Review
Mathematics, Applied
Grigori A. Karagulyan
Summary: This paper extends sharp inequalities for martingale differences to general multiplicative systems of random variables, using a technique that reduces the general case to Rademacher random variables without changing the constants in the inequalities.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Vladimir E. Bening, Victor Y. Korolev
Summary: This paper introduces a new approach to comparing the distributions of sums of random variables using the notion of deficiency from mathematical statistics. The approach is used to determine the distribution of a separate random variable in the sum that guarantees a desired quantile or probability, with the fewest possible number of summands. The paper also considers the case of comparing distributions when the number of summands is random, and applies the approach to determining the distribution of insurance payments for minimum portfolio size under specified risk or non-ruin probability.
Article
Engineering, Electrical & Electronic
Shahana Ibrahim, Xiao Fu
Summary: Learning the joint probability of random variables is essential in statistical signal processing and machine learning, but direct estimation of high-dimensional joint probability is usually impossible. Recent research has proposed a method for recovering the joint probability mass function from three-dimensional marginals, and a new framework using only pairwise marginals has been developed to lower sample complexity. The coupled nonnegative matrix factorization framework has been analyzed for joint PMF recovery guarantees under various conditions, showing promising results in terms of accuracy and efficiency.
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2021)
Article
Mathematics, Interdisciplinary Applications
Zareen A. Khan, Hijaz Ahmad
Summary: Discrete fractional calculus is utilized to interpret neural schemes with memory impacts. This study formulates discrete fractional nonlinear inequalities for numerical solutions of discrete fractional differential equations and discusses aspects such as boundedness, uniqueness, and continuous dependency. The methodology employed proves the leading consequences for solutions of discrete fractional difference equations.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Mathematics
Carlos Granados
Summary: The study introduces neutrosophic random variables and applies various discrete random distributions, providing a new approach to dealing with issues involving unspecified data.
HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
(2022)
Article
Mechanics
Pierre Mergny, Marc Potters
Summary: In this study, we investigate the large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices A and B as their dimensions tend to infinity. We consider a general framework that includes various cases, such as when A and/or B are selected from an invariant ensemble or are fixed diagonal matrices. Our findings demonstrate that the tilting method introduced by Guionnet and Maida (2020 Electron. J. Probab. 25 1-24) can be extended to our general setting and is equivalent to studying a spherical spin glass model specific to the operation-sum/product of symmetric matrices/sum of rectangular matrices that we are examining.
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
(2022)
Article
Mechanics
Pierre Mergny, Satya N. Majumdar
Summary: In this study, the probability of stability of a large complex system within the framework of a generalized May model is investigated, focusing on the impact of inhomogeneities in intrinsic damping rates on system stability. It is found that as the interaction strength increases, the system undergoes a phase transition from a stable phase to an unstable phase.
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
(2021)
Article
Mathematics, Applied
Tran Loc Hung
Summary: This paper studies the convergence rates of compound random sums of arrays of row-wise independent random variables and their relation to classical results.
Article
Mathematics, Applied
Susanna Spektor
Summary: We derived a non-commutative Khinchine-type inequality when the Rademacher random variables are dependent and the sum of them equals some integer M.
FORUM MATHEMATICUM
(2023)
Article
Mathematics
Naomi Dvora Feldheim, Ohad Noy Feldheim
Summary: The study shows that for independent non-compactly supported random variables X, Y on [0, infinity), the limit probability of P(min(X, Y) > m | X + Y > 2m) as m approaches infinity is 0. Furthermore, there are conjectures of multivariate and weighted extensions of this result, which have been proven under the additional assumption that the random variables are identically distributed.
ISRAEL JOURNAL OF MATHEMATICS
(2021)
Article
Mathematics, Applied
Matthew Kwan, Lisa Sauermann
Summary: The study focuses on the magnitude of random symmetric matrices and proves a specific probability distribution for the magnitude of the matrices, introducing techniques through the study of evolution of permanents of submatrices.
SELECTA MATHEMATICA-NEW SERIES
(2022)
Article
Mathematics, Applied
A. A. El-Deeb, Saima Rashid, Zareen A. Khan, S. D. Makharesh
Summary: In this paper, dynamic Hilbert-type inequalities in two independent variables on time scales are established using the Fenchel-Legendre transform. These inequalities are then applied to discrete and continuous calculus to derive new inequalities as specific cases. The results provide more general forms of several previously established inequalities.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Mathematics, Applied
Ya-Hui Zhu, Wei-Cai Peng, Yong-Jin Zhang, Zhong-Zhi Wang
Summary: In this paper, the concept of generalized relative entropy is introduced as a random measure between two probability measures. Then, a class of strong deviation theorem for dependent random variables is established, and based on this theorem, strong deviation theorems and strong law of large numbers for row-wise negatively dependent random variables are obtained.
JOURNAL OF MATHEMATICAL INEQUALITIES
(2023)
Article
Physics, Multidisciplinary
Antonio Mandarino, Giovanni Scala
Summary: The theorem developed by John Bell marked the beginning of a revolution in the field of quantum information technologies, transforming a philosophical question about reality into a broad and intensive area of research. In this study, we examined the behavior of nonlocality in a system of two qubits prepared in a random mixed state using the CHSH-Bell inequality. We also investigated to what extent states close to those with a high degree of nonclassicality could violate local realism, accounting for inefficiency in state preparation.
Article
Mathematics, Applied
A. Talebi, Gh Sadeghi, M. S. Moslehian
Summary: The paper establishes some Freedman inequalities for martingales in noncommutative probability spaces by utilizing a result of Lieb-Araki and constructing special projections. As an application, a noncommutative Bernstein-type inequality is provided.
COMPLEX ANALYSIS AND OPERATOR THEORY
(2022)
Article
Mathematics, Applied
Joel A. Tropp
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
(2018)
Article
Physics, Multidisciplinary
M. Guta, J. Kahn, R. Kueng, J. A. Tropp
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2020)
Article
Statistics & Probability
De Huang, Joel A. Tropp
Summary: This paper deduces exponential matrix concentration using a short, conceptual argument based on a Poincare inequality. The theory also applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on subadditivity of Poincare inequalities and a chain rule inequality for the trace of the matrix Dirichlet form, avoiding difficulties associated with a direct extension of the classic scalar argument through symmetrization technique.
Article
Computer Science, Theory & Methods
De Huang, Jonathan Niles-Weed, Joel A. Tropp, Rachel Ward
Summary: This paper presents nonasymptotic growth and concentration bounds for a product of independent random matrices, relying on the uniform smoothness properties of the Schatten trace classes. These results refine and extend previous work and are akin to other results on sums of independent random matrices.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Joel A. Tropp
Summary: Randomized block Krylov subspace methods are powerful algorithms for computing extreme eigenvalues of symmetric matrices or extreme singular values of general matrices. This paper develops new theoretical bounds on the performance of these methods, showing that for matrices with polynomial spectral decay, accurate spectral norm estimates can be obtained with a constant number of steps. The analysis also indicates that algorithm behavior is sensitive to block size, a finding confirmed by numerical evidence.
NUMERISCHE MATHEMATIK
(2022)
Proceedings Paper
Computer Science, Artificial Intelligence
Aviad Levis, Daeyoung Lee, Joel A. Tropp, Charles F. Gammie, Katherine L. Bouman
Summary: We propose a method to recover the underlying properties of fluid-dynamical processes from sparse measurements, demonstrated through simulations of black hole evolution. Our approach shows advantages over state-of-the-art dynamic black hole imaging techniques.
2021 IEEE/CVF INTERNATIONAL CONFERENCE ON COMPUTER VISION (ICCV 2021)
(2021)
Article
Mathematics, Applied
Lijun Ding, Alp Yurtsever, Volkan Cevher, Joel A. Tropp, Madeleine Udell
Summary: This paper introduces a new storage-optimal algorithm that can solve almost all SDPs, particularly effective for weakly constrained SDPs. By formulating an approximate complementarity principle, the algorithm significantly reduces the search space for the primal solution. Numerical experiments demonstrate the success of this approach for a variety of large-scale SDPs.
SIAM JOURNAL ON OPTIMIZATION
(2021)
Article
Quantum Science & Technology
Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, Joel A. Tropp
Summary: This study explores the use of a simple and powerful randomized method called QDRIFT to accelerate quantum simulation, finding that it can generate random product formulas that approximate the ideal evolution. The gate complexity is shown to be independent of the number of terms in the Hamiltonian, with the same random evolution producing shorter circuits depending on the input state. The proofs rely on concentration inequalities for vector and matrix martingales, and the results are applicable to other randomized product formulas.
Article
Mathematics, Applied
Richard Kueng, Joel A. Tropp
Summary: This paper investigates the decomposition of a low-rank positive-semidefinite matrix into symmetric factors with binary entries and provides answers to fundamental questions regarding the existence and uniqueness of these decompositions. The research also introduces tractable factorization algorithms that work under a mild deterministic condition.
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2021)
Article
Mathematics, Applied
Alp Yurtsever, Joel A. Tropp, Olivier Fercoq, Madeleine Udell, Volkan Cevher
Summary: Semidefinite programming (SDP) is a powerful framework in convex optimization with potential for data science. The paper introduces a randomized algorithm for solving large SDP problems, which is cost-effective. Numerical evidence shows the algorithm's effectiveness in various applications.
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2021)
Article
Statistics & Probability
De Huang, Joel A. Tropp
Summary: Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the '2 operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities, with the main result being that the classical Bakry-Emery curvature criterion implies subgaussian concentration for matrix Lipschitz functions, overcoming technical obstacles in previous attempts. The approach unifies and extends much of the previous work on matrix concentration, reproducing the matrix Efron-Stein inequalities and handling matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.
ELECTRONIC JOURNAL OF PROBABILITY
(2021)
Article
Mathematics, Applied
Yiming Sun, Yang Guo, Charlene Luo, Joel Tropp, Madeleine Udell
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2020)
Article
Mathematics
Per-Gunnar Martinsson, Joel A. Tropp
Article
Mathematics, Applied
Joel A. Tropp, Alp Yurtsever, Madeleine Udell, Volkan Cevher
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Samet Oymak, Joel A. Tropp
INFORMATION AND INFERENCE-A JOURNAL OF THE IMA
(2018)
