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Signaling game problems investigate communication scenarios where encoder(s) and decoder(s) have misaligned objectives due to the fact that they either employ different cost functions or have inconsistent priors. We investigate a signaling game problem where an encoder observes a multi-dimensional source and conveys a message to a decoder, and the quadratic objectives of the encoder and decoder are misaligned due to a bias vector. We first provide a set of geometry conditions that needs to be satisfied in equilibrium considering any multi-dimensional source. Then, we consider independent and identically distributed sources and completely characterize conditions under which an informative linear Nash equilibrium exists. In particular, we show that if the components of the bias vector are not equal in magnitude, then there exists a linear equilibrium if and only if the source distribution is Gaussian. On the other hand, for a linear equilibrium to exist in the case of equal bias components, it is required that the source density is symmetric around its mean. Moreover, in the case of Gaussian sources, our results have a rate-distortion theoretic implication that achievable rates and distortions in the considered game theoretic setup can be obtained from their team theoretic counterpart.

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the $\Gamma$-convergence of Onsager-Machlup functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

In this paper, we present a spectral-based approach to study the linear approximation of two-layer neural networks. We first consider the case of single neuron and show that the linear approximability, quantified by the Kolmogorov width, is controlled by the eigenvalue decay of an associate kernel. Then, we show that similar results also hold for two-layer neural networks. This spectral-based approach allows us to obtain upper bounds, lower bounds, and explicit hard examples in a united manner. In particular, these bounds imply that for networks activated by smooth functions, restricting the norms of inner-layer weights may significantly impair the expressiveness. By contrast, for non-smooth activation functions, such as ReLU, the network expressiveness is independent of the inner-layer weight norms. In addition, we prove that for a family of non-smooth activation functions, including ReLU, approximating any single neuron with random features suffers from the \emph{curse of dimensionality}. This provides an explicit separation of expressiveness between neural networks and random feature models.

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager--Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the $\Gamma$-convergence of Onsager--Machlup functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

In a minimum cost submodular cover problem (MinSMC), given a monotone non-decreasing submodular function $f\colon 2^V \rightarrow \mathbb{Z}^+$, a cost function $c: V\rightarrow \mathbb R^{+}$, an integer $k\leq f(V)$, the goal is to find a subset $A\subseteq V$ with the minimum cost such that $f(A)\geq k$. MinSMC has a lot of applications in machine learning and data mining. In this paper, we design a parallel algorithm for MinSMC which obtains a solution with approximation ratio at most $\frac{H(\min\{\Delta,k\})}{1-5\varepsilon}$ with probability $1-3\varepsilon$ in $O(\frac{\log m\log n\log^2 mn}{\varepsilon^4})$ rounds, where $\Delta=\max_{v\in V}f(v)$, $H(\cdot)$ is the Hamornic number, $n=f(V)$, $m=|V|$ and $\varepsilon$ is a constant in $(0,\frac{1}{5})$. This is the first paper obtaining a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to $H(\min\{\Delta,k\})$.

In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lam\'e parameters and on the underlying mesh.

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of L\^e (2020). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is $H\in(0,1)$ and the drift is $\mathcal{C}^\alpha$, $\alpha\in[0,1]$ and $\alpha>1-1/(2H)$, we show the strong $L_p$ and almost sure rates of convergence to be $((1/2+\alpha H)\wedge 1) -\varepsilon$, for any $\varepsilon>0$. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier, Gubinelli (2016). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $1/2-\varepsilon$ of the Euler-Maruyama scheme for $\mathcal{C}^\alpha$ drift, for any $\varepsilon,\alpha>0$.

This survey is meant to provide an introduction to linear models and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to ordinary least squares. In machine learning, the output is usually a nonlinear function of the input. Deep learning even aims to find a nonlinear dependence with many layers which require a large amount of computation. However, most of these algorithms build upon simple linear models. We then describe linear models from different views and find the properties and theories behind the models. The linear model is the main technique in regression problems and the primary tool for it is the least squares approximation which minimizes a sum of squared errors. This is a natural choice when we're interested in finding the regression function which minimizes the corresponding expected squared error. This survey is primarily a summary of purpose, significance of important theories behind linear models, e.g., distribution theory, minimum variance estimator. We first describe ordinary least squares from three different points of view upon which we disturb the model with random noise and Gaussian noise. By Gaussian noise, the model gives rise to the likelihood so that we introduce a maximum likelihood estimator. It also develops some distribution theories via this Gaussian disturbance. The distribution theory of least squares will help us answer various questions and introduce related applications. We then prove least squares is the best unbiased linear model in the sense of mean squared error and most importantly, it actually approaches the theoretical limit. We end up with linear models with the Bayesian approach and beyond.

Despite many applications, dimensionality reduction in the $\ell_1$-norm is much less understood than in the Euclidean norm. We give two new oblivious dimensionality reduction techniques for the $\ell_1$-norm which improve exponentially over prior ones: 1. We design a distribution over random matrices $S \in \mathbb{R}^{r \times n}$, where $r = 2^{\tilde O(d/(\varepsilon \delta))}$, such that given any matrix $A \in \mathbb{R}^{n \times d}$, with probability at least $1-\delta$, simultaneously for all $x$, $\|SAx\|_1 = (1 \pm \varepsilon)\|Ax\|_1$. Note that $S$ is linear, does not depend on $A$, and maps $\ell_1$ into $\ell_1$. Our distribution provides an exponential improvement on the previous best known map of Wang and Woodruff (SODA, 2019), which required $r = 2^{2^{\Omega(d)}}$, even for constant $\varepsilon$ and $\delta$. Our bound is optimal, up to a polynomial factor in the exponent, given a known $2^{\sqrt d}$ lower bound for constant $\varepsilon$ and $\delta$. 2. We design a distribution over matrices $S \in \mathbb{R}^{k \times n}$, where $k = 2^{O(q^2)}(\varepsilon^{-1} q \log d)^{O(q)}$, such that given any $q$-mode tensor $A \in (\mathbb{R}^{d})^{\otimes q}$, one can estimate the entrywise $\ell_1$-norm $\|A\|_1$ from $S(A)$. Moreover, $S = S^1 \otimes S^2 \otimes \cdots \otimes S^q$ and so given vectors $u_1, \ldots, u_q \in \mathbb{R}^d$, one can compute $S(u_1 \otimes u_2 \otimes \cdots \otimes u_q)$ in time $2^{O(q^2)}(\varepsilon^{-1} q \log d)^{O(q)}$, which is much faster than the $d^q$ time required to form $u_1 \otimes u_2 \otimes \cdots \otimes u_q$. Our linear map gives a streaming algorithm for independence testing using space $2^{O(q^2)}(\varepsilon^{-1} q \log d)^{O(q)}$, improving the previous doubly exponential $(\varepsilon^{-1} \log d)^{q^{O(q)}}$ space bound of Braverman and Ostrovsky (STOC, 2010).

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.