Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDuality for optimal couplings in free probability
77
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras and probability measures on $mathbb{R}^m$ are replaced by non-commutative laws of $m$-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescus non-commutative $L^2$-Wasserstein distance using a new type of convex functions. As a consequence, we show that if $(X,Y)$ is a pair of optimally coupled $m$-tuples of non-commutative random variables in a tracial $mathrm{W}^*$-algebra $mathcal{A}$, then $mathrm{W}^*((1 - t)X + tY) = mathrm{W}^*(X,Y)$ for all $t in (0,1)$. Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of $m$-tuples is not separable with respect to the Wasserstein distance for $m > 1$.
rate research
Read More
Let $(X_1,dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $mathrm{W}^*(X_1,dots,X_m) to mathrm{W}^*(S_1,dots,S_m)$ which maps $mathrm{W}^*(X_1,dots,X_k)$ to $mathrm{W}^*(S_1,dots,S_k)$ for each $k = 1, dots, m$, and which also witnesses the Talagrand inequality for the law of $(X_1,dots,X_m)$ relative to the law of $(S_1,dots,S_m)$.
As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.
Regularization in Optimal Transport (OT) problems has been shown to critically affect the associated computational and sample complexities. It also has been observed that regularization effectively helps in handling noisy marginals as well as marginals with unequal masses. However, existing works on OT restrict themselves to $phi$-divergences based regularization. In this work, we propose and analyze Integral Probability Metric (IPM) based regularization in OT problems. While it is expected that the well-established advantages of IPMs are inherited by the IPM-regularized OT variants, we interestingly observe that some useful aspects of $phi$-regularization are preserved. For example, we show that the OT formulation, where the marginal constraints are relaxed using IPM-regularization, also lifts the ground metric to that over (perhaps un-normalized) measures. Infact, the lifted metric turns out to be another IPM whose generating set is the intersection of that of the IPM employed for regularization and the set of 1-Lipschitz functions under the ground metric. Also, in the special case where the regularization is squared maximum mean discrepancy based, the proposed OT variant, as well as the corresponding Barycenter formulation, turn out to be those of minimizing a convex quadratic subject to non-negativity/simplex constraints and hence can be solved efficiently. Simulations confirm that the optimal transport plans/maps obtained with IPM-regularization are intrinsically different from those obtained with $phi$-regularization. Empirical results illustrate the efficacy of the proposed IPM-regularized OT formulation. This draft contains the main paper and the Appendices.
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescus fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescus fully matricial function theory. In the end, we study relations between certain convolutions and transforms in $C^*$-operator valued probability.
With the development of human space exploration, the space environment is gradually filled with abandoned satellite debris and unknown micrometeorites, which will seriously affect capture motion of space robot. Hence, a novel fast collision-avoidance trajectory planning strategy for a dual-arm free-floating space robot (FFSR) with predefined-time pose feedback will be mainly studied to achieve micron-level tracking accuracy of end-effector in this paper. However, similar to control, the exponential feedback results in larger initial joint angular velocity relative to proportional feedback. Firstly, a pose-error-based kinematic model of the FFSR will be derived from a control perspective. Then, a cumulative dangerous field (CDF) collision-avoidance algorithm is applied in predefined-time trajectory planning to achieve micron-level collision-avoidance trajectory tracking precision. In the end, a GA-based optimization algorithm is used to optimize the predefined-time parameter to obtain a motion trajectory of low joint angular velocity of robotic arms. The simulation results verify our conjecture and conclusion.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
