African University of Science and TechnologyThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.http://repository.aust.edu.ng:80/xmlui2019-03-11T11:05:26Z2019-03-11T11:05:26ZGen: A General-Purpose Probabilistic Programming System with Programmable Inferencehttp://repository.aust.edu.ng/xmlui/handle/1721.1/1192552018-11-28T10:19:48Z2018-11-26T00:00:00ZGen: A General-Purpose Probabilistic Programming System with Programmable Inference
Probabilistic modeling and inference are central to many fields. A key challenge for wider adoption of probabilistic programming languages is designing systems that are both flexible and performant. This paper introduces Gen, a new probabilistic programming system with novel language con- structs for modeling and for end-user customization and optimization of inference. Gen makes it practical to write probabilistic programs that solve problems from multiple fields. Gen programs can combine generative models written in Julia, neural networks written in TensorFlow, and custom inference algorithms based on an extensible library of Monte Carlo and numerical optimization techniques. This paper also presents techniques that enable Gen’s combination of flexibility and performance: (i) the generative function inter- face, an abstraction for encapsulating probabilistic and/or differentiable computations; (ii) domain-specific languages with custom compilers that strike different flexibility/per- formance tradeoffs; (iii) combinators that encode common patterns of conditional independence and repeated compu- tation, enabling speedups from caching; and (iv) a standard inference library that supports custom proposal distributions also written as programs in Gen. This paper shows that Gen outperforms state-of-the-art probabilistic programming systems, sometimes by multiple orders of magnitude, on problems such as nonlinear state-space modeling, structure learning for real-world time series data, robust regression, and 3D body pose estimation from depth images.
2018-11-26T00:00:00ZThe Calderón problem for connectionshttp://repository.aust.edu.ng/xmlui/handle/123456789/40072018-11-24T23:27:35Z2017-10-03T00:00:00ZThe Calderón problem for connections
This thesis is concerned with the inverse problem of determining a
unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over
a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann
(DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$.
The connection is to be determined up to a unitary gauge equivalence
equal to the identity at the boundary.
In our first approach to the problem, we restrict our attention to
conformally transversally anisotropic (cylindrical) manifolds $M \Subset
\mathbb{R}\times M_0$. Our strategy can be described as follows: we
construct the special Complex Geometric Optics solutions oscillating in
the vertical direction, that concentrate near geodesics and use their
density in an integral identity to reduce the problem to a suitable
$X$-ray transform on $M_0$. The construction is based on our proof of
existence of Gaussian Beams on $M_0$, which are a family of smooth
approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau
\in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure
along geodesics when $\tau \to \infty$, whereas the small remainder
(that makes the solution exact) can be shown to exist by using suitable
Carleman estimates.
In the case $m = 1$, we prove the recovery of the connection given the
injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For
$m > 1$ and $M_0$ simple we reduce the problem to a certain two
dimensional $\textit{new non-abelian ray transform}$.
In our second approach, we assume that the connection $A$ is a
$\textit{Yang-Mills connection}$ and no additional assumption on $M$. We
construct a global gauge for $A$ (possibly singular at some points) that
ties well with the DN map and in which the Yang-Mills equations become
elliptic. By using the unique continuation property for elliptic systems
and the fact that the singular set is suitably small, we are able to
propagate the gauges globally. For the case $m = 1$ we are able to
reconstruct the connection, whereas for $m > 1$ we are forced to make
the technical assumption that $(M, g)$ is analytic in order to prove the
recovery.
Finally, in both approaches we are using the vital fact that is proved
in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$
acting on sections of $E|_{\partial M}$, whose full symbol determines
the full Taylor expansion of $A$ at the boundary.
2017-10-03T00:00:00ZSymmetry in monotone Lagrangian Floer theoryhttp://repository.aust.edu.ng/xmlui/handle/123456789/40062018-11-24T23:27:35Z2017-10-01T00:00:00ZSymmetry in monotone Lagrangian Floer theory
In this thesis we study the self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, working over an enriched Novikov ring. This imposes constraints on the differentials in the spectral sequence which force them to vanish in certain situations. We then specialise to the case where $L$ is $K$-homogeneous for a compact Lie group $K$, meaning roughly that $X$ is Kaehler, $K$ acts on $X$ by holomorphic automorphisms, and $L$ is a Lagrangian orbit. By studying holomorphic discs with boundary on $L$ we compute the image of low codimension $K$-invariant subvarieties of $X$ under the length zero closed-open string map. This places restrictions on the self-Floer cohomology of $L$ which generalise and refine the Auroux-Kontsevich-Seidel criterion. These often result in the need to work over fields of specific positive characteristics in order to obtain non-zero cohomology. The disc analysis is then developed further, with the introduction of the notion of poles and a reflection mechanism for completing holomorphic discs into spheres. This theory is applied to two main families of examples. The first is the collection of four Platonic Lagrangians in quasihomogeneous threefolds of $\mathrm{SL}(2, \mathbb{C})$, starting with the Chiang Lagrangian in $\mathbb{CP}^3$. These were previously studied by Evans and Lekili, who computed the self-Floer cohomology of the latter. We simplify their argument, which is based on an explicit construction of the Biran-Cornea pearl complex, and deal with the remaining three cases. The second is a family of $\mathrm{PSU}(n)$-homogeneous Lagrangians in products of projective spaces. Here the presence of both discrete and continuous symmetries leads to some unusual properties: in particular we obtain non-displaceable monotone Lagrangians which are narrow in a strong sense. We also discuss related examples including applications of Perutz's symplectic Gysin sequence and quilt functors. The thesis concludes with a discussion of directions for further research and a collection of technical appendices.
2017-10-01T00:00:00ZFano Varieties in Mori Fibre Spaceshttp://repository.aust.edu.ng/xmlui/handle/123456789/40042018-11-24T23:27:34Z2016-01-01T00:00:00ZFano Varieties in Mori Fibre Spaces
2016-01-01T00:00:00Z