arXiv research

Random representations of surface groups approach asymptotic freeness in large $n$ limit.

Expected centre of mass for random embeddings is constant.

Quantum neural networks converge to Gaussian processes as they grow.

QCNNs avoid barren plateaus, making them trainable.

Improved bounds on moments of word measures on unitary groups.

Random flat bundles on surfaces have least eigenvalues at least 1/4.

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free…

Quantum walks blend patterns into splines when averaged.

We show that for any positive integer $n$, the maps $x \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of emb…

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group $\mathcal{U}\left(n…

This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.

The paper proves a distribution claim for neural network Jacobians.

Graph Neural Networks (GNNs) have become a topic of intense research recently due to their powerful capability in high-dimensional classification and regression tasks for graph-structured data. However, as GNNs typically define the graph convolution by the orthonormal basis for the graph Laplacian, they suffer from hig…

We consider the volumes of classical supermanifolds such as the supersphere, complex projective superspace, and Stiefel and Grassmann supermanifolds, with respect to the natural metrics or symplectic structures. We show that the formulas for the volumes, upon certain universal normalization, can be obtained by an analy…

Pairs of elements in quaternionic hyperbolic space have zero measure of being strongly doubly reversible.

The study finds effective lower bounds for spectra of random surfaces and bundles.

Novel Haar-Laplacian for directed graphs enhances spectral graph applications.

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connecti…

Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.

Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.

Deep Graph Neural Networks (GNNs) are useful models for graph classification and graph-based regression tasks. In these tasks, graph pooling is a critical ingredient by which GNNs adapt to input graphs of varying size and structure. We propose a new graph pooling operation based on compressive Haar transforms -- HaarPo…

Hyperbolic groups' infinite orbits spread evenly in spaces.

This paper proposes a new methodology to compute Value at Risk (VaR) for quantifying losses in credit portfolios. We approximate the cumulative distribution of the loss function by a finite combination of Haar wavelets basis functions and calculate the coefficients of the approximation by inverting its Laplace transfor…

Paper presents a new VMBQC model with fewer parameters for better generative modeling.

In this article we propose building general-purpose function approximators on top of Haar Scattering Networks. We advocate that this architecture enables a better comprehension of feature extraction, in addition to its implementation simplicity and low computational costs. We show its approximation and feature extracti…

We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat con…

The aim of this paper is to discuss the use of Haar scattering networks, which is a very simple architecture that naturally supports a large number of stacked layers, yet with very few parameters, in a relatively broad set of pattern recognition problems, including regression and classification tasks. This architecture…

Study symmetry breaking in quantum mechanics to understand many-body physics.

In this letter we exhibit the relation between the isometries of a Riemannian contraction of a sub-Riemannian manifold and those of the sub-Riemannian metric, for to use this relation with two goals: establishing a result about the existence of fixed points of isometries groups; and the other, defining a Multiresolutio…

We analyze in this paper a random feature map based on a theory of invariance I-theory introduced recently. More specifically, a group invariant signal signature is obtained through cumulative distributions of group transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as …

New MBQC algorithm uses randomness for generative modeling.

Random matrix ensembles yield uniform distributions on manifolds.

Optimized sampling scheme for compressed sensing combining randomness and determinism.

Modeling functional data, this study uncovers the size-and-shape of functions under noisy observations.

New invariants derived from random matrices for words in free groups.

One-bit clustering method for two-component sub-Gaussian mixture models

Paper improves tensor completion by reducing sample entries needed.

Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.

This paper presents a basic property of region dividing of ReLU (rectified linear unit) deep learning when new layers are successively added, by which two new perspectives of interpreting deep learning are given. The first is related to decision trees and forests; we construct a deep learning structure equivalent to a …

Quantum methods model uncertain volatility in financial markets.

HD algorithm simulates dynamics on random matrix ensembles without generating full matrices.

Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.

Automatic continuity of polynomial maps and cocycles proved.

The abstract theorem extends a Lie group result to Lie groupoids.

In this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups.

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

Contact group retracts to unitary subgroup.

Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.