This paper approximates the Gerber-Shiu function using phase-type Levy processes.
problem Measuring the risk of insurance companies through the Gerber-Shiu function.
method Approximate the Gerber-Shiu function by fitting the underlying process with phase-type Levy processes.
result A closed-form approximation of the Gerber-Shiu function is derived.
The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We con…
PH-VAE models heavy-tailed data with flexible Phase-Type distributions.
problem Standard VAEs fail to capture heavy-tailed behavior in real-world data.
method PH-VAE uses Phase-Type distributions defined by continuous-time Markov chains to adaptively model tail behavior.
result PH-VAE significantly outperforms existing heavy-tail-aware VAEs in approximating diverse heavy-tailed distributions.
The paper models stochastic interest rates for life insurance using phase-type distributions.
problem Modeling stochastic interest rates in life insurance with matrix approach.
method Integrates piecewise deterministic interest rates into a Markov jump process framework.
result Explicit formulas for reserves and future payments can be derived.
A hybrid model combines BPH and HE distributions for better heavy-tailed distribution approximation.
problem Accurate modeling of heavy-tailed distributions in various applications.
method A hybrid model of Bernstein phase-type and hyperexponential distributions with optimized parameters.
result Significant improvement in capturing both body and tail of heavy-tailed distributions.
The paper calculates ruin probabilities for insurers with phase-type distributed claims.
problem Calculating ruin probabilities for insurers with specific claim distributions.
method Change-of-measure technique applied to phase-type distributed claim amounts.
result The mixture of Erlangs best fits real-world loss data, improving risk assessment.
We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Levy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Levy process at a constant horizon and hence t…
The Wiener-Hopf factorization is obtained in closed form for a phase type approximation to the CGMY Lévy process. This allows, for the approximation, exact computation of first passage times to barrier levels via Laplace transform inversion. Calibration of the CGMY model to market option prices defines the risk neutral…
Markov chain decoders improve generative models' ability to produce heavy-tailed data.
problem Generative models struggle with heavy-tailed distributions.
method Replaced Gaussian decoder with Markov chain-based Phase-Type distributions.
result Significantly reduced tail Kolmogorov-Smirnov distance and extreme quantile error.
The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.
problem Existence of non-trivial harmonic maps into higher-dimensional target manifolds.
method Perturbative argument, refined neck-analysis, energy identity, min-max problems.
result Construction of an infinite family of new null-homotopic n-harmonic n-spheres. In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching Lévy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem…
Paper extends cepstral distance for stable, minimum-phase models.
problem Quantifying similarity between data objects in deterministic systems.
method Combines insights from systems theory and machine learning to extend weighted cepstral distance.
result Extended weighted cepstral distance can be interpreted in terms of poles and zeros of the model.
The paper develops methods to approximate quantities of interest in insurance models using deterministic integration.
problem Computing quantities of interest in insurance models, such as the probability of ruin and insurance company value.
method Adapting the problem to allow for deterministic numerical integration algorithms, including quasi-Monte Carlo rules and smoothing techniques.
result Convergence result justifying phase-type approximations on the process level.
Paper connects neural network score approximation to reverse diffusion model distribution approximation.
problem Quantifying the relationship between neural network score approximation and the distribution generated by reverse diffusion models.
method Combines Hornik's universal approximation theorem, Girsanov's theorem, and data processing inequality.
result Neural network score approximation guarantees distribution approximation in reverse diffusion models.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Paper proposes a new adaptive multiscale value function approximation for reinforcement learning.
problem Value function approximation in reinforcement learning with varying complexity.
method Adaptive multiscale approximation using multiresolution analysis and tree approximation.
result Convergence rate of the multiscale approximation is independent of basis function regularity.
Optimal function approximation with Relu neural networks achieves minimal error.
problem Finding the minimal error in approximating convex functions with Relu networks.
method Established necessary and sufficient conditions for optimal approximations, presented neural network architectures, and proposed an algorithm for convergence.
result Proved the convergence of the proposed algorithm and validated it with experimental results.
Paper proposes MCMA architecture for neural approximate computing with higher invocation rate and energy savings.
problem Limited invocation rate of neural approximators leading to suboptimal energy efficiency.
method Introduces MCMA architecture with a multiclass classifier and multiple approximators, sharing hardware resources and efficiently swapping approximators.
result Significantly higher invocation rate and energy savings compared to existing methods.
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Deep learning networks are approximated using dynamical systems theory.
problem Understanding the approximation capabilities of deep learning networks.
method Modeling deep residual networks as continuous-time dynamical systems and using approximation theories in Lp. result Established general sufficient conditions for universal approximation of deep residual networks.
Method approximates Riemannian barycenter on manifolds.
problem Computing the exact Riemannian barycenter is computationally expensive.
method Uses under- and over-approximations of Riemannian distance to compute an approximate barycenter.
result Approximation method is more efficient than exact methods and steepest descent.
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such …
Softmax attention approximates complex functions and subsumes many known universal approximators.
problem Universal approximation of continuous sequence-to-sequence functions.
method Interpolation-based analysis of attention's internal mechanism, showing its ability to approximate ReLU functions.
result Softmax attention is a universal approximator for continuous sequence-to-sequence functions.
Deviation inequalities for stochastic approximation methods.
problem Establishing bounds on the deviation of stochastic approximation methods.
method Martingale approximation method for separately Lipschitz functions.
result Established various deviation inequalities for stochastic approximation by averaging and minimization.
Improved matrix approximation using randomized algorithms.
problem Finding better approximations of given matrices.
method Randomized algorithms to compute (HT) as an improved approximation. result Computed (HT) provides a better approximation than given F∗. AXNet combines two neural networks into one for efficient approximate computing.
problem Efficient approximate computing for error-resilient applications.
method End-to-end trainable AXNet architecture that fuses approximator and predictor.
result Significant improvement in invocation rate and reduction in training time.
Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…
Novel method uses MCMC to improve approximation networks.
problem Approximating complex, intractable distributions.
method Amortized MCMC with iterative refinement of approximation network.
result Improved quality of deep generative model training.
We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…
New algorithms minimize non-zero entries in low-rank approximations.
problem Minimizing non-zero entries in low-rank approximations of matrices.
method Approximation algorithms for minimizing ℓ0-norm of rank-k matrices. result First provable guarantees for ℓ0-Low Rank Approximation for k>1. Transformers use ReLUs to approximate softmax efficiently.
problem Analyzing resource usage in softmax transformer models.
method Translating ReLU approximation results to softmax attention mechanisms.
result Economic resource bounds for softmax attention mechanisms.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…
Adaptive approximations improve variational inference for complex models.
problem Efficiently approximate marginal distributions and partition functions in complex probabilistic models.
method Two classes of adaptive approximations that include Bethe, tree-reweighted, and convex free energies.
result Proposed approximations automatically adapt to a given model and outperform existing methods.
Non-negative L1-approximating polynomials for Gaussian distributions are proven for certain classes of sets.
problem Existence of non-negative L1-approximating polynomials for Gaussian distributions. method Proving the existence of degree-k non-negative polynomials that approximate indicator functions of sets with Gaussian surface area in L1-norm. result Proves the existence of non-negative L1-approximating polynomials for certain classes of sets with Gaussian surface area. Variational boosting refines posterior approximations through iterative optimization.
problem Approximating intractable distributions with rich approximations.
method Iteratively solves optimization problems to refine variational approximations.
result Posterior inferences using variational boosting are more accurate and efficient.
Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.
problem Non-asymptotic bounds for accuracy of normal approximation in linear two-timescale stochastic approximation algorithms.
method Established bounds for normal approximation in terms of convex distance, focusing on last iterate and Polyak-Ruppert averaging.
result Normal approximation rate for the last iterate improves with increased timescale separation, while it decreases in the averaged setting.
One-pass algorithm finds small subset for ℓp subspace approximation with additive error.
problem Finding a small subset of data points for ℓp subspace approximation. method One-pass subset selection with additive approximation guarantee for p∈[1,∞). result First one-pass algorithm with additive error for ℓp subspace approximation. Paper introduces new approximations for lognormal sums, matching comonotonicity and moments.
problem Approximating sums of lognormal random variables accurately.
method Introduces new approximations based on weighted distribution theory, emphasizing comonotonicity and moment matching.
result Approximations perform better than classical methods, especially in the right tail of the distribution.
We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is a classical problem of bilinear approximation. In the case of approximation in the L2 space the bili…
A new method for efficient Gaussian process inference using sparse approximations.
problem Scalable and accurate inference for latent Gaussian processes.
method Variational approximation with sparse inverse Cholesky factors and double Kullback-Leibler minimization.
result The proposed method can achieve highly accurate approximations with polylogarithmic time complexity.
In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low-rank approximation in order to exploit the possible sp…
Low-precision quantization improves kernel approximation under memory constraints.
problem Training kernel approximation methods efficiently with limited memory.
method Low-precision quantization of random Fourier features (LP-RFFs).
result LP-RFFs can match the performance of full-precision RFFs and Nyström method with significantly less memory.
The paper approximates supply curves using a one-step basis method.
problem Computing supply curves accurately and efficiently.
method Derives L2 approximation expression and proposes node selection procedure.
result Illustrates the approach with European electricity market bid curves.
The paper defines a new concept of approximability for Lagrangian submanifolds.
problem Understanding the approximability of Lagrangian submanifolds.
method Introducing a new notion of categorical approximability for metric spaces, showing it applies to specific types of Lagrangian submanifolds.
result Examples of Lagrangian submanifolds are found that are approximable but not precompact.
Boosting Nyström improves accuracy of matrix approximations.
problem Generating low-rank approximations of large matrices efficiently.
method Iteratively generate multiple weak Nyström approximations, combine them to form a strong approximation.
result Boosting Nyström yields more efficient and accurate low-rank approximations.