This article provides a case study for a recently introduced diffusion in the space of probability measures over the reals, namely rearranged stochastic heat, which solves a stochastic partial differential equation valued in the set of symmetrised quantile functions over the unit circle. This contribution studies probability measure-valued flows perturbed by this noise with a special focus on gradient flows. This is done by introducing a drift to the rearranged stochastic heat equation by means of a vector field from the set of random variables over the unit circle into itself. When the flow is a gradient flow, the vector field may coincide with the Wasserstein derivative of a mean-field potential function. The resulting equation reads as a sort of McKean-Vlasov stochastic differential equation with an infinite-dimensional common noise. This article provides conditions on the drift under which solutions exist uniquely for any time horizon and converge exponentially fast towards a unique equilibrium. When the drift derives from a potential, some metastability properties are obtained as the intensity of the noise is tuned to zero: it is shown that under a particular scaling regime, the gradient flow lingers near local minimisers for expected times of the same order as in the finite-dimensional setting.
Excursion sets of Poisson shot noise processes are a prominent class of random sets. We consider a specific class of Poisson shot noise processes whose excursion sets within compact convex observation windows are almost surely polyconvex. This class contains, for example, the Boolean model. In this paper, we analyse the behaviour of geometric functionals such as the intrinsic volumes of these excursion sets for growing observation windows. In particular, we study the asymptotics of the expectation and the variance, derive a lower variance bound and show qualitative and quantitative central
The Gapeev-Shiryaev conjecture (originating in Stochastics 83 (2011) 519-535 and Adv. Appl. Probab. 45 (2013) 164-185) can be broadly stated as follows: Monotonicity of the signal-to-noise ratio implies monotonicity of the optimal stopping boundaries. The conjecture was originally formulated both within (i) sequential testing problems for diffusion processes (where one needs to decide which of the two drifts is being indirectly observed) and (ii) quickest detection problems for diffusion processes (where one needs to detect when the initial drift changes to a new drift). In this paper we present proofs of the Gapeev-Shiryaev conjecture both in (i) the sequential testing setting (under Lipschitz/H & ouml;lder coefficients of the underlying SDEs) and (ii) the quickest detection setting (under analytic coefficients of the underlying SDEs). The method of proof in the sequential testing setting relies upon a stochastic time change and pathwise comparison arguments. Both arguments break down in the quickest detection setting and get replaced by arguments arising from a stochastic maximum principle for hypoelliptic equations (satisfying H & ouml;rmander's condition) that is of independent interest. Verification of the Gapeev-Shiryaev conjecture establishes the fact that sequential testing and quickest detection problems with monotone signal-to-noise ratios are amenable to known methods of solution.
Mathematical methods are developed to characterize the asymptotics ofrecurrent neural networks (RNN) as the number of hidden units, data samplesin the sequence, hidden state updates, and training steps simultaneously growto infinity. In the case of an RNN with a simplified weight matrix, we provethe convergence of the RNN to the solution of an infinite-dimensional ODEcoupled with the fixed point of a random algebraic equation. The analysisrequires addressing several challenges which are unique to RNNs. In typicalmean-field applications (e.g., feedforward neural networks), discrete updatesare of magnitude & Oscr;(1/N )and the number of updates is & Oscr;(N). Therefore,the system can be represented as an Euler approximation of an appropriateODE/PDE, which it will converge to asN ->infinity. However, the RNN hiddenlayer updates are & Oscr;(1). Therefore, RNNs cannot be represented as a dis-cretization of an ODE/PDE and standard mean-field techniques cannot beapplied. Instead, we develop a fixed point analysis for the evolution of theRNN memory states, with convergence estimates in terms of the number ofupdate steps and the number of hidden units. The RNN hidden layer is stud-ied as a function in a Sobolev space, whose evolution is governed by the datasequence (a Markov chain), the parameter updates, and its dependence on theRNN hidden layer at the previous time step. Due to the strong correlation be-tween updates, a Poisson equation must be used to bound the fluctuations ofthe RNN around its limit equation. These mathematical methods give rise tothe neural tangent kernel (NTK) limits for RNNs trained on data sequencesas the number of data samples and size of the neural network grow to infinity.
The constrained minimization-respectively maximization-of dissimilarity-modeling divergences (i.e., generally nonsymmetric, directed distances) and of related generalized entropies is a fundamental task in many areas of quantitative sciences. On R-K of arbitrary dimension K, we derive a method which tackles such kind of constrained optimization problems-and beyond-by limits of sequences of appropriately constructed, dimension-free, typically comfortably simulable random vectors; almost no assumptions (like convexity) on the set of constraints are needed. This very largely extends our recent results on f-divergences which can be connected to light-tailed probability distributions in a certain manner (cf. IEEE Trans. Inform. Theory (2023) 69 3062-3120). For instance, in the current paper we cover constrained optimizations of arbitrary f-divergences, Bregman distances, scaled Bregman distances and weighted & ell;(r)-distances.
