A new kernel improves tensor classification accuracy and reduces computation time.
problem Challenges in classifying high-dimensional tensor data.
method Proposes a weighted subspace exponential kernel based on Tucker decomposition.
result The new kernel outperforms existing methods in accuracy and computational efficiency.
A novel Laplace-approximated Bayesian Tensor Network Kernel Machine (LA-TNKM) provides principled uncertainty estimates.
problem How to provide principled uncertainty estimates for tensor network kernel machines.
method Employing a linearized Laplace approximation for Bayesian inference.
result Consistently matches or surpasses Gaussian Processes and BNNs across diverse UCI regression benchmarks.
Estimates spatio-temporal Hawkes processes using tensor recovery.
problem Estimating influence functions for spatio-temporal Hawkes processes.
method Formulates influence function as a tensor kernel, assumes low-rank structure, solves as convex optimization problem.
result Provides theoretical guarantees and demonstrates efficiency with simulations.
Tensor networks constrain kernel machines to Gaussian processes.
problem Speeding up kernel machines with reduced model complexity.
method Proving CPD and TT-constrained models recover Gaussian processes with i.i.d. priors.
result TT-constrained models exhibit more Gaussian process behavior than CPD for the same parameters.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation for high-dimensional functional MRI and dynamic graph recovery.
method Reformulates imputation as RKHS regression with TT-constrained coefficients and Hadamard overparameterization. Optimizes TT coefficients and kernel matrices on Riemannian manifolds.
result Consistently outperforms state-of-the-art methods in modeling accuracy.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation in high-dimensional spaces.
method Reformulates imputation as RKHS regression with TT-constrained coefficients, optimized on manifold frameworks.
result Consistently outperforms state-of-the-art methods in accuracy.
In this paper we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel-Rockafellar duality theory, we give explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving…
A new method reduces Volterra kernel complexity and uncertainty quantification.
problem Challenges in modeling nonlinear systems with Volterra series due to high model order.
method Bayesian Tensor Network Volterra kernel machines (BTN-V) using canonical polyadic decomposition.
result Competitive accuracy, enhanced uncertainty quantification, and reduced computational cost.
Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. Traditional machine learning approaches are vector- or matrix-based, and cannot handle tensorial data directly. In this paper, we propose a tensor train (TT)-based kernel technique for the first time, and apply it…
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
Bayesian tensor train kernel machine uses Laplace approximation for scalable GP regression.
problem Scalability limitations of Gaussian process regression.
method Bayesian tensor train kernel machine with Laplace approximation and variational inference.
result VI replaces cross-validation and offers up to 65x faster training.
Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.
problem Asymptotic behavior of Bergman kernels for high tensor powers of positive line bundles.
method Analyzing the Schwartz kernel of the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector, proving exponential estimates and asymptotic expansions.
result Explicit asymptotic expansions for the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector.
Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. T…
This work improves fair tensor decomposition using a kernel criterion.
problem Learning fair low-rank tensor decompositions with statistical parity.
method Regularizes Canonical Polyadic Decomposition with KHSIC to ensure approximate statistical parity.
result The proposed algorithm achieves better fairness and fit than state-of-the-art FATR.
Efficient tensor kernel method reduces memory usage and computational cost for sparse regression.
problem Memory and computational limitations in tensor kernel methods for sparse regression.
method Proposes a new tensor data layout and Nystrom subsampling approach to reduce memory and computational requirements.
result Improvements lead to more efficient tensor kernel methods for sparse regression.
The X-ray transform on the periodic slab [0,1]×Tn, n≥0, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless n=0. We characterize t…
Bayesian TNKMs automatically infer model complexity and feature relevance.
problem Manual tuning of TN rank and feature dimensions is error-prone and computationally expensive.
method Bayesian approach with hierarchical priors on TN factors for automatic rank and feature selection.
result Superior performance in prediction accuracy, uncertainty quantification, interpretability, and scalability.
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However, these techniques require tensor decomposition which could lead to the loss of infor…
In this paper, we discuss how a suitable family of tensor kernels can be used to efficiently solve nonparametric extensions of ℓp regularized learning methods. Our main contribution is proposing a fast dual algorithm, and showing that it allows to solve the problem efficiently. Our results contrast recent finding…
Proposes a new method for high-dimensional density estimation.
problem Estimating high-dimensional probability density functions efficiently.
method Tensorizing flow method combining tensor-train and flow-based generative modeling.
result Efficiently constructs an approximate density in tensor-train form and trains a flow model to match empirical distribution.
A new method reduces the complexity of tensor products from cubic to quadratic, improving both speed and accuracy.
problem Efficiently computing high-dimensional tensor products for polynomial kernels.
method Complex-to-Real (CtR) modification of sketches using complex random projections.
result Achieves state-of-the-art performance in accuracy and speed.
Efficiently fine-tunes patient-independent seizure detection models with tensor kernel machine.
problem Improving seizure detection accuracy for wearable devices.
method Transfer learning with tensor kernel machine using canonical polyadic decomposition.
result Patient fine-tuned model achieves high performance with smaller model size.
BKTR models spatiotemporal data with scalable tensor regression.
problem High computational cost in applying STVC to large-scale spatiotemporal data.
method Summarize STVC coefficients in a tensor, reformulate as low-rank tensor regression, incorporate GP priors for local dependencies.
result BKTR efficiently models large spatiotemporal datasets with reduced parameters and local dependencies.
A Python package for GPU-accelerated signature kernel computation.
problem Efficient computation of signature kernels for sequential data.
method GPU-accelerated algorithms and tensor sketches.
result New algorithm outperforms existing methods.
New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
BKTF uses tensor factorization for Bayesian optimization of complex functions.
problem Complex functions with nonstationary, nonseparable, and multimodal features.
method Bayesian Kernelized Tensor Factorization (BKTF) approximates complex functions using a low-rank tensor CP decomposition with GP priors.
result BKTF provides flexible and effective surrogate modeling with uncertainty quantification.
Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
Kernelized cumulants improve statistical analysis in high-dimensional spaces.
problem Statistical analysis in high-dimensional spaces with low variance estimators.
method Extending cumulants to RKHS using tensor algebra and kernel trick.
result Kernelized cumulants provide new all-purpose statistics with computational tractability.
We study the asymptotic of the Bergman kernel of the spinc Dirac operator on high tensor powers of a line bundle.
Accelerates signature kernel computation for sequences.
problem Severe computational bottleneck in computing signature kernel.
method Random Fourier features to accelerate signature kernel computation.
result Uniform approximation guarantees for unbiased estimator with linear computation time.
Paper learns meaningful state and action representations from MDP trajectories.
problem Learning good state and action representations from MDP trajectories.
method Tensor decomposition, kernelization, importance sampling, low-Tucker-rank approximation.
result The learned state/action abstractions provide accurate approximations to latent block structures.
Paper tackles tensor decomposition for unaligned observations using RKHS and novel loss functions.
problem Tackles tensor decomposition for unaligned observations.
method Uses functions in RKHS to represent mode with unaligned observations, introduces versatile loss function, proposes optimization algorithm and stochastic gradient method.
result Demonstrates improved tensor decomposition efficiency and effectiveness with synthetic and real data.
We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the re…
GLSKF improves tensor completion by capturing both global and local variations.
problem Tensor completion with missing entries, especially in data with spatial or temporal side information.
method Integrates smoothness-constrained low-rank factorization with a locally correlated residual process.
result GLSKF achieves superior performance and scalability on real-world datasets.
Uniform proof for Ricci flows on complete manifolds.
problem Proving short-time existence, uniqueness, and continuous dependence for Ricci flows.
method Using Koch-Lamm framework and tensor heat kernel estimates, with a new continuous dependence estimate.
result Uniform proof of short-time existence, uniqueness, and continuous dependence for Ricci flows.
We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}t>0, are also studied. We show that these he…
Tensor network surrogate for efficient option pricing in large portfolios.
problem Large-scale portfolio revaluation problems in market risk management.
method Tensor-train (TT) approximation for high-dimensional price surfaces, direct inference using Laplacian kernel and TT representations.
result Tensor surrogate achieves lower test error and faster evaluation times compared to standard GPR.
In this paper we study the problem of learning the weights of a deep convolutional neural network. We consider a network where convolutions are carried out over non-overlapping patches with a single kernel in each layer. We develop an algorithm for simultaneously learning all the kernels from the training data. Our app…
Tensor programs prove neural network limits for any architecture.
problem Understanding the limits of neural networks of any architecture.
method Prove convergence of neural network's Tangent Kernel (NTK) to a deterministic limit as network widths increase.
result Identify conditions for correct NTK limit calculation based on gradient independence assumption.
An increasing amount of collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever-present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data moti…
In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.
Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of…
We outline an inherent weakness of tensor factorization models when latent factors are expressed as a function of side information and propose a novel method to mitigate this weakness. We coin our method \textit{Kernel Fried Tensor}(KFT) and present it as a large scale forecasting tool for high dimensional data. Our re…
New method uses tensor decompositions to overcome the curse of dimensionality for large-scale learning.
problem Large-scale machine learning problems with kernel methods.
method Deterministic Fourier features combined with low-rank tensor decomposition for tensor product structure.
result Demonstrated consistent performance and superior results compared to random Fourier features.
The emerging edge computing has promoted immense interests in compacting a neural network without sacrificing much accuracy. In this regard, low-rank tensor decomposition constitutes a powerful tool to compress convolutional neural networks (CNNs) by decomposing the 4-way kernel tensor into multi-stage smaller ones. Bu…
Explicit formula for Bergman kernel of abelian varieties proved.
problem Explicit formula for Bergman kernel of polarized abelian varieties.
method Explicit formula for Bergman kernel of polarized abelian varieties.
result Explicit formula for Bergman kernel of polarized abelian varieties.
Researchers calculate entropy of heat kernel on manifolds for very small times.
problem Estimating entropy of heat kernel on compact Riemannian manifolds for small times.
method Asymptotic expansion, polynomial expressions in curvature tensor components.
result First three coefficients of entropy expansion computed and expressed as polynomials.
In many problems of supervised tensor learning (STL), real world data such as face images or MRI scans are naturally represented as matrices, which are also called as second order tensors. Most existing classifiers based on tensor representation, such as support tensor machine (STM) need to solve iteratively which occu…