshold is discussed according to extensive data simulation. The result shows that fixed failure rate approach is a more reasonable ambiguity validation method with proper stochastic model.

Analysis of stream cipher based authenticated encryption schemes

Authenticated Encryption (AE) is the cryptographic process of providing simultaneous confidentiality and integrity protection to messages. This approach is more efficient than applying a two-step process of providing confidentiality for a message by encrypting the message, and in a separate pass providing integrity protection by generating a Message Authentication Code (MAC). AE using symmetric ciphers can be provided by either stream ciphers with built in authentication mechanisms or block ciphers using appropriate modes of operation. However, stream ciphers have the potential for higher performance and smaller footprint in hardware and/or software than block ciphers. This property makes stream ciphers suitable for resource constrained environments, where storage and computational power are limited. There have been several recent stream cipher proposals that claim to provide AE. These ciphers can be analysed using existing techniques that consider confidentiality or integrity separately; however currently there is no existing framework for the analysis of AE stream ciphers that analyses these two properties simultaneously. This thesis introduces a novel framework for the analysis of AE using stream cipher algorithms. This thesis analyzes the mechanisms for providing confidentiality and for providing integrity in AE algorithms using stream ciphers. There is a greater emphasis on the analysis of the integrity mechanisms, as there is little in the public literature on this, in the context of authenticated encryption. The thesis has four main contributions as follows. The first contribution is the design of a framework that can be used to classify AE stream ciphers based on three characteristics. The first classification applies Bellare and Namprempre's work on the the order in which encryption and authentication processes take place. The second classification is based on the method used for accumulating the input message (either directly or indirectly) into the into the internal states of the cipher to generate a MAC. The third classification is based on whether the sequence that is used to provide encryption and authentication is generated using a single key and initial vector, or two keys and two initial vectors. The second contribution is the application of an existing algebraic method to analyse the confidentiality algorithms of two AE stream ciphers; namely SSS and ZUC. The algebraic method is based on considering the nonlinear filter (NLF) of these ciphers as a combiner with memory. This method enables us to construct equations for the NLF that relate the (inputs, outputs and memory of the combiner) to the output keystream. We show that both of these ciphers are secure from this type of algebraic attack. We conclude that using a keydependent SBox in the NLF twice, and using two different SBoxes in the NLF of ZUC, prevents this type of algebraic attack. The third contribution is a new general matrix based model for MAC generation where the input message is injected directly into the internal state. This model describes the accumulation process when the input message is injected directly into the internal state of a nonlinear filter generator. We show that three recently proposed AE stream ciphers can be considered as instances of this model; namely SSS, NLSv2 and SOBER-128. Our model is more general than a previous investigations into direct injection. Possible forgery attacks against this model are investigated. It is shown that using a nonlinear filter in the accumulation process of the input message when either the input message or the initial states of the register is unknown prevents forgery attacks based on collisions. The last contribution is a new general matrix based model for MAC generation where the input message is injected indirectly into the internal state. This model uses the input message as a controller to accumulate a keystream sequence into an accumulation register. We show that three current AE stream ciphers can be considered as instances of this model; namely ZUC, Grain-128a and Sfinks. We establish the conditions under which the model is susceptible to forgery and side-channel attacks.

Fast formulas for computing cryptographic pairings

The most powerful known primitive in public-key cryptography is undoubtedly elliptic curve pairings. Upon their introduction just over ten years ago the computation of pairings was far too slow for them to be considered a practical option. This resulted in a vast amount of research from many mathematicians and computer scientists around the globe aiming to improve this computation speed. From the use of modern results in algebraic and arithmetic geometry to the application of foundational number theory that dates back to the days of Gauss and Euler, cryptographic pairings have since experienced a great deal of improvement. As a result, what was an extremely expensive computation that took several minutes is now a high-speed operation that takes less than a millisecond. This thesis presents a range of optimisations to the state-of-the-art in cryptographic pairing computation. Both through extending prior techniques, and introducing several novel ideas of our own, our work has contributed to recordbreaking pairing implementations.

Modelling sea water intrusion in coastal aquifers using heterogeneous computing

The objective of this PhD research program is to investigate numerical methods for simulating variably-saturated flow and sea water intrusion in coastal aquifers in a high-performance computing environment. The work is divided into three overlapping tasks: to develop an accurate and stable finite volume discretisation and numerical solution strategy for the variably-saturated flow and salt transport equations; to implement the chosen approach in a high performance computing environment that may have multiple GPUs or CPU cores; and to verify and test the implementation. The geological description of aquifers is often complex, with porous materials possessing highly variable properties, that are best described using unstructured meshes. The finite volume method is a popular method for the solution of the conservation laws that describe sea water intrusion, and is well-suited to unstructured meshes. In this work we apply a control volume-finite element (CV-FE) method to an extension of a recently proposed formulation (Kees and Miller, 2002) for variably saturated groundwater flow. The CV-FE method evaluates fluxes at points where material properties and gradients in pressure and concentration are consistently defined, making it both suitable for heterogeneous media and mass conservative. Using the method of lines, the CV-FE discretisation gives a set of differential algebraic equations (DAEs) amenable to solution using higher-order implicit solvers. Heterogeneous computer systems that use a combination of computational hardware such as CPUs and GPUs, are attractive for scientific computing due to the potential advantages offered by GPUs for accelerating data-parallel operations. We present a C++ library that implements data-parallel methods on both CPU and GPUs. The finite volume discretisation is expressed in terms of these data-parallel operations, which gives an efficient implementation of the nonlinear residual function. This makes the implicit solution of the DAE system possible on the GPU, because the inexact Newton-Krylov method used by the implicit time stepping scheme can approximate the action of a matrix on a vector using residual evaluations. We also propose preconditioning strategies that are amenable to GPU implementation, so that all computationally-intensive aspects of the implicit time stepping scheme are implemented on the GPU. Results are presented that demonstrate the efficiency and accuracy of the proposed numeric methods and formulation. The formulation offers excellent conservation of mass, and higher-order temporal integration increases both numeric efficiency and accuracy of the solutions. Flux limiting produces accurate, oscillation-free solutions on coarse meshes, where much finer meshes are required to obtain solutions with equivalent accuracy using upstream weighting. The computational efficiency of the software is investigated using CPUs and GPUs on a high-performance workstation. The GPU version offers considerable speedup over the CPU version, with one GPU giving speedup factor of 3 over the eight-core CPU implementation.

Generalized-positioning for mixed-frequency of mixed-GNSS and its preliminary applications

Modernized GPS and GLONASS, together with new GNSS systems, BeiDou and Galileo, offer code and phase ranging signals in three or more carriers. Traditionally, dual-frequency code and/or phase GPS measurements are linearly combined to eliminate effects of ionosphere delays in various positioning and analysis. This typical treatment method has imitations in processing signals at three or more frequencies from more than one system and can be hardly adapted itself to cope with the booming of various receivers with a broad variety of singles. In this contribution, a generalized-positioning model that the navigation system independent and the carrier number unrelated is promoted, which is suitable for both single- and multi-sites data processing. For the synchronization of different signals, uncalibrated signal delays (USD) are more generally defined to compensate the signal specific offsets in code and phase signals respectively. In addition, the ionospheric delays are included in the parameterization with an elaborate consideration. Based on the analysis of the algebraic structures, this generalized-positioning model is further refined with a set of proper constrains to regularize the datum deficiency of the observation equation system. With this new model, uncalibrated signal delays (USD) and ionospheric delays are derived for both GPS and BeiDou with a large dada set. Numerical results demonstrate that, with a limited number of stations, the uncalibrated code delays (UCD) are determinate to a precision of about 0.1 ns for GPS and 0.4 ns for BeiDou signals, while the uncalibrated phase delays (UPD) for L1 and L2 are generated with 37 stations evenly distributed in China for GPS with a consistency of about 0.3 cycle. Extra experiments concerning the performance of this novel model in point positioning with mixed-frequencies of mixed-constellations is analyzed, in which the USD parameters are fixed with our generated values. The results are evaluated in terms of both positioning accuracy and convergence time.

A family of Alltop functions that are EA-inequivalent to the cubic function

Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (\emph{IEEE Trans. Inf. Theory} 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases (MUBs), a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called \emph{Alltop functions}. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.