Strassen's Theorem Research Articles

Coupling capacity in C*-algebras

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

Multi‐disease classification model using deep neural network and Strassen's rectilinear fine‐tune bouncing training algorithm

AbstractA deep neural network (DNN) is being used in the healthcare industry to improve care delivery at a lower cost and in less time. A DNN is well‐known for its diagnostic applications. However, it is also increasingly being utilized to guide healthcare management decisions. DNNs have been applied to real‐life disease classification problems. The performances of DNN are improved by reducing the training time, performing fast classification, and improving the parameters such as accuracy and error rate, and so forth. We have proposed a hybrid DNN named Strassen's rectilinear fine‐tune bouncing training (SRFBT) Algorithm by combining Strassen's theorem and bouncing training algorithm. The simulation result shows that the SRFBT algorithm outperforms both the DNN, support vector machine, radial basis function network and deep belief network algorithms in terms of network training, testing time and accuracy on various UCI machine learning healthcare datasets. The SRFBT has improved performance with a minimum average training time of 43.6249 and a maximum accuracy of 96.86.

Applications of weak transport theory

Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali (J. Funct. Anal. 273 (2017) 3327–3405) introduced a transport problem for ‘weak’ cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality. In this article, we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schrödinger problem, the Brenier–Strassen theorem, optimal mechanism design, linear transfers, semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.

Some remarks on associated random fields, random measures and point processes

In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space (POP), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in [Lindqvist 1988]. We use these results to show on Polish spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in [Poinas et al. 2017] and [Lyons 2014] which restrict to point processes in Euclidean spaces and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.

Existence, duality, and cyclical monotonicity for weak transport costs

The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on mathbb {R}^d under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

Strassen's theorem for quantum couplings

Strassen's theorem for probabilistic couplings is a fundamental theorem in probability theory that can be used to bound the probability of an event in a distribution by the probability of an event in another distribution coupled with the first. It has been widely applied in computer science for analysis of random algorithms, machine learning and verification of security and privacy protocols. We extend the coupling techniques in probability theory to quantum systems. A quantum generalisation of the notion of lifting, a coupling under certain constraints, is introduced. Several interesting examples and basic properties of quantum couplings and liftings are presented. Finally, a quantum extension of Strassen's theorem is established.

Stochastic monotonicity from an Eulerian viewpoint

Stochastic monotonicity is a well-known partial order relation between probability measures defined on the same partially ordered set. Strassen theorem establishes equivalence between stochastic monotonicity and the existence of a coupling compatible with respect to the partial order. We consider the case of a countable set and introduce the class of finitely decomposable flows on a directed acyclic graph associated to the partial order. We show that a probability measure stochastically dominates another probability measure if and only if there exists a finitely decomposable flow having divergence given by the difference of the two measures. We illustrate the result with some examples.

On a mixture of Brenier and Strassen Theorems

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33]. Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem [14].

Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem

We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.

A generalization of Strassen's law and Lévy's modulus of continuity for $\boldsymbol{G}$-Brownian motion

The Strassen theorem and Levy's modulus of continuity for Brownian motion case play important roles in the probability theory. Muller gave the unification form on them in 1981. In this paper, the Strassen's law and Levy's modulus of continuity for $G$-Brownian motion are extended to the general case. It can be seen as the unification of Strassen's law and Levy's modulus of continuity for $G$-Brownian motion.

Conditional convex orders and measurable martingale couplings

Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.

Complexity of multiplication in commutative group algebras over fields of prime characteristic

The objective of this research is to study the complexity of multiplication in commutative group algebras over arbitrary fields of prime characteristic. In order to solve this problem, we suggest a method to find the structure of group algebras which allows us to use the Alder–Strassen theorem to obtain lower bounds and the Bläser theorem describing all algebras of minimal rank to obtain upper bounds.

On a Theorem of Strassen for Vector-Valued Measures

The aim of this note is to extend a type of Strassen's theorem for the existence of probability measures with given marginals to positive measures taking values in a Dedekind complete vector lattice with a sequentially Lebesgue locally convex topology.

Neyman–Pearson testing under interval probability by globally least favorable pairs: Reviewing Huber–Strassen theory and extending it to general interval probability

The paper studies the extension of one of the basic problems of classical statistics to a generalized concept of probability. It is concerned with the generalized Neyman–Pearson problem, i.e. hypothesis testing when both hypotheses are described by interval probability (also called interval-valued probability or lower and upper probability). First the Huber–Strassen theorem and the literature based on it are reviewed. The assumption of two-monotonicity underlying all that work, however, excludes many models of practical interest. Therefore, the second part of the paper goes beyond Huber–Strassen theory. Results not relying on this severe restriction are derived, showing that a neat theory of hypothesis testing can be achieved in the framework of general interval probability.

On the optimal stopping values induced by general dependence structures

The optimal stopping value of random variables X 1,…,X n depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F 1,…,F n . They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

On the optimal stopping values induced by general dependence structures

The optimal stopping value of random variables X1,…,Xn depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F1,…,Fn. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

Existence and Application of Optimal Markovian Coupling with Respect to Non-Negative Lower Semi-Continuous Functions

In this paper, for two given transition probabilities, the existence of the optimal Markovian coupling with respect to a non-negative lower semi-continuous function is proved. As an application of this result, the well-known Strassen's theorem is generalized. Moreover, it is proved that the existence of an order-preserving Markovian coupling of two given jump processes is equivalent to their stochastical comparability.

On Strassen's Theorem on Stochastic Domination

The purpose of this note is to make available a reasonably complete and straightforward proof of Strassen's theorem on stochastic domination, and to draw attention to the original paper. We also point out that the maximal possible value of $P(Z = Z')$ is actually not reduced by the requirement $Z \leq Z'$. Here, $Z,Z'$ are stochastic elements that Strassen's theorem states exist under a stochastic domination condition. The consequence of that observation to stochastically monotone Markov chains is pointed out. Usually the theorem is formulated with the assumption that $\leq$ is a partial ordering; the proof reveals that a pre-ordering suffices.

A version of Strassen’s Theorem for vector-valued measures

A formulation of is given for measures taking values in a Banach lattice. The main result (Theorem 2) corrects earlier work of the second author. In joint work of M. Marz and the second author [5, 3.7], a version of the theorem known in probability theory as Strassen's Theorem (see [8], [2, 11.6]) was generalized to the context of measures assuming values in a reflexive Banach lattice. In [7, 3.2], the second author announced an extension of the result to all order-complete Banach lattices. However, the proof given in [7, p.816] contains an error, leaving the validity of the result an open question. In 2 below, a result of this general type is proved for measures taking values in Banach lattices of a certain type: the so-called KB-spaces. The KBspaces occupy a position between the reflexive and the complete Banach lattices (reflexive =X KB =* complete), so that 2 is a generalization of [5, 3.7], but is not as strong as what was asserted in [7]. Our technique makes use of a Strassen-type result (given below as 1 and proved in [3]) for finitely additive measures with values in a complete Banach lattice. As a general reference for basic facts about vector measures, we recommend the text of Diestel and Uhl [1]. For Banach lattices, see [4], [6]. If A and B are fields on sets X and Y, respectively, then A x B is the field on X x Y generated by all rectangles E x F for E e A and F e S. The following result is 2.1 in [3]. It replaces the imprecisely stated and incompletely proved 2.2 of [7]. 1. Let A and B be countable fields on sets X and Y respectively and let A: G+ and i : B -G+ be finitely additive measures taking values in the positive cone of a divisible, a-complete, partially ordered group G. We assume that p(X) _= v(Y) = a for some a E G+. Let S be an arbitrary subset of X x Y and let C be the field on X x Y generated by S and the sets in A x B. For an element v E G with 0 < v < a, we consider the following conditions: i) There is a finitely additive measure p: C -+ G+ such that p(E x Y) = /1(E) and p(X x F) = v(F) for all E e A and F E B (i.e. p has marginals ,u and v) and such that p(S) = v. ii) Whenever E x F C S for E e A and F E B, then p(E) + v(F) < c + v. Received by the editors July 24, 1996 and, in revised form, October 28, 1996. 1991 Mathematics Subject Classification. Primary 28B05; Secondary 30C62, 46B42. (D1998 American Mathematical Society

Strassen theorems for a class of iterated processes

A general direct Strassen theorem is proved for a class of stochastic processes and applied for iterated processes such as W ( L t ) W(L_t) , where W ( ⋅ ) W(\cdot ) is a standard Wiener process and L . L_. is a local time of a Lévy process independent from W ( ⋅ ) W(\cdot ) .