Paper introduces RKHM for more explicit variable structures analysis.
problem Explicitly analyzing structures among variables.
method Orthonormal systems in Hilbert C∗-modules, RKHM. result Theoretical and practical procedures for RKHM orthonormalization.
Improves Bayesian optimization efficiency for mixed variable spaces.
problem Boosting sample efficiency in Bayesian optimization for mixed variable spaces.
method Proposes frequency modulated (FM) kernels to model complex dependencies across different types of variables.
result BO-FM outperforms competitors in various optimization problems.
Study the module structure of homology of Artin kernels.
problem Characterize the module structure of homology of Artin kernels.
method Use flag complex and double covers of toric complexes to analyze properties of torsion part.
result Determine dimensions and sizes of Jordan forms of the torsion part.
Paper introduces RKHM and KME for richer data analysis.
problem Lack of rich data structures in kernel methods.
method Proposes RKHM and KME for functional data analysis.
result RKHM captures structural properties in functional data.
New rough stochastic volatility models using log-modulated fractional Brownian motion.
problem Analyzing rough stochastic volatility models over the range 0≤H<1/2. method Introducing log-modulated fractional Brownian motion (log-fBm) to handle H=0 and analyze over the full range. result Obtained skew asymptotics of log(1/T)−pTH−1/2 as To0 for H≥0, no flattening of skew as Ho0. KM method reduces ConvNet parameters to 9% higher accuracy with minimal additional memory.
problem Expensive memory usage for training ConvNets on embedded devices.
method Kernel Modulation (KM) method that adapts all network parameters for each task.
result KM delivers up to 9% higher accuracy than other parameter-efficient methods.
New approach to supervised learning in RKHS and vvRKHS using C∗-algebras.
problem Traditional supervised learning in RKHS and vvRKHS.
method Generalizing supervised learning to RKHM using C∗-algebras. result Constructing RKHMs with enhanced representation power.
We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial unipotent Tg-module for all g≥4 and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when g≥6. We do this by proving that, for a finitely generated group G satisfying an assumpti…
Proves conjecture simplifying mapping class group action on Steinberg module.
problem Characterizing action of mapping class group on Steinberg module.
method Simple characterisation of mapping class group action on Harvey's complex of curves.
result Kernel of mapping class group action on Steinberg module is trivial.
Paper generalizes kernel mean embedding to von Neumann-algebra-valued measures.
problem Analyzing complex multivariate distributions and quantum mechanics.
method Generalizes kernel mean embedding to von Neumann-algebra-valued measures in reproducing kernel Hilbert modules.
result Injectivity and universality of the generalized KME are confirmed.
For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skei…
Extends Kauffman's formula to 3-manifolds with markings.
problem Generalizing Jones polynomial to 3-manifolds with markings.
method Epimorphism between skein modules of tangles in 3-manifolds.
result Defines new skein modules measuring the difference between Jones and bracket skein modules.
A commuting n-tuple (T1,…,Tn) of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module H over C[z1,…,zn] in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…
The paper studies the structure of a specific homology group related to mapping class groups.
problem Understanding the structure of a specific homology group of the Johnson kernel.
method Constructing and analyzing abelian cycles to describe the module structure.
result Described the structure of the subgroup of the homology group generated by simplest abelian cycles and found relations between them.
Modular deep learning framework using pairwise labels without backpropagation.
problem Efficiently training deep neural networks with limited supervision.
method Stacked linear models in feature spaces, provably optimal modular learning framework.
result High accuracy (94.88%) achieved with minimal labeled examples (1 per class).
Complex analysis techniques link Gaussian RBF kernels to quantum mechanics.
problem Understanding the Gaussian RBF kernel in machine learning and SVMs.
method Using Fock space and Segal-Bargmann theories in complex analysis.
result Proves connections between Gaussian RBF kernels and quantum mechanics operators.
Mode decomposition is a prototypical pattern recognition problem that can be addressed from the (a priori distinct) perspectives of numerical approximation, statistical inference and deep learning. Could its analysis through these combined perspectives be used as a Rosetta stone for deciphering mechanisms at play in de…
The problem of accurately measuring the similarity between graphs is at the core of many applications in a variety of disciplines. Graph kernels have recently emerged as a promising approach to this problem. There are now many kernels, each focusing on different structural aspects of graphs. Here, we present GraKeL, a …
The aim of this note is to introduce the notion of a D-Lie algebra and to prove some elementary properties of D-Lie algebras, the category of D-Lie algebras, the category of modules on a D-Lie algebra and extensions of D-Lie algebras. …
The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) G-module, where G-modules are struc…
In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module A→1. The cohomology of the Lie 2-groups corresponding to the universal crossed modules $G\to \Aut(G)$ and $G\to \Aut^+(G)$ is the abutment of …
Efficiently accelerates attention calculation for Transformers with relative positional encoding.
problem Quadratic complexity of attention in long sequences.
method Kernelized attention with Fast Fourier Transform (FFT) for RPE.
result Achieves O(n log n) time complexity, mitigates training instability, and outperforms other models.
Proves an equivariant version of Heegaard Floer link surgery formula.
problem Equivariant knot surgery in S3. method Naturality theorem for bordered modules.
result Kernel of forgetful map contains a Z∞-summand. Classifies and constructs intertwining differential operators between vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between vector bundles over RP2. method Utilizes SL(3,R)-intertwining differential operators, BGG resolution, and representation theory. result Irreducible unitary highest weight modules of SU(1,2) at reduction points classified by Cartan and PRV operators. Deep learning framework for kernel methods using RKHM and Perron-Frobenius operators.
problem Kernel methods in deep learning with potential overfitting issues.
method Combining RKHM and Perron-Frobenius operator to derive a new Rademacher bound and analyze deep kernel methods.
result Theoretical interpretation of benign overfitting and milder dependency on output dimension.
This paper improves GNNs' generalization by adding a Low-Rank Global Attention module.
problem Improving the generalization power of Graph Neural Networks (GNNs).
method Incorporating a Low-Rank Global Attention (LRGA) module into GNNs.
result Augmenting GNNs with LRGA aligns them with a powerful graph isomorphism test, 2-Folklore Weisfeiler-Lehman (2-FWL).
Establishes a connection between skein modules and algebraic sets.
problem Understanding the structure of stated SLn-skein modules. method Uses algebraic homomorphisms and isomorphisms to relate skein modules to algebraic structures.
result Proves the isomorphism between stated skein modules and universal representation algebras.
Random forests with attention and self-attention improve regression performance.
problem Improving regression model performance on various datasets.
method Proposes new models using attention and self-attention mechanisms to solve regression problems.
result The models improve model performance on many datasets.
We propose a novel class of Gaussian processes (GPs) whose spectra have compact support, meaning that their sample trajectories are almost-surely band limited. As a complement to the growing literature on spectral design of covariance kernels, the core of our proposal is to model power spectral densities through a rect…
A novel algorithm for unbiased graph kernel estimation with subquadratic time complexity.
problem Efficient estimation of graph kernels for large networks.
method Random walk-based algorithm with modulation function parameterized by neural network.
result Higher-quality kernel estimates and efficient scalable learning on larger networks.
Spectral mixture (SM) kernels comprise a powerful class of generalized kernels for Gaussian processes (GPs) to describe complex patterns. This paper introduces model compression and time- and phase (TP) modulated dependency structures to the original (SM) kernel for improved generalization of GPs. Specifically, by adop…
Inference of space-time varying signals on graphs emerges naturally in a plethora of network science related applications. A frequently encountered challenge pertains to reconstructing such dynamic processes, given their values over a subset of vertices and time instants. The present paper develops a graph-aware kernel…
In classical Hawkes process, the baseline intensity and triggering kernel are assumed to be a constant and parametric function respectively, which limits the model flexibility. To generalize it, we present a fully Bayesian nonparametric model, namely Gaussian process modulated Hawkes process and propose an EM-variation…
DoRA improves adaptation efficiency for large models by factoring norms and fusing kernels.
problem High-rank DoRA is computationally expensive and infeasible on common GPUs.
method Factored norms and fused Triton kernels to reduce memory and speed up computation.
result Fused implementation is up to 2.0x faster for inference and 1.9x faster for gradient computation.
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…
UT module refines VAE latent space, improving disentanglement and interpretability.
problem Irregular latent distributions cause posterior collapse and misalignment in VAEs.
method UT module uses G-KDE clustering, GM modeling, and PIT to transform latent space into uniform distribution.
result UT module enhances disentanglement and interpretability of latent representations.
New non-separable covariance kernels for spatiotemporal data derived from harmonic oscillator physics.
problem Capturing complex spatiotemporal dependencies in Gaussian processes.
method Hybrid spectral method based on the harmonic oscillator, deriving explicit covariance kernels.
result Explicit non-separable covariance kernels with space-time interactions.
New homomorphisms from homology cylinders to torsion modules via LMO functor.
problem Understanding the abelian quotients of homology cylinders.
method Constructing homomorphisms via LMO functor and analyzing their properties.
result Abelianization of Johnson kernel has torsion elements.
In this paper we propose the use of continuous residual modules for graph kernels in Graph Neural Networks. We show how both discrete and continuous residual layers allow for more robust training, being that continuous residual layers are those which are applied by integrating through an Ordinary Differential Equation …
AaSP improves audio self-supervised learning by addressing aliasing issues.
problem Alias issues in audio spectrogram transformers.
method AaSP combines aliasing-aware patch representation, teacher-student masked modeling, cross-attention predictor, and contrastive regularization.
result AaSP learns more stable representations that integrate high-frequency cues.
LOBRM model recreates limit order books from trade and quote data.
problem Lack of LOB data and limitations in LOBRM model.
method Extended LOBRM with time-weighted z-score standardization and exponential decay kernel, conducted in chronological order.
result LOBRM with decay kernel outperforms traditional models and module ensembling is effective.
ML4Chem offers a user-friendly platform for developing and deploying machine learning models in chemistry.
problem Developing and deploying machine learning models in chemistry and materials science.
method User-experience design, six core building blocks: data, featurization, models, model optimization, inference, and visualization.
result Ease of use and functionality of the atomistic module for neural networks and kernel ridge regression.
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism φfrom F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invari…
For n an even number, we study representations of the mapping class group of the n-punctured sphere arising from SU(2)-TQFT when all punctures are colored by the same integer N≥1. We prove that the conjecture of Andersen, Masbaum and Ueno holds for the 4-punctured sphere for all N≥2. In t…
Stability of catenoid in hyperbolic space proven without symmetry assumptions.
problem Stability of catenoid in hyperbolic space.
method Profile construction, modulation analysis, integrated local energy decay, vectorfield method.
result Nonlinear asymptotic stability of catenoid for n≥5 without symmetry assumptions. Deep neural-kernel models combine neural networks and kernel machines for scalable large datasets.
problem Combining neural networks and kernel machines for efficient large-scale learning.
method Hybrid neural-kernel architecture using explicit feature mapping and pooling layers.
result The deep neural-kernel models are effective and scalable on benchmark datasets.
FFN addresses spectral bias in neural value approximation, improving reinforcement learning performance.
problem Spectral bias in neural value approximation, leading to slow convergence and poor performance.
method Proposes Fourier feature networks (FFN) to overcome spectral bias by using a composite neural tangent kernel.
result FFN achieves state-of-the-art performance on challenging continuous control domains with faster convergence and better stability.
In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of J…