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Verification of successive convexification algorithm
In this report, I describe a technique which allows a non-convex optimal control problem to be expressed and solved in a convex manner. I then verify the resulting solution to ensure its physical feasibility and its optimality. The original, non-convex problem is the fuel-optimal powered landing problem with aerodynamic drag. The non-convexities present in this problem include mass depletion dynamics, aerodynamic drag, and free final time. Through the use of lossless convexification and successive convexification, this problem can be formulated as a series of iteratively solved convex problems that requires only a guess of a final time of flight. The solution’s physical feasibility is verified through a nonlinear simulation built in Simulink, while its optimality is verified through the general nonlinear optimal control software GPOPS-II.Aerospace Engineerin
Recovery from Non-Decomposable Distance Oracles
A line of work has looked at the problem of recovering an input from distance
queries. In this setting, there is an unknown sequence , and one chooses a set of queries and
receives for a distance function . The goal is to make as few
queries as possible to recover . Although this problem is well-studied for
decomposable distances, i.e., distances of the form for some function , which includes the important cases of
Hamming distance, -norms, and -estimators, to the best of our
knowledge this problem has not been studied for non-decomposable distances, for
which there are important special cases such as edit distance, dynamic time
warping (DTW), Frechet distance, earth mover's distance, and so on. We initiate
the study and develop a general framework for such distances. Interestingly,
for some distances such as DTW or Frechet, exact recovery of the sequence
is provably impossible, and so we show by allowing the characters in to be
drawn from a slightly larger alphabet this then becomes possible. In a number
of cases we obtain optimal or near-optimal query complexity. We also study the
role of adaptivity for a number of different distance functions. One motivation
for understanding non-adaptivity is that the query sequence can be fixed and
the distances of the input to the queries provide a non-linear embedding of the
input, which can be used in downstream applications involving, e.g., neural
networks for natural language processing.Comment: This work has been presented at conference The 14th Innovations in
Theoretical Computer Science (ITCS 2023) and accepted for publishing in the
journal IEEE Transactions on Information Theor
Fine-grained complexity and algorithm engineering of geometric similarity measures
Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the Fréchet distance, the Fréchet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous Fréchet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür
Sparse Graph Learning from Spatiotemporal Time Series
Outstanding achievements of graph neural networks for spatiotemporal time
series analysis show that relational constraints introduce an effective
inductive bias into neural forecasting architectures. Often, however, the
relational information characterizing the underlying data-generating process is
unavailable and the practitioner is left with the problem of inferring from
data which relational graph to use in the subsequent processing stages. We
propose novel, principled - yet practical - probabilistic score-based methods
that learn the relational dependencies as distributions over graphs while
maximizing end-to-end the performance at task. The proposed graph learning
framework is based on consolidated variance reduction techniques for Monte
Carlo score-based gradient estimation, is theoretically grounded, and, as we
show, effective in practice. In this paper, we focus on the time series
forecasting problem and show that, by tailoring the gradient estimators to the
graph learning problem, we are able to achieve state-of-the-art performance
while controlling the sparsity of the learned graph and the computational
scalability. We empirically assess the effectiveness of the proposed method on
synthetic and real-world benchmarks, showing that the proposed solution can be
used as a stand-alone graph identification procedure as well as a graph
learning component of an end-to-end forecasting architecture.Comment: updated and extended versio
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