VIBNN accelerates Bayesian Neural Networks on FPGAs for efficient inference.
problem Overfitting and small-data training issues in BNNs.
method Hardware accelerator design for variational inference on BNNs, using novel Gaussian random number generators.
result VIBNN achieves high throughput and energy efficiency on FPGA, matching software performance.
Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.
problem Spectral number variance convergence for twisted Laplacians and Dirac operators.
method Extends Rudnick's approach to Gaussian ensembles for twisted Laplacians and Dirac operators.
result Convergence to Gaussian ensembles for twisted Laplacians and Dirac operators.
Method generates random numbers from sensor noise, improving accuracy and speed.
problem Improving accuracy and speed of Monte Carlo integration.
method Sampling a physical process in a controlled environment.
result Reduces error of Monte Carlo integration by 10^68 times while doubling speed.
New methods reduce computational cost for Gaussian Markov Random Fields with sparse constraints.
problem Inference and simulation of GMRFs are computationally prohibitive with many constraints.
method Proposes a basis transformation into blocks of constrained and non-constrained subspaces.
result Significantly outperforms existing alternatives in computational cost.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
Improved perturbation reduces matrix condition number to O(n) with minimal storage.
problem Reducing the condition number of deterministic matrices for efficient algorithmic use.
method Introduced pattern matrices and sparse perturbations with dependent entries.
result Condition number reduced to O(n) with O(n) random numbers in O(log n) precision.
New method for ancestral inference in branching processes with random environments.
problem Determining ancestor distribution parameters in branching processes with random environments.
method Generalized method of moments for ancestral inference.
result Limiting distribution of ancestor and offspring estimators decouple and converge to independent Gaussian variables under certain conditions.
The paper proves that Gaussian field critical points have finite moments.
problem Proving the finiteness of moments for Gaussian field critical points.
method General approach not specific to critical points, using Taylor polynomial non-degeneracy.
result The finiteness of moments of the number of critical points of Gaussian fields.
Paper connects neural networks to Gaussian processes for understanding double-descent.
problem Understanding the double-descent phenomenon in neural networks.
method Uses techniques from random matrix theory and Gaussian processes.
result Establishes a connection between NNGP and random matrix theory for neural networks.
The paper analyzes why Gaussianization slows down with higher dimensions and proposes a solution.
problem The convergence rate of Gaussianization slows down as the dimension increases.
method Analytical and empirical analysis of Gaussianization with random rotations.
result The number of required layers scales linearly with the dimension for Gaussian input.
A scalable deep GMRF model for general graphs improves predictions and uncertainty estimates.
problem Handling generally structured data on graphs efficiently.
method A new multi-layer structure of Deep GMRFs designed for general graphs, enabling efficient training and close-to-exact Bayesian inference.
result Close-to-exact Bayesian inference for latent field predictions with uncertainty estimates.
HMC improves Gaussian sampling efficiency with long, random steps.
problem Efficiently sampling from high-dimensional Gaussian distributions.
method Hamiltonian Monte Carlo with long and random integration times.
result HMC achieves ε \varepsilon ε -closeness in total variation distance with O ~ ( κ d 1 / 4 log ( 1 / ε ) ) \widetilde{O}(\sqrt{\kappa} d^{1/4} \log(1/\varepsilon)) O ( κ d 1/4 log ( 1/ ε )) gradient queries. New method controls false discovery rate in learning Gaussian MRF structures.
problem Learning the structure of Gaussian MRFs from data, especially when p >> n, leads to false edges.
method Proposes nsSLOPE using sorted l1-norm regularization to control false discovery rate.
result Controls false discovery rate in learning the structure of Gaussian MRFs.
A new method for LDA using randomized Kaczmarz improves accuracy for large datasets.
problem Efficiently performing LDA on large datasets.
method Randomized Kaczmarz method applied to linear discriminant analysis.
result The method achieves comparable accuracy to full data LDA.
Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.
problem Distribution of zeros of Gaussian sections on semipositive line bundles.
method Analysis of Bergman kernels and random zeros in high tensor powers.
result Equidistribution, large deviation estimates, central limit theorem, and number variances for zeros in the semi-classical limit.
New method improves Gaussian kernel approximations for high-frequency data.
problem Limited scalability of kernel-based models to large data sets.
method Local random feature approximations using Maclaurin expansions and polynomial sketches.
result Significant improvement in kernel approximations and downstream performance for high-frequency data.
The study of topological properties of random smooth maps, focusing on Kac-Rice formula and Betti numbers.
problem Topological and geometric properties of random smooth maps.
method Developed a general framework for differential geometric and topological issues of smooth Gaussian Random Fields, generalized Kac-Rice formula, applied to Kostlan random polynomials, and proved an original theorem in Differential Topology.
result The Betti numbers of the solution of a system of regular equations cannot decrease under a C 0 \mathcal{C}^0 C 0 -small perturbation of the equations. Bayes-optimal learning of deep random networks with Gaussian weights is studied.
problem Learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights.
method Closed-form expressions for Bayes-optimal test error, ridge regression, kernel and random features regression are computed.
result Optimally regularized ridge regression and kernel regression achieve Bayes-optimal performances, while logistic loss yields a near-optimal test error for classification.
Two randomized algorithms improve regret bounds for generalized linear bandits.
problem Improving regret bounds for generalized linear bandits.
method Two randomized algorithms: GLM-TSL and GLM-FPL.
result Upper bounds of O ( d n log K ) O(d \sqrt{n \log K}) O ( d n log K ) on regret for both algorithms. We analyze learning curves of RF models with convex regularization and derive precise asymptotic expressions.
problem Understanding the learning curves of RF models with general convex regularization.
method Novel multi-level application of the convex Gaussian min max theorem (CGMT) to compute precise asymptotic expressions.
result Precise asymptotic expressions for learning curves of RF models with separable strongly convex regularization or ℓ 1 \ell_1 ℓ 1 regularization. Paper proves sufficient conditions for tensor recovery using t-RIP with random measurements.
problem Establish robust recovery guarantees for low-tubal-rank tensors.
method Probabilistic arguments and random sub-Gaussian distributions to ensure t-RIP conditions.
result Minimal number of linear measurements nearly optimal for tensor recovery.
Two parties estimate cross-correlation matrix from i.i.d. samples, achieving near-optimal variance.
problem Estimating cross-correlation matrix between two parts of a random vector.
method Constructive unbiased estimators for jointly and vector Gaussian cases, leveraging i.i.d. samples and bit transmission.
result Achieves variance of ( 1 − ρ 2 ) / ( 2 k ln 2 ) (1-ρ^2)/(2k\ln 2) ( 1 − ρ 2 ) / ( 2 k ln 2 ) for jointly Gaussian scalar random variables, and uniformly better for vector Gaussian case. Bayesian optimization improves performance with common random numbers.
problem Optimizing expensive stochastic functions with common random numbers.
method Proposes a novel Gaussian process model and Knowledge Gradient for Common Random Numbers.
result Significant performance improvements with moderate computational cost.
Region detection in Gaussian Markov fields with limited samples.
problem Consistent graph recovery in sample deficient scenarios.
method Partitioning the graph into spatial regions with similar edge parameters and regular boundaries, developing new sample complexity bounds, and introducing an efficient region growing algorithm.
result A bounded number of samples can be sufficient for consistent region recovery.
BART's performance improves with more trees, converging to a Gaussian process.
problem Understanding and explaining BART's superior performance in prediction and causal inference.
method Analyzing BART as the number of trees grows towards infinity, showing convergence to a Gaussian process.
result BART converges to a Gaussian process with favorable inferential properties, explaining its excellent performance.
AMP with Gaussian initialization shows weak-recovery threshold for phase retrieval.
problem Phase retrieval with noiseless data.
method Approximate message passing with random initialization.
result Random initialization attains weak-recovery threshold \( \delta_{ ext{weak}} = 1/2 \).
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
New algorithm speeds up image denoising to linear time.
problem Bayesian image denoising with Gaussian Markov Random Field.
method Proposes a new algorithm solving in O(n) time.
result Effective in practice with hyperparameter estimation.
Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve defined as the average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inf…
McCullagh and Yang (2006) suggest a family of classification algorithms based on Cox processes. We further investigate the log Gaussian variant which has a number of appealing properties. Conditioned on the covariates, the distribution over labels is given by a type of conditional Markov random field. In the supervised…
Quantum algorithm estimates mean with sub-Gaussian error.
problem Estimating mean of quantum-computed random variables.
method Quantum mean estimation algorithm with sub-Gaussian error rate.
result Achieves nearly-optimal quadratic speedup over classical methods.
The paper compares Bayesian uncertainty to MAP estimator in random features regression.
problem Comparing Bayesian uncertainty to MAP estimator in random features regression.
method Analyzing the variance of the posterior predictive distribution and comparing it to the risk of the MAP estimator.
result Asymptotic agreement between Bayesian uncertainty and MAP estimator under specific signal-to-noise ratios and sample sizes.
Random neural networks with ReLU activations are non-Gaussian processes.
problem Understanding the behavior of neural networks with random initialization and rectified linear units.
method Proving these networks are non-Gaussian processes and deriving their properties.
result These networks can converge to non-Gaussian processes under certain conditions.
Study on products of large random matrices and neural network gradients stability.
problem Understanding exploding and vanishing gradient problem in deep neural networks.
method Analyzes products of many large random matrices and applies to neural network gradients stability.
result Precise information about the stability of gradients in randomly initialized deep neural networks.
We extend Kac-Rice formula to compute expected intersections of random submanifolds.
problem Computing expected intersections of random submanifolds.
method Generalized Kac-Rice formula using measure theory and integration.
result Formula computes expected cardinality of preimages of submanifolds via random maps.
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…
FGN models networks with fractal structures using Gaussian Multiplicative Chaos.
problem Modeling networks with fractal structures.
method FGN model based on Gaussian Multiplicative Chaos.
result FGNs reveal distinct scaling patterns in edge and clique counts.
New algorithm converts data into sub-gaussian designs efficiently.
problem Efficiently converting large datasets into sub-gaussian random designs for robust performance.
method Algorithmic Gaussianization through sketching and averaging, using LESS embeddings.
result Efficient data sketches nearly indistinguishable from sub-gaussian designs.
A new method for anomaly detection using random subspaces and Gaussian mixture models.
problem Anomaly detection in high-dimensional data.
method Statistical estimation of probability density using random subspaces combined with geometric averaging.
result The method achieves competitive AUC scores and is interpretable.
New framework uses deep generative priors for robust phase retrieval.
problem Highly ill-posed and non-linear phase retrieval problem.
method Regularization through deep generative priors with gradient descent.
result Effective for random Gaussian and Fourier friendly measurements.
Enhanced Gaussian process models accelerate optimization and posterior approximation.
problem Improving the accuracy and speed of Gaussian process models for optimization and inference.
method Introduces a random exploration step to classical GP-UCB algorithms, facilitating faster convergence.
result New algorithms achieve nearly optimal convergence rates and provide bounds for Hellinger distance.
Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct bou…
Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, calle…
Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.
problem Failure of Gaussian equivalence in polynomial feature embeddings under quadratic scaling.
method Introduced Conditional Gaussian Equivalent (CGE) model to capture non-Gaussian behavior.
result Correct asymptotics derived for training and test errors in CGE model.
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
Random Fourier Features reduce kernel matrix reconstruction error without dimensionality dependence.
problem Error reduction in kernel matrix reconstruction for high-dimensional data.
method Random Fourier Features with theoretical error bounds.
result Error probability is independent of data dimensionality.
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…
We derive Gaussian approximations for random forest predictions using region-based stabilization.
problem Improving the accuracy of random forest predictions for Poisson process data.
method Region-based stabilization and Malliavin-Stein method for multivariate Gaussian approximation.
result Established Gaussian approximation bounds for random forest predictions under Poisson process.