4,286 research outputs found
On the modular decomposition of the spin representation of Sn indexed by the partition (n − 2, 2) and the combinatorics of bar-core partitions
This thesis consists of two research projects on the spin representation theory of the symmetric group. In Chapters 2 and 3, we determine the modular decomposition of the spin representation of Sn indexed by the partition (n − 2, 2). Whilst James provided a characteristic-free construction of the linear representations of the symmetric group Sn, there is no analogous construction for the spin (or projective) representations of Sn, i.e. the linear representations of a double cover Sn+ of Sn. The most crucial open problem in the spin representation theory of Sn is determining the number of times each prime characteristic irreducible appears in the decomposition of the modular reduction of a characteristic 0 irreducible. Inspired by James’ description of the linear representations of Sn in terms of submodules and induced modules, recovering the Specht modules, Wales showed that inducing the basic representation from Sn-1+ to Sn+ provides an irreducible 2-modular representation other than the basic representation, leading to a description of the modular decomposition of the spin representations denoted by the partitions (n) and (n − 1, 1). We extend this method to determine the decomposition of the spin representation corresponding to (n − 2, 2). In Chapter 4, we establish combinatorial results about bar-core partitions. When p and q are coprime odd integers no less than 3, Olsson proved that if λ is a p-bar-core partition, then the q-bar-core of λ is again a p-bar-core. We establish a generalisation of this theorem: that the p-bar-weight of the q-bar-core of any bar partition λ is at most the p-bar-weight of λ. We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type C ̃(p−1)/2 × C ̃(q−1)/2. We also provide an algorithm for constructing a bar partition in this set with a given p-bar-core and q-bar-core
Spectral properties of random matrices
In the first part of this thesis, we give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution we will be working with. The main technical tool of the first paper is also defined rigorously and analyzed deeply, which is the Stieltjes transform method. We then use this tool to prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix as well as about the rate of their convergence. In the second part of this thesis, we define another important and more general technical tool which works additionally well with non-Hermitian random matrices and that is the Dyson equation method which was used in the second paper. Just like the Stieltjes transform method, it is also defined rigorously and analyzed deeply. We then prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can now be considered universal, which means that they are independent of the initial distribution of the random matrix entries. We finally give an overview of our new results and provide new directions of study
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Pointless Global Bundle Adjustment With Relative Motions Hessians
Bundle adjustment (BA) is the standard way to optimise camera poses and to
produce sparse representations of a scene. However, as the number of camera
poses and features grows, refinement through bundle adjustment becomes
inefficient. Inspired by global motion averaging methods, we propose a new
bundle adjustment objective which does not rely on image features' reprojection
errors yet maintains precision on par with classical BA. Our method averages
over relative motions while implicitly incorporating the contribution of the
structure in the adjustment. To that end, we weight the objective function by
local hessian matrices - a by-product of local bundle adjustments performed on
relative motions (e.g., pairs or triplets) during the pose initialisation step.
Such hessians are extremely rich as they encapsulate both the features' random
errors and the geometric configuration between the cameras. These pieces of
information propagated to the global frame help to guide the final optimisation
in a more rigorous way. We argue that this approach is an upgraded version of
the motion averaging approach and demonstrate its effectiveness on both
photogrammetric datasets and computer vision benchmarks
Deformation theory of G-valued pseudocharacters and symplectic determinant laws
We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and
Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are
very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group,
where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are
so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a
generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion
of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters.
We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant
laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from
small characteristics.
We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously,
we obtain that tensor products of determinant laws exist and are compatible with tensor products
of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor
product of Taylor’s pseudocharacters.
We generalize some of the results of [Che14] to general reductive groups, in particular we show that
the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely
generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show,
that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Φ_p.
One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp
be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and
concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater
of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension
n(2n + 1)[F : Qp].
We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected
reductive groups G come from a semisimple representation. We introduce a formal scheme
and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on
our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition
as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to
conjugacy and Frobenius automorphisms.
At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic
groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under
an involution. We prove the expected basic properties in that we show that symplectic determinant laws
over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and
that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings
are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s
pseudocharacters and show that it is an isomorphism over reduced rings
Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings
This paper is devoted to the construction of approximations of the propagator
associated with a semi-classical matrix-valued Schr\"odinger operator with
symbol presenting smooth eigenvalues crossings. Inspired by the approach of the
theoretical chemists Herman and Kluk who propagated continuous superpositions
of Gaussian wave-packets for scalar equations, we consider frozen and thawed
Gaussian initial value representations that incorporate classical transport and
branching processes along a hopping hypersurface. Based on the Gaussian
wave-packet framework, our result relies on an accurate analysis of the
solutions of the associated Schr\"odinger equation for data that are
vector-valued wave-packets. We prove that these solutions are asymptotic to
wavepackets at any order in terms of the semi-classical parameter
Learning and Control of Dynamical Systems
Despite the remarkable success of machine learning in various domains in recent years, our understanding of its fundamental limitations remains incomplete. This knowledge gap poses a grand challenge when deploying machine learning methods in critical decision-making tasks, where incorrect decisions can have catastrophic consequences. To effectively utilize these learning-based methods in such contexts, it is crucial to explicitly characterize their performance. Over the years, significant research efforts have been dedicated to learning and control of dynamical systems where the underlying dynamics are unknown or only partially known a priori, and must be inferred from collected data. However, much of these classical results have focused on asymptotic guarantees, providing limited insights into the amount of data required to achieve desired control performance while satisfying operational constraints such as safety and stability, especially in the presence of statistical noise.
In this thesis, we study the statistical complexity of learning and control of unknown dynamical systems. By utilizing recent advances in statistical learning theory, high-dimensional statistics, and control theoretic tools, we aim to establish a fundamental understanding of the number of samples required to achieve desired (i) accuracy in learning the unknown dynamics, (ii) performance in the control of the underlying system, and (iii) satisfaction of the operational constraints such as safety and stability. We provide finite-sample guarantees for these objectives and propose efficient learning and control algorithms that achieve the desired performance at these statistical limits in various dynamical systems. Our investigation covers a broad range of dynamical systems, starting from fully observable linear dynamical systems to partially observable linear dynamical systems, and ultimately, nonlinear systems.
We deploy our learning and control algorithms in various adaptive control tasks in real-world control systems and demonstrate their strong empirical performance along with their learning, robustness, and stability guarantees. In particular, we implement one of our proposed methods, Fourier Adaptive Learning and Control (FALCON), on an experimental aerodynamic testbed under extreme turbulent flow dynamics in a wind tunnel. The results show that FALCON achieves state-of-the-art stabilization performance and consistently outperforms conventional and other learning-based methods by at least 37%, despite using 8 times less data. The superior performance of FALCON arises from its physically and theoretically accurate modeling of the underlying nonlinear turbulent dynamics, which yields rigorous finite-sample learning and performance guarantees. These findings underscore the importance of characterizing the statistical complexity of learning and control of unknown dynamical systems.</p
A lift of West's stack-sorting map to partition diagrams
We introduce a lifting of West's stack-sorting map to partition diagrams,
which are combinatorial objects indexing bases of partition algebras. Our
lifting of is such that behaves in the same way
as when restricted to diagram basis elements in the order- symmetric
group algebra as a diagram subalgebra of the partition algebra
. We then introduce a lifting of the notion of
-stack-sortability, using our lifting of . By direct analogy with Knuth's
famous result that a permutation is -stack-sortable if and only if it avoids
the pattern , we prove a related pattern-avoidance property for partition
diagrams, as opposed to permutations, according to what we refer to as
stretch-stack-sortability.Comment: Submitted for publicatio
Euclidean Contractivity of Neural Networks with Symmetric Weights
This paper investigates stability conditions of continuous-time Hopfield and
firing-rate neural networks by leveraging contraction theory. First, we present
a number of useful general algebraic results on matrix polytopes and products
of symmetric matrices. Then, we give sufficient conditions for strong and weak
Euclidean contractivity, i.e., contractivity with respect to the norm,
of both models with symmetric weights and (possibly) non-smooth activation
functions. Our contraction analysis leads to contraction rates which are
log-optimal in almost all symmetric synaptic matrices. Finally, we use our
results to propose a firing-rate neural network model to solve a quadratic
optimization problem with box constraints.Comment: 16 pages, 2 figure
An inverse spectral problem for non-self-adjoint Jacobi matrices
We consider the class of bounded symmetric Jacobi matrices J with positive off-diagonal elements and complex diagonal elements. With each matrix J from this class, we associate the spectral data, which consists of a pair (ν, ψ). Here ν is the spectral measure of |J| = √ J ∗J and ψ is a phase function on the real line satisfying |ψ| ≤ 1 almost everywhere with respect to the measure ν. Our main result is that the map from J to the pair (ν, ψ) is a bijection between our class of Jacobi matrices and the set of all spectral data
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