Continuous Probability Measure Research Articles

Distributionally Robust Memory Evolution With Generalized Divergence for Continual Learning.

Continual learning (CL) aims to learn a non-stationary data distribution and not forget previous knowledge. The effectiveness of existing approaches that rely on memory replay can decrease over time as the model tends to overfit the stored examples. As a result, the model's ability to generalize well is significantly constrained. Additionally, these methods often overlook the inherent uncertainty in the memory data distribution, which differs significantly from the distribution of all previous data examples. To overcome these issues, we propose a principled memory evolution framework that dynamically adjusts the memory data distribution. This evolution is achieved by employing distributionally robust optimization (DRO) to make the memory buffer increasingly difficult to memorize. We consider two types of constraints in DRO: f-divergence and Wasserstein ball constraints. For f-divergence constraint, we derive a family of methods to evolve the memory buffer data in the continuous probability measure space with Wasserstein gradient flow (WGF). For Wasserstein ball constraint, we directly solve it in the Euclidean space. Extensive experiments on existing benchmarks demonstrate the effectiveness of the proposed methods for alleviating forgetting. As a by-product of the proposed framework, our method is more robust to adversarial examples than compared CL methods.

Doubly Intermittent Full Branch Maps with Critical Points and Singularities

We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild assumptions we prove the existence of a unique invariant mixing absolutely continuous probability measure, study its rate of decay of correlation and prove a number of limit theorems.

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure f mathrm {d}x by a discrete probability measure sum _i m_i delta _{z_i}, subject to a constraint on the particle sizes m_i. The locations z_i of the particles, their sizes m_i, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

Effective randomness for continuous measures

We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n n , all but countably many reals are n n -random for such a measure, where n n indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof rests upon an application of Borel determinacy. Therefore, the proof presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G G such that, for any n n , the statement “All but countably many reals are G ( n ) G(n) -random with respect to a continuous probability measure” cannot be proved in Z F C n − \mathsf {ZFC}^-_n . Here Z F C n − \mathsf {ZFC}^-_n stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n n -many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.

Estimates of heat kernels of non-symmetric Lévy processes

Abstract We investigate densities of vaguely continuous convolution semigroups of probability measures on ℝ d {{\mathbb{R}^{d}}} . First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.

Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary

We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.

Fundamental Theorem of Asset Pricing under fixed and proportional transaction costs

We show that the absence of arbitrage in a model with both fixed and proportional transaction costs is equivalent to the existence of a family of absolutely continuous single-step probability measures, together with an adapted process with values within the bid-ask intervals that satisfies the martingale property with respect to each of the measures. This extends Harrison and Pliska’s classical Fundamental Theorem of Asset Pricing to the case of combined fixed and proportional transaction costs.

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers p1,. .. , pr there exists a continuous probability measure µ on the unit circle T such that inf k 1 ≥0,...,kr ≥0 | µ(p k 1 1. .. p kr r)| > 0. This results applies in particular to the Furstenberg set F = {2 k 3 k ; k ≥ 0, k ≥ 0}, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous ×2-×3 conjecture. We also estimate the modified Kazhdan constant of F and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.

Large deviations of radial SLE$_{\infty }$

We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname {SLE}$) on the unit disk with parameter $\kappa \rightarrow \infty $. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{\phi _{t}^{2} (\zeta )\, d\zeta \}_{t \in [0,1]}$ on the unit circle and equals $\int _{0}^{1} \int _{S^{1}} |\phi _{t}'|^{2}/2\,d\zeta \,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.

The limit distribution of the maximum probability nearest-neighbour ball

AbstractLet X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

Scaling properties of the thue–morse measure

The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.

On representations of the set of supermartingale measures and applications in continuous time

In a continuous time setting, we consider a set of -absolutely continuous probability measures, containing at least one equivalent to , and denote by , the smallest m-stable closed convex set of probability measures containing . We shall prove mainly under the martingale representation property with respect to a continuous vector valued martingale S, that is the set of supermartingale measures for a family , where is a set of predictable vector valued processes. We state as an immediate consequence of Föllmer-Kramkov Theorem in H. Föllmer and D. Kramkov (Optional decompositions under constraints, Probab. Theory Relat. Fields 109 (1997), pp. 1–25), the decomposition of -supermartingales.

Global-local mixing for the Boole map

In the context of ‘infinite-volume mixing’ we prove global-local mixing for the Boole map, a.k.a. Boole transformation, which is the prototype of a non-uniformly expanding map with two neutral fixed points. Global-local mixing amounts to the decorrelation of all pairs of global and local observables. In terms of the equilibrium properties of the system it means that the evolution of every absolutely continuous probability measure converges, in a certain precise sense, to an averaging functional over the entire space.

Invariance of nonatomic measures on effect algebras

Abstract The present paper deals with invariance of nonatomic measures defined on effect algebras. Firstly, it is proved that if μ is a nonatomic and continuous probability measure defined on a σ-complete effect algebra L, then it satisfies para-Darboux property. Then, the invariance between continuous probability measures m and μ defined on a σ-complete effect algebra L is established when μ is nonatomic satisfying para-Darboux property on L.

Optimal Quantizers for Some Absolutely Continuous Probability Measures

The representation of a given quantity with less information is often referred to as ‘quantization’ and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit discs, squares, and the real line. For these probability measures the optimal sets of \(n\)-means and the \(n\)th quantization errors are calculated for some positive integers \(n\).

Curvatures of homogeneous sub-Riemannian manifolds

The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented.

A theorem of uniqueness for characteristic functions

We discuss a uniqueness property of the characteristic function of an absolutely continuous probability measure. Our study is initiated by the question posed by N.G. Ushakov: is it true that, for any interval [a,b]⊂R, 0∉[a,b], there exists a characteristic function f such that f≢e−t2/2, but f(t)=e−t2/2 for all t∈[a,b]?

On the measure of Voronoi cells

Abstract We study the measure of a typical cell in a Voronoi tessellation defined by n independent random points X1, . . ., Xn drawn from an absolutely continuous probability measure μ with density f in ℝd. We prove that the asymptotic distribution of the measure – with respect to dμ = f(x)dx – of the cell containing X1 given X1 = x is independent of x and the density f. We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as d becomes large. In particular, we show that the variance converges to 0 exponentially fast in d. We also obtain a bound independent of the density for the rate of convergence of the diameter of a typical Voronoi cell.

Dynamical Localization for Discrete Anderson Dirac Operators

We establish dynamical localization for random Dirac operators on the d-dimensional lattice, with \(d\in \left\{ 1, 2, 3\right\} \), in the three usual regimes: large disorder, band edge and 1D. These operators are discrete versions of the continuous Dirac operators and consist in the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by different scalar potentials, which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support. We prove the exponential decay of fractional moments of the Green function for such models in each of the above regimes, i.e., (j) throughout the spectrum at larger disorder, (jj) for energies near the band edges at arbitrary disorder and (jjj) in dimension one, for all energies in the spectrum and arbitrary disorder. Dynamical localization in theses regimes follows from the fractional moments method. The result in the one-dimensional regime contrast with one that was previously obtained for 1D Dirac model with Bernoulli potential.

Accessing the Appropriateness of a Spatial Regression Using Generalized (<i>h</i><sub>1</sub><i>h</i><sub>2</sub>-)Slepian Random Field

In this study we derived asymptotic goodness-of-fit test (model check) for spatial regression where the critical region as well as the p-value of the tests are approximated based on the distribution of a type of the integral functional of the generalized (h1h2-)-Slepian field and the set-indexed Gaussian white noise. Such random fields are obtained as the limit process of the moving and the cumulative sums processes of the sequence of random matrices consist of independent and identically distributed random variables indexed by the points of a design constructed by means of a given continuous probability measure. Although the common approach in model diagnostic for regression is based on the functional of the residuals, in this study a new different idea is proposed by directly investigating the moving and the cumulative sums of the array of the observations. It is shown that these approaches are mathematically tractable and practically more applicable. Simulation study is conducted for investigating the finite sample size behavior of the tests. An application of the procedure to a mining data is also discussed, where from the perspective of geology and geophysics, polynomial model is reasonable and suitable for the data.