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"Matrix theory"
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Congruence Lattices of Ideals in Categories and (Partial) Semigroups
This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations,
diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain
normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several
specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions;
Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations
are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid
categories.
eBook
HIGH-DIMENSIONAL ASYMPTOTICS OF PREDICTION
We provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model. We work in a high-dimensional asymptotic regime where p,n → ∞ and p/n → γ > 0, and allow for arbitrary covariance among the features. For both methods, we provide an explicit and efficiently computable expression for the limiting predictive risk, which depends only on the spectrum of the feature-covariance matrix, the signal strength and the aspect ratio γ. Especially in the case of regularized discriminant analysis, we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix, suggesting that analyses based on the operator norm of the covariance matrix may not be sharp. Our results also uncover an exact inverse relation between the limiting predictive risk and the limiting estimation risk in high-dimensional linear models. The analysis builds on recent advances in random matrix theory.
Journal Article
THE SPECTRUM OF KERNEL RANDOM MATRICES
We place ourselves in the setting of high-dimensional statistical inference where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. We consider the spectrum of certain kernel random matrices, in particular n × n matrices whose (i, j)th entry is $f(X_{i}^{\\prime }X_{j}/p)$ or f(∥X i – X j ∥²/p) where p is the dimension of the data, and X i are independent data vectors. Here f is assumed to be a locally smooth function. The study is motivated by questions arising in statistics and computer science where these matrices are used to perform, among other things, nonlinear versions of principal component analysis. Surprisingly, we show that in high-dimensions, and for the models we analyze, the problem becomes essentially linear—which is at odds with heuristics sometimes used to justify the usage of these methods. The analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model high-dimensional data encountered in practice.
Journal Article
DISTRIBUTED LINEAR REGRESSION BY AVERAGING
Distributed statistical learning problems arise commonly when dealing with large datasets. In this setup, datasets are partitioned over machines, which compute locally, and communicate short messages. Communication is often the bottleneck. In this paper, we study one-step and iterative weighted parameter averaging in statistical linear models under data parallelism. We do linear regression on each machine, send the results to a central server and take a weighted average of the parameters. Optionally, we iterate, sending back the weighted average and doing local ridge regressions centered at it. How does this work compared to doing linear regression on the full data? Here, we study the performance loss in estimation and test error, and confidence interval length in high dimensions, where the number of parameters is comparable to the training data size.
We find the performance loss in one-step weighted averaging, and also give results for iterative averaging. We also find that different problems are affected differently by the distributed framework. Estimation error and confidence interval length increases a lot, while prediction error increases much less. We rely on recent results from random matrix theory, where we develop a new calculus of deterministic equivalents as a tool of broader interest.
Journal Article
Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model
Spectral correlations are a powerful tool to study the dynamics of quantum many-body systems. For Hermitian Hamiltonians, quantum chaotic motion is related to random matrix theory spectral correlations. Based on recent progress in the application of spectral analysis to non-Hermitian quantum systems, we show that local level statistics, which probe the dynamics around the Heisenberg time, of a non-Hermitianq-body Sachdev-Ye-Kitev (nHSYK) model withNMajorana fermions, and its chiral and complex-fermion extensions, are also well described by random matrix theory forq>2, while forq=2, they are given by the equivalent of Poisson statistics. For that comparison, we combine exact diagonalization numerical techniques with analytical results obtained for some of the random matrix spectral observables. Moreover, depending onqandN, we identify 19 out of the 38 non-Hermitian universality classes in the nHSYK model, including those corresponding to the tenfold way. In particular, we realize explicitly 14 out of the 15 universality classes corresponding to non-pseudo-Hermitian Hamiltonians that involve universal bulk correlations of classesAI†andAII†, beyond the Ginibre ensembles. These results provide strong evidence of striking universal features in nonunitary many-body quantum chaos, which in all cases can be captured by nHSYK models withq>2.
Journal Article
Phase separation in fluids with many interacting components
Fluids in natural systems, like the cytoplasm of a cell, often contain thousands of molecular species that are organized into multiple coexisting phases that enable diverse and specific functions. How interactions between numerous molecular species encode for various emergent phases is not well understood. Here, we leverage approaches from random-matrix theory and statistical physics to describe the emergent phase behavior of fluid mixtures with many species whose interactions are drawn randomly from an underlying distribution. Through numerical simulation and stability analyses, we show that these mixtures exhibit staged phase-separation kinetics and are characterized by multiple coexisting phases at steady state with distinct compositions. Random-matrix theory predicts the number of coexisting phases, validated by simulations with diverse component numbers and interaction parameters. Surprisingly, this model predicts an upper bound on the number of phases, derived from dynamical considerations, that is much lower than the limit from the Gibbs phase rule, which is obtained from equilibrium thermodynamic constraints. We design ensembles that encode either linear or nonmonotonic scaling relationships between the number of components and coexisting phases, which we validate through simulation and theory. Finally, inspired by parallels in biological systems, we show that including nonequilibrium turnover of components through chemical reactions can tunably modulate the number of coexisting phases at steady state without changing overall fluid composition. Together, our study provides a model framework that describes the emergent dynamical and steady-state phase behavior of liquid-like mixtures with many interacting constituents.
Journal Article
Scattering of ultrashort electron wave packets: optical theorem, differential phase contrast and angular asymmetries
Recent advances in electron microscopy allowed the generation of high-energy electron wave packets of ultrashort duration. Here we present a non-perturbative S -matrix theory for scattering of ultrashort electron wave packets by atomic targets. We apply the formalism to a case of elastic scattering and derive a generalized optical theorem for ultrashort wave-packet scattering. By numerical simulations with 1 fs wave packets, we find in angular distributions of electrons on a detector one-fold and anomalous two-fold azimuthal asymmetries. We discuss how the asymmetries relate to the coherence properties of the electron beam, and to the magnitude and phase of the scattering amplitude. The essential role of the phase of the exact scattering amplitude is revealed by comparison with results obtained using the first-Born approximation. Our work paves a way for controlling electron-matter interaction by the lateral and transversal coherence properties of pulsed electron beams.
Journal Article
KLT factorization of nonrelativistic string amplitudes
A
bstract
We continue our study of the Kawai-Lewellen-Tye (KLT) factorization of winding string amplitudes in [1]. In a toroidal compactification, amplitudes for winding closed string states factorize into products of amplitudes for open strings ending on an array of D-branes localized in the compactified directions; the specific D-brane configuration is determined by the closed string data. In this paper, we study a zero Regge slope limit of the KLT relations between winding string amplitudes. Such a limit of string theory requires a critically tuned Kalb-Ramond field in a compact direction, and leads to a self-contained corner called nonrelativistic string theory. This theory is unitary, ultraviolet complete, and its string spectrum and spacetime S-matrix satisfy nonrelativistic symmetry. Moreover, the asymptotic closed string states in nonrelativistic string theory necessarily carry nonzero windings. First, starting with relativistic string theory, we construct a KLT factorization of amplitudes for winding closed strings in the presence of a critical Kalb-Ramond field. Then, in the zero Regge limit, we uncover a KLT relation for amplitudes in nonrelativistic string theory. Finally, we show how such a relation can be reproduced from first principles in a purely nonrelativistic string theory setting. We will also discuss connections to the amplitudes of string theory in the discrete light cone quantization (DLCQ), a method that is relevant for Matrix theory.
Journal Article
Chaotic and thermal aspects in the highly excited string S-matrix
A
bstract
We compute tree level scattering amplitudes involving more than one highly excited states and tachyons in bosonic string theory. We use these amplitudes to understand the chaotic and thermal aspects of the excited string states lending support to the Susskind-Horowitz-Polchinski correspondence principle. The unaveraged amplitudes exhibit chaos in the resonance distribution as a function of the kinematic parameters, which can be described by random matrix theory. Upon coarse-graining, these amplitudes are shown to exponentiate, and capture various thermal features, including features of a stringy version of the eigenstate thermalization hypothesis as well as notions of typicality. Further, we compute the effective string form factor corresponding to the highly excited states, and argue for the random walk behaviour of the long strings.
Journal Article