Study on robust Dynkin game over singular probabilities.
problem A game over mutually singular probabilities with singular technical difficulties.
method Proved value process coincidence, used approximations for stopping times.
result Value process is a submartingale up to first meeting time.
Study on survival probability of insurance companies using integro-differential equations.
problem Survival probability of insurance companies over infinite time.
method Analytical and numerical methods for solving integro-differential equations with singularities.
result Existence and uniqueness of solutions to the integro-differential equation.
Kernel embeddings separate distinct probability distributions, simplifying testing.
problem Testing equality of non-atomic probability distributions.
method Kernel covariance embeddings and Gaussian measures in reproducing kernel Hilbert spaces.
result Testing for singularity between Gaussian measures is equivalent to testing for equality of non-atomic probability distributions.
HADES detects data singularities quickly and accurately.
problem Detecting singularities in data efficiently.
method Kernel goodness-of-fit test based on differential geometry and optimal transport theory.
result Correctly detects singularities with high probability.
The paper tackles optimal stopping problems using reinforcement learning and singular control.
problem Continuous-time and state-space optimal stopping problems.
method Formulated as a singular control problem with randomized stopping times and penalized cumulative residual entropy.
result Identified unique optimal exploratory strategy through dynamic programming.
We consider a finitely generated torsion free Kleinian group H and a random walk on H with respect to a symmetric nondegenerate probability measure μ with finite support. When H is geometrically infinite without parabolics or when H is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan me…
New method uses free probability to improve deep ResNet initialization speed.
problem Improving initialization speed of deep ResNets.
method Introducing a novel analytical tool in free probability for non-Hermitian random matrices.
result Proposed initialization scheme learns at a speed of orders of magnitudes faster.
Study optimal stopping with random maturity under nonlinear expectations.
problem Optimal stopping problem with random maturity under nonlinear expectation.
method Analysis of optimal stopping problem with random maturity under a nonlinear expectation with respect to a weakly compact set of mutually singular probabilities.
result The optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility.
Investigates portfolio selection for rank-dependent utilities in incomplete markets.
problem Portfolio selection for agents with rank-dependent utility in incomplete financial markets.
method Characterizes deterministic strict equilibrium strategies for constant-coefficient and time-invariant probability weighting functions. Addresses the issue of selecting an optimal strategy from multiple equilibrium strategies for time-variant probability weighting functions.
result Characterizes deterministic strict equilibrium strategies and identifies optimal strategies from multiple equilibrium strategies.
Study financial bubbles in a model with multiple probability measures.
problem Understanding financial bubbles in markets with multiple probability measures.
method Introduced robust bubble and fundamental value concepts, investigated no dominance under uncertainty.
result Provided concrete examples of the introduced concepts.
The paper proposes a test to determine the number of latent classes in ordinal categorical data.
problem Determining the correct number of latent classes in latent class models with ordinal categorical data.
method The test statistic centers the largest singular value of a normalized residual matrix by a simple sample-size adjustment.
result The test statistic converges to zero under the null hypothesis and exceeds a fixed positive constant under an under-fitted alternative.
Deep sigmoidal networks can achieve dynamical isometry with orthogonal weight initialization, significantly speeding up learning.
problem Ensuring efficient learning in deep neural networks, especially with nonlinear activation functions.
method Employing free probability theory to compute the singular value distribution of a deep network's input-output Jacobian.
result Deep sigmoidal networks can achieve dynamical isometry with orthogonal weight initialization, leading to faster learning.
A novel kernel-based test detects equality versus singularity of two probability measures.
problem Detecting equality versus singularity of two probability distributions.
method Combines kernel mean and kernel covariance embeddings to construct a likelihood ratio test statistic.
result The test statistic satisfies a '0/\infty' law, vanishing under the null and diverging under the alternative.
The paper derives an equation linking WAIC and WBIC for singular models.
problem In singular models, conventional criteria fail due to likelihood and posterior breakdown.
method Theoretical derivation linking WAIC and WBIC.
result An asymptotic equation linking WAIC and WBIC for singular models.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating p(x,y) with its first r singular values. result The new MMD distance measures the difference of two distributions with respect to r∗ local moments, where r∗ depends on singular values decay rate. New method approximates high-dimensional probability densities efficiently.
problem Approximating high-dimensional probability densities accurately and efficiently.
method Hierarchical tensor-network approach using randomized SVD and linear equations.
result The method effectively approximates high-dimensional densities with linear complexity.
Adaptive PINNs improve accuracy by adding points where solutions are uncertain.
problem Inadequate sampling in PINNs leads to inaccurate solutions, especially near singularities.
method FI-PINNs use failure probability to dynamically add points, improving numerical accuracy.
result FI-PINNs achieve better accuracy through adaptive sampling, as proven by rigorous error bounds.
Bootstrapping regularizes singular correlation matrices, reducing the need for complex regularization.
problem Singular correlation matrices in large datasets.
method Averaging bootstrapped correlation matrices to ensure positive-definiteness.
result The averaged correlation matrix is almost surely positive-definite with a sufficient number of bootstraps.
LS improves model selection for singular statistical models.
problem Challenges in model selection for singular statistical models.
method Integrates empirical loss from WAIC and sBIC penalty term.
result Enhanced utility for model selection without regularity constraints.
Study on random matrices in deep neural networks with IID entries.
problem Distribution of singular values in product of random matrices for deep neural networks.
method Random matrix theory with a streamlined approach for non-Gaussian data.
result Generalization of macroscopic universality property to non-Gaussian data.
We study a robust optimal stopping problem with respect to a set $\cP$ of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its pay-off while an adverse player wants to minimize this payoff by choosing an evaluation criteria from $\…
Study on random matrices in deep neural networks using Gaussian data.
problem Distribution of singular values in product of random matrices in deep learning.
method Free probability theory combined with standard techniques of random matrix theory.
result Justification for applying free probability theory to non-independent random data matrices.
This paper provides a functional analytic foundation for singular value decomposition of RKHS operators.
problem Singular value decomposition of operators on RKHSs.
method Functional analytic approach, extending matrix eigenvalue problems to RKHS operators.
result Solid foundation and extension of singular value decomposition to RKHS operators.
Unified framework for singular statistical models using observable charts.
problem Non-identifiability and breakdown of classical asymptotic theory in singular models.
method Invariant framework based on observable charts to define local coordinate systems in model space.
result Observable order provides a lower bound on KL divergence vanishing rate in singular models.
The paper proves a distribution claim for neural network Jacobians.
problem Distribution of singular values in deep neural networks.
method Free probability and random matrix theory techniques.
result Singular value distribution matches for specific cases.
We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming…
The paper shows that random frames have full spark with high probability.
problem The probability of a random frame having full spark.
method Relating frame spaces to toric symplectic manifolds to analyze geometric and spectral properties.
result The probability of a random frame having full spark is one.
We use matricial free energy to regularize autoencoders, producing Gaussian-like codes.
problem Generating Gaussian-like codes for autoencoders.
method Define a differentiable loss function based on singular values of the code matrix, minimizing matricial free energy.
result Minimizing matricial free energy results in Gaussian-like codes that generalize.
Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.
problem Solving Monge-Ampère equations on compact Hessian manifolds.
method Perron method, compactness properties of normalized quasi-convex functions, local and global comparison principles for twisted Monge-Ampère operators.
result Solves Monge-Ampère equations involving arbitrary probability measures.
Study simplicial volume via foliated simplices and duality.
problem Calculate simplicial volume using foliated simplices and duality.
method Defined real singular foliated homology, constructed foliated fundamental class, and established isometric isomorphism with measurable bounded cohomology.
result Norm of foliated fundamental class equals simplicial volume of M. We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted versions to more singular situations. Applications to Monge-Ampère equations of mean fi…
A new method for robust PCA using nonconvex rank approximation.
problem Recovering a matrix of minimal rank in data mining and machine learning.
method Proposes a nonconvex rank approximation to the nuclear norm, solving the associated nonconvex minimization problem with an efficient algorithm.
result Our method outperforms current state-of-the-art algorithms in both accuracy and efficiency.
This paper considers nonlinear regular-singular stochastic optimal control of large insurance company. The company controls the reinsurance rate and dividend payout process to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. However, if the optimal dividend barrier is too low t…
Entropy for uniform hypergraphs defined via tensor theory.
problem Entropy calculation for uniform hypergraphs.
method Probability distribution of generalized singular values from Laplacian tensors, Shannon entropy formula.
result Tensor entropy is a measure of regularity for uniform hypergraphs.
Unified framework for convergence of discrete diffusion models without state space size dependence.
problem Fundamental limitations in existing convergence theory for discrete diffusion models, especially under singular priors and large vocabularies.
method Unified adjoint-equation-based framework that establishes dimension-free convergence guarantees in any integral probability metric (IPM).
result First dimension-free convergence bounds applicable to both masked and uniform priors, free of state space size S. A statistical model or a learning machine is called regular if the map taking a parameter to a probability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus…
Study minimal supersolutions for BSDEs with infinite terminal values, solving portfolio liquidation problems.
problem Existence of minimal supersolutions for BSDEs with infinite terminal values.
method Generalized BSDEs on a general filtered probability space with singular terminal condition.
result Solved optimal portfolio liquidation problems using minimal supersolutions.
Flexible model for pairwise comparisons relaxes strong parametric assumptions.
problem Limitations of strong parametric models for pairwise comparison data.
method Stochastically transitive models, non-trivial minimax-optimal estimation.
result Flexible models can be estimated at the same rate as parametric models but require computationally tractable alternatives for optimal estimation.
Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular hom…
Paper investigates separating times for general diffusions, providing new insights.
problem Understanding phase transitions between equivalence and singularity in diffusions.
method Representation of separating time as hitting time of a deterministic set, characterized by speed and scale.
result Explicit and easy-to-check conditions for absolute continuity and singularity of diffusions.
We consider fundamental questions of arbitrage pricing arising when the uncertainty model is given by a set of possible mutually singular probability measures. With a single probability model, essential equivalence between the absence of arbitrage and the existence of an equivalent martingale measure is a folk theorem,…
The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, gener…
New method improves option pricing for non-smooth functions.
problem Inefficiency of Fourier techniques with non-smooth probability density functions.
method Singular Fourier-Padé (SFP) method
result Restores global spectral convergence rate and fast error convergence.
A new method for VAEs improves latent space disentanglement without violating probability laws.
problem Improving latent space disentanglement in VAEs without violating probability laws.
method Developed a Renyi VAE with a conditional distribution not learned, using Singular Value Decomposition for evaluation.
result Improved latent space disentanglement without violating probability laws.
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c >= rank(A) columns depend on the right singular vector matrix of A. For a Monte-…
Random feature matrices' singular values concentrate near their full expectation in high dimensions.
problem Characterizing the spectra of random feature matrices for regression problems.
method Analyzing two settings of input variables (random or well-separated) with conditions on dimension, complexity ratio, and sampling variance.
result The singular values of random feature matrices concentrate near their full expectation and near one with high probability.
We introduce an infectious default and recovery model for N obligors. Obligors are assumed to be exchangeable and their states are described by N Bernoulli random variables S_{i} (i=1,...,N). They are expressed by multiplying independent Bernoulli variables X_{i},Y_{ij},Y'_{ij}, and default and recovery infections are …