306 research outputs found
Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations
Fast linear transforms are ubiquitous in machine learning, including the
discrete Fourier transform, discrete cosine transform, and other structured
transformations such as convolutions. All of these transforms can be
represented by dense matrix-vector multiplication, yet each has a specialized
and highly efficient (subquadratic) algorithm. We ask to what extent
hand-crafting these algorithms and implementations is necessary, what
structural priors they encode, and how much knowledge is required to
automatically learn a fast algorithm for a provided structured transform.
Motivated by a characterization of fast matrix-vector multiplication as
products of sparse matrices, we introduce a parameterization of
divide-and-conquer methods that is capable of representing a large class of
transforms. This generic formulation can automatically learn an efficient
algorithm for many important transforms; for example, it recovers the Cooley-Tukey FFT algorithm to machine precision, for dimensions up to
. Furthermore, our method can be incorporated as a lightweight
replacement of generic matrices in machine learning pipelines to learn
efficient and compressible transformations. On a standard task of compressing a
single hidden-layer network, our method exceeds the classification accuracy of
unconstrained matrices on CIFAR-10 by 3.9 points---the first time a structured
approach has done so---with 4X faster inference speed and 40X fewer parameters
Solution of partial differential equations on vector and parallel computers
The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed
Digital Filters and Signal Processing
Digital filters, together with signal processing, are being employed in the new technologies and information systems, and are implemented in different areas and applications. Digital filters and signal processing are used with no costs and they can be adapted to different cases with great flexibility and reliability. This book presents advanced developments in digital filters and signal process methods covering different cases studies. They present the main essence of the subject, with the principal approaches to the most recent mathematical models that are being employed worldwide
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Efficient FPGA implementation and power modelling of image and signal processing IP cores
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Field Programmable Gate Arrays (FPGAs) are the technology of choice in a number ofimage
and signal processing application areas such as consumer electronics, instrumentation,
medical data processing and avionics due to their reasonable energy consumption, high performance, security, low design-turnaround time and reconfigurability. Low power FPGA
devices are also emerging as competitive solutions for mobile and thermally constrained platforms. Most computationally intensive image and signal processing algorithms also consume a lot of power leading to a number of issues including reduced mobility, reliability concerns and increased design cost among others. Power dissipation has become one of the most important challenges, particularly for FPGAs. Addressing this problem requires optimisation and awareness at all levels in the design flow. The key achievements of the
work presented in this thesis are summarised here. Behavioural level optimisation strategies have been used for implementing matrix product and inner product through the use of mathematical techniques such as Distributed Arithmetic (DA) and its variations including offset binary coding, sparse factorisation and novel vector level transformations. Applications to test the impact of these algorithmic and arithmetic transformations include the fast Hadamard/Walsh transforms and Gaussian mixture models. Complete design space exploration has been performed on these cores, and where appropriate, they have been shown to clearly outperform comparable existing implementations. At the architectural level, strategies such as parallelism, pipelining and systolisation have been successfully applied for the design and optimisation of a number of
cores including colour space conversion, finite Radon transform, finite ridgelet transform and circular convolution. A pioneering study into the influence of supply voltage scaling for FPGA based designs, used in conjunction with performance enhancing strategies such as parallelism and pipelining has been performed. Initial results are very promising and indicated significant potential for future research in this area.
A key contribution of this work includes the development of a novel high level power macromodelling technique for design space exploration and characterisation of custom IP cores for FPGAs, called Functional Level Power Analysis and Modelling (FLPAM). FLPAM
is scalable, platform independent and compares favourably with existing approaches. A hybrid, top-down design flow paradigm integrating FLPAM with commercially available design tools for systematic optimisation of IP cores has also been developed
Measurement, Decoherence and Master Equations
In the first part of this thesis we concern ourselves with the problem of generating
pseudo-random circuits. These are a series of quantum gates chosen at
random, with the overall effect of implementing unitary operations with statistical
properties close to that of unitaries drawn at random with respect to the
Haar measure. Such circuits have a growing number of applications in quantum-information
processing, but all known algorithms require an external input of
classical randomness. We suggest a scheme to implement random circuits in a
weighted graph state. The input state is entangled with the weighted graph state
and a random circuit is implemented by performing local measurements in one
fixed basis only. A central idea in the analysis of this proposal is the average
bipartite entanglement generated by the repeated application of such circuits on
a large number of randomly chosen input product states. For a truly random circuit,
this should agree with that obtained by applying unitaries at random chosen
uniformly with respect to the Haar measure, values which can be calculated using
Pages Conjecture.
Part II is largely concerned with continuous variables (CV) systems. In particular,
we are interested in two descriptions. That of the class of Gaussian
states, and that of systems which can be adequately described through the use
of Markovian master equations. In the case of the latter, there are a number of
approaches one may take in order to derive a suitable equation, all of which require
some sort of approximation. These approximations can be made based on a
mixture of mathematical and physical grounds. However, unfortunately it is not
always clear how justified we are in making a particular choice, especially when
the test system we wish to describe includes its own internal interactions. In an
attempt to clarify this situation, we derive Markovian master equations for single
and interacting harmonic systems under different scenarios, including strong
internal coupling. By comparing the dynamics resulting from the corresponding
master equations with numerical simulations of the global systems evolution, we
assess the robustness of the assumptions usually made in the process of deriving
the reduced Markovian dynamics. This serves to clarify the general properties of
other open quantum system scenarios subject to treatment within a Markovian
approximation.
Finally, we extend the notions of the smooth min- and smooth max-entropies
to the continuous variable setting. Specifically, we have provided expressions to
evaluate these measures on arbitrary Gaussian states. These expressions rely
only on the symplectic eigenvalues of the corresponding covariance matrix. As
an application, we have considered their use as a suitable measure for detecting
thermalisation
An Algorithmic Interpretation of Quantum Probability
The Everett (or relative-state, or many-worlds) interpretation of quantum mechanics has come under fire for inadequately dealing with the Born rule (the formula for calculating quantum probabilities). Numerous attempts have been made to derive this rule from the perspective of observers within the quantum wavefunction. These are not really analytic proofs, but are rather attempts to derive the Born rule as a synthetic a priori necessity, given the nature of human observers (a fact not fully appreciated even by all of those who have attempted such proofs). I show why existing attempts are unsuccessful or only partly successful, and postulate that Solomonoff's algorithmic approach to the interpretation of probability theory could clarify the problems with these approaches. The Sleeping Beauty probability puzzle is used as a springboard from which to deduce an objectivist, yet synthetic a priori framework for quantum probabilities, that properly frames the role of self-location and self-selection (anthropic) principles in probability theory. I call this framework "algorithmic synthetic unity" (or ASU). I offer no new formal proof of the Born rule, largely because I feel that existing proofs (particularly that of Gleason) are already adequate, and as close to being a formal proof as one should expect or want. Gleason's one unjustified assumption--known as noncontextuality--is, I will argue, completely benign when considered within the algorithmic framework that I propose. I will also argue that, to the extent the Born rule can be derived within ASU, there is no reason to suppose that we could not also derive all the other fundamental postulates of quantum theory, as well. There is nothing special here about the Born rule, and I suggest that a completely successful Born rule proof might only be possible once all the other postulates become part of the derivation. As a start towards this end, I show how we can already derive the essential content of the fundamental postulates of quantum mechanics, at least in outline, and especially if we allow some educated and well-motivated guesswork along the way. The result is some steps towards a coherent and consistent algorithmic interpretation of quantum mechanics
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