Geometry and Arithmetic around Hypergeometric Functions

[no abstract available

Cryptographic approaches to security and optimization in machine learning

Modern machine learning techniques have achieved surprisingly good standard test accuracy, yet classical machine learning theory has been unable to explain the underlying reason behind this success. The phenomenon of adversarial examples further complicates our understanding of what it means to have good generalization ability. Classifiers that generalize well to the test set are easily fooled by imperceptible image modifications, which can often be computed without knowledge of the classifier itself. The adversarial error of a classifier measures the error under which each test data point can be modified by an algorithm before it is given as input to the classifier. Followup work has showed that a tradeoff exists between optimizing for standard generalization error versus for adversarial error. This calls into question whether standard generalization error is the correct metric to measure. We try to understand the generalization capability of modern machine learning techniques through the lens of adversarial examples. To reconcile the apparent tradeoff between the two competing notions of error, we create new security definitions and classifier constructions which allow us to prove an upper bound on the adversarial error that decreases as standard test error decreases. We introduce a cryptographic proof technique by defining a security assumption in a simpler attack setting and proving a security reduction from a restricted black-box attack problem to this security assumption. We then investigate the double descent curve in the interpolation regime, where test error can continue to decrease even after training error has reached zero, to give a natural explanation for the observed tradeoff between adversarial error and standard generalization error. The second part of our work investigates further this notion of a black-box model by looking at the separation between being able to evaluate a function and being able to actually understand it. This is formalized through the notion of function obfuscation in cryptography. Given some concrete implementation of a function, the implementation is considered obfuscated if a user cannot produce the function output on a test input without querying the implementation itself. This means that a user cannot actually learn or understand the function even though all of the implementation details are presented in the clear. As expected this is a very strong requirement that does not exist for all functions one might be interested in. In our work we make progress on providing obfuscation schemes for simple, explicit function classes. The last part of our work investigates non-statistical biases and algorithms for nonconvex optimization problems. We show that the continuous-time limit of stochastic gradient descent does not converge directly to the local optimum, but rather has a bias term which grows with the step size. We also construct novel, non-statistical algorithms for two parametric learning problems by employing lattice basis reduction techniques from cryptography

Near-capacity MIMOs using iterative detection

In this thesis, Multiple-Input Multiple-Output (MIMO) techniques designed for transmission over narrowband Rayleigh fading channels are investigated. Specifically, in order to providea diversity gain while eliminating the complexity of MIMO channel estimation, a Differential Space-Time Spreading (DSTS) scheme is designed that employs non-coherent detection. Additionally, in order to maximise the coding advantage of DSTS, it is combined with Sphere Packing (SP) modulation. The related capacity analysis shows that the DSTS-SP scheme exhibits a higher capacity than its counterpart dispensing with SP. Furthermore, in order to attain additional performance gains, the DSTS system invokes iterative detection, where the outer code is constituted by a Recursive Systematic Convolutional (RSC) code, while the inner code is a SP demapper in one of the prototype systems investigated, while the other scheme employs a Unity Rate Code (URC) as its inner code in order to eliminate the error floor exhibited by the system dispensing with URC. EXIT charts are used to analyse the convergence behaviour of the iteratively detected schemes and a novel technique is proposed for computing the maximum achievable rate of the system based on EXIT charts. Explicitly, the four-antenna-aided DSTSSP system employing no URC precoding attains a coding gain of 12 dB at a BER of 10-5 and performs within 1.82 dB from the maximum achievable rate limit. By contrast, the URC aidedprecoded system operates within 0.92 dB from the same limit.On the other hand, in order to maximise the DSTS system’s throughput, an adaptive DSTSSP scheme is proposed that exploits the advantages of differential encoding, iterative decoding as well as SP modulation. The achievable integrity and bit rate enhancements of the system are determined by the following factors: the specific MIMO configuration used for transmitting data from the four antennas, the spreading factor used and the RSC encoder’s code rate.Additionally, multi-functional MIMO techniques are designed to provide diversity gains, multiplexing gains and beamforming gains by combining the benefits of space-time codes, VBLASTand beamforming. First, a system employing Nt=4 transmit Antenna Arrays (AA) with LAA number of elements per AA and Nr=4 receive antennas is proposed, which is referred to as a Layered Steered Space-Time Code (LSSTC). Three iteratively detected near-capacity LSSTC-SP receiver structures are proposed, which differ in the number of inner iterations employed between the inner decoder and the SP demapper as well as in the choice of the outer code, which is either an RSC code or an Irregular Convolutional Code (IrCC). The three systems are capable of operating within 0.9, 0.4 and 0.6 dB from the maximum achievable rate limit of the system. A comparison between the three iteratively-detected schemes reveals that a carefully designed two-stage iterative detection scheme is capable of operating sufficiently close to capacity at a lower complexity, when compared to a three-stage system employing a RSC or a two-stage system using an IrCC as an outer code. On the other hand, in order to allow the LSSTC scheme to employ less receive antennas than transmit antennas, while still accommodating multiple users, a Layered Steered Space-Time Spreading (LSSTS) scheme is proposed that combines the benefits of space-time spreading, V-BLAST, beamforming and generalised MC DS-CDMA. Furthermore, iteratively detected LSSTS schemes are presented and an LLR post-processing technique is proposed in order to improve the attainable performance of the iteratively detected LSSTS system.Finally, a distributed turbo coding scheme is proposed that combines the benefits of turbo coding and cooperative communication, where iterative detection is employed by exchanging extrinsic information between the decoders of different single-antenna-aided users. Specifically, the effect of the errors induced in the first phase of cooperation, where the two users exchange their data, on the performance of the uplink in studied, while considering different fading channel characteristics

Heights on elliptic curves over number fields, period lattices, and complex elliptic logarithms

This thesis presents some major improvements in the following computations: a lower bound for the canonical height, period lattices, and elliptic logarithms. On computing a lower bound for the canonical height, we have successfully generalised the existing algorithm of Cremona and Siksek [CS06] to elliptic curves over totally real number fields, and then to elliptic curves over number fields in general. Both results, which are also published in [Tho08] and [Tho10] respectively, will be fully explained in Chapter 2 and 3. In Chapter 4, we give a complete method on computing period lattices of elliptic curves over C, whereas this was only possible for elliptic curves over R in the past. Our method is based on the concept of arithmetic-geometric mean (AGM). In addition, we extend our method further to find elliptic logarithms of complex points. This work is done in collaboration with Professor John E. Cremona; another version of this chapter has been submitted for publication [CT]. In Chapter 5, we finally illustrate the applications of our main results towards certain computations which did not work well in the past due to lack of some information on elliptic curves. This includes determining a Mordell{Weil basis, finding integral points on elliptic curves over number fields [SS97], and finding elliptic curves with everywhere good reduction [CL07]. A number of computer programs have been implemented for the purpose of illustration and verification. Their source code (written in MAGMA) can be found in Appendix A.EThOS - Electronic Theses Online ServiceInstitute for the Promotion of Teaching Science and Technology (Thailand)GBUnited Kingdo