The paper defines Haar system preserving morphisms and applies them to groupoid C∗-algebras.
problem Understanding and constructing inverse systems of groupoids.
method Defining Haar system preserving morphisms and using them to induce *-morphisms between convolution algebras.
result Inverse systems of groupoids with Haar system preserving bonding maps have limits, and corresponding direct systems of groupoid C∗-algebras. This paper introduces Haar convolution for GNNs to reduce computational cost.
problem High computational cost in GNNs for large graph sizes.
method Introduces Haar basis for graph convolution and Fast Haar Transforms.
result State-of-the-art results on graph-based regression and node classification tasks.
Proposes Haar Scattering Networks for better feature extraction.
problem Improving feature extraction in various signal processing and system identification tasks.
method Building function approximators on top of Haar Scattering Networks.
result Demonstrates the architecture's effectiveness in multiple fields.
HaarPooling compresses graphs by Haar transforms, improving graph classification and regression.
problem Handling graphs of varying size and structure in GNNs.
method HaarPooling, a cascade of clusterings and compressive Haar transforms.
result HaarPooling synthesizes graph features into uniform size, achieving state-of-the-art performance.
Novel Haar-Laplacian for directed graphs enhances spectral graph applications.
problem Lack of suitable Laplacian for directed graphs in spectral graph theory.
method Inspired by Haar-like transformation, introduces a Hermitian matrix preserving direction and weight.
result HaarNet outperforms in weight prediction and denoising on directed graphs.
Haar scattering networks improve pattern recognition across various tasks.
problem Improving pattern recognition in diverse tasks like regression and classification.
method Stacking convolutional filters based on Haar wavelets followed by non-linear operators.
result Outperformed best algorithms in 4 out of 18 data classification problems.
This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
problem Proving the asymptotic freeness of layerwise Jacobians in multilayer perceptrons (MLPs).
method Replacing each layer's parameter matrix with itself multiplied by a Haar orthogonal matrix, and using the invariance of the MLP.
result Proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
Paper interprets deep learning using decision trees and Haar wavelets.
problem Understanding the function approximation capabilities of ReLU deep learning.
method Constructing a deep learning structure equivalent to a forest and approximating Haar wavelet functions with ReLU deep learning.
result ReLU deep learning can be considered as decision trees and approximates Haar wavelet functions with arbitrary precision.
Hyperbolic groups' infinite orbits spread evenly in spaces.
problem Equidistribution of hyperbolic groups in homogeneous spaces.
method Averaging measures along spheres in Cayley graphs converges to Haar measure.
result Infinite orbits of hyperbolic groups equidistribute in homogeneous 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…
Quantum walks blend patterns into splines when averaged.
problem Understanding the asymptotic patterns of quantum random walks.
method Averaging over quantum coins using the Haar measure.
result Patterns blend into splines, showing a unified behavior.
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…
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…
Introduces new algebraic structures for relational groupoids and proves a reduction theorem.
problem Developing algebraic tools for relational groupoids.
method Introduces relational groupoids and convolution algebras, provides examples, and proves a reduction theorem.
result Establishes a reduction theorem recovering the usual convolution of Lie groupoids.
We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid e…
New proof of graph covering theorem for graphs with fins.
problem Graph covering theorem for graphs with fins.
method New proof using Haar measure.
result Pattern rigidity for free groups with line patterns.
Quantum neural networks converge to Gaussian processes as they grow.
problem Understanding the convergence of quantum neural networks to Gaussian processes.
method Analyzing Haar random unitary and orthogonal deep QNNs, considering input states, measurement observables, and non-independence of unitary matrix entries.
result Quantum neural networks outputs converge to Gaussian processes in the limit of large Hilbert space dimension.
Expected centre of mass for random embeddings is constant.
problem Understanding the expected centre of mass for random embeddings.
method Analyzing the Haar measure and Gaussian unitary ensemble on SL(N, C).
result The expectation of the centre of mass is a constant multiple of the identity matrix.
The paper proves a distribution claim for neural network Jacobians.
problem Distribution of singular values in deep neural networks.
method Free probability and random matrix theory techniques.
result Singular value distribution matches for specific cases.
Paper uses deep learning to suppress bones on chest X-rays.
problem Improving pathologies classification by suppressing bones on chest X-rays.
method Conditional Generative Adversarial Network (GAN) and Haar 2D wavelet decomposition.
result Achieves state-of-the-art performance on bone suppression.
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
The abstract theorem extends a Lie group result to Lie groupoids.
problem Expressing functions on Lie groupoids as convolutions of two functions.
method Using a lemma from Dixmier-Malliavin, Lie algebroids, and exponential map.
result Every smooth, compactly-supported function on a Lie groupoid can be expressed as a finite sum of convolutions of two such functions.
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.
Random representations of surface groups approach asymptotic freeness in large n limit.
problem Asymptotic freeness of Haar unitary matrices for surface groups.
method Interplay between Dehn's work and classical invariant theory.
result Expected value of trace of a fixed non-identity element is bounded as no∞. Paper introduces Laplace-HDC for better binary hyperdimensional computing.
problem Improving binary hyperdimensional computing for spatial information.
method Develops Laplace-HDC using the Laplace kernel and Haar convolutional features.
result Laplace-HDC outperforms previous methods in encoding spatial information.
QCNNs avoid barren plateaus, making them trainable.
problem Exponentially vanishing gradients in QNNs.
method Graph-based method to analyze Haar-distributed unitaries.
result QCNNs do not exhibit barren plateaus, implying trainability.
We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms…
MathNet uses wavelets for graph representation and learning.
problem Graph Neural Networks (GNNs) for graph classification and regression.
method Multiresolution Haar-like wavelets, graph convolution, and pooling.
result MathNet achieves notable accuracy gains on graph classification and regression tasks.
Improved bounds on moments of word measures on unitary groups.
problem Analyzing the moments of word measures on unitary groups.
method Using Haar-uniform sampling and algebraic invariants to study asymptotic behavior.
result Found a new algebraic invariant related to the moments of word measures.
The paper reformulates U-Nets as wavelet-based models and applies this to hierarchical VAEs.
problem Theoretical understanding and regularization properties of U-Nets and their relationship to wavelets.
method Formulating a multi-resolution framework to identify U-Nets as finite-dimensional truncations of infinite-dimensional models, proving average pooling corresponds to projection, and identifying HVAEs as discretizations of multi-resolution diffusion processes.
result HVAEs learn a time representation allowing for improved parameter efficiency through weight-sharing.
Counting spheres in hyperbolic space with effective methods.
problem Counting spheres in Apollonian and Kleinian packings.
method Spectral methods and orbit counting, extending Kontorovich and Lax-Phillips techniques.
result Best-known effective error rate for sphere packing counting problems.
One-bit clustering method for two-component sub-Gaussian mixture models
problem Clustering in sub-Gaussian mixture models
method One-bit clustering using dithered quantization
result Decaying misclassification rate with exponential signal-to-noise ratio
Pairs of elements in quaternionic hyperbolic space have zero measure of being strongly doubly reversible.
problem Characterizing pairs of elements in quaternionic hyperbolic space that are strongly doubly reversible.
method Analyzing conjugacy conditions and using Haar measure.
result The set of strongly doubly reversible pairs has Haar measure zero in $\PSp(n,1) imes \PSp(n,1)$.
Develops new e-processes and confidence sequences for Gaussian means with unknown variance.
problem Constructing valid t-tests and confidence sequences for Gaussian means with unknown variance.
method Explores generalized nonintegrable martingales and extended Ville's inequality, developing two new e-processes and confidence sequences.
result Analyzes the width of resulting confidence sequences with a polynomial dependence on error probability, proving it to be unavoidable and even better than classical fixed-sample t-tests.
Study proves existence of regions minimizing perimeter in specific geometric structures.
problem Existence of isoperimetric regions in sub-Finsler nilpotent groups.
method Analyzes nilpotent Lie groups with a bracket-generating distribution and asymmetric norms.
result Proves existence of minimizers of perimeter under volume constraint.
Deep neural networks' Jacobian spectrum becomes well-conditioned with orthogonal weights.
problem Understanding and handling the Jacobian spectrum of deep neural networks.
method Applying free probability theory to show almost sure asymptotic freeness of Jacobians in the wide limit.
result Layer-wise Jacobians of deep neural networks with orthogonal weights are almost surely asymptotically free.
The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a …
Polynomial-time algorithm forecasts TV-bounded sequences with optimal error rate.
problem Online forecasting of sequences with bounded total variation under noisy observations.
method Designing an O(nlogn)-time algorithm leveraging Haar wavelet basis and adaptivity. result Achieves optimal O(n1/3) cumulative square error with high probability. We perform wavelet decomposition of high frequency financial time series into large and small time scale components. Taking the FTSE100 index as a case study, and working with the Haar basis, it turns out that the small scale component defined by most (≃ 99.6%) of the wavelet coefficients can be neglected for th…
HD algorithm simulates dynamics on random matrix ensembles without generating full matrices.
problem Simulating dynamics on dense random matrix ensembles with high space and time complexity.
method Householder reflectors for adaptive and recursive construction, deferring decisions.
result Significant reductions in runtime and memory footprint for practical T≪n. Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su(m) and pro…
New theory connects random matrices to surface graphs and mapping class groups.
problem Understanding moments of measures induced by free words on unitary matrices.
method Study measures induced by free words on U(n) and relate to surfaces and mapping class groups.
result Every moment of the measure on U(n) is determined by pairs (Σ, f) involving surfaces and maps.
New neural network outperforms existing methods in scene matching.
problem Automated scene matching with high accuracy and low false positives.
method Convolutional hashing using a new loss function and training scheme.
result Significantly higher true positive rate and 100-fold reduction in false positives.
The paper improves on existing algorithms for minimizing different types of regret in online learning.
problem Minimizing external, internal, and swap regret in online learning with multiple experts.
method Develops a single algorithm using φ-regret minimization and Haar-wavelet-inspired matrix features to achieve optimal bounds in various scenarios.
result Achieves optimal bounds for external, internal, and swap regrets in different expert scenarios.
One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe h…
Formula calculates higher moments of Siegel-Veech transform over Hecke triangle groups.
problem Computing higher moments of Siegel-Veech transform over specific groups.
method Geometric results and linear algebra to create integration formulas.
result Explicit integration formulas for densities of vector orbits.
A method based on wavelet transform and genetic programming is proposed for characterizing and modeling variations at multiple scales in non-stationary time series. The cyclic variations, extracted by wavelets and smoothened by cubic splines, are well captured by genetic programming in the form of dynamical equations. …
Proposes PT-MMD for evaluating generative models.
problem Evaluating generative models under implementation constraints.
method Combines MMD and PT resampling for statistical evaluation.
result Demonstrates effectiveness in model selection and image fidelity.