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In this paper, we consider a distributed lossy compression network with $L$ encoders and a decoder. Each encoder observes a source and compresses it, which is sent to the decoder. Moreover, each observed source can be written as the sum of a target signal and a noise which are independently generated from two symmetric multivariate Gaussian distributions. The decoder jointly constructs the target signals given a threshold on the mean squared error distortion. We are interested in the minimum compression rate of this network versus the distortion threshold which is known as the \emph{rate-distortion function}. We derive a lower bound on the rate-distortion function by solving a convex program, explicitly. The proposed lower bound matches the well-known Berger-Tung's upper bound for some values of the distortion threshold. The asymptotic expressions of the upper and lower bounds are derived in the large $L$ limit. Under specific constraints, the bounds match in the asymptotic regime yielding the characterization of the rate-distortion function.

The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the true system response. We provide a probabilistic error analysis for a prototypical statFEM setup based on a Gaussian process prior under the assumption that the noisy measurement data are generated by a deterministic true system response function that satisfies a second-order elliptic partial differential equation for an unknown true source term. In certain cases, properties such as the smoothness of the source term may be misspecified by the Gaussian process model. The error estimates we derive are for the expectation with respect to the measurement noise of the $L^2$-norm of the difference between the true system response and the mean of the statFEM posterior. The estimates imply polynomial rates of convergence in the numbers of measurement points and finite element basis functions and depend on the Sobolev smoothness of the true source term and the Gaussian process model. A numerical example for Poisson's equation is used to illustrate these theoretical results.

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function \[ W_{\rm magic}^+(F)=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-\log\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+\log\det F=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+2\log\lambda_{\rm min}\,, \] where $\lambda_{\rm max}\geq\lambda_{\rm min}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.

In this letter, we analyze the performance of covert communications under faster-than-Nyquist (FTN) signaling in the Rayleigh block fading channel. Both Bayesian criterion- and Kullback-Leibler (KL) divergence-based covertness constraints are considered. Especially, for KL divergence-based one, we prove that both the maximum transmit power and covert rate under FTN signaling are higher than those under Nyquist signaling. Numerical results coincide with our analysis and validate the advantages of FTN signaling to realize covert data transmission.

Despite the many recent practical and theoretical breakthroughs in computational game theory, equilibrium finding in extensive-form team games remains a significant challenge. While NP-hard in the worst case, there are provably efficient algorithms for certain families of team game. In particular, if the game has common external information, also known as A-loss recall -- informally, actions played by non-team members (i.e., the opposing team or nature) are either unknown to the entire team, or common knowledge within the team -- then polynomial-time algorithms exist (Kaneko & Kline 1995). In this paper, we devise a completely new algorithm for solving team games. It uses a tree decomposition of the constraint system representing each team's strategy to reduce the number and degree of constraints required for correctness (tightness of the mathematical program). Our approach has the bags of the tree decomposition correspond to team-public states. Our algorithm reduces the problem of solving team games to a linear program with at most $O(NW^{w+1})$ nonzero entries in the constraint matrix, where $N$ is the size of the game tree, $w$ is a parameter that depends on the amount of uncommon external information, and $W$ is the treewidth of the tree decomposition. In public-action games, our program size is bounded by the tighter $2^{O(nt)}N$ for teams of $n$ players with $t$ types each. Our algorithm is based on a new way to write a custom, concise tree decomposition, and its fast run time does not assume that the decomposition has small treewidth. Since our algorithm describes the polytope of correlated strategies directly, we get equilibrium finding in correlated strategies for free -- instead of, say, having to run a double oracle algorithm. We show via experiments on a standard suite of games that our algorithm achieves state-of-the-art performance on all benchmark game classes except one.

In this paper, we consider a discrete-time Stackelberg mean field game with a leader and an infinite number of followers. The leader and the followers each observe types privately that evolve as conditionally independent controlled Markov processes. The leader commits to a dynamic policy and the followers best respond to that policy and each other. Knowing that the followers would play a mean field game based on her policy, the leader chooses a policy that maximizes her reward. We refer to the resulting outcome as a Stackelberg mean field equilibrium (SMFE). In this paper, we provide a master equation of this game that allows one to compute all SMFE. Based on our framework, we consider two numerical examples. First, we consider an epidemic model where the followers get infected based on the mean field population. The leader chooses subsidies for a vaccine to maximize social welfare and minimize vaccination costs. In the second example, we consider a technology adoption game where the followers decide to adopt a technology or a product and the leader decides the cost of one product that maximizes his returns, which are proportional to the people adopting that technology

In simplicial complexes it is well known that many of the global properties of the complex, can be deduced from expansion properties of its links. This phenomenon was first discovered by Garland [G]. In this work we develop a local to global machinery for general posets. We first show that the basic localization principle of Garland generalizes to more general posets. We then show that notable local to global theorems for simplicial complexes arise from general principles for general posets with expanding links. Specifically, we prove the following theorems for general posets satisfying some assumptions: Expanding links (one sided expansion) imply fast convergence of high dimensional random walks (generalization [KO,AL]); Expanding links imply Trickling down theorem (generalizing [O]); and a poset has expanding links (with two sided expansion) iff it satisfies a global random walk convergence property (generalization [DDFH]). We axiomatize general conditions on posets that imply local to global theorems. By developing this local to global machinery for general posets we discover that some posets behave better than simplicial complexes with respect to local to global implications. Specifically, we get a trickling down theorem for some posets (e.g. the Grassmanian poset) which is better behaved than the trickling down theorem known for simplicial complexes. In addition to this machinery, we also present a method to construct a new poset out of a pair of an initial poset and an auxiliary simplicial complex. By applying this procedure to the case where the pair is the Grassmanian poset and a bounded degree high dimensional expander, we obtain a bounded degree Grassmanian poset. We prove, using the tools described above, that this poset is a bounded degree expanding Grassmanian poset, partially proving a conjecture of [DDFH].

In every state of a quantum particle, Wigner's quasidistribution is the unique quasidistribution on the phase space with the correct marginal distributions for position, momentum, and all their linear combinations.

The sequence $(Sm(n))_{n\geqslant 0}$: $1$, $12$, $123$, $\ldots$ formed by concatenating the first $n+1$ positive integers is often called Smarandache consecutive numbers. We consider the more general case of concatenating arithmetic progressions and establish formulas to compute them. Three types of concatenation are taken into account: the right-concatenation like $(Sm(n))_{n\geqslant0}$ or the concatenation of odd integers: $1$, $13$, $135$, $\ldots$; the left-concatenation like the reverse of Smarandache consecutive numbers $(Smr(n))_{n\geqslant 0}$: $1$, $21$, $321$, $\ldots$; and the concatenation of right-concatenation and left-concatenation like $1$, $121$, $12321$, $1234321$,$\ldots$ formed by $Sm(n)$ and $Smr(n-1)$ for $n\geqslant1$, with the initial term $Sm(0)$. The resulting formulas enable fast computations of asymptotic terms of these sequences. In particular, we use our implementation in the Computer Algebra System Maple to compute billionth terms of $(Sm(n))_{n\geqslant0}$ and $(Smr(n))_{n\geqslant0}$.

Optimal zero-delay coding (quantization) of $\mathbb{R}^d$-valued linearly generated Markov sources is studied under quadratic distortion. The structure and existence of deterministic and stationary coding policies that are optimal for the infinite horizon average cost (distortion) problem are established. Prior results studying the optimality of zero-delay codes for Markov sources for infinite horizons either considered finite alphabet sources or, for the $\mathbb{R}^d$-valued case, only showed the existence of deterministic and non-stationary Markov coding policies or those which are randomized. In addition to existence results, for finite blocklength (horizon) $T$ the performance of an optimal coding policy is shown to approach the infinite time horizon optimum at a rate $O(\frac{1}{T})$. This gives an explicit rate of convergence that quantifies the near-optimality of finite window (finite-memory) codes among all optimal zero-delay codes.