SKI accelerates GP inference with sparse grids to handle higher dimensions.
problem SKI scales poorly in high dimensions due to dense grid size.
method Sparse grids within SKI framework, novel matrix-vector multiplication algorithm.
result SKI can be scaled to higher dimensions while maintaining accuracy.
In sparse-view Computed Tomography (CT), only a small number of projection images are taken around the object, and sinogram interpolation method has a significant impact on final image quality. When the amount of sparsity (the amount of missing views in sinogram data) is not high, conventional interpolation methods hav…
A new image interpolation model using sparse representation and nonlocal linear regression.
problem Image interpolation without blurring and noise.
method Sparse representation, nonlocal self-similarity, nonlocal linear regression, adaptive sub-dictionary learning, weighted encoding.
result Our method outperforms state-of-the-art methods in quantitative measures and visual quality.
This study uses neural networks to solve interpolation problems with sparse, infinitely wide layers.
problem Exact data interpolation using sparse, infinitely wide neural networks.
method Atomic norm framework to derive convex hulls and equivalent convex formulations.
result Simple characterizations of convex hulls for different constraints on network weights and biases.
Gaussian Processes improve data interpolation from diverse experiments.
problem Interpolation of sparse and inconsistent datasets from various experiments.
method Used Gaussian Processes (GP) for data interpolation, including uncertainty quantification.
result GPs successfully interpolate data and quantify uncertainties, demonstrating consistency across different sources.
Study tightens bounds for interpolating noisy data using minimum l1-norm.
problem Predicting noisy data with minimum l1-norm interpolation.
method Provided matching upper and lower bounds for prediction error.
result Tight consistency up to negligible terms for d≫n. New model leads to optimal test loss in sparse linear regression.
problem Sparse linear regression with low test loss despite interpolating training data.
method Developed a new parametrization of the model that combines benefits of ℓ1 and ℓ2 norms.
result Training via gradient descent leads to an interpolator with near-optimal test loss.
Efficiently prices American options with multiple assets using sparse grids.
problem Pricing American options with multiple underlying assets efficiently.
method Dynamic programming formulation followed by sparse grid interpolation.
result Sparse grids reduce the number of interpolation points and maintain function smoothness.
Lower bounds show OLS outperforms basis pursuit in overparameterized linear regression.
problem Excess risk of sparse interpolating procedures in overparameterized linear regression.
method Proved lower bounds on excess risk for OLS and basis pursuit.
result Excess risk of basis pursuit can converge at an exponentially slower rate than OLS.
CNNs predict spatial fields from sparse data.
problem Predicting complete spatial fields from limited observations.
method Convolutional Neural Networks (CNNs) trained on a single partially observed field.
result CNNs can flexibly capture local spatial patterns without explicit covariance modeling.
Study finds exact limits for sparse regression with fewer observations than usual.
problem Understanding sparse linear regression with sublinear sparsity.
method Adaptive interpolation method and modified AMP algorithm.
result Exact asymptotic expressions for mutual information and MMSE in sublinear sparsity.
In this paper, we present a new deep learning architecture for addressing the problem of supervised learning with sparse and irregularly sampled multivariate time series. The architecture is based on the use of a semi-parametric interpolation network followed by the application of a prediction network. The interpolatio…
Noise affects the effectiveness of interpolating models, especially those with strong inductive biases.
problem The impact of noise on interpolating models with strong inductive biases.
method Analyzing linear and classification models with sparse ground truths, proving fast rates for interpolators.
result Strong inductive biases can lead to faster but noisier interpolators, contrary to intuition.
The paper studies the minimum ℓ₁-norm interpolator's risk behavior in over-parameterized settings.
problem Understanding the risk behavior of minimum ℓ₁-norm interpolators in high-dimensional settings.
method Exact characterization of the risk behavior through a system of two non-linear equations.
result Observation of a multi-descent phenomenon in the generalization risk of the minimum ℓ₁-norm interpolator.
A new GP inference method using simplices for high-dimensional data.
problem Scalable Gaussian Processes in high dimensions.
method Developed a Simplex-GP method using a sparse simplicial grid to accelerate MVMs.
result Significantly faster GP inference in high dimensions compared to SKI.
Framework learns continuous dynamics from sparse trajectories.
problem Learning dynamics from sparsely sampled and high-dimensional trajectories.
method Interpolative Multi-Marginal Flow Matching (IMMFM) framework.
result IMMFM outperforms existing methods in forecasting and downstream tasks.
DGPs improve air quality inference from sparse data.
problem Accurate air quality monitoring in unmonitored areas.
method Deep Gaussian Processes with Doubly Stochastic Variational Inference.
result DGPs outperform state-of-the-art models in AQ inference.
Adversarial training improves linear regression solutions, revealing sparsity and abrupt interpolation.
problem Adversarial attacks on linear regression models.
method Formulated as a convex problem, adversarial training is used to find robust solutions that are sparse and interpolate data.
result Adversarial training with small disturbances gives the solution with the minimum-norm that interpolates the training data, revealing abrupt transition into interpolation.
Improved trajectory prediction for team sports using sparse outputs.
problem Challenging to train deep learning models for player trajectory prediction.
method Sparse trajectory prediction and constant acceleration interpolation.
result Interpolation improves performance for all tested models.
A continuing mystery in understanding the empirical success of deep neural networks is their ability to achieve zero training error and generalize well, even when the training data is noisy and there are more parameters than data points. We investigate this overparameterized regime in linear regression, where all solut…
Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the di…
We present a signal representation framework called the sparse manifold transform that combines key ideas from sparse coding, manifold learning, and slow feature analysis. It turns non-linear transformations in the primary sensory signal space into linear interpolations in a representational embedding space while maint…
This paper finds sparsest ReLU networks for interpolating data.
problem Finding the sparsest neural network that fits a dataset.
method Proposes a continuous, differentiable objective function based on ℓp quasinorms. result Global minimizers of the proposed objective correspond to sparsest ReLU networks.
Method interpolates option prices and volatilities without arbitrage.
problem Interpolating option prices and volatilities without arbitrage.
method Sparse modeling approach based on integral equations and SVD.
result Flexible and efficient framework for arbitrage-free interpolation.
Deep network clusters hospital patients' vital signs.
problem Sparse and irregularly collected vital sign data.
method Deep interpolation network for latent representation extraction.
result Extracted 7 distinct clusters from vital sign data.
Signal processing tasks as fundamental as sampling, reconstruction, minimum mean-square error interpolation and prediction can be viewed under the prism of reproducing kernel Hilbert spaces. Endowing this vantage point with contemporary advances in sparsity-aware modeling and processing, promotes the nonparametric basi…
Deep learning improves model discovery from sparse sensor data.
problem Improving physical understanding and predictions from coarse, non-grid sampled data.
method Physics-informed neural networks and automatic differentiation.
result Deep learning can recover underlying equations from sparse, non-grid data.
The paper explores how different network architectures learn logical functions under GOTU, finding that a min-degree-interpolator is learned.
problem Learning logical functions with a focus on generalization on the unseen.
method Study of different network architectures trained by SGD under GOTU.
result For sparse functions and certain network models, a min-degree-interpolator is learned on the unseen.
While the volume of electronic health records (EHR) data continues to grow, it remains rare for hospital systems to capture dense physiological data streams, even in the data-rich intensive care unit setting. Instead, typical EHR records consist of sparse and irregularly observed multivariate time series, which are wel…
For a wide range of clinical applications, such as adaptive treatment planning or intraoperative image update, feature-based deformable registration (FDR) approaches are widely employed because of their simplicity and low computational complexity. FDR algorithms estimate a dense displacement field by interpolating a sp…
The paper improves GP regression for sparse sensor data in structural mode shape reconstruction.
problem Reconstructing full-field structural mode shapes from sparse sensor data.
method Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework.
result The proposed method provides more accurate and reliable mode shapes.
Super-resolution is a classical problem in image processing, with numerous applications to remote sensing image enhancement. Here, we address the super-resolution of irregularly-sampled remote sensing images. Using an optimal interpolation as the low-resolution reconstruction, we explore locally-adapted multimodal conv…
Deep learning improves nearshore bathymetry estimation from sparse data.
problem Estimating nearshore bathymetry from limited and often sparse data.
method Deep Neural Network (DNN) and Conditional Generative Adversarial Network (cGAN) for posterior estimates; Kriging for comparison.
result DNN-based methods outperform traditional Kriging in predicting nearshore bathymetry with sharp gradients.
Traditionally in regression one minimizes the number of fitting parameters or uses smoothing/regularization to trade training (TE) and generalization error (GE). Driving TE to zero by increasing fitting degrees of freedom (dof) is expected to increase GE. However modern big-data approaches, including deep nets, seem to…
DAISI improves data assimilation for complex systems with noisy observations.
problem Limited accuracy of classical DA methods in complex, nonlinear systems.
method Generative models with inverse sampling for flexible probabilistic inference.
result DAISI achieves accurate filtering results in challenging nonlinear systems.
A semi-supervised framework using stochastic interpolation and latent representations.
problem Challenges in conditional generative modeling with scarce labeled data.
method Combines conditional stochastic interpolation with low-dimensional latent representations.
result Significantly improves sample complexity and achieves faster convergence rate.
The paper examines how kernel approximations affect Gaussian process regression in large data applications.
problem Effect of kernel approximations on Gaussian process regression in large data applications.
method Unified framework to analyze Gaussian process regression under computational and epistemic misspecification.
result Theoretical analysis of Gaussian process regression under various misspecifications.
New method solves sparse approximation problem using trimmed lasso and generalized soft-min penalties.
problem Sparse approximation or best subset selection problem.
method Regularized approach with trimmed lasso and generalized soft-min penalties.
result The trimmed lasso provides sparse recovery guarantees and a practical optimization algorithm.
Sparse RSP routing improves graph exploration and classification.
problem Optimal randomized routing and distance measures on weighted graphs.
method Tsallis divergence regularization for sparse RSP.
result Sparse random walk converges to least-cost graph as temperature decreases.
Sparse neural networks can match dense models on Lipschitz functions.
problem Sparse networks are more efficient but lack theoretical guarantees.
method Formal model of sparse networks, LSH-based routing function, Lipschitz function approximation.
result Sparse networks can approximate dense networks on Lipschitz functions.
SKI speeds up Toeplitz Neural Networks by avoiding explicit decay bias and using frequency response.
problem Efficiently compute and update Toeplitz matrices in neural networks.
method Sparse plus low-rank decomposition, asymmetric SKI, frequency response modeling.
result Achieved significant speedup with minimal performance loss.
New research shows the maximum ℓ1-margin classifier doesn't adapt to sparse ground truths.
problem Understanding the limitations of the maximum ℓ1-margin classifier in high-dimensional settings.
method Analyzing convergence and prediction error rates of the maximum ℓ1-margin classifier.
result Proves tight upper and lower bounds for prediction error, showing benign overfitting.
Task shift from classification to regression is possible in overparameterized linear models with limited additional data.
problem Transferability of latent knowledge from classification to regression in overparameterized linear models.
method Investigation of task shift in overparameterized linear regression, zero-shot and few-shot cases, with a focus on minimum-norm interpolation.
result Minimum-norm interpolators can transfer latent knowledge from classification to regression with limited additional data.
Study resolves conjecture on overparameterized linear models' generalization.
problem Asymptotic generalization of multiclass classification with overparameterized models.
method Gaussian covariates bi-level model, Hanson-Wright inequality variant.
result Min-norm interpolating classifier can be suboptimal compared to noninterpolating classifiers.
Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by constructing an interpolation function that maps from the input space to the low-dimen…
Combines deep and statistical learning for structured data.
problem Structured high-dimensional data challenges.
method Generates nonlinear features via sparse regularization and stochastic optimisation, uses probabilistic output layer for uncertainty.
result Achieves best of scalability and uncertainty quantification.
This work shows how penalising bias terms in norm regularisation leads to sparse solutions.
problem Understanding the relation between parameter norm regularization and the sparsity of neural network solutions.
method Analyzes one hidden ReLU layer networks with unidimensional data, showing the norm required for function representation and the importance of the bias term's norm.
result Penalising the bias terms in regularisation leads to sparse solutions, enforcing the uniqueness and sparsity of the minimal norm interpolator.
Model shows loss curve with two distinct exponents due to sparse activations.
problem Sparse activations impact neural network scaling laws.
method Introduced a model for neural scaling laws under sparse activations, derived asymptotic population loss, and analyzed gradient-descent dynamics.
result Loss curve exhibits double-descent peak near interpolation threshold with two distinct scaling exponents.