12,693 research outputs found
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
Intersection matrices for the minimal regular model of and applications to the Arakelov canonical sheaf
Let be an integer coprime to such that and let
be the genus of the modular curve . We compute the
intersection matrices relative to special fibres of the minimal regular model
of . Moreover we prove that the self-intersection of the Arakelov
canonical sheaf of is asymptotic to , for .Comment: 27 pages. The main results have been improved for any N coprime to 6.
Moreover there is an Appendix with the drawings of the special fibre
Introduction to Facial Micro Expressions Analysis Using Color and Depth Images: A Matlab Coding Approach (Second Edition, 2023)
The book attempts to introduce a gentle introduction to the field of Facial
Micro Expressions Recognition (FMER) using Color and Depth images, with the aid
of MATLAB programming environment. FMER is a subset of image processing and it
is a multidisciplinary topic to analysis. So, it requires familiarity with
other topics of Artifactual Intelligence (AI) such as machine learning, digital
image processing, psychology and more. So, it is a great opportunity to write a
book which covers all of these topics for beginner to professional readers in
the field of AI and even without having background of AI. Our goal is to
provide a standalone introduction in the field of MFER analysis in the form of
theorical descriptions for readers with no background in image processing with
reproducible Matlab practical examples. Also, we describe any basic definitions
for FMER analysis and MATLAB library which is used in the text, that helps
final reader to apply the experiments in the real-world applications. We
believe that this book is suitable for students, researchers, and professionals
alike, who need to develop practical skills, along with a basic understanding
of the field. We expect that, after reading this book, the reader feels
comfortable with different key stages such as color and depth image processing,
color and depth image representation, classification, machine learning, facial
micro-expressions recognition, feature extraction and dimensionality reduction.
The book attempts to introduce a gentle introduction to the field of Facial
Micro Expressions Recognition (FMER) using Color and Depth images, with the aid
of MATLAB programming environment.Comment: This is the second edition of the boo
The Role of Entropy and Reconstruction in Multi-View Self-Supervised Learning
The mechanisms behind the success of multi-view self-supervised learning
(MVSSL) are not yet fully understood. Contrastive MVSSL methods have been
studied through the lens of InfoNCE, a lower bound of the Mutual Information
(MI). However, the relation between other MVSSL methods and MI remains unclear.
We consider a different lower bound on the MI consisting of an entropy and a
reconstruction term (ER), and analyze the main MVSSL families through its lens.
Through this ER bound, we show that clustering-based methods such as
DeepCluster and SwAV maximize the MI. We also re-interpret the mechanisms of
distillation-based approaches such as BYOL and DINO, showing that they
explicitly maximize the reconstruction term and implicitly encourage a stable
entropy, and we confirm this empirically. We show that replacing the objectives
of common MVSSL methods with this ER bound achieves competitive performance,
while making them stable when training with smaller batch sizes or smaller
exponential moving average (EMA) coefficients.
Github repo: https://github.com/apple/ml-entropy-reconstruction.Comment: 18 pages: 9 of main text, 2 of references, and 7 of supplementary
material. Appears in the proceedings of ICML 202
Gaussian Control Barrier Functions : A Gaussian Process based Approach to Safety for Robots
In recent years, the need for safety of autonomous and intelligent robots has increased. Today, as robots are being increasingly deployed in closer proximity to humans, there is an exigency for safety since human lives may be at risk, e.g., self-driving vehicles or surgical robots. The objective of this thesis is to present a safety framework for dynamical systems that leverages tools from control theory and machine learning. More formally, the thesis presents a data-driven framework for designing safety function candidates which ensure properties of forward invariance. The potential benefits of the results presented in this thesis are expected to help applications such as safe exploration, collision avoidance problems, manipulation tasks, and planning, to name some.
We utilize Gaussian processes (GP) to place a prior on the desired safety function candidate, which is to be utilized as a control barrier function (CBF). The resultant formulation is called Gaussian CBFs and they reside in a reproducing kernel Hilbert space. A key concept behind Gaussian CBFs is the incorporation of both safety belief as well as safety uncertainty, which former barrier function formulations did not consider. This is achieved by using robust posterior estimates from a GP where the posterior mean and variance serve as surrogates for the safety belief and uncertainty respectively. We synthesize safe controllers by framing a convex optimization problem where the kernel-based representation of GPs allows computing the derivatives in closed-form analytically.
Finally, in addition to the theoretical and algorithmic frameworks in this thesis, we rigorously test our methods in hardware on a quadrotor platform. The platform used is a Crazyflie 2.1 which is a versatile palm-sized quadrotor. We provide our insights and detailed discussions on the hardware implementations which will be useful for large-scale deployment of the techniques presented in this dissertation.Ph.D
Fuzzy Natural Logic in IFSA-EUSFLAT 2021
The present book contains five papers accepted and published in the Special Issue, âFuzzy Natural Logic in IFSA-EUSFLAT 2021â, of the journal Mathematics (MDPI). These papers are extended versions of the contributions presented in the conference âThe 19th World Congress of the International Fuzzy Systems Association and the 12th Conference of the European Society for Fuzzy Logic and Technology jointly with the AGOP, IJCRS, and FQAS conferencesâ, which took place in Bratislava (Slovakia) from September 19 to September 24, 2021. Fuzzy Natural Logic (FNL) is a system of mathematical fuzzy logic theories that enables us to model natural language terms and rules while accounting for their inherent vagueness and allows us to reason and argue using the tools developed in them. FNL includes, among others, the theory of evaluative linguistic expressions (e.g., small, very large, etc.), the theory of fuzzy and intermediate quantifiers (e.g., most, few, many, etc.), and the theory of fuzzy/linguistic IFâTHEN rules and logical inference. The papers in this Special Issue use the various aspects and concepts of FNL mentioned above and apply them to a wide range of problems both theoretically and practically oriented. This book will be of interest for researchers working in the areas of fuzzy logic, applied linguistics, generalized quantifiers, and their applications
Numerical Methods for Convex Multistage Stochastic Optimization
Optimization problems involving sequential decisions in a stochastic
environment were studied in Stochastic Programming (SP), Stochastic Optimal
Control (SOC) and Markov Decision Processes (MDP). In this paper we mainly
concentrate on SP and SOC modelling approaches. In these frameworks there are
natural situations when the considered problems are convex. Classical approach
to sequential optimization is based on dynamic programming. It has the problem
of the so-called ``Curse of Dimensionality", in that its computational
complexity increases exponentially with increase of dimension of state
variables. Recent progress in solving convex multistage stochastic problems is
based on cutting planes approximations of the cost-to-go (value) functions of
dynamic programming equations. Cutting planes type algorithms in dynamical
settings is one of the main topics of this paper. We also discuss Stochastic
Approximation type methods applied to multistage stochastic optimization
problems. From the computational complexity point of view, these two types of
methods seem to be complimentary to each other. Cutting plane type methods can
handle multistage problems with a large number of stages, but a relatively
smaller number of state (decision) variables. On the other hand, stochastic
approximation type methods can only deal with a small number of stages, but a
large number of decision variables
Limit Theory under Network Dependence and Nonstationarity
These lecture notes represent supplementary material for a short course on
time series econometrics and network econometrics. We give emphasis on limit
theory for time series regression models as well as the use of the
local-to-unity parametrization when modeling time series nonstationarity.
Moreover, we present various non-asymptotic theory results for moderate
deviation principles when considering the eigenvalues of covariance matrices as
well as asymptotics for unit root moderate deviations in nonstationary
autoregressive processes. Although not all applications from the literature are
covered we also discuss some open problems in the time series and network
econometrics literature.Comment: arXiv admin note: text overlap with arXiv:1705.08413 by other author
DiffFacto: Controllable Part-Based 3D Point Cloud Generation with Cross Diffusion
While the community of 3D point cloud generation has witnessed a big growth
in recent years, there still lacks an effective way to enable intuitive user
control in the generation process, hence limiting the general utility of such
methods. Since an intuitive way of decomposing a shape is through its parts, we
propose to tackle the task of controllable part-based point cloud generation.
We introduce DiffFacto, a novel probabilistic generative model that learns the
distribution of shapes with part-level control. We propose a factorization that
models independent part style and part configuration distributions and presents
a novel cross-diffusion network that enables us to generate coherent and
plausible shapes under our proposed factorization. Experiments show that our
method is able to generate novel shapes with multiple axes of control. It
achieves state-of-the-art part-level generation quality and generates plausible
and coherent shapes while enabling various downstream editing applications such
as shape interpolation, mixing, and transformation editing. Project website:
https://difffacto.github.io
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