26,678 research outputs found
A Local-to-Global Theorem for Congested Shortest Paths
Amiri and Wargalla (2020) proved the following local-to-global theorem in
directed acyclic graphs (DAGs): if is a weighted DAG such that for each
subset of 3 nodes there is a shortest path containing every node in ,
then there exists a pair of nodes such that there is a shortest
-path containing every node in .
We extend this theorem to general graphs. For undirected graphs, we prove
that the same theorem holds (up to a difference in the constant 3). For
directed graphs, we provide a counterexample to the theorem (for any constant),
and prove a roundtrip analogue of the theorem which shows there exists a pair
of nodes such that every node in is contained in the union of a
shortest -path and a shortest -path.
The original theorem for DAGs has an application to the -Shortest Paths
with Congestion (()-SPC) problem. In this problem, we are given a
weighted graph , together with node pairs ,
and a positive integer . We are tasked with finding paths such that each is a shortest path from to , and every
node in the graph is on at most paths , or reporting that no such
collection of paths exists.
When the problem is easily solved by finding shortest paths for each
pair independently. When , the -SPC problem recovers
the -Disjoint Shortest Paths (-DSP) problem, where the collection of
shortest paths must be node-disjoint. For fixed , -DSP can be solved in
polynomial time on DAGs and undirected graphs. Previous work shows that the
local-to-global theorem for DAGs implies that -SPC on DAGs whenever
is constant. In the same way, our work implies that -SPC can be
solved in polynomial time on undirected graphs whenever is constant.Comment: Updated to reflect reviewer comment
Computing Motion Plans for Assembling Particles with Global Control
We investigate motion planning algorithms for the assembly of shapes in the
\emph{tilt model} in which unit-square tiles move in a grid world under the
influence of uniform external forces and self-assemble according to certain
rules. We provide several heuristics and experimental evaluation of their
success rate, solution length, runtime, and memory consumption.Comment: 20 pages, 12 figure
Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications
Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system.
The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods.
Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms.
In this thesis we study various consequences and the broad applicability of facial reduction.
The thesis is organized in two parts.
In the first part, we show the instabilities accompanied by the absence
of strict feasibility through the lens of facially reduced systems.
In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity.
This leads to the two-step facial reduction and two novel related notions of singularity.
For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound.
For the area of linear programming, we reveal degeneracies caused by the implicit redundancies.
Furthermore, we propose a preprocessing tool that uses the simplex method.
In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points.
We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function.
We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method.
We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution.
Facial reduction continues to play an important role for
providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances
Two string theory flavours of generalised Eisenstein series
Generalised Eisenstein series are non-holomorphic modular invariant functions
of a complex variable, , subject to a particular inhomogeneous Laplace
eigenvalue equation on the hyperbolic upper-half -plane. Two infinite
classes of such functions arise quite naturally within different string theory
contexts. A first class can be found by studying the coefficients of the
effective action for the low-energy expansion of type IIB superstring theory,
and relatedly in the analysis of certain integrated four-point functions of
stress tensor multiplet operators in supersymmetric
Yang-Mills theory. A second class of such objects is known to contain all
two-loop modular graph functions, which are fundamental building blocks in the
low-energy expansion of closed-string scattering amplitudes at genus one. In
this work, we present a Poincar\'e series approach that unifies both classes of
generalised Eisenstein series and manifests certain algebraic and differential
relations amongst them. We then combine this technique with spectral methods
for automorphic forms to find general and non-perturbative expansions at the
cusp . Finally, we find intriguing connections between the
asymptotic expansion of these modular functions as and the
non-trivial zeros of the Riemann zeta function.Comment: 44 pages + 3 figure
TabR: Unlocking the Power of Retrieval-Augmented Tabular Deep Learning
Deep learning (DL) models for tabular data problems are receiving
increasingly more attention, while the algorithms based on gradient-boosted
decision trees (GBDT) remain a strong go-to solution. Following the recent
trends in other domains, such as natural language processing and computer
vision, several retrieval-augmented tabular DL models have been recently
proposed. For a given target object, a retrieval-based model retrieves other
relevant objects, such as the nearest neighbors, from the available (training)
data and uses their features or even labels to make a better prediction.
However, we show that the existing retrieval-based tabular DL solutions provide
only minor, if any, benefits over the properly tuned simple retrieval-free
baselines. Thus, it remains unclear whether the retrieval-based approach is a
worthy direction for tabular DL.
In this work, we give a strong positive answer to this question. We start by
incrementally augmenting a simple feed-forward architecture with an
attention-like retrieval component similar to those of many (tabular)
retrieval-based models. Then, we highlight several details of the attention
mechanism that turn out to have a massive impact on the performance on tabular
data problems, but that were not explored in prior work. As a result, we design
TabR -- a simple retrieval-based tabular DL model which, on a set of public
benchmarks, demonstrates the best average performance among tabular DL models,
becomes the new state-of-the-art on several datasets, and even outperforms GBDT
models on the recently proposed ``GBDT-friendly'' benchmark (see the first
figure).Comment: Code: https://github.com/yandex-research/tabular-dl-tab
Signed tropicalization of polar cones
We study the tropical analogue of the notion of polar of a cone, working over
the semiring of tropical numbers with signs. We characterize the cones which
arise as polars of sets of tropically nonnegative vectors by an invariance
property with respect to a tropical analogue of Fourier-Motzkin elimination. We
also relate tropical polars with images by the nonarchimedean valuation of
classical polars over real closed nonarchimedean fields and show, in
particular, that for semi-algebraic sets over such fields, the operation of
taking the polar commutes with the operation of signed valuation (keeping track
both of the nonarchimedean valuation and sign). We apply these results to
characterize images by the signed valuation of classical cones of matrices,
including the cones of positive semidefinite matrices, completely positive
matrices, completely positive semidefinite matrices, and their polars,
including the cone of co-positive matrices, showing that hierarchies of
classical cones collapse under tropicalization. We finally discuss an
application of these ideas to optimization with signed tropical numbers.Comment: 24 pages, 1 figure. Changes with respect to Version 2: we improved
Introduction and added Examples 3.24 and 3.25 illustrating that "bend
addition" can be considered as a tropical analogue of the Fourier-Motzkin
eliminatio
Reinforcement learning in large state action spaces
Reinforcement learning (RL) is a promising framework for training intelligent agents which learn to optimize long term utility by directly interacting with the environment. Creating RL methods which scale to large state-action spaces is a critical problem towards ensuring real world deployment of RL systems. However, several challenges limit the applicability of RL to large scale settings. These include difficulties with exploration, low sample efficiency, computational intractability, task constraints like decentralization and lack of guarantees about important properties like performance, generalization and robustness in potentially unseen scenarios.
This thesis is motivated towards bridging the aforementioned gap. We propose several principled algorithms and frameworks for studying and addressing the above challenges RL. The proposed methods cover a wide range of RL settings (single and multi-agent systems (MAS) with all the variations in the latter, prediction and control, model-based and model-free methods, value-based and policy-based methods). In this work we propose the first results on several different problems: e.g. tensorization of the Bellman equation which allows exponential sample efficiency gains (Chapter 4), provable suboptimality arising from structural constraints in MAS(Chapter 3), combinatorial generalization results in cooperative MAS(Chapter 5), generalization results on observation shifts(Chapter 7), learning deterministic policies in a probabilistic RL framework(Chapter 6). Our algorithms exhibit provably enhanced performance and sample efficiency along with better scalability. Additionally, we also shed light on generalization aspects of the agents under different frameworks. These properties have been been driven by the use of several advanced tools (e.g. statistical machine learning, state abstraction, variational inference, tensor theory).
In summary, the contributions in this thesis significantly advance progress towards making RL agents ready for large scale, real world applications
Moments of Dirichlet L-functions in Function Fields
In this thesis, we compute several moments and mean values of Dirichlet L-functions in function fields, in both the odd and even characteristic setting.Leverhulme Trus
Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems
Unstable periodic orbits (UPOs), exact periodic solutions of the evolution equation, offer a very
powerful framework for studying chaotic dynamical systems, as they allow one to dissect their
dynamical structure. UPOs can be considered the skeleton of chaotic dynamics, its essential
building blocks. In fact, it is possible to prove that in a chaotic system, UPOs are dense in
the attractor, meaning that it is always possible to find a UPO arbitrarily near any chaotic
trajectory. We can thus think of the chaotic trajectory as being approximated by different
UPOs as it evolves in time, jumping from one UPO to another as a result of their instability.
In this thesis we provide a contribution towards the use of UPOs as a tool to understand and
distill the dynamical structure of chaotic dynamical systems. We will focus on two models,
characterised by different properties, the Lorenz-63 and Lorenz-96 model.
The process of approximation of a chaotic trajectory in terms of UPOs will play a central role
in our investigation. In fact, we will use this tool to explore the properties of the attractor of
the system under the lens of its UPOs.
In the first part of the thesis we consider the Lorenz-63 model with the classic parameters’ value.
We investigate how a chaotic trajectory can be approximated using a complete set of UPOs
up to symbolic dynamics’ period 14. At each instant in time, we rank the UPOs according to
their proximity to the position of the orbit in the phase space. We study this process from
two different perspectives. First, we find that longer period UPOs overwhelmingly provide the
best local approximation to the trajectory. Second, we construct a finite-state Markov chain
by studying the scattering of the trajectory between the neighbourhood of the various UPOs.
Each UPO and its neighbourhood are taken as a possible state of the system. Through the
analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a
different interpretation of the mixing processes occurring in the system by taking advantage of
the concept of quasi-invariant sets.
In the second part of the thesis we provide an extensive numerical investigation of the variability
of the dynamical properties across the attractor of the much studied Lorenz ’96 dynamical
system. By combining the Lyapunov analysis of the tangent space with the study of the
shadowing of the chaotic trajectory performed by a very large set of unstable periodic orbits,
we show that the observed variability in the number of unstable dimensions, which shows a
serious breakdown of hyperbolicity, is associated with the presence of a substantial number of
finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times
are considered
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