1 
Sketchbased Path DesignMcCrae, James 30 July 2008 (has links)
We first present a novel approach to sketching 2D curves with minimally varying curvature as piecewise clothoids. A stable and efficient algorithm fits a sketched piecewise linear curve using a number of clothoid segments with G2 continuity based on a specified error tolerance.
We then present a system for conceptually sketching 3D layouts for road and other
path networks. Our system makes four key contributions. First, we generate paths
with piecewise linear curvature by fitting 2D clothoid curves to strokes sketched on a
terrain. Second, the height of paths above the terrain is automatically determined using
a new constraint optimization formulation of the occlusion relationships between sketched
strokes. Third, we present the breakout lens, a novel widget inspired by breakout views used in engineering visualization, to facilitate the incontext and interactive manipulation of paths from alternate view points. Finally, our path construction is terrain sensitive.

2 
On Hierarchies of Strong SDP Relaxations for Combinatorial Optimization ProblemsBenabbas, Siavosh 10 December 2012 (has links)
Studying the approximation threshold of NPhard optimization problems, i.e. the ratio of the objective value achievable by a polynomial time algorithm to that of the optimal solution is an important field in theoretical computer science. In the past two decades there has been significant development both in terms of finding good approximation algorithms, e.g. through the use of semidefinite programming and hardness of approximation, e.g. the development of Probabilistically Checkable Proofs and Unique Games Conjecture.
Trying to prove lower bounds for the approximation threshold of an optimization problem, one could take one of two approaches. In the first approach, one proves such lower bounds under a complexity assumption like P =/= NP or Unique Games Conjecture. In the second approach, one studies the behaviour of prominent algorithmic schemes, such as Linear Programming (LP) and Semidefinite Programming (SDP) relaxations for the problem of interest. There, the measure of efficiency is the integrality gap which sets the approximation limitation of the algorithms based on these relaxations.
In this work we study the integrality gap of families of strong LP and SDP relaxations for a number of combinatorial optimization problems. The relaxations come from the context of LiftandProject systems. There one starts from a standard (weak) relaxation from a problem and iteratively adds more constraints to make the relaxation stronger, i.e. closer to an exact formulation of the problem. We mostly study the performance of the SheraliAdams SDP relaxations. Specifically, the main contributions of this thesis are as follows.
• We show optimal integrality gaps for the level\Theta(n) SheraliAdams SDP relaxation of MAX kCSP_q(P) for any predicate P:[q]^k > {0,1} satisfying a technical condition, which we call being promising. Our result show that for such predicates MAX kCSP_q (P) cannot be approximated (beyond a trivial approximation) by the SheraliAdams SDP relaxations of even very high level. Austrin and Håstad (SIAM J. Comput., 2011) show that a random predicate is almost surely promising.
• We complement the above result by showing that for some class of predicates which are not promising MAX kCSP_q(P) can be approximated (beyond the trivial approximation) by its canonical SDP relaxation.
• We show optimal integrality gap lower bounds for levelpoly(n) SheraliAdams SDP relaxations of Quadratic Programming. We also present superconstant integrality gap lower bounds for superconstant levels of the same hierarchy for MaxCutGain.
• We show optimal integrality gap lower bounds for the level5 SheraliAdams SDP relaxation of Vertex Cover. We also conjecture a positivity condition on the Taylor expansion of a certain function which, if proved, shows optimal integrality gaps for any constant level of the SheraliAdams SDP hierarchy for Vertex Cover.
• We revisit the connection between integrality gap lower bounds and the FranklRödl theorem (Trans. of the AMS, 1987). We prove a new density version of that theorem which can be interpreted as a new isoperimetric inequality of the Hamming cube. Using this inequality we prove integrality gap lower bounds for the LovászSchrijver SDP (resp. SheraliAdams LP) relaxation of Vertex Cover (resp. Independent Set) in degree bounded graphs.

3 
Sketchbased Path DesignMcCrae, James 30 July 2008 (has links)
We first present a novel approach to sketching 2D curves with minimally varying curvature as piecewise clothoids. A stable and efficient algorithm fits a sketched piecewise linear curve using a number of clothoid segments with G2 continuity based on a specified error tolerance.
We then present a system for conceptually sketching 3D layouts for road and other
path networks. Our system makes four key contributions. First, we generate paths
with piecewise linear curvature by fitting 2D clothoid curves to strokes sketched on a
terrain. Second, the height of paths above the terrain is automatically determined using
a new constraint optimization formulation of the occlusion relationships between sketched
strokes. Third, we present the breakout lens, a novel widget inspired by breakout views used in engineering visualization, to facilitate the incontext and interactive manipulation of paths from alternate view points. Finally, our path construction is terrain sensitive.

4 
On Hierarchies of Strong SDP Relaxations for Combinatorial Optimization ProblemsBenabbas, Siavosh 10 December 2012 (has links)
Studying the approximation threshold of NPhard optimization problems, i.e. the ratio of the objective value achievable by a polynomial time algorithm to that of the optimal solution is an important field in theoretical computer science. In the past two decades there has been significant development both in terms of finding good approximation algorithms, e.g. through the use of semidefinite programming and hardness of approximation, e.g. the development of Probabilistically Checkable Proofs and Unique Games Conjecture.
Trying to prove lower bounds for the approximation threshold of an optimization problem, one could take one of two approaches. In the first approach, one proves such lower bounds under a complexity assumption like P =/= NP or Unique Games Conjecture. In the second approach, one studies the behaviour of prominent algorithmic schemes, such as Linear Programming (LP) and Semidefinite Programming (SDP) relaxations for the problem of interest. There, the measure of efficiency is the integrality gap which sets the approximation limitation of the algorithms based on these relaxations.
In this work we study the integrality gap of families of strong LP and SDP relaxations for a number of combinatorial optimization problems. The relaxations come from the context of LiftandProject systems. There one starts from a standard (weak) relaxation from a problem and iteratively adds more constraints to make the relaxation stronger, i.e. closer to an exact formulation of the problem. We mostly study the performance of the SheraliAdams SDP relaxations. Specifically, the main contributions of this thesis are as follows.
• We show optimal integrality gaps for the level\Theta(n) SheraliAdams SDP relaxation of MAX kCSP_q(P) for any predicate P:[q]^k > {0,1} satisfying a technical condition, which we call being promising. Our result show that for such predicates MAX kCSP_q (P) cannot be approximated (beyond a trivial approximation) by the SheraliAdams SDP relaxations of even very high level. Austrin and Håstad (SIAM J. Comput., 2011) show that a random predicate is almost surely promising.
• We complement the above result by showing that for some class of predicates which are not promising MAX kCSP_q(P) can be approximated (beyond the trivial approximation) by its canonical SDP relaxation.
• We show optimal integrality gap lower bounds for levelpoly(n) SheraliAdams SDP relaxations of Quadratic Programming. We also present superconstant integrality gap lower bounds for superconstant levels of the same hierarchy for MaxCutGain.
• We show optimal integrality gap lower bounds for the level5 SheraliAdams SDP relaxation of Vertex Cover. We also conjecture a positivity condition on the Taylor expansion of a certain function which, if proved, shows optimal integrality gaps for any constant level of the SheraliAdams SDP hierarchy for Vertex Cover.
• We revisit the connection between integrality gap lower bounds and the FranklRödl theorem (Trans. of the AMS, 1987). We prove a new density version of that theorem which can be interpreted as a new isoperimetric inequality of the Hamming cube. Using this inequality we prove integrality gap lower bounds for the LovászSchrijver SDP (resp. SheraliAdams LP) relaxation of Vertex Cover (resp. Independent Set) in degree bounded graphs.

5 
Spectral Methods in Extremal CombinatoricsFilmus, Yuval 09 January 2014 (has links)
Extremal combinatorics studies how large a collection of objects can be if it satisfies a given set of restrictions. Inspired by a classical theorem due to Erdos, Ko and Rado, Simonovits and Sos posed the following problem: determine how large a collection of graphs on the vertex set {1,...,n} can be, if the intersection of any two of them contains a triangle. They conjectured that the largest possible collection, containing 1/8 of all graphs, consists of all graphs containing
a fixed triangle (a trianglestar). The first major contribution of this thesis is a confirmation of this conjecture.
We prove the Simonovits–Sos conjecture in the following strong form: the only triangleintersecting families of measure at least 1/8 are trianglestars (uniqueness), and every triangleintersecting family of measure 1/8−e is O(e)close to a trianglestar (stability). In order to prove the stability part of our theorem, we utilize a structure theorem for Boolean functions on {0,1}^m whose Fourier expansion is concentrated on the first t+1 levels, due to Kindler and Safra. The second major contribution of this thesis consists of two analogs of this theorem for Boolean functions on S_m whose Fourier expansion is concentrated on the first two levels. In the same way that the Kindler–Safra theorem is useful for studying triangleintersecting
families, our structure theorems are useful for studying intersecting families of permutations, which are families in which any two permutations agree on the image of at least one point.
Using one of our theorems, we give a simple proof of the following result of Ellis, Friedgut and Pilpel: an intersecting family of permutations on S_m of size (1−e)(m−1)! is O(e)close to a double coset, a family which consists of all permutations sending some point i to some point j.

6 
Spectral Methods in Extremal CombinatoricsFilmus, Yuval 09 January 2014 (has links)
Extremal combinatorics studies how large a collection of objects can be if it satisfies a given set of restrictions. Inspired by a classical theorem due to Erdos, Ko and Rado, Simonovits and Sos posed the following problem: determine how large a collection of graphs on the vertex set {1,...,n} can be, if the intersection of any two of them contains a triangle. They conjectured that the largest possible collection, containing 1/8 of all graphs, consists of all graphs containing
a fixed triangle (a trianglestar). The first major contribution of this thesis is a confirmation of this conjecture.
We prove the Simonovits–Sos conjecture in the following strong form: the only triangleintersecting families of measure at least 1/8 are trianglestars (uniqueness), and every triangleintersecting family of measure 1/8−e is O(e)close to a trianglestar (stability). In order to prove the stability part of our theorem, we utilize a structure theorem for Boolean functions on {0,1}^m whose Fourier expansion is concentrated on the first t+1 levels, due to Kindler and Safra. The second major contribution of this thesis consists of two analogs of this theorem for Boolean functions on S_m whose Fourier expansion is concentrated on the first two levels. In the same way that the Kindler–Safra theorem is useful for studying triangleintersecting
families, our structure theorems are useful for studying intersecting families of permutations, which are families in which any two permutations agree on the image of at least one point.
Using one of our theorems, we give a simple proof of the following result of Ellis, Friedgut and Pilpel: an intersecting family of permutations on S_m of size (1−e)(m−1)! is O(e)close to a double coset, a family which consists of all permutations sending some point i to some point j.

7 
Nonparametric Bayesian Methods for Extracting Structure from DataMeeds, Edward 01 August 2008 (has links)
One desirable property of machine learning
algorithms is the ability to balance
the number of parameters in a model
in accordance with the amount of available data.
Incorporating nonparametric Bayesian priors into models is
one approach of automatically
adjusting model capacity to the amount of available data: with small
datasets, models are less complex
(require storing fewer parameters in memory), whereas with larger datasets, models
are implicitly more complex
(require storing more parameters in memory).
Thus, nonparametric Bayesian priors satisfy frequentist intuitions
about model complexity within a fully Bayesian framework.
This thesis presents several novel
machine learning models and applications that use
nonparametric Bayesian priors.
We introduce two novel models that use flat,
Dirichlet process priors. The first is an infinite mixture
of experts model, which builds
a fully generative, joint density model of the input and output space.
The second is a Bayesian
biclustering model, which simultaneously
organizes a
data matrix into
blockconstant biclusters.
The model capable of efficiently processing very large, sparse matrices,
enabling cluster analysis on incomplete data matrices.
We introduce binary matrix factorization,
a novel matrix factorization model that, in contrast to
classic factorization methods, such as singular value decomposition,
decomposes a matrix using latent binary matrices.
We describe two nonparametric Bayesian priors
over tree structures. The first is an infinitely exchangeable
generalization of the nested
Chinese restaurant process that generates
datavectors at a single node in the tree.
The second is a novel, finitely exchangeable
prior generates trees by first partitioning data indices into groups
and then by randomly
assigning groups to a tree.
We present two applications of the tree priors: the first
automatically learns probabilistic stickfigure models of motioncapture
data that recover
plausible structure and are robust to missing
marker data.
The second learns hierarchical
allocation models based on the latent Dirichlet allocation
topic model for document corpora,
where nodes in a topictree
are latent ``supertopics", and nodes
in a documenttree are latent
categories.
The thesis concludes
with a summary of contributions, a discussion
of the models and their limitations, and a brief outline
of potential future research
directions.

8 
Firstorder Decisiontheoretic Planning in Structured Relational EnvironmentsSanner, Scott 28 July 2008 (has links)
We consider the general framework of firstorder decisiontheoretic planning in structured relational environments. Most traditional solution approaches to these planning problems ground the relational specification w.r.t. a specific domain instantiation and apply a solution approach directly to the resulting ground Markov decision process (MDP). Unfortunately, the space and time complexity of these solution algorithms scale linearly with the domain size in the best case and exponentially in the worst case. An alternate approach to grounding a relational planning problem is to lift it to a firstorder MDP (FOMDP) specification. This FOMDP can then be solved directly, resulting in a domainindependent solution whose space and time complexity either do not scale with domain size or can scale sublinearly in the domain size. However, such generality does not come without its own set of challenges and the first purpose of this thesis is to explore exact and approximate solution techniques for practically solving FOMDPs. The second purpose of this thesis is to extend the FOMDP specification to succinctly capture factored actions and additive rewards while extending the exact and approximate solution techniques to directly exploit this structure. In addition, we provide a proof of correctness of the firstorder symbolic dynamic programming approach w.r.t. its wellstudied ground MDP counterpart.

9 
Computational Prediction of Target Genes of MicroRNAsRadfar, Hossein 01 April 2014 (has links)
MicroRNAs (miRNAs) are a class of short (2125 nt) noncoding endogenous RNAs that mediate the
expression of their direct target genes posttranscriptionally. The goal of this thesis is to identify the
target genes of miRNAs using computational methods. The most popular computational target prediction
methods rely on sequence based determinants to predict targets. However, these determinants are neither
sufficient nor necessary to identify functional target sites,
and commonly ignore the cellular conditions in which miRNAs interact with their targets \emph{in vivo}.
Since miRNAs activity reduces the steadystate
abundance of mRNA targets, the main goal of this thesis is to augment large scale gene expression profiles as
a supplement to sequencebased computational miRNA target prediction techniques.
We develop two computational miRNA target prediction methods: InMiR and BayMiR; in addition, we study the interaction between miRNAs and lncRNAs using long RNA expression data.
InMiR is a computational method that predicts the targets of intronic miRNAs based on the expression profiles of their host
genes across a large number of datasets. InMiR can also predict
which host genes have expression profiles that are good surrogates for those of their intronic miRNAs. Host
genes that InMiR predicts are bad surrogates contain significantly more miRNA target sites in their 3 UTRs
and are significantly more likely to have predicted Pol IIIII promoters in their introns.
We also develop BayMiR that scores miRNAmRNA pairs based on the endogenous footprint of miRNAs on gene expression in a genomewide scale. BayMiR provides an ``endogenous target repression" index, that identifies the contribution of each miRNA in repressing a target gene in presence of other targeting miRNAs.
This thesis also addresses the interactions between miRNAs and lncRNAs. Our analysis on expression abundance of long RNA transcripts (mRNA and lncRNA) shows that the lncRNA target set of some miRNAs have relatively low abundance in the tissues that these miRNAs are highly active. We also found lncRNAs and mRNAs that shared many targeting miRNAs are significantly positively correlated, indicating that these set of highly expressed lncRNAs may act as miRNA sponges to promote mRNA regulation.

10 
Computational Prediction of Target Genes of MicroRNAsRadfar, Hossein 01 April 2014 (has links)
MicroRNAs (miRNAs) are a class of short (2125 nt) noncoding endogenous RNAs that mediate the
expression of their direct target genes posttranscriptionally. The goal of this thesis is to identify the
target genes of miRNAs using computational methods. The most popular computational target prediction
methods rely on sequence based determinants to predict targets. However, these determinants are neither
sufficient nor necessary to identify functional target sites,
and commonly ignore the cellular conditions in which miRNAs interact with their targets \emph{in vivo}.
Since miRNAs activity reduces the steadystate
abundance of mRNA targets, the main goal of this thesis is to augment large scale gene expression profiles as
a supplement to sequencebased computational miRNA target prediction techniques.
We develop two computational miRNA target prediction methods: InMiR and BayMiR; in addition, we study the interaction between miRNAs and lncRNAs using long RNA expression data.
InMiR is a computational method that predicts the targets of intronic miRNAs based on the expression profiles of their host
genes across a large number of datasets. InMiR can also predict
which host genes have expression profiles that are good surrogates for those of their intronic miRNAs. Host
genes that InMiR predicts are bad surrogates contain significantly more miRNA target sites in their 3 UTRs
and are significantly more likely to have predicted Pol IIIII promoters in their introns.
We also develop BayMiR that scores miRNAmRNA pairs based on the endogenous footprint of miRNAs on gene expression in a genomewide scale. BayMiR provides an ``endogenous target repression" index, that identifies the contribution of each miRNA in repressing a target gene in presence of other targeting miRNAs.
This thesis also addresses the interactions between miRNAs and lncRNAs. Our analysis on expression abundance of long RNA transcripts (mRNA and lncRNA) shows that the lncRNA target set of some miRNAs have relatively low abundance in the tissues that these miRNAs are highly active. We also found lncRNAs and mRNAs that shared many targeting miRNAs are significantly positively correlated, indicating that these set of highly expressed lncRNAs may act as miRNA sponges to promote mRNA regulation.

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